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Mathematical Model of Nonlinear Coupling and its
Effect on Rotation Sensitivity of Semiconductor
Ring Laser Gyroscope
Arpit Khandelwal1, Azeemuddin Syed1, Jagannath Nayak2
1CVEST IIIT Hyderabad, 2RCI DRDO Hyderabad, India.
Email: arpit.khandelwal@research.iiit.ac.in
Abstract—The moving population inversion gratings formed
due to spatial hole burning inside a semiconductor ring laser
gyroscope causes nonlinear coupling of the counter-traveling
modes. We mathematically demonstrate this phenomenon using
Perturbation Theory and calculate the limit imposed by it on the
sensitivity of inertial rotation.
I. INTRODUCTION
Integrated on-chip semiconductor ring laser gyroscopes
(SRLG) hold the promise of reliable, compact, low cost and
low power alternative to the bulky He-Ne ring laser gyros [1].
These devices tend to reduce the overall size of the attitude
and heading reference system (AHRS) of missiles, aircrafts
and satellites, thus increasing their payload carrying capability
and reducing the cost simultaneously [2]. Many designs of
integrated SRLG have been proposed and implemented, but
have failed to achieve the high performance levels required for
military applications because of fairly high lock-in threshold
as compared to He-Ne RLG [3]. The lock-in threshold in any
gyro depends upon the strength of backscattering and coupling
between the clockwise (CW) and counterclockwise (CCW)
modes inside the ring laser.
In this paper, we mathematically model the nonlinear cou-
pling between the CW and CCW waves inside a SRLG, which
occurs due to backscattering from the moving population
inversion gratings. Also, the effect of coupling on the rotation
sensitivity of the SRLG is also analyzed. The analysis shows
that, nonlinear coupling leads to frequency synchronization
of the CW and CCW modes causing the suppression of beat
signal and high value of lock-in threshold.
II. MATH EM ATIC AL ANALYS IS
In the presence of two counter-traveling modes inside the
ring laser gyro, the temporal and spatial variations of the re-
sultant intensity causes similar variations in the carrier density
leading to the formation of grating like structure as shown in
Fig 1. As the gyro rotates, the gratings start moving because
of the frequency difference between the CW and CCW modes.
The dependence of refractive index of the semiconductor gain
medium on its carrier density causes the counter-traveling
modes to backscatter and they couple to each other as shown
in Fig. 1 [4]. Thus, the overall electric field inside the ring
cavity is given by
Fig. 1: Backscattering of modes from population inversion
grating formed inside the semiconductor gain medium
¯
E=U(x, y)[Efej(ω1t−β1z)+Ebej(ω2t+β2z)](1)
where U(x, y)is the transverse mode distribution, Efand
Eb,ω1and ω2,β1and β2are the amplitudes, frequency and
wave vector of forward and backreflected wave respectively.
In order to get the coupling between the forward and reflected
wave, we substitute Eq. (1) into wave equation
∆2¯
E−
c2∂2¯
E
∂t2= 0 (2)
Next, we apply Perturbation Theory [5] and let the dielectric
constant and optical mode to vary by small amounts by
substituting →+ ∆and U→U+ ∆Uin Eq. (2). While
solving for the time varying coupling effects, we neglect the
terms that lead to unperturbed wave equation and consider
only the first order perturbation terms, dropping the higher
order perturbations. Also, to focus on the time varying effects
of rotation of a gyro on electric field, we consider only the
temporal derivatives. The coupled equations are then given as
dEf
dt =−jβ2
2c2
2ω1
Ebej(ω2−ω1)te−j(2π/T )tRδ(x, y )|U|2dA
R(+δ)|U|2dA
(3)
dEb
dt =−jβ2
1c2
2ω2
Efej(ω1−ω2)te−j(2π/T )tRδ(x, y )|U|2dA
R(+δ)|U|2dA
(4)
where the variations in dielectric constant are assumed to
be sinusoidal with a time period Ti.e.
∆=δ(x, y)
2ej(2πt/T )+δ(x, y )
2e−j(2πt/T )(5)
Now, assuming perfectly normalized transverse modes and
expressing the dielectric variations in terms of group refractive
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(a) (b)
(c) (d)
Fig. 2: (a) Electric field distribution, (b) instantaneous phase,
(c) time domain beat signal, (d) frequency domain beat signal
for no lock-in case i.e. Ωr>> ΩL
index variations (δng) as δ(x, y)=2ngδng(x, y), we get the
magnitude of generalized coupling coefficient (κ) from Eq. (3)
and (4) as
κ=β2c2
2ωngδng(x, y)
+ngδng(x, y)(6)
The rate equations of the CW and CCW electric fields inside
an integrated SRLG considering the nonlinear coupling can be
written as [6]
dE1,2
dt =vg(gc−αi)E1,2−j(ω1,2−Ω) −jκE2,1(7)
where vgis group velocity, gcis gain coefficient, αiis
average internal loss and Ωis cold cavity resonance frequency
of the ring laser. Substituting E1=Ae−jφ1,E2=Ae−jφ2
in Eq. (7) and separating into real and imaginary parts,
instantaneous frequency difference is given as
dφ
dt = (ω2−ω1)−jκsin(φ)(8)
The above equation shows the instantaneous beat frequency
of the rotating SRLG in the presence of nonlinear coupling of
CW and CCW modes. When expressed in terms of gyro scale
factor (S) and external rotation rate (Ωr), Eq. (8) becomes
dφ
dt = 2πSΩr−jκsin(φ)(9)
Lock-in threshold of SRLG, which is the value of rotation
rate below which the instantaneous frequency difference be-
comes zero, is given from Eq. (9) as
ΩL=κ
2πS (10)
(a) (b)
(c) (d)
Fig. 3: (a) Electric field distribution, (b) instantaneous phase,
(c) time domain beat signal, (d) frequency domain beat signal
for lock-in case i.e. Ωr<ΩL
III. SIMULATION RESULTS AN D CONCLUSIONS
Fig. (2) and (3) show the comparison of the electric field
distribution ( ¯
E), instantaneous phase (φ) and output of SRLG
in time and frequency domain when the gyro is operating in
lock-free regime (Ωr>> ΩL) and locked regime (Ωr<ΩL).
As seen from Fig. (3c) and (3d), the beat signal is completely
lost in the locked zone of the gyro.
In conclusion, nonlinear backscattering from the popula-
tion inversion grating and the resultant coupling of counter-
traveling waves imposes a severe limitation on the operation
of integrated SRLG. For a typical on-chip SRLG, the lock-
in threshold as calculated from Eq. (10) can be as high as
1×108 0/h, thus making it unsuitable for use in military appli-
cations. In order to achieve high performance from SRLG, the
nonlinear coupling needs to be eliminated by use of methods
such as orthogonal polarization of counter-traveling waves and
external biasing using a phase modulator.
REFERENCES
[1] M. N. Armenise, V. M. Passaro, F. De Leonardis, and M. Armenise,
“Modeling and design of a novel miniaturized integrated optical sensor
for gyroscope systems,” Journal of lightwave technology, vol. 19, no. 10,
p. 1476, 2001.
[2] O. J. Woodman, “An introduction to inertial navigation,” University of
Cambridge, Computer Laboratory, Tech. Rep. UCAMCL-TR-696, vol. 14,
p. 15, 2007.
[3] C. Ciminelli, F. DellOlio, C. E. Campanella, and M. N. Armenise,
“Photonic technologies for angular velocity sensing,” Advances in Optics
and Photonics, vol. 2, no. 3, pp. 370–404, 2010.
[4] A. E. Siegman, “Lasers,” University Science Books, Mill Valley, CA,
vol. 37, 1986.
[5] L. A. Coldren, S. W. Corzine, and M. L. Mashanovitch, Diode lasers and
photonic integrated circuits. John Wiley & Sons, 2012, vol. 218.
[6] G. P. Agrawal and N. K. Dutta, “Long wavelength semiconductor lasers,”
1986.
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