Sensing Rotation with Light: From Fiber Optic Gyroscope to...

Stanford University 1November 7–8, 2018 Stanford's 2018 PNT Symposium

Sensing Rotation with Light:�From Fiber Optic Gyroscope�

to Exceptional Points

Michel Digonnet

Applied Physics DepartmentStanford University


!  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

The Sagnac Effect in Vacuum

φS =2πΔLλ

=8π 2R2Ω

cλ= Scale factor x Ω

The two beams experience a Sagnac phase shift proportional to the rotation rate and the coil area


Inertial-navigation-grade FOGs must be able to measure ~10-3 of Earth rate

How Strong is the Sagnac Effect?

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-16

10-15

10-14

10-13

10-12

10-11

10-10

10-4 10-3 10-2 10-1 100 101 102

Phas

e sh

ift (r

ad)

Path difference (m)

Rotation rate (deg/hour)

FOG parameters:R = 5 cmN = 1000 turnsL = 314 mλ = 1.55 µm

Inertialnavigation

5.1 10-13 m

Diameter of thehydrogen atom

Earth rate

5.1 10 m-15

!  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

!  Requirement for strategic grade (navigating a submarine): –  A drift < 0.001 degree/hour (a 360-degree turn in ~40 days)–  Measuring a path length difference less than the size of a proton


!  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

The Sagnac Interferometer and Reciprocity


!  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 LiNbO3 circuit (MIOC)–  Broadband phase modulator (for square-wave phase biasing)–  High extinction ratio polarizer (~70 dB) (for reciprocity)

“Minimum configuration” of Fiber Optic Gyroscope


!  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?

Main Nonreciprocal Sources of Noise and Drift


!  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!

Errors Due To Coherent Backscattering

J. Mackintosh, and B. Culshaw, �J. of Lightwave Technol. 7(9), 1323-1328 (1989)


To eliminate polarization-induced drift, use:–  broadband light–  polarizer with high extinction ratio–  fiber with high holding parameter h

Errors Due To Nonreciprocal Polarization Coupling

!  Similar mechanism, except that coupling occurs between two polarizations!  Only detrimental component is again the coherent scattered fields


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!” (Anthony Lawrence, Modern Inertial Technology)

« Nature is rarely that cooperative » A. Lawrence« Murphy’s laws do not apply to fiber gyro » H. C. Lefèvre


!  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

World’s Most Sensitive Fiber Optic Gyroscope (1)


World’s Most Sensitive Fiber Optic Gyroscope (2)

Paturel et al., Gyroscopy and Navigation 5(1), 1–8, 2014

Longitude error after 38 days in a temperature-controlled environment is under 1/2 nautical mile,

or a drift of ~9 µdeg/hour!


Laser-Driven FOG


1.  Mean wavelength of a broadband source is difficult to stabilize "  Scale factor stability is limited to 10-100 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

Limitations of the SFS-Driven FOG

Broadband source makes it difficult for the FOG to be used�for inertial navigation of aircraft


Solution: The Laser-Driven FOG

Benefits of a semiconductor laser1. 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


!  Polarization-coupling drift dominates at large laser linewidth

Predicted Polarization-Coupling Errors

To achieve a drift low enough for aircraft inertial navigation, need to design a laser with a linewidth greater than ~40 GHz


!  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 Vrm–  Carrier suppression (important to reduce temporal coherence) �

is optimum for a specific Vrms

Gaussian White Noise Modulation Principle


!  Good agreement between polarization/backscattering models �over orders of magnitude of laser linewidths!

!  Lowest measured drift of 0.025 deg/h is at the navigation-grade requirement!

Measured Drift Dependence on Linewidth

A laser-driven FOG has almost the same low drift �as a conventional FOG!


Wavelength Stability of PRBS-Modulated Laser

!  Broadened laser mean wavelength stability is ~0.06 ppm!  Limited by the stability of the measurement instrument (OSA)!

Allan deviation of 10-GHz PRBS-broadened laser measured with optical spectrum analyzer

Laser-driven FOG has much better scale-factor stability than a conventional FOG!


Can the Sagnac Effect be Enhanced?


!  When index of medium is increased from 1 (vacuum) to n (silica):

1. Light takes longer to travel around the loopa  Time difference between cw and ccw signals increases as n2

2. Light is pulled by the moving medium (relativistic Fresnel-Fizeau drag) and takes less time to travel around the loopa  Time difference between cw and ccw signals decreases as 1/n2

!  The two effects cancel each other exactly: Sagnac phase shift is independent of n:

Sagnac Effect in Medium of Index n

H. Arditty, et al., Opt. Lett. 6, 401, 1981φS =

8π 2R2

cλΩ

H. Fizeau, Comp. Rend. 33, 349 (1851)

Scale factor{Slow or fast light do not affect the Sagnac phase shift


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


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, 023805, 2006

Great strain or temperature sensor, but detrimental for rotation sensing


Structural Slow Light in Resonant FOG (RFOG)

Sensitivity = dTdΩ

=dTdλ

dλdΩ

= Slope x Spectral shift

ACCUMULATED Sagnac phase shift IS enhanced because light travels through the loop multiple times


Sensitivity Comparison: FOG vs. RFOG

!  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)

2.07x only!


Does Coupling Resonators Improve Sensitivity?

No enhancement over a single-ring resonatorfor any combination of coupling κ (yet bigger and harder to stabilize)!

K. Zamani Aghaie et al., JOSA B 32, 339 (2015)


Many Other Coupled-Resonator Gyroscopes

Same conclusion:No enhancement over a single-ring resonator


Sagnac Phase Shift at an Exceptional Point!  Two coupled ring resonators, one with a gain g1 and the other with a loss g2,

constitute a parity-time-symmetric system

–  EP exists for a particular condition �on the coupling κ between rings:

–  Sagnac phase shift is proportional to √Ω

J. Ren et al., Opt. Lett. 42, 1556 (2017)

κ = g1 + g2( ) 2Satoshi Sunada, Phys. Rev. A 96, 033842 (2017)


!  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

!  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)

A Simple Interpretation of EP in Rotation Sensing

Finesse = 2πLoss


Huge Sagnac Phase Shifts at an Exceptional Point

At the exceptional point, �Sagnac frequency shift is enhanced by ~108!!


Conclusions


!  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


!  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 Dependence on Linewidth

Measured noise below aircraft navigation requirement�with a 3-GHz laser linewidth!

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Laser-Driven FOG Best Performance To Date

Laser-driven FOG

Aircraft navigationrequirement

Dominant residualcontribution

Random walk 5.5 10-4 deg/√h 10-3 deg/√h Backscattering

Drift 0.0068 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

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