Laser Gyroscope

Some Description

Team members

Darren e0303300@u.nus.edu

Wen Xin e0309271@u.nus.edu

Idea

This project aims to measure some rotation by using the Sagnac effect, by using a loop of fiber as a ring interferometer.

Sagnac effect

When two beams of light are sent into the two ports of a ring interferometer, they travel clockwise (CW) and counter-clockwise (CCW) paths respectively. For a stationary ring interferometer, since the path length traversed by each beam is the same, this leads to interference at the output port.

In the following derivation, we assume the interferometer consist of a singular circular loop. Let's say the interferometer is spinning in the CW direction with rotation rate $\mathrm{\Omega }$. Then there will be a change in path length due to the rotation given by

$\mathrm{\Delta }L=R\mathrm{\Omega }{t}_{CW/CCW}$,

where R is the radius of the loop and $t_{CW/CCW}$ is the time it takes for the light to travel clockwise or counter clockwise in the loop.

which results in the following $t_{CW/CCW}$

$t_{CW}=\frac{2\pi nR+nR\mathrm{\Omega }{t}_{CW}}{c},t_{CCW}=\frac{2\pi nR-nR\mathrm{\Omega }{t}_{CCW}}{c}$

where n is the refractive index of the medium. Simplifying,

$t_{CW}=\frac{L_0}{c+nR\mathrm{\Omega }},t_{CCW}=\frac{L_0}{c-nR\mathrm{\Omega }},L_0=2\pi nR,\mathrm{\Delta }t=t_{CW}-t_{CCW}=\frac{L_0\cdot 2R\mathrm{\Omega }}{c^2}\cdot \frac{1}{1+{\left(\frac{R\mathrm{\Omega }}{c}\right)}^2}$

as $R\mathrm{\Omega }<<c$,

$\mathrm{\Delta }t\approx \frac{2R{L}_0}{c^2}\cdot \mathrm{\Omega }$

If we use multiple loops N instead,

$\mathrm{\Delta }t=\frac{4N\pi n{R}^2}{c^2}\cdot \mathrm{\Omega }$

Thus, for light of frequency f, we can calculate the Sagnac phase shift:

$\mathrm{\Delta }{\varphi }_S=2\pi f\cdot \mathrm{\Delta }t=\frac{8\pi nNA}{\lambda c}\cdot \mathrm{\Omega }$

The change in the phase difference between the two paths changes the degree of constructive or destructive interference that happens, resulting in a change in the intensity detected. This is illustrated by the sketch of the graph below. As such, the change in the voltage across the detector corresponds to $\mathrm{\Omega }$.

The derivation so far only assumes that we only have 1 polarization of light. However in non-polarizing maintaining fiber, the birefrigence in the fiber causes a change in the polarization of the light when it travels through the fiber, so the resulting light that travels in the CW and CCW direction will in general have different polarizations, resulting in the inability to achieve complete constructive or destructive interference. We will try to ignore effects from polarization first and try to overcome problems with it if necessary.

Parts

50m smf28 fiber spools

20km smf28 fiber spools

2x Thorlabs FGA01FC photodiodes

1550nm Fiber circulator

Fiber paddle

Laser driver

1550nm Laser diode

1550nm Fiber isolator

1st Setup

11 and 14 March The main setup will be a fiber based Sagnac interferometer. The setup is as shown below:

Parts used are: 1x laser driver 1x 1550nm DFB laser diode 1x fiber isolator to prevent backreflections to the laser diode 1x 50:50 fiber beam splitter at 1550nm 1x 50m smf28 fiber coil 1x FGA01FC photodiode a lot of fiber to fiber connectors 1x 1m smf28 fiber for connecting beam splitter to the photodiode 2x apc to pc smf28 fiber

All components are taped to the fiber loop and the fiber loop is placed on top of a rotating chair. The interferometer output is sent to a photodiode which is connected to an oscilloscope.

During out 1st iteration, rotating the 50m fiber coil causes a change in the voltage output of the photodiode. However, the change in the voltage output remains even after the fiber coil is stationary, which should not happen since the voltage change we think we should observed should only come from movement and should not depend on the final orientation.

To further investigate, we removed the fiber loop and connected only the 2 ends of the fiber beam splitter together. In practice, rotating this setup should show a very small change or no change since the area enclosed by the fiber is much smaller as compared to the fiber coil. However, we still observe the same voltage change as before. Suspected that the voltage change we observe might be due to birefrigence in the fiber instead. Fiddling with the fiber, the voltage change was observed as well. So the voltage change is likely due to the polarization change when the fiber was moving instead. This would cause a change in the intensity of the interfereometer output.

Stray fibers were also taped down to prevent them moving and changing the polarization of the light but it does not seem to work.

Planned to investigate this further using a circulator and a seperate photodiode to conver the sagnac configuration into a mach-zender configuration and look at the output of both ports.

2nd setup

18 March

File:SecondSetup picture.jpeg

We moved all the components to an optical breadboard. All fibers are taped down to the breadboard with only 2 fibers untaped. 1 from the 1550nm laser diode and the other leading to the photodiode. With this new setup, we seem to get the expected behaviour? A voltage change from the photodiode is only observed when the fiber coil is moving and returns to its original value when its stationary. So with this setup, we minimized the amount of polarization change in the fiber that occurs when we rotate the chair.

In order to verify the rotation rate of the fiber coil, we placed a phone with a gyroscope readout enabled (phyphox). We then turn the chair clockwise and counter-clockwise continuously, and plot the results below:

The orange trace represents the output from the phone's gyroscope, while the blue trace and green trace represents the voltage readings from PD1 (connected to the 50:50 beam splitter) and PD2 (connected to the circulator) respectively. Clearly, we do observe correlation between when the chair is rotated, and when fluctuations in the PD voltage outputs are observed. We also see that the fluctuations from PD1 and PD2 are anti-correlated, which verifies the observation of interference.

However, there are clearly several issues from this measurement. First, we observe much greater noise, and much lower sensitivity from PD2's output. We suspect that the circulator might be the cause of this. Given that, we will mainly be looking at the PD1 voltage readings. Second, we expect the voltage signal to change one way when the chair is rotated CW, and to change the other way when the chair is rotated CCW. The plot of PD1 voltage above shows that when the rotation rate $\mathrm{\Omega }$ is negative (CW), the voltage change is positive, whereas when $\mathrm{\Omega }$ is negative (CCW), the voltage change can be both positive or negative. We suspect that this might be due to the operating point being close to a minima in the interference pattern (the sketch in the idea section), and the CCW rotation is sufficient to push the signal over the minima.

We also observe fluctuations at the peaks of the rotation rate when $\mathrm{\Omega }<0$, which does not agree with the relatively smooth rotations as indicated by the phone gyroscope readings.

Third setup

File:ThirdSetup image.jpeg

The results from the previous setup seems to indicate some improvements can still be made. The first improvement is to include a polarization paddle at one side of the fiber coil, which should allow us to tune the polarization of the light interfering at the 50:50 BS to some extent. We also noticed that the fiber from the laser source to the isolator was hanging, and lightly disturbing this fiber led to small but visible fluctuations in the PD readings. So we squeezed the laser source onto the chair (this seemed to be the cause of the fluctuations discussed at the end of the previous section).

For our first measurement, we adjusted the polarization in the interferometer such that the voltage reading the photodiode is midway between the maximum and minimum voltage. With just this change, we already see a large improvement compared to the second iteration. We can clearly correlate both CW and CCW rotations.

Next, we would like to verify our claim during our second iteration that we are at the minima or maxima of the interference pattern, causing rotations in either direction to create the same voltage change. This was done by adjusting the polarization till the voltage of the photodiode was at its minimum or maximum. In both cases, we see that both CW and CCW rotations causes either only a positive or negative change in voltage, verifying our claim. The difference in voltage change for CW and CCW rotations can be attributed to how far away we are from actual maximum or minimum voltage.

Lastly, we would like to see how sensitive our laser gyroscope is. With a slow CCW rotation, we can see that our setup is sensitive to rotation on the order of 0.1rad/s or 5.7deg/s.

Fourth Setup

Since we have a 20km fiber loop lying around, we decided to replace the 50m fiber loop with the 20km fiber loop.