Sagnac interferometer

Mahesh Gandikota Lab partner: Srijit Paul

March 12, 2014

Contents

1 History 2 1.1 ...... 2 1.2 Sagnac’s interferometer ...... 2 1.3 Michelson-Gale interferometer ...... 3 1.4 ...... 4

2 Theory 5 2.1 Laser ...... 5 2.2 Fabry-Perot etalon ...... 7 2.3 Interferometer ...... 8 2.3.1 Kinematic analysis ...... 8 2.3.2 Langevin method ...... 9 2.3.3 Sagnac’s formula ...... 10 2.3.4 Beats ...... 10

3 Procedure 11

4 Analysis 11 4.1 Beats ...... 11 4.2 Fourier transform graphs ...... 14

5 Observations 17

6 Conclusions 18

7 Acknowledgement 18

8 References 18

1 1 History

1.1 Absolute rotation A person moving with a uniform velocity in a vehicle over a road can find out that he is moving if he looks out of the window. However, by performing any experiment (mechanical/optical) inside the vehicle, he cannot conclude that he is in a state of uniform motion with respect to the ground. There are a set of frames which are called the inertial frames which are physically equivalent to each other. The co-ordinate transformations between such frames are given by the Lorentz transformations. However, when a disk is rotating, the motion is taken to be absolute1. An experimenter on the disk can conclude by the results of experiments that he is in a non-inertial rotating frame. He can also measure the of the disk. Foucault[10], in 1852, standing on earth could conclude that the earth is rotating and measured it’s angular speed using his pendulum. Sagnac interferometer is an optical analogue of Foucault’s pendulum which was used by Michelson and Gale to measure the angular velocity of the earth.

1.2 Sagnac’s interferometer In 1911, Franz Haress was working on his doctoral thesis and was trying to measure the Fresnel drag of light propagating through moving glass. He was getting an unexplained bias in his measurements which he could not account for.

Figure 1: Haress’s ring interferometer[2]

Sagnac[1] in 1913 did an experiment where he had a rotating table of diameter of 50 cm which firmly anchored various parts of an interferometer: an electric lamp and a telescope which focusses the fringes onto a fine-grained photographic plate which records the image. The table rotated about a vertical axis with a maximum frequency of 2Hz.

1I guess that’s the reason no one says that a disk is rotating with respect to a given inertial frame. They simply say that the disk is rotating.

2 Figure 2: Sagnac’s original interferometer[2]

For a rotational frequency of 2 turns per second, and an area enclosed by the circuit to be 860cm, for indigo light2, he found the fringe shift to be 0.07. He concluded that the results constitute a proof for the existence of a stationary aether. However Laue[6] had already showed that the results can alternately be explained by too which doesn’t suppose the existence of aether.

1.3 Michelson-Gale interferometer

Figure 3: Michelson-Gale interferometer[2]

2His lamp produced multi-coloured fringes. Indigo, having the least wavelength should be expected to show the best fringe shift.

3 Michelson[5] wrote,

Suppose it were possible to transmit two pencils of light in opposite directions around the earth parallel to the equator, returning the pencils to the starting- point. If the rotation of the earth does not entrain the aether, it is clear that one of the two pencils will be accelerated and the other retarded (relatively to the observing apparatus) by a quantity proportional to the velocity of the earth’s surface, and to the length of the parallel of latitude at the place; so that a measurement of the difference of time required for the two pencils to traverse the circuit would furnish a quantitative test of the entrainement.

The experiment results gave one more proof against aether drag hypothesis. However, this experiment is consistent with both stationary aether concept and relativity. The experiment was a major optical achievement considering that the whole of the 1.2 miles of trajectory was maintained at vacuum.

1.4 Gyroscopes is a device for measuring or maintaining the orientation of a vehicle, satellite, etc[11]. In the 19th century, with the advent of electric motors, gyroscopes were built by using the principle of conservation of . A ship for example had a heavy fly-wheel which was rotated by an electric motor with an angular velocity whose direction matched with the needle of the ship. Further into the sea, if the ship losed it’s orientation (needle not matching the axis of flywheel), then the sailor would know that the ship is moving the wrong way. The axis of the fly-wheel wouldn’t change to conserve the angular momentum.

Figure 4: A ship moving away from the shore

4 In the 21st century, the principle of constancy of is used in making more precise and much lighter gyroscopes. These gyroscopes are Sagnac interferometers.

2 Theory

2.1 Laser Charged particles emit electromagnetic radiation when they accelerate. Thermal radia- tion is such EM radiation produced due to the thermal motion of charged particles. By keeping a radiating source inside, say a cubic box; at thermodynamic equilibrium, the radiation field E~ is stationary. The field can be described by the superposition of plane waves. By the virtue of interference of these plane waves, a stationary field configuration may be ensured only when the plane-waves superpose to give standing waves[9]. This imposes certain boundary conditions on the wavelengths that can occur in this square cavity at thermal equilibrium. The permitted wavelengths are: 2L λ = (1) p 2 2 2 n1 + n2 + n3

where L is the length of the cube and n1, n2, n3 are positive integers. These standing waves are called cavity modes. At large L, an almost continuous distribution of modes are possible. The density of modes that occur at a given ν can be derived to be, 8πν2 n(ν) = (2) c3 The radiation can be seen to be isotropic in space as no particular direction is pre- ferred. Planck introduced the concept of photons to explain the experimentally observed energy density for cavities which have attained thermal equilibrium. The energy density corresponding to different frequencies is derived to be, 8πν2 hν ρ(ν) = 3 hν (3) c e kT − 1 Lasers on the other hand represent an extreme case of a non-thermal and an anisotropic radiation source. The radiation field is concentrated in a few modes and most of the ra- diation energy is emitted into a small solid angle. Lasers consist of an active medium which contains a broad band of absorption states, a metastable state and a ground state. This forms a three level laser. The active medium is contained between two mirrors, one of them partially silvered. This geometry is called the resonant cavity. Through optical pumping, population inversion from ground state to the metastable state is achieved. A few photons which are spontaneously emitted due to electrons jumping from the metastable to ground state, stimulate more electrons in the metastable state to decay and emit photons. This is called stimulated emission which causes a chain reaction and an intense laser beam comes out from the partially silvered mirror. Like a radiating source in a box which is in thermal equilibrium has certain modes, even a laser has modes called resonant modes. These are the modes that survive after

5 the diffraction losses of other modes which initially exist but die after repeated internal reflections in the cavity. The condition for a standing wave configuration is

mλm = 2L

where L is the length of the cavity and m is a positive integer. Thus, the possible longitudinal modes in the resonant cavity are, 2L λ = (4) m m From the Kirchoff diffraction theory, it can be shown that transverse modes are possible too in the resonant cavity.

Figure 5: The three level system [8]

6 Figure 6: Few TEM modes in a laser[8]

2.2 Fabry-Perot etalon To select a particular mode, a Fabry-Perot etalon is used. By changing the angle which the normal makes with the laser beam, the T00 mode may be selected.

Figure 7: Etalon tuned to resonance - cw and ccw beam have same frequency [7]

The cw and the ccw frequency are the same at the resonance orientation of the etalon. When the interferometer rotates, the frequencies of both change due to Doppler shift.

7 2.3 Interferometer In this section, the working formula of Sagnac effect will be derived by kinematic analysis and Langevin’s[3] approach. The notations used are as in Post[2].

2.3.1 Kinematic analysis

Figure 8: Simplified Sagnac configuration [2]

A platform of radius R is rotating with angular velocity Ω in clockwise direction. Two beams of light, one clockwise(cw), other counter clock-wise(ccw) beam leave the beam- splitter whose initial position is at C. We use here that the speed of light is independent of the source speed. The ccw beam has to travel a lesser distance to meet the beam-splitter at C0. The cw beam travels more distance to meet the beam-splitter at C00.

2πR − ∆s0 = cτ 0 (5) 2πR + ∆s00 = cτ 00

Also, ∆s0 = vτ 0 = ΩRτ 0 (6) ∆s00 = vτ 00 = ΩRτ 00 00 0 Defining ∆τs ≡ τ − τ to be the time-difference between the two beams reaching the beam-splitter, a little algebra gives 4AΩ τ = (7) s c2 − (ΩR)2

A = πR2 is the area of the circle. Subscripts s, m stand for stationary and moving frame respectively. When the instrument which measures the time-difference is aboard the rotating appa- ratus, then an appropriate co-ordinate transformation has to be made. To comply with

8 relativity, a is to be done according to which the time-difference measured in stationary frame is dilated by a factor γ. So, ∆τ ∆τ = s (8) m γ As r (ωR)2 γ = 1 − (9) c2 we get 4AΩ 4AΩ  (ΩR)2 −1 ∆τ = = 1 − (10) m γ(c2 − (ΩR)2) c2 c2 3 From (8), we can see that, at non-relativistic speeds, ∆τm(NR) = ∆τs(NR) . We don’t have to consider when we transform to the co-ordinates of the rotating frame of reference. To explain the results as measured in a rotating frame at non-relativistic speeds, Galilean transformations are enough. At non-relativistic speeds, the Sagnac effect is a simple consequence of the source independence of the speed of light. In other words, the Sagnac experiment does not distinguish between pre-relativistic and relativistic physics[4].

2.3.2 Langevin method The line-element in space-time in polar co-ordinates is, ds2 = c2dt2 − dr2 − r2dφ2 (11) The Lorentz transformation between a rotating and a stationary frame is (super-script is for rotating frame), dt = γdt0 dr = dr0 (12) dφ = dφ0 + γΩdt0 The line-element vanishes for a light-like path. So, in our case γ2c2dt02 − R2(dφ0 + γΩdt0)2 = 0 (13) Solving for dt0, Rdφ0 dt0 = (14) (c ∓ ΩR)γ The ∓ corresponds to the cw and ccw propagation of light. By integrating from 0 → −2π and 0 → 2π, 2πR t0 = 1 (c − ΩR)γ (15) 2πR t0 = 2 (c + ΩR)γ 0 0 ∆τm ≡ t1 − t2 4AΩ ∆τ = (16) m γ(c2 − Ω2R2)

(10)=(16), the results of both methods result in the same expression for ∆τm. 3NR denotes non-relativistic.

9 2.3.3 Sagnac’s formula Though the results were derived by taking a circular path of light, the same result holds true for any path (even for Sagnac’s original setup - Fig(2)). The corresponding area enclosed (A) by the path should be inserted. The path difference may be calculated, c ∆φ = 2π ∆τ (17) λo

λo is the wavelength of light in an inertial frame. The number of fringes shifted is just,

∆φs AΩc ∆zs = = 2 2 (18) 2π λo(c − v ) v = ΩR. At non-relativistic speeds, by an approximation v << c, AΩ ∆zm = (19) λoz This is the form of the formula that appears in Sagnac’s paper[1].

2.3.4 Beats For optical rings, detection of the frequency shift is easier than the detection of a fringe pattern shift. The frequency of beats generated due to Doppler effect is big to be detected easily by electronic means (photodetector). The frequency shift of a reflected beam from a moving mirror (with velocity v) is v ∆ν = 2ν c For two waves circulating in opposite directions in a square geometry, the frequency difference between the two beams would become 4vν 4v 4Lω 4L2ω 4Aω ∆ν = = = = = c λ λ Lλ Lλ For the given geometry of an isosceles triangle (of length l) of our experiment, the working formula is, l ∆ν = √ ω = βω (20) 3λ The value of β in our experiment is 6.6753 Hz/grad.

10 3 Procedure

Other than the necessary procedures mentioned in the instruction manual, we did the following:

1. A majority of the light was not falling on the photo-diode due to it’s improper height. We adjusted the height of the photo-diodes using extra screws underneath them.

2. The FFT of the apparatus’ software was not working. We were trying to get the FFT done in the oscilloscope instead. As that too was not much helpful, we FFTed the waveforms ourselves using MATLAB.

3. Once the laser is ignited, a Fabry-Perot etalon is used to select a particular mode for the experiment. However, finding the right orientation for the Fabry-Perot is difficult by trial and error. A good way to find the right orientation is by re-installing the green-laser and making the bright spots on the mirror meet by adjusting the etalon.

Figure 9: Experimental setup[7]

4 Analysis

4.1 Beats

11 Figure 10: X scale: Time (s), Y scale: Voltage (mV); Angular velocity = 1 deg/s (Data 1)

Figure 11: X scale: Time (s), Y scale: Voltage (mV); Angular velocity = 1 deg/s (Data 2)

12 Figure 12: X scale: Time (s), Y scale: Voltage (mV) Angular velocity = 4 deg/s

Figure 13: X scale: Time (s), Y scale: Voltage (mV) Angular velocity = 8 deg/s

13 • The red + signs are the data points of Channel 1, pink square boxes are the data points of channel 2.

• Each set of data points are connected by lines. Also a smooth bezier is fitted through it in GNUplot.

• It can be seen that both the channels differ by phase π. This is expected as the ccw beam gets reflected at M4 to go to PD1 and acquires a phase difference of π with respect to cw beam which directly goes to PD2. Refer Fig.(14).

Figure 14: [7]

4.2 Fourier transform graphs The fast Fourier transform of the beat graphs were done using MATLAB.

14 20 'fft' 'fft' 18 'fft' u 3:4 'fft' u 3:4

16

14

12

10 Amplitude 8

6

4

2

0 0 2000 4000 6000 8000 10000 12000 Frequency (Hz)

Figure 15: Angular velocity = 1 deg/s (Data 1)

14 'fft' u 5:6 'fft' u 5:6 'fft' u 7:8 12 'fft' u 7:8

10

8

Amplitude 6

4

2

0 0 2000 4000 6000 8000 10000 12000 Frequency (Hz)

Figure 16: Angular velocity = 1 deg/s (Data 2)

15 20 'fft' u 9:10 'fft' u 9:10 18 'fft' u 11:12 'fft' u 11:12

16

14

12

10 Amplitude 8

6

4

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0 0 5000 10000 15000 20000 25000 Frequency (Hz)

Figure 17: Angular velocity = 4 deg/s

10 'fft' u 13:14 'fft' u 13:14 9 'fft' u 15:16 'fft' u 15:16

8

7

6

5 Amplitude 4

3

2

1

0 0 5000 10000 15000 20000 25000 Frequency (Hz)

Figure 18: Angular velocity = 8 deg/s

16 • The major (non-spurious) peaks have been identified (sheet attached with report).

• The FFT for angular speed 8 deg/s, is expected to show a peak at double the frequency at which the peak of the angular speed 4 deg/s occurs. This is ≈ 44kHz. However, as the FFT’s frequency axis doesn’t stretch till there, we were not able to investigate the peak in that neighbourhood.

• We are able to get only two unique data points for the experiment. This was for angular speeds 1deg/s and 4deg/s. A straight line is plotted connecting these two points.

20000

15000

10000 \Delta \nu (Hz)

5000

0 0 1 2 3 4 5 Angular speed (deg/s)

Figure 19: Straight line ‘fit’

The slope of the line is β = 6.04 kHz s/deg. The calculated value is β = 6.675 khZ s/deg.

5 Observations

• While aligning the etalon, we could see that the laser was igniting for more than one orientation. Each position of the etalon resulted in the selection of different modes.

• The interference of the beams may be seen on the wall by putting a beam expander at any of the photo-diodes. One can see stripes of red and darkness. The quality of this interference pattern is important in making a judgement of the mixing of the two beams.

• By not using the setup for a weekend, the laser won’t ignite due to the dust accu- mulation on Brewster windows. Many times the laser re-ignites after cleaning the windows. Re-adjusting of the setup should be done only after this is tried.

• It is beneficial to spend the whole day, from morning till night, to get usable results. Otherwise, one would be stuck in igniting the laser and putting the etalon everyday.

17 6 Conclusions

The laser of the active interferometer was lit, gaussian mode selected by using an etalon and the beams were made to interfere at the photo-diodes. The beats were obtained for three different angular velocities: 1, 4 and 8 deg/s. However, only the first two data points could be used as the frequency axis of the FFT of the the data set of 8 deg/s does not extend to the required neighbourhood. The measured β was 6.04 kHz s/deg which has an error of 9.5% compared to the calculated β. More data (atleast one extra data point) could have enabled to estimate the variance of our measurements. Lock-in-threshold was not studied due to lack of data.

7 Acknowledgement

We thank Ritwick sir for his guidance. We also thank Gaurav Nirala for his experience and labour; Sourabh for doing FFT of the graphs.

8 References

1. G.Sagnac - in French: L’ ´etherlumineux ’emontr´epar l’effet du vent relatif d’´ether dans un interf´erom`etreen rotation uniforme (The demonstrated by the effect of the wind relative to the aether in a uniformly rotating interferome- ter), Comptes Rendus, 157: 708-710 2. E.J.Post - Sagnac effect, Reviews Of Modern Physics, Vol39, 2, April 1967 3. Langevin P., 1921, Compt. Rend. 173, 831 4. Brown, Kevin. “The Sagnac Effect”. MathPages 5. A.A. Michelson, Philosophical Magazine, S. 6, Vol. 8. No. 48, Dec. 1904, pp. 716-719 6. - in German: Uber¨ einen Versuch zur Optik der bewegten K¨orper (On an Experiment on the Optics of Moving Bodies), M¨unchener Sitzungsberichte 7. Luhs - He Ne laser gyroscope, instruction manual 8. Hecht - Optics (Fourth edition) 9. Demtr¨oder- Laser Spectroscopy - Vol.1 (Fourth edition) 10. L. Foucault, 1852 - in French: ‘Sur les ph´enom`enesdorientation des corps tournants entran´espar un axe fixe `ala surface de la terre Nouveaux signes sensibles du mouvement diurne (On the phenomena of the orientation of rotating bodies carried along by an axis fixed to the surface of the earth New perceptible signs of the daily movement), Comptes rendus hebdomadaires des s´eancesde l’Acad´emiedes Sciences (Paris), vol. 35, pages 424427. 11. Wikipedia

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