Precision Measurement of Material and Optical Properties Using Interferometry

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 $\lambda$ and field amplitude $E_0$. 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 $L_1$ and $L_2$, the output field:

$E_{out}=E_1+E_2=\frac{1}{2}{E}_0{e}^{i2k{L}_1}(1+e^{i2k\mathrm{\Delta }L})$

where $\mathrm{\Delta }L=L_2-L_1$ and $k=2\pi /\lambda$. Field maxima occur whenever

$\frac{4\pi \mathrm{\Delta }L}{\lambda }=2m\pi ,$

and minima when

$\frac{4\pi \mathrm{\Delta }L}{\lambda }=(2m+1)\pi ,$

where $m$ 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 $L$ 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 $n$. The presence of the solution changes the optical path length in that arm. The additional optical path difference introduced by the solution is

$\mathrm{\Delta }\text{OPL}=2(n-1)L,$

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

The corresponding phase change is

$\mathrm{\Delta }\varphi =\frac{2\pi }{\lambda }\cdot 2(n-1)L.$

As the concentration of the solution changes, the refractive index $n$ 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 $n$ 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:

Change → Phase → OPL → Diffractive Index → Salt Concentration

Let us assume the container originally contains pure water, which has a refractive index of $n_0=1$. The change in OPL can be expressed by:

$\mathrm{\Delta }OPL=2(n-1)L$

Where n is the solution diffractive index, L is the physical path length. The constant 2 in this equation is due to the fact that light travels in both the forward and the backward direction through the solution.

Experiment setup & procedures

A basic Michelson interferometer will be constructed for the experiment.

Interferometer Alignment and Stability

Figure 1: Schematic of the Michelson interferometer. OI: optical isolator; M: mirror; BS: beam splitter; PM: powermeter

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.

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.

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.

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.

$\mathrm{\Delta }\varphi =\frac{2\pi }{\lambda }\cdot 2\mathrm{\Delta }nL.$

$\mathrm{\Delta }n=\frac{\lambda \mathrm{\Delta }\varphi }{4\pi L}$

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

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