Sensing Rotation with Light: From Fiber Optic Gyroscope to Exceptional Points

Transcription

1 Sensing Rotation with Light: From Fiber Optic Gyroscope to Exceptional Points Michel Digonnet Applied Physics Department Stanford University Stanford University 1

2 The Sagnac Effect in Vacuum! The fiber optic gyroscope (FOG) is based on the Sagnac effect Light beams propagating in opposite directions in a rotating frame experience a different optical path length! The relative phase difference is the Sagnac phase shift φ S = 2πΔL λ = 8π 2 R 2 Ω cλ = Scale factor x Ω The two beams experience a Sagnac phase shift proportional to the rotation rate and the coil area Stanford University 2

3 How Strong is the Sagnac Effect? Phase shift (rad) FOG parameters: R = 5 cm N = 1000 turns L = 314 m λ = 1.55 µm Inertial navigation Diameter of the hydrogen atom Rotation rate (deg/hour) m Earth rate! Requirement for inertial grade (navigating an aircraft): A drift < 0.01 degree/hour Measuring a path length change of 0.005% of the diameter of hydrogen Inertial-navigation-grade FOGs must be able to measure ~10-3 of Earth rate! Requirement for strategic grade (navigating a submarine): A drift < degree/hour (a 360-degree turn in ~40 days) Measuring a path length difference less than the size of a proton Path difference (m) Stanford University 3

4 The Sagnac Interferometer and Reciprocity! Sagnac phase shift is measured with a Sagnac interferometer! Reciprocity is the single most important feature of a Sagnac interferometer Common path prevents the cw and ccw signals from seeing different phase shifts (other than the Sagnac phase shift)! To guarantee reciprocity, use: a single-mode fiber throughout a circulator to tap the reciprocal return signal out of the input fiber a polarization-maintaining fiber so that both signals have the same polarization a polarizer to select the same input and output polarization Stanford University 4

5 Minimum configuration of Fiber Optic Gyroscope! Push-pull phase modulators provide a differential phase shift between the cw and ccw signals that biases the interferometer for maximum sensitivity! Y junction, polarizer, and phase modulators are fabricated on a compact LiNbO 3 circuit (MIOC) Broadband phase modulator (for square-wave phase biasing) High extinction ratio polarizer (~70 db) (for reciprocity) Stanford University 5

6 Main Nonreciprocal Sources of Noise and Drift! Three fundamental non-reciprocal effects taking place in the fiber overwhelm the Sagnac effect Backscattering Polarization coupling Optical Kerr effect! When FOG is interrogated with a laser, they all induce significant noise in the gyroscope output! They also induce significant drift, indistinguishable from a rotation-induced signal How to eliminate them? Stanford University 6

7 Errors Due To Coherent Backscattering J. Mackintosh, and B. Culshaw, J. of Lightwave Technol. 7(9), (1989)! Fiber defects backscatter fields that add to the two main signals Interference converts light frequency noise into output intensity noise Fiber temperature fluctuations produce output drift To eliminate backscattering noise and drift, use a broadband light source! Stanford University 7

8 Errors Due To Nonreciprocal Polarization Coupling! Similar mechanism, except that coupling occurs between two polarizations! Only detrimental component is again the coherent scattered fields To eliminate polarization-induced drift, use: broadband light polarizer with high extinction ratio fiber with high holding parameter h Stanford University 8

9 The SFS-Driven FOG! One solution a broadband light source solved all three problems and was instrumental in the success of the FOG «Nature is rarely that cooperative» A. Lawrence «Murphy s laws do not apply to fiber gyro» H. C. Lefèvre Nature is rarely that cooperative! (Anthony Lawrence, Modern Inertial Technology) Stanford University 9

10 World s Most Sensitive Fiber Optic Gyroscope (1)! In inertial navigation mode, gyros follows Earth s movement! At 48 of latitude in Paris, the rotation s tangential speed of Earth is 1,100 km/h! Test performed over 38 days at rest.. which means traveling over one million kilometers!! Position (longitude) is found from measurement of rotation rate, which gives the value of tangential speed! Experiment used a prototype IMU fiber gyro with 3-km coils on a 20-cm-diameter spool, temperature stabilized to ~ 0.2 C Borrowed from Hervé Lefèvre, ixblue, France Stanford University 10

11 World s Most Sensitive Fiber Optic Gyroscope (2) Longitude error after 38 days in a temperaturecontrolled environment is under 1/2 nautical mile, or a drift of ~9 µdeg/hour! Paturel et al., Gyroscopy and Navigation 5(1), 1 8, 2014 Stanford University 11

12 Laser-Driven FOG Stanford University 12

13 Limitations of the SFS-Driven FOG 1. Mean wavelength of a broadband source is difficult to stabilize " Scale factor stability is limited to ppm (aircraft navigation requires ~1 ppm) 2. Broadband light sources have large excess intensity noise " Limits noise to 5-20 times the aircraft-navigation requirement Broadband source makes it difficult for the FOG to be used for inertial navigation of aircraft Stanford University 13

14 Solution: The Laser-Driven FOG Benefits of a semiconductor laser 1. Highly stable wavelength (< 0.1 ppm) # Excellent scale-factor stability (<0.1 ppm) 2. Negligible excess noise # Reduced noise and higher sensitivity for the FOG 3. More efficient, fewer components, cheaper than a broadband source # Lower cost and power consumption Stanford University 14

15 Predicted Polarization-Coupling Errors! Polarization-coupling drift dominates at large laser linewidth To achieve a drift low enough for aircraft inertial navigation, need to design a laser with a linewidth greater than ~40 GHz Stanford University 15

16 Gaussian White Noise Modulation Principle! To broaden a laser to the tens of GHz range, modulate its phase externally with an electro-optic modulator (EOM) driven by noise! Produces a laser spectrum with a Gaussian spectrum Linewidth ~4 times the EOM bandwidth Linewidth increases with increasing V rm Carrier suppression (important to reduce temporal coherence) is optimum for a specific V rms Stanford University 16

17 Measured Drift Dependence on Linewidth! Good agreement between polarization/backscattering models over orders of magnitude of laser linewidths!! Lowest measured drift of deg/h is at the navigation-grade requirement! A laser-driven FOG has almost the same low drift as a conventional FOG! Stanford University 17

18 Wavelength Stability of PRBS-Modulated Laser Allan deviation of 10-GHz PRBS-broadened laser measured with optical spectrum analyzer! Broadened laser mean wavelength stability is ~0.06 ppm! Limited by the stability of the measurement instrument (OSA)! Laser-driven FOG has much better scale-factor stability than a conventional FOG! Stanford University 18

19 Can the Sagnac Effect be Enhanced? Stanford University 19

20 Sagnac Effect in Medium of Index n! When index of medium is increased from 1 (vacuum) to n (silica): 1. Light takes longer to travel around the loop a Time difference between cw and ccw signals increases as n 2 2. Light is pulled by the moving medium (relativistic Fresnel-Fizeau drag) and takes less time to travel around the loop a Time difference between cw and ccw signals decreases as 1/n 2! The two effects cancel each other exactly: Sagnac phase shift is independent of n: φ S = 8π 2 R 2 cλ Ω H. Arditty, et al., Opt. Lett. 6, 401, 1981 { Scale factor H. Fizeau, Comp. Rend. 33, 349 (1851) Slow or fast light do not affect the Sagnac phase shift Stanford University 20

21 Experimental Proof of Independence on Index! Measure phase shift induced by moving a portion of a Sagnac loop made either with Conventional fiber (n 1.44) Air-core fiber (n 0.95)! Observation: same phase shift for both fibers, equal to the Sagnac phase shift R. Wang, et al., Phys. Rev. Lett. 93, 14, 2004 Stanford University 21

22 Atomic Slow Light in Non-reciprocal Sagnac Loops! Light is slowed down in a rubidium cell only in one direction Cw sees a much longer delay Differential phase shift now proportional to group index! Observations: Huge differential phase delay Used to measure EIT dispersion! Impact on rotation sensing: No effect on Sagnac phase shift No longer reciprocal Greatly increases temperature sensitivity G. Purves, et al., Phys. Rev. A 74, , 2006 Great strain or temperature sensor, but detrimental for rotation sensing Stanford University 22

23 Structural Slow Light in Resonant FOG (RFOG) Sensitivity = dt dω = dt dλ dλ dω = Slope x Spectral shift ACCUMULATED Sagnac phase shift IS enhanced because light travels through the loop multiple times Stanford University 23

24 Sensitivity Comparison: FOG vs. RFOG 2.07x only!! RFOG s maximum sensitivity twice as high as the FOG s Only because RFOG utilizes two outputs and the FOG only one! Main saving offered by RFOG is significantly shorter fiber (up to x10) Stanford University 24

25 Does Coupling Resonators Improve Sensitivity? K. Zamani Aghaie et al., JOSA B 32, 339 (2015) No enhancement over a single-ring resonator for any combination of coupling κ (yet bigger and harder to stabilize)! Stanford University 25

26 Many Other Coupled-Resonator Gyroscopes Same conclusion: No enhancement over a single-ring resonator Stanford University 26

27 Sagnac Phase Shift at an Exceptional Point! Two coupled ring resonators, one with a gain g 1 and the other with a loss g 2, constitute a parity-time-symmetric system J. Ren et al., Opt. Lett. 42, 1556 (2017) EP exists for a particular condition on the coupling κ between rings: κ = ( g 1 + g 2 ) 2 Sagnac phase shift is proportional to Ω Satoshi Sunada, Phys. Rev. A 96, (2017) Stanford University 27

28 A Simple Interpretation of EP in Rotation Sensing! In RFOG, light travels N times around the ring, where N is approximately the finesse of the resonator Sagnac phase accumulated by light is N times larger than in a FOG For equal loop radius, RFOG is ~N times more sensitive than a FOG! Finesse is limited by the loss Finesse = 2π Loss! If an amplifier is added to the ring, effective loss is reduced, finesse and sensitivity increase! A similar behavior occurs near an exceptional point: When gain = loss, lasing condition becomes very sensitive to feedback from the loss loop (a very narrowband reflector), and to phase changes (e.g., Sagnac phase shift) Stanford University 28

29 Huge Sagnac Phase Shifts at an Exceptional Point At the exceptional point, Sagnac frequency shift is enhanced by ~10 8!! Stanford University 29

30 Conclusions Stanford University 30

31 ! Broadband FOGs Can detect one rotation in 45 centuries! Needs improved scale-factor stability, compactness, cost! Laser-driven FOGs All metrics at inertial-navigation grade, approaching strategic grade! What enhances the accumulated Sagnac phase shift? Single resonator (RFOG): ~10-fold in a silica fiber Coupling resonators of equal loss does not do better than RFOG Atomic slow light does not! Enticing near-term prospects Exceptional points do (possibly orders of magnitude) Fast light Summary Stanford University 31

32 Measured Noise Dependence on Linewidth THIS SLIDE COULD GO! Measured noise agrees with combined models of backscattering and polarization coupling Backscattering noise dominates For large linewidths, noise decreases as linewidth -1/2 Measured noise below aircraft navigation requirement with a 3-GHz laser linewidth! Stanford University 32

33 Laser-Driven FOG Best Performance To Date THIS SLIDE COULD GO Laser-driven FOG Aircraft navigation requirement Dominant residual contribution Random walk deg/ h 10-3 deg/ h Backscattering Drift deg/h 0.01 deg/h Polarization coupling Scale-factor stability 0.15 ppm 1 5 ppm Measurement instrument (OSA) Exceeds requirements for aircraft navigation Stanford University 33