pc5271AY2526wiki - User contributions [en]

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-21T17:00:42Z

Zifang: /* Phase Extraction and Refractive Index Conversion */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermocouple (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

===Phase Extraction and Refractive Index Conversion===

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shift:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|600px]]

Figure 5: Change in refractive index as a function of the number of added drops of salt solution.

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

===Discussions===

The main difficulty in this experiment lies in maintaining optical path length stability while varying the concentration of the solution. When drops are added to adjust the concentration, the solution requires time to become homogeneous. During this mixing process, refractive index gradients can exist within the liquid, leading to additional phase fluctuations in the interferometer signal.

At the same time, environmental perturbations such as mechanical vibrations and air currents introduce further phase noise. As a result, it becomes challenging to distinguish the phase shift caused by the change in concentration from fluctuations due to external disturbances. We suspect that the outliers observed in the results arise from excessive disturbance of the water during the addition of drops, which leads to transient and non-uniform refractive index changes and consequently inaccurate phase measurements.

Several improvements can be made to enhance the reliability of the measurement. First, a reference interferometer could be implemented to monitor and subtract environmental phase fluctuations, thereby isolating the contribution from the solution. Second, the method of introducing the solution can be improved by using a controlled delivery system to regulate the rate and volume of added liquid, reducing mechanical disturbances. Finally, allowing sufficient settling time after each adjustment, or introducing gentle mixing methods, can help ensure the solution reaches a homogeneous state before measurement.

With these improvements, the accuracy and repeatability of the concentration-dependent refractive index measurement can be significantly enhanced.

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

===Collection of data & analysis===

During this experiment, a video recording was taken to simultaneously capture the oscilloscope trace and the temperature reading from the thermocouple. Due to the large file size, the full video is not included here but is available upon request. An example segment is presented below to illustrate the method used for counting fringe shifts.

[[File: 17APR-01 segment start to -160 shifted.png|800px]]

[[File: 17APR-01 segment start to -160 shifted fringe count.png |800px]]

Figure 6: Example of interference signal during mirror heating. The raw oscilloscope trace (gray) is overlaid with a low-pass filtered signal (blue) to highlight the fringe evolution. Successive maxima and minima are labeled and used to count fringe shifts (N), corresponding to changes in optical path length as the mirror expands with increasing temperature.

[[File:thermal_expansion.jpg|600px]]

Figure 7: Measured displacement as a function of temperature during mirror heating.

During the experiment, a time trace of the detector signal was recorded while the mirror was being heated. The recorded signal shows a slow but sustained drift in voltage, superimposed with oscillatory behavior. This reflects the gradual expansion of the mirror assembly, which changes the optical path length and causes the interference signal to cycle through successive bright and dark fringes.

The detector voltage is interpreted using the standard interferometric model:

<math>
V_{det}(t) = V_{mid} + A \cos(\Delta \phi(t)),
</math>

where <math>V_{mid}</math> is the midpoint voltage and <math>A</math> is the fringe amplitude. As the mirror expands, the phase difference <math>\Delta \phi</math> evolves continuously in time.

In a Michelson interferometer, a physical displacement <math>\Delta x</math> of the mirror produces an optical path change of <math>2\Delta x</math> due to the round-trip propagation of light. The corresponding phase shift is

<math>
\Delta \phi = \frac{4\pi}{\lambda} \Delta x.
</math>

Fringe counting provides a robust method to extract displacement from the measured signal. One complete fringe corresponds to a phase change of <math>2\pi</math>, which is equivalent to a mirror displacement of <math>\frac{\lambda}{2}.</math>

Thus, the total displacement can be determined from the number of counted fringes <math>\Delta N</math>:

<math>
\Delta x = \Delta N \frac{\lambda}{2}.
</math>

For the laser wavelength used in this experiment (<math>\lambda = 780\ \mathrm{nm}</math>), each fringe corresponds to a displacement of 390 nm.

The displacement as a function of temperature is then obtained by accumulating the fringe count over time. The thermal expansion relation is given by

<math>
\Delta x = \alpha L_{\mathrm{eff}} \Delta T,
</math>

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L_{\mathrm{eff}}</math> is the effective length of the heated structure. Taking the derivative with respect to temperature gives

<math>
\alpha = \frac{1}{L_{\mathrm{eff}}} \frac{d(\Delta x)}{dT}.
</math>

From the linear fit of the displacement–temperature data, the slope is estimated to be

<math>
\frac{d(\Delta x)}{dT} \approx 501.43\ \mathrm{nm/^\circ C}.
</math>

Assuming an effective length of (from Thorlabs)

<math>
L_{\mathrm{eff}} \approx 6\ \mathrm{mm},
</math>

the thermal expansion coefficient is calculated as

<math>
\alpha \approx \frac{501.43 \times 10^{-9}}{6 \times 10^{-3}} = 8.36 \times 10^{-5}\ \mathrm{^\circ C^{-1}}.
</math>

===Discussion===

Several factors may influence the measured thermal expansion coefficient:

'''Mirror mount and composite structure:'''
The measured thermal expansion coefficient is significantly larger than that of silica (by approximately two orders of magnitude), indicating that the expansion is dominated by the mirror mount and surrounding structure rather than the mirror substrate alone. The measured displacement therefore includes contributions from the mirror, mount, and thermal interfaces. Since these components typically have larger expansion coefficients (e.g., aluminum), the extracted value represents an ''effective'' expansion of the entire assembly rather than that of the mirror alone.

'''Thermal gradients:'''
Thermal gradients across the mirror assembly can lead to non-uniform expansion. The Peltier element heats the mirror from one side, so different parts of the structure may be at different temperatures. As a result, the measured displacement may not correspond to a single well-defined temperature, introducing systematic error.

'''Thermocouple response and placement:'''
There can be a time lag and spatial offset between the thermocouple reading and the actual temperature of the reflecting surface, especially during continuous heating. While the linear fit reduces random fluctuations, it does not fully eliminate this systematic mismatch.

'''Mechanical drift and environmental perturbations:'''
Similar to the previous experiment, mechanical drift and residual environmental perturbations (e.g., vibrations or air currents) can introduce slow phase shifts that are not related to thermal expansion. These effects may contribute to deviations from perfect linearity in the displacement–temperature relation.

Despite these limitations, the approximately linear trend observed in the data suggests that thermal expansion is the dominant effect. The extracted coefficient therefore provides a reasonable estimate of the effective expansion of the mirror assembly.

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-21T16:58:36Z

Zifang: /* Thermal expansion of the mirror */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermocouple (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

===Phase Extraction and Refractive Index Conversion===

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shift:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|600px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

===Discussions===

The main difficulty in this experiment lies in maintaining optical path length stability while varying the concentration of the solution. When drops are added to adjust the concentration, the solution requires time to become homogeneous. During this mixing process, refractive index gradients can exist within the liquid, leading to additional phase fluctuations in the interferometer signal.

At the same time, environmental perturbations such as mechanical vibrations and air currents introduce further phase noise. As a result, it becomes challenging to distinguish the phase shift caused by the change in concentration from fluctuations due to external disturbances. We suspect that the outliers observed in the results arise from excessive disturbance of the water during the addition of drops, which leads to transient and non-uniform refractive index changes and consequently inaccurate phase measurements.

Several improvements can be made to enhance the reliability of the measurement. First, a reference interferometer could be implemented to monitor and subtract environmental phase fluctuations, thereby isolating the contribution from the solution. Second, the method of introducing the solution can be improved by using a controlled delivery system to regulate the rate and volume of added liquid, reducing mechanical disturbances. Finally, allowing sufficient settling time after each adjustment, or introducing gentle mixing methods, can help ensure the solution reaches a homogeneous state before measurement.

With these improvements, the accuracy and repeatability of the concentration-dependent refractive index measurement can be significantly enhanced.

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

===Collection of data & analysis===

During this experiment, a video recording was taken to simultaneously capture the oscilloscope trace and the temperature reading from the thermocouple. Due to the large file size, the full video is not included here but is available upon request. An example segment is presented below to illustrate the method used for counting fringe shifts.

[[File: 17APR-01 segment start to -160 shifted.png|800px]]

[[File: 17APR-01 segment start to -160 shifted fringe count.png |800px]]

Figure 6: Example of interference signal during mirror heating. The raw oscilloscope trace (gray) is overlaid with a low-pass filtered signal (blue) to highlight the fringe evolution. Successive maxima and minima are labeled and used to count fringe shifts (N), corresponding to changes in optical path length as the mirror expands with increasing temperature.

[[File:thermal_expansion.jpg|600px]]

Figure 7: Measured displacement as a function of temperature during mirror heating.

During the experiment, a time trace of the detector signal was recorded while the mirror was being heated. The recorded signal shows a slow but sustained drift in voltage, superimposed with oscillatory behavior. This reflects the gradual expansion of the mirror assembly, which changes the optical path length and causes the interference signal to cycle through successive bright and dark fringes.

The detector voltage is interpreted using the standard interferometric model:

<math>
V_{det}(t) = V_{mid} + A \cos(\Delta \phi(t)),
</math>

where <math>V_{mid}</math> is the midpoint voltage and <math>A</math> is the fringe amplitude. As the mirror expands, the phase difference <math>\Delta \phi</math> evolves continuously in time.

In a Michelson interferometer, a physical displacement <math>\Delta x</math> of the mirror produces an optical path change of <math>2\Delta x</math> due to the round-trip propagation of light. The corresponding phase shift is

<math>
\Delta \phi = \frac{4\pi}{\lambda} \Delta x.
</math>

Fringe counting provides a robust method to extract displacement from the measured signal. One complete fringe corresponds to a phase change of <math>2\pi</math>, which is equivalent to a mirror displacement of <math>\frac{\lambda}{2}.</math>

Thus, the total displacement can be determined from the number of counted fringes <math>\Delta N</math>:

<math>
\Delta x = \Delta N \frac{\lambda}{2}.
</math>

For the laser wavelength used in this experiment (<math>\lambda = 780\ \mathrm{nm}</math>), each fringe corresponds to a displacement of 390 nm.

The displacement as a function of temperature is then obtained by accumulating the fringe count over time. The thermal expansion relation is given by

<math>
\Delta x = \alpha L_{\mathrm{eff}} \Delta T,
</math>

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L_{\mathrm{eff}}</math> is the effective length of the heated structure. Taking the derivative with respect to temperature gives

<math>
\alpha = \frac{1}{L_{\mathrm{eff}}} \frac{d(\Delta x)}{dT}.
</math>

From the linear fit of the displacement–temperature data, the slope is estimated to be

<math>
\frac{d(\Delta x)}{dT} \approx 501.43\ \mathrm{nm/^\circ C}.
</math>

Assuming an effective length of (from Thorlabs)

<math>
L_{\mathrm{eff}} \approx 6\ \mathrm{mm},
</math>

the thermal expansion coefficient is calculated as

<math>
\alpha \approx \frac{501.43 \times 10^{-9}}{6 \times 10^{-3}} = 8.36 \times 10^{-5}\ \mathrm{^\circ C^{-1}}.
</math>

===Discussion===

Several factors may influence the measured thermal expansion coefficient:

'''Mirror mount and composite structure:'''
The measured thermal expansion coefficient is significantly larger than that of silica (by approximately two orders of magnitude), indicating that the expansion is dominated by the mirror mount and surrounding structure rather than the mirror substrate alone. The measured displacement therefore includes contributions from the mirror, mount, and thermal interfaces. Since these components typically have larger expansion coefficients (e.g., aluminum), the extracted value represents an ''effective'' expansion of the entire assembly rather than that of the mirror alone.

'''Thermal gradients:'''
Thermal gradients across the mirror assembly can lead to non-uniform expansion. The Peltier element heats the mirror from one side, so different parts of the structure may be at different temperatures. As a result, the measured displacement may not correspond to a single well-defined temperature, introducing systematic error.

'''Thermocouple response and placement:'''
There can be a time lag and spatial offset between the thermocouple reading and the actual temperature of the reflecting surface, especially during continuous heating. While the linear fit reduces random fluctuations, it does not fully eliminate this systematic mismatch.

'''Mechanical drift and environmental perturbations:'''
Similar to the previous experiment, mechanical drift and residual environmental perturbations (e.g., vibrations or air currents) can introduce slow phase shifts that are not related to thermal expansion. These effects may contribute to deviations from perfect linearity in the displacement–temperature relation.

Despite these limitations, the approximately linear trend observed in the data suggests that thermal expansion is the dominant effect. The extracted coefficient therefore provides a reasonable estimate of the effective expansion of the mirror assembly.

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-21T16:58:14Z

Zifang: /* Experiment Procedures & Discussion */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermocouple (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

===Phase Extraction and Refractive Index Conversion===

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shift:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|600px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

===Discussions===

The main difficulty in this experiment lies in maintaining optical path length stability while varying the concentration of the solution. When drops are added to adjust the concentration, the solution requires time to become homogeneous. During this mixing process, refractive index gradients can exist within the liquid, leading to additional phase fluctuations in the interferometer signal.

At the same time, environmental perturbations such as mechanical vibrations and air currents introduce further phase noise. As a result, it becomes challenging to distinguish the phase shift caused by the change in concentration from fluctuations due to external disturbances. We suspect that the outliers observed in the results arise from excessive disturbance of the water during the addition of drops, which leads to transient and non-uniform refractive index changes and consequently inaccurate phase measurements.

Several improvements can be made to enhance the reliability of the measurement. First, a reference interferometer could be implemented to monitor and subtract environmental phase fluctuations, thereby isolating the contribution from the solution. Second, the method of introducing the solution can be improved by using a controlled delivery system to regulate the rate and volume of added liquid, reducing mechanical disturbances. Finally, allowing sufficient settling time after each adjustment, or introducing gentle mixing methods, can help ensure the solution reaches a homogeneous state before measurement.

With these improvements, the accuracy and repeatability of the concentration-dependent refractive index measurement can be significantly enhanced.

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

===Collection of data & analysis===

During this experiment, a video recording was taken to simultaneously capture the oscilloscope trace and the temperature reading from the thermocouple. Due to the large file size, the full video is not included here but is available upon request. An example segment is presented below to illustrate the method used for counting fringe shifts.

[[File: 17APR-01 segment start to -160 shifted.png|800px]]

[[File: 17APR-01 segment start to -160 shifted fringe count.png |800px]]

Figure 6: Example of interference signal during mirror heating. The raw oscilloscope trace (gray) is overlaid with a low-pass filtered signal (blue) to highlight the fringe evolution. Successive maxima and minima are labeled and used to count fringe shifts (N), corresponding to changes in optical path length as the mirror expands with increasing temperature.

[[File:thermal_expansion.jpg|600px]]

Figure 7: Measured displacement as a function of temperature during mirror heating.

During the experiment, a time trace of the detector signal was recorded while the mirror was being heated. The recorded signal shows a slow but sustained drift in voltage, superimposed with oscillatory behavior. This reflects the gradual expansion of the mirror assembly, which changes the optical path length and causes the interference signal to cycle through successive bright and dark fringes.

The detector voltage is interpreted using the standard interferometric model:

<math>
V_{det}(t) = V_{mid} + A \cos(\Delta \phi(t)),
</math>

where <math>V_{mid}</math> is the midpoint voltage and <math>A</math> is the fringe amplitude. As the mirror expands, the phase difference <math>\Delta \phi</math> evolves continuously in time.

In a Michelson interferometer, a physical displacement <math>\Delta x</math> of the mirror produces an optical path change of <math>2\Delta x</math> due to the round-trip propagation of light. The corresponding phase shift is

<math>
\Delta \phi = \frac{4\pi}{\lambda} \Delta x.
</math>

Fringe counting provides a robust method to extract displacement from the measured signal. One complete fringe corresponds to a phase change of <math>2\pi</math>, which is equivalent to a mirror displacement of <math>\frac{\lambda}{2}.</math>

Thus, the total displacement can be determined from the number of counted fringes <math>\Delta N</math>:

<math>
\Delta x = \Delta N \frac{\lambda}{2}.
</math>

For the laser wavelength used in this experiment (<math>\lambda = 780\ \mathrm{nm}</math>), each fringe corresponds to a displacement of 390 nm.

The displacement as a function of temperature is then obtained by accumulating the fringe count over time. The thermal expansion relation is given by

<math>
\Delta x = \alpha L_{\mathrm{eff}} \Delta T,
</math>

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L_{\mathrm{eff}}</math> is the effective length of the heated structure. Taking the derivative with respect to temperature gives

<math>
\alpha = \frac{1}{L_{\mathrm{eff}}} \frac{d(\Delta x)}{dT}.
</math>

From the linear fit of the displacement–temperature data, the slope is estimated to be

<math>
\frac{d(\Delta x)}{dT} \approx 501.43\ \mathrm{nm/^\circ C}.
</math>

Assuming an effective length of (from Thorlabs)

<math>
L_{\mathrm{eff}} \approx 6\ \mathrm{mm},
</math>

the thermal expansion coefficient is calculated as

<math>
\alpha \approx \frac{501.43 \times 10^{-9}}{6 \times 10^{-3}} = 8.36 \times 10^{-5}\ \mathrm{^\circ C^{-1}}.
</math>

===Discussion===

Several factors may influence the measured thermal expansion coefficient:

'''Mirror mount and composite structure:'''
The measured thermal expansion coefficient is significantly larger than that of silica (by approximately two orders of magnitude), indicating that the expansion is dominated by the mirror mount and surrounding structure rather than the mirror substrate alone. The measured displacement therefore includes contributions from the mirror, mount, and thermal interfaces. Since these components typically have larger expansion coefficients (e.g., aluminum), the extracted value represents an ''effective'' expansion of the entire assembly rather than that of the mirror alone.

'''Thermal gradients:'''
Thermal gradients across the mirror assembly can lead to non-uniform expansion. The Peltier element heats the mirror from one side, so different parts of the structure may be at different temperatures. As a result, the measured displacement may not correspond to a single well-defined temperature, introducing systematic error.

'''Thermocouple response and placement:'''
There can be a time lag and spatial offset between the thermocouple reading and the actual temperature of the reflecting surface, especially during continuous heating. While the linear fit reduces random fluctuations, it does not fully eliminate this systematic mismatch.

'''Mechanical drift and environmental perturbations:'''
Similar to the previous experiment, mechanical drift and residual environmental perturbations (e.g., vibrations or air currents) can introduce slow phase shifts that are not related to thermal expansion. These effects may contribute to deviations from perfect linearity in the displacement–temperature relation.

Despite these limitations, the approximately linear trend observed in the data suggests that thermal expansion is the dominant effect. The extracted coefficient therefore provides a reasonable estimate of the effective expansion of the mirror assembly.

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-21T16:55:09Z

Zifang: /* Thermal expansion of the mirror */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermocouple (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shift:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|600px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

'''Discussions'''

The main difficulty in this experiment lies in maintaining optical path length stability while varying the concentration of the solution. When drops are added to adjust the concentration, the solution requires time to become homogeneous. During this mixing process, refractive index gradients can exist within the liquid, leading to additional phase fluctuations in the interferometer signal.

At the same time, environmental perturbations such as mechanical vibrations and air currents introduce further phase noise. As a result, it becomes challenging to distinguish the phase shift caused by the change in concentration from fluctuations due to external disturbances. We suspect that the outliers observed in the results arise from excessive disturbance of the water during the addition of drops, which leads to transient and non-uniform refractive index changes and consequently inaccurate phase measurements.

Several improvements can be made to enhance the reliability of the measurement. First, a reference interferometer could be implemented to monitor and subtract environmental phase fluctuations, thereby isolating the contribution from the solution. Second, the method of introducing the solution can be improved by using a controlled delivery system to regulate the rate and volume of added liquid, reducing mechanical disturbances. Finally, allowing sufficient settling time after each adjustment, or introducing gentle mixing methods, can help ensure the solution reaches a homogeneous state before measurement.

With these improvements, the accuracy and repeatability of the concentration-dependent refractive index measurement can be significantly enhanced.

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

'''Collection of data & analysis'''

During this experiment, a video recording was taken to simultaneously capture the oscilloscope trace and the temperature reading from the thermocouple. Due to the large file size, the full video is not included here but is available upon request. An example segment is presented below to illustrate the method used for counting fringe shifts.

[[File: 17APR-01 segment start to -160 shifted.png|800px]]

[[File: 17APR-01 segment start to -160 shifted fringe count.png |800px]]

Figure 6: Example of interference signal during mirror heating. The raw oscilloscope trace (gray) is overlaid with a low-pass filtered signal (blue) to highlight the fringe evolution. Successive maxima and minima are labeled and used to count fringe shifts (N), corresponding to changes in optical path length as the mirror expands with increasing temperature.

[[File:thermal_expansion.jpg|600px]]

Figure 7: Measured displacement as a function of temperature during mirror heating.

During the experiment, a time trace of the detector signal was recorded while the mirror was being heated. The recorded signal shows a slow but sustained drift in voltage, superimposed with oscillatory behavior. This reflects the gradual expansion of the mirror assembly, which changes the optical path length and causes the interference signal to cycle through successive bright and dark fringes.

The detector voltage is interpreted using the standard interferometric model:

<math>
V_{det}(t) = V_{mid} + A \cos(\Delta \phi(t)),
</math>

where <math>V_{mid}</math> is the midpoint voltage and <math>A</math> is the fringe amplitude. As the mirror expands, the phase difference <math>\Delta \phi</math> evolves continuously in time.

In a Michelson interferometer, a physical displacement <math>\Delta x</math> of the mirror produces an optical path change of <math>2\Delta x</math> due to the round-trip propagation of light. The corresponding phase shift is

<math>
\Delta \phi = \frac{4\pi}{\lambda} \Delta x.
</math>

Fringe counting provides a robust method to extract displacement from the measured signal. One complete fringe corresponds to a phase change of <math>2\pi</math>, which is equivalent to a mirror displacement of

<math>
\frac{\lambda}{2}.
</math>

Thus, the total displacement can be determined from the number of counted fringes <math>\Delta N</math>:

<math>
\Delta x = \Delta N \frac{\lambda}{2}.
</math>

For the laser wavelength used in this experiment (<math>\lambda = 780\ \mathrm{nm}</math>), each fringe corresponds to a displacement of 390 nm.

The displacement as a function of temperature is then obtained by accumulating the fringe count over time. The thermal expansion relation is given by

<math>
\Delta x = \alpha L_{\mathrm{eff}} \Delta T,
</math>

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L_{\mathrm{eff}}</math> is the effective length of the heated structure. Taking the derivative with respect to temperature gives

<math>
\alpha = \frac{1}{L_{\mathrm{eff}}} \frac{d(\Delta x)}{dT}.
</math>

From the linear fit of the displacement–temperature data, the slope is estimated to be

<math>
\frac{d(\Delta x)}{dT} \approx 501.43\ \mathrm{nm/^\circ C}.
</math>

Assuming an effective length of (from Thorlabs)

<math>
L_{\mathrm{eff}} \approx 6\ \mathrm{mm},
</math>

the thermal expansion coefficient is calculated as

<math>
\alpha \approx \frac{501.43 \times 10^{-9}}{6 \times 10^{-3}} = 8.36 \times 10^{-5}\ \mathrm{^\circ C^{-1}}.
</math>

'''Discussion'''

Several factors may influence the measured thermal expansion coefficient:

- Mirror mount and composite structure
The measured thermal expansion coefficient is significantly larger than that of silica (by approximately two orders of magnitude), indicating that the expansion is dominated by the mirror mount and surrounding structure rather than the mirror substrate alone. The measured displacement therefore includes contributions from the mirror, mount, and thermal interfaces. Since these components typically have larger expansion coefficients (e.g., aluminum), the extracted value represents an ''effective'' expansion of the entire assembly rather than that of the mirror alone.

- Thermal gradients
Thermal gradients across the mirror assembly can lead to non-uniform expansion. The Peltier element heats the mirror from one side, so different parts of the structure may be at different temperatures. As a result, the measured displacement may not correspond to a single well-defined temperature, introducing systematic error.

- Thermocouple response and placement
There can be a time lag and spatial offset between the thermocouple reading and the actual temperature of the reflecting surface, especially during continuous heating. While the linear fit reduces random fluctuations, it does not fully eliminate this systematic mismatch.

- Mechanical drift and environmental perturbations
Similar to the previous experiment, mechanical drift and residual environmental perturbations (e.g., vibrations or air currents) can introduce slow phase shifts that are not related to thermal expansion. These effects may contribute to deviations from perfect linearity in the displacement–temperature relation.

Despite these limitations, the approximately linear trend observed in the data suggests that thermal expansion is the dominant effect. The extracted coefficient therefore provides a reasonable estimate of the effective expansion of the mirror assembly.

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

File:Thermal expansion.jpg

2026-04-21T16:42:15Z

Zifang:

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-21T16:37:22Z

Zifang: /* Thermal expansion of the mirror */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermocouple (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shift:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|600px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

'''Discussions'''

The main difficulty in this experiment lies in maintaining optical path length stability while varying the concentration of the solution. When drops are added to adjust the concentration, the solution requires time to become homogeneous. During this mixing process, refractive index gradients can exist within the liquid, leading to additional phase fluctuations in the interferometer signal.

At the same time, environmental perturbations such as mechanical vibrations and air currents introduce further phase noise. As a result, it becomes challenging to distinguish the phase shift caused by the change in concentration from fluctuations due to external disturbances. We suspect that the outliers observed in the results arise from excessive disturbance of the water during the addition of drops, which leads to transient and non-uniform refractive index changes and consequently inaccurate phase measurements.

Several improvements can be made to enhance the reliability of the measurement. First, a reference interferometer could be implemented to monitor and subtract environmental phase fluctuations, thereby isolating the contribution from the solution. Second, the method of introducing the solution can be improved by using a controlled delivery system to regulate the rate and volume of added liquid, reducing mechanical disturbances. Finally, allowing sufficient settling time after each adjustment, or introducing gentle mixing methods, can help ensure the solution reaches a homogeneous state before measurement.

With these improvements, the accuracy and repeatability of the concentration-dependent refractive index measurement can be significantly enhanced.

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

'''Collection of data & analysis'''

During this experiment, a video recording was taken to simultaneously capture the oscilloscope trace and the temperature reading from the thermocouple. Due to the large file size, the full video is not included here but is available upon request. An example segment is presented below to illustrate the method used for counting fringe shifts.

[[File: 17APR-01 segment start to -160 shifted.png|800px]]

[[File: 17APR-01 segment start to -160 shifted fringe count.png |800px]]

Figure 6: Example of interference signal during mirror heating. The raw oscilloscope trace (gray) is overlaid with a low-pass filtered signal (blue) to highlight the fringe evolution. Successive maxima and minima are labeled and used to count fringe shifts (N), corresponding to changes in optical path length as the mirror expands with increasing temperature.

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-21T16:32:53Z

Zifang: /* List of Apparatus */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermocouple (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shift:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|600px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

'''Discussions'''

The main difficulty in this experiment lies in maintaining optical path length stability while varying the concentration of the solution. When drops are added to adjust the concentration, the solution requires time to become homogeneous. During this mixing process, refractive index gradients can exist within the liquid, leading to additional phase fluctuations in the interferometer signal.

At the same time, environmental perturbations such as mechanical vibrations and air currents introduce further phase noise. As a result, it becomes challenging to distinguish the phase shift caused by the change in concentration from fluctuations due to external disturbances. We suspect that the outliers observed in the results arise from excessive disturbance of the water during the addition of drops, which leads to transient and non-uniform refractive index changes and consequently inaccurate phase measurements.

Several improvements can be made to enhance the reliability of the measurement. First, a reference interferometer could be implemented to monitor and subtract environmental phase fluctuations, thereby isolating the contribution from the solution. Second, the method of introducing the solution can be improved by using a controlled delivery system to regulate the rate and volume of added liquid, reducing mechanical disturbances. Finally, allowing sufficient settling time after each adjustment, or introducing gentle mixing methods, can help ensure the solution reaches a homogeneous state before measurement.

With these improvements, the accuracy and repeatability of the concentration-dependent refractive index measurement can be significantly enhanced.

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

[[File: 17APR-01 segment start to -160 shifted.png|800px]]

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

[[File: 17APR-01 segment start to -160 shifted fringe count.png |800px]]

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-21T16:24:54Z

Zifang: /* Refractive index of the solution */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermistor (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shift:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|600px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

'''Discussions'''

The main difficulty in this experiment lies in maintaining optical path length stability while varying the concentration of the solution. When drops are added to adjust the concentration, the solution requires time to become homogeneous. During this mixing process, refractive index gradients can exist within the liquid, leading to additional phase fluctuations in the interferometer signal.

At the same time, environmental perturbations such as mechanical vibrations and air currents introduce further phase noise. As a result, it becomes challenging to distinguish the phase shift caused by the change in concentration from fluctuations due to external disturbances. We suspect that the outliers observed in the results arise from excessive disturbance of the water during the addition of drops, which leads to transient and non-uniform refractive index changes and consequently inaccurate phase measurements.

Several improvements can be made to enhance the reliability of the measurement. First, a reference interferometer could be implemented to monitor and subtract environmental phase fluctuations, thereby isolating the contribution from the solution. Second, the method of introducing the solution can be improved by using a controlled delivery system to regulate the rate and volume of added liquid, reducing mechanical disturbances. Finally, allowing sufficient settling time after each adjustment, or introducing gentle mixing methods, can help ensure the solution reaches a homogeneous state before measurement.

With these improvements, the accuracy and repeatability of the concentration-dependent refractive index measurement can be significantly enhanced.

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

[[File: 17APR-01 segment start to -160 shifted.png|800px]]

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

[[File: 17APR-01 segment start to -160 shifted fringe count.png |800px]]

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T18:03:40Z

Zifang: /* Discussion on Fringe Visibility */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermistor (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shit:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|800px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

[[File: 17APR-01 trimmed -160 to 100s.png|800px]]

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

[[File: Fringe count plot.png|800px]]

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T18:03:15Z

Zifang: /* Interference fringes */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermistor (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

===Discussion on Fringe Visibility===

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, resulting in imperfect spatial overlap when the two beams recombine at the beam splitter. Consequently, only partial interference occurs, which reduces the observed visibility.

Another contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms and further degrading the interference contrast.

We also observe that the fringe visibility varies over time. Each time the setup is restarted and realigned, the measured visibility changes slightly. This behavior is likely due to the lack of a stabilized optical table, making the system sensitive to small mechanical perturbations and environmental disturbances that affect alignment.

Despite the relatively low and time-varying visibility, this does not significantly impact the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the powermeter, the phase variation can still be determined with sufficient accuracy.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shit:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|800px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

[[File: 17APR-01 trimmed -160 to 100s.png|800px]]

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

[[File: Fringe count plot.png|800px]]

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T17:58:31Z

Zifang: /* List of Apparatus */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth), power supply

Peltier, thermistor (for heating the mirror and measuring its temperature)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode quality of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, which results in imperfect spatial overlap when the two beams recombine at the beam splitter. As a consequence, only partial interference occurs, reducing the observed visibility.

A secondary contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms, which further degrades the interference contrast.

Despite the low visibility, this does not significantly affect the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the power meter, the phase variation can still be accurately determined.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shit:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|800px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

[[File: 17APR-01 trimmed -160 to 100s.png|800px]]

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

[[File: Fringe count plot.png|800px]]

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T17:55:16Z

Zifang: /* Interferometer alignment and stability */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

{| style="width:80%; text-align:center;"
| [[File:interferometer_photo.jpeg|400px]]
| [[File:interferometer_expan.jpeg|400px]]
|-
| ''(a) Setup for refractive index measurement''
| ''(b) Setup for thermal expansion measurement''
|}

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode quality of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, which results in imperfect spatial overlap when the two beams recombine at the beam splitter. As a consequence, only partial interference occurs, reducing the observed visibility.

A secondary contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms, which further degrades the interference contrast.

Despite the low visibility, this does not significantly affect the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the power meter, the phase variation can still be accurately determined.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shit:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|800px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

[[File: 17APR-01 trimmed -160 to 100s.png|800px]]

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

[[File: Fringe count plot.png|800px]]

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

File:Interferometer expan.jpeg

2026-04-20T17:52:05Z

Zifang:

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T17:49:21Z

Zifang: /* Experiment setup & procedures */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment Procedures & Discussion=
==Constructing a Michelson Interferometer==

A basic Michelson interferometer will be constructed for the experiment.

===Interferometer alignment and stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer used for measuring the refractive index of the solution. For the thermal expansion measurement, the solution cell is removed, and one of the mirrors is attached to a Peltier element mounted on the mirror holder. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

[[Image: interferometer_photo.jpeg|450px]]

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic for measuring the refractive index of the solution.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference fringes===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode quality of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, which results in imperfect spatial overlap when the two beams recombine at the beam splitter. As a consequence, only partial interference occurs, reducing the observed visibility.

A secondary contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms, which further degrades the interference contrast.

Despite the low visibility, this does not significantly affect the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the power meter, the phase variation can still be accurately determined.

==Refractive index of the solution==

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shit:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|800px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

==Thermal expansion of the mirror==

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

[[File: 17APR-01 trimmed -160 to 100s.png|800px]]

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

[[File: Fringe count plot.png|800px]]

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T17:41:10Z

Zifang: /* Theory */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: optical isolator (Thorlabs IO-3D-780-VLP), beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Analog powermeter

Oscilloscope (Rohde & Schwarz RTB2004, 100MHz bandwidth)

=Idea=
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentrations by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrate the advantages of high-precision experimental technique.

=Theory=

===Michelson Interferometer===

An interferometer is a precision scientific instrument in which a light wave is split into multiple paths and later recombined. The resulting interference depends on the relative phase difference accumulated along the different paths. Because the phase is sensitive to path length and optical properties, interferometers are widely used for precision measurements. A common example is the '''Michelson interferometer''', which illustrates the basic principles of optical interference.

The basic version of the Michelson interferometer consists of a 50:50 beam splitter (BS) and two mirrors placed in separate arms of the interferometer. Assume that the input beam is a linearly polarized monochromatic source of wavelength <math>\lambda</math> and field amplitude <math>E_0</math>. The light is incident on the input port of the BS, where it is divided into two beams that propagate along the two arms toward the mirrors. Each beam is reflected by its respective mirror and recombined at BS, producing an interference pattern at the output port.

The output electric field can be described by summing the two contributions from the two arms. If the path lengths are <math>L_1</math> and <math>L_2</math>, the output field:

<math>
E_{out} = E_1 + E_2 = \frac{1}{2}E_0 e^{i2kL_1}(1+e^{i2k\Delta L})
</math>

where <math>\Delta L=L_2-L_1</math> and <math>k=2\pi/\lambda</math>. Field maxima occur whenever

<math>
\frac{4\pi\Delta L}{\lambda} = 2m\pi,
</math>

and minima when

<math>
\frac{4\pi\Delta L}{\lambda} = (2m+1)\pi,
</math>

where <math>m</math> is an integer.

Changes in the optical path length of either arm caused by displacement of a mirror, or variation of refractive index in a medium will change the relative phase difference, leading to a shift of the interference fringes. he high sensitivity of the interference pattern to such changes forms the basis for precision measurements using a Michelson interferometer.

For example, a transparent container of length <math>L</math> can be placed in one arm of the interferometer. The container is initially filled with air (refractive index approximately 1), and is then gradually filled with a solution of refractive index <math>n</math>. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

<math>
\Delta \text{OPL} = 2(n-1)L,
</math>

where the factor of 2 arises because the light passes through the container twice (forward and backward).

The corresponding phase change is

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

As the concentration of the solution changes, the refractive index <math>n</math> varies, leading to a continuous shift of the interference fringes. By counting the fringe shifts at the output port, the change in optical path length can be determined, allowing us to measure the dependence of the refractive index <math>n</math> on the solution concentration.

===Measuring Salt Solution Concentration===

This part mainly focuses on how the Michelson interferometer can quantitatively relate fringe shifts to salt concentration in water, via refractive index and optical path length.

When a cup of solution is placed in one of the arms of the interferometer, the refractive index relative to air, pure water, and the container wall changes, altering the optical path length (OPL) and, ultimately, resulting in a measurable phase shift via a powermeter when interference occurs.

In summary, the measurement chain is:

Fringe Shift → Phase → OPL → Refractive Index → Salt Concentration

Beginning form the equation we introduced above:

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(n-1)L.
</math>

At room temperature, when the solution concentration is reasonably small, the relationship between the solution's refractive index and concentration can be approximated by a linear function, say

<math>
n = \alpha C + \beta T + n_0,
</math>

where C is concentration, T is temperature (with <math>\beta << \alpha</math> at least for salt and alchoho), and <math>n_0</math> is the refractive index of pure water.

Substituting the second equation into the first, we have

<math>
\Delta \phi = \frac{2\pi}{\lambda} \cdot 2(\alpha C + \beta T + n_0 - 1)L,
</math>

which leads to the conclusion that

<math>
C(\Delta\phi) = \frac{1}{\alpha}(\frac{\lambda}{4\pi L}\Delta \phi - \beta T - (n_0 - 1)).
</math>

In practice, we can assume the temperature effect is negligible due to the small <math>\beta</math> approximation. Also, we only measure the relative phase shift, so the constant terms are negligible. After that, the final function relating <math>C</math> and <math>\phi</math> is

<math>
C(\Delta\phi) = \frac{\lambda}{4\pi \alpha L}\Delta \phi.
</math>

For salt, the value of <math>\alpha</math> is approximately 0.002. Using this value for estimation, we only need about 1 milligram of salt per liter of water to see a significant phase shift, as long as the container shape is not eccentric. Instead of adding salt crystals directly to water, for the simplicity of experiment, we prefer adding a high-concentration salt solution to the measured solution to better control the final concentration.

===Thermal Expansion of the Mirror===

In a Michelson interferometer, a change in the position of a mirror leads to a variation in the optical path length of that arm, resulting in a phase shift between the two beams and a corresponding shift in the interference fringes.

For a temperature change <math>\Delta T</math>, the linear thermal expansion of a material is given by

<math>\Delta L = \alpha L \Delta T</math>,

where <math>\alpha</math> is the linear thermal expansion coefficient and <math>L</math> is the initial length.

In the interferometer, a displacement <math>\Delta L</math> of the mirror produces an optical path difference of

<math>\Delta \mathrm{OPL} = 2\Delta L</math>,

due to the round-trip propagation of light. Since one fringe shift corresponds to a path change of one wavelength <math>\lambda</math>, the number of fringes shifted is

<math>N = \frac{2\Delta L}{\lambda}</math>.

Combining the above expressions, the thermal expansion coefficient can be written as

<math>\alpha = \frac{N\lambda}{2L\Delta T}</math>.

=Experiment setup & procedures=
A basic Michelson interferometer will be constructed for the experiment.

===Interferometer Alignment and Stability===

[[Image: interferometer_setup.png|350px]]

Figure 1: Schematic of the Michelson interferometer. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

[[Image: interferometer_photo.jpeg|450px]]

Figure 2: Photo of the actual Michelson interferometer setup constructed according to the schematic.

After constructing the interferometer, the beams reflected from both mirrors were carefully aligned so that they recombined at the beam splitter. The resulting interference signal was monitored using an analog power meter.

During the early stage of the interferometer construction, several sources of fluctuations that could affect the stability of the interference signal were identified:

'''Optical feedback into the laser diode:''' Reflected light entering the laser cavity can cause intensity fluctuations and frequency instability. 
Counter: An optical isolator was added to suppress back reflections.

'''Mechanical vibrations:''' Vibrations from the environment (for example, people walking near the setup) could be observed as fluctuations in the signal on the oscilloscope. 
Counter: Sorbothane vibration-isolation feet (Thorlabs AV4) were placed under the aluminum breadboard to provide basic vibration and acoustic isolation.

'''Air currents:''' Air movement around the optical paths can change the optical path length through refractive index variations. 
Counter: The entire interferometer setup was enclosed inside a styrofoam box to reduce airflow and improve thermal stability.

===Interference Signal Measurement===

For simplicity, no active elements were included to stabilize or control the optical path length of the interferometer. The interferometer was allowed to free run, and the interference signal was recorded using the oscilloscope.

Two datasets are presented here: one at the maximum intensity and one at the minimum intensity of the interference signal. Each measurement corresponds to a 12-second-long trace.

[[File:24MAR-05 peak troughn.png|800px]]

Figure 3: Oscilloscope trace of the interference signal at minimum intensity.

This figure shows the trace corresponding to the minimum intensity that can be achieved. The average voltage is 1.66 V. The red region in the plot represents intentionally induced noise spikes that were excluded from the averaging process. These excluded points account for approximately 8.8% of the dataset.

[[File:24MAR-06 peak troughn.png|800px]]

Figure 4: Oscilloscope trace of the interference signal at maximum intensity.

This shows the trace corresponding to the maximum intensity of the interference signal. The average voltage measured in this case is 2.06 V.

'''Discussion on Fringe Visibility'''

The relatively low fringe visibility observed in this interferometer is primarily attributed to the poor spatial mode quality of the laser diode. The emitted beam is neither perfectly Gaussian nor symmetric, which results in imperfect spatial overlap when the two beams recombine at the beam splitter. As a consequence, only partial interference occurs, reducing the observed visibility.

A secondary contribution may arise from imperfect power splitting at the beam splitter, leading to unequal intensities in the two arms, which further degrades the interference contrast.

Despite the low visibility, this does not significantly affect the objective of the experiment. Since the measurement relies on extracting phase information, it is sufficient to maintain a clear and distinguishable difference between the maximum and minimum signals. As long as the interference fringes can be reliably resolved using the power meter, the phase variation can still be accurately determined.

===Refractive index of the solution===

Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

The analysis begins by reading each oscilloscope CSV as a time-voltage trace, with the detector voltage treated as a proxy for the Michelson output intensity. A separate calibration was used to establish the bright and dark interference levels. The calibration trace was low-pass filtered so that the bright and dark levels were estimated from the underlying fringe swing rather than from point-to-point fluctuation.

The interferometer output was modeled with the standard cosine response

<math>
V = V_{\mathrm{mid}} + A\cos(\Delta \phi)
</math>

where V is the measured detector voltage, <math>V_{\mathrm{mid}}</math> is the midpoint voltage between bright and dark fringe, A is half of the fringe swing, and Delta phi is the phase difference between the two interferometer arms. From the calibration trace, the midpoint and amplitude were defined as:

<math>
V_{\mathrm{mid}} = \frac{V_{\mathrm{bright}} + V_{\mathrm{dark}}}{2}
</math>

<math>
A = \frac{V_{\mathrm{bright}} - V_{\mathrm{dark}}}{2}
</math>

For the present dataset, those calibration values were:

Dark level = 1.074943 V

Bright level = 1.393521 V

Midpoint = 1.234232 V

Amplitude = 0.159289 V

Each salt solution dataset was then low-pass filtered and represented by its median filtered voltage, <math>V_{\mathrm{data}}</math>. This representative voltage was normalized according to:

<math>
x = \frac{V_{\mathrm{data}} - V_{\mathrm{mid}}}{A}
</math>

so that x corresponds to the cosine interference term and, in the ideal model, lies between -1 and +1. This step connects the raw oscilloscope voltage directly to the interferometric phase response.

'''Phase Extraction and Refractive Index Conversion'''

Once the normalized interference variable x has been obtained, the phase was extracted from the inverse cosine relation:

<math>
\phi = \cos^{-1}(x)
</math>

Because intensity-only detection measures the cosine of the phase rather than the signed phase itself, a second mathematically valid branch also exists:

<math>
\phi_{\mathrm{alt}} = 2\pi - \phi
</math>

The water only dataset was used as the phase reference. For each subsequent drop count, the smallest phase change consistent with the measured voltage was taken relative to that reference.

The phase shift was then converted into refractive-index change through the Michelson optical path relation. As the beam traverses the sample on the forward and return paths, the optical path change is:

<math>
\Delta \mathrm{OPL} = 2L\,\Delta n
</math>

and the corresponding phase shift is:

<math>
\Delta \phi = \frac{2\pi}{\lambda}\Delta \mathrm{OPL} = \frac{4\pi L}{\lambda}\Delta n
</math>

Rearranging gives the working equation used to calculate the refractive-index change from the measured phase shit:

<math>
\Delta n = \frac{\Delta \phi\lambda}{4\pi L}
</math>

With wavelength = 780nm and L = 55mm, one full fringe corresponds to a refractive index change of approximately:

<math>
\Delta n_{\mathrm{one\ fringe}} = \frac{\lambda}{2L} \approx 7.091e-06
</math>

[[File:Delta n vs drops upto6 linear fit 2.png|800px]]

<math>
|\Delta n| = \left(2.176 \times 10^{-7}\right) N_{\mathrm{drops}} - 1.997 \times 10^{-7}
</math>

===Thermal expansion measurement===

In this part, a Peltier element is used to heat the mirror assembly. The hot side of the Peltier is placed in contact with the back of the mirror, while the cold side is attached to the mirror mount. In this configuration, both the mirror and parts of the mount are heated simultaneously.

As the temperature increases, the thermal expansion of the mirror assembly causes a displacement of the reflecting surface, leading to a shift in the interference fringes. The temperature is monitored using a thermocouple, and the fringe movement is recorded using the interferometer signal.

In practice, the measured expansion corresponds to an ''effective'' thermal expansion coefficient, as contributions may arise not only from the mirror substrate but also from the mount and the thermal interface between them.

[[File: 17APR-01 trimmed -160 to 100s.png|800px]]

The CSV trace above shows that the heating the mirror produces a slow but sustained drift in the output. The signal does not remain at one fixed voltage but instead it moves through repeated oscillatory cycles. As the heated mirror or its mount expands, the optical path in that arm changes, and the detector voltage passes successively through brighter and darker interference states.

The measured detector voltage is interpreted using the standard interferometric model in which the observed signal is a cosine function of the phase difference between the two interferometer arms:

<math>
V_{det}(t) = V_{mid} + A\cos(\Delta \phi(t))
</math>

where <math>V_{mid}</math> is the midpoint voltage and A is the fringe amplitude. For the mirror-heating experiment, the relevant physical mechanism is displacement of the mirror along the beam path. In a Michelson interferometer, if the mirror moves by <math>\Delta x</math>, the optical path length changes by <math>2\Delta x</math> because the the beam travels to the mirror and return. The corresponding phase shift is therefore:

<math>
\Delta \phi = \frac{4\pi}{\lambda}\Delta x
</math>

Fringe counting provides the most robust route from the observed signal to displacement. One complete fringe corresponds to a phase change of <math>2\pi</math>, which in the Michelson geometry is equivalent to a mirror displacement of <math>\frac{\lambda}{2}</math>. This then gives:

<math>
\Delta x = \Delta N\frac{\lambda}{2}
</math>

With the laser wavelength as 780nm, one fringe corresponds to 390nm of mirror displacement.

[[File: Fringe count plot.png|800px]]

=References=
M. Fox, "Quantum Optics: An Introduction", (Oxford University Press, 2006) Chap. 2.

Shang, J., Ping, L., Ling, Y. et al. A instrument for measuring solution concentration. Opt Rev 30, 310–321 (2023).

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T17:39:24Z

Zifang: /* Thermal expansion measurement */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T17:28:33Z

Zifang: /* Theory */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-20T17:27:11Z

Zifang: /* Measuring Salt Solution Concentration */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-10T03:23:47Z

Zifang: /* Interference Signal Measurement */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-10T03:14:27Z

Zifang: /* Interferometer Alignment and Stability */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-10T03:13:27Z

Zifang: /* Archived */

File:Interferometer setup.png

2026-04-10T03:12:44Z

Zifang: Zifang uploaded a new version of File:Interferometer setup.png

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-10T03:06:58Z

Zifang: /* Interferometer Alignment and Stability */

File:Interferometer photo.jpeg

2026-04-10T03:03:12Z

Zifang: Zifang uploaded a new version of File:Interferometer photo.jpeg

File:Interferometer photo.jpeg

2026-04-10T02:44:33Z

Zifang:

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-07T03:29:57Z

Zifang: /* Experiment setup & procedures */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-07T03:05:01Z

Zifang: /* List of Apparatus */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-07T02:57:25Z

Zifang: /* Experiment setup & procedures */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-07T02:25:25Z

Zifang: /* Archived */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-07T02:25:13Z

Zifang: /* Experiment setup & procedures */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-07T02:21:46Z

Zifang:

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-05T09:34:33Z

Zifang: /* References */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-05T09:01:21Z

Zifang: /* Introduction */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-05T08:57:38Z

Zifang: /* Theory */

Precision Measurement of Material and Optical Properties Using Interferometry

2026-04-05T08:38:03Z

Zifang: /* Background */

File:Interferometer setup.png

2026-03-24T13:26:00Z

Zifang: Zifang uploaded a new version of File:Interferometer setup.png

Precision Measurement of Material and Optical Properties Using Interferometry

2026-03-24T13:22:16Z

Zifang: /* Experiment setup & procedures */

File:Interferometer setup.png

2026-03-24T13:21:49Z

Zifang: Zifang uploaded a new version of File:Interferometer setup.png

Precision Measurement of Material and Optical Properties Using Interferometry

2026-03-24T13:16:54Z

Zifang:

File:Interferometer setup.png

2026-03-24T13:12:39Z

Zifang:

Precision Measurement of Material and Optical Properties Using Interferometry

2026-03-13T03:13:25Z

Zifang:

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: beamsplitter, two 25.4mm broadband dielectric mirrors (BB1-E02-10)

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Powermeter

=Introduction=
==Idea==
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentration by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrates the advantages of high-precision experimental technique.

==Background==

==Experiment setup & procedures==
A basic Michelson interferometer will be constructed for the experiment.

Procedure:

1. Construct the Michelson interferometer and verify the presence of interference fringes.

2. '''Refractive index measurement''' Place the test solution in one arm of the interferometer. Gradually vary the solution concentration by adding either distilled water or a higher-concentration solution to the container, and observe the resulting changes in the interference pattern.

3. '''Thermal expansion measurement''' To investigate the thermal expansion of the mirror, slowly heat one of the mirrors using a heating resistor. The temperature will be monitored with a thermocouple. Record how the interference pattern changes as the temperature increases.

=References=
[To be added]

Precision Measurement of Material and Optical Properties Using Interferometry

2026-03-13T02:55:12Z

Zifang: /* List of Apparatus */

=Team Members=
Yang SangUk

Zhang ShunYang

Xu Zifang

=List of Apparatus=

Light source: 780nm laser diode (GH0781RA2C SHARP diode) with LT230-B collimation tube

Optics: beamsplitter, 2 mirrors

Salt water of different concentrations in a transparent container (for determining the refractive index of solutions of different concentrations)

Powermeter

=Introduction=
==Idea==
We will be constructing an interferometer as a tool for precision measurement. One primary objective is to determine the refractive index of the solutions of different salt concentration by analyzing the resulting shift in interference fringes. Additionally, the thermal expansion of the metal sample will be measured by monitoring changes in the optical path length as the temperature of the sample varies. The project will highlight the relationship between wave optics and measurable physical parameters and illustrates the advantages of high-precision experimental technique.

==Background==

=References=