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Sagnac Interferometer Sagnac interferometer Mahesh Gandikota Lab partner: Srijit Paul March 12, 2014 Contents 1 History 2 1.1 Absolute rotation . 2 1.2 Sagnac's interferometer . 2 1.3 Michelson-Gale interferometer . 3 1.4 Gyroscopes . 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 angular velocity 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 special relativity 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 Gyroscope 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 angular momentum. 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 speed of light 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.
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