Asymmetric ring laser gyroscope and method for detecting rotation with acentric photorefractive crystal

An asymmetric ring laser gyroscope includes a laser gain medium for generating a clockwise beam of coherent light and a counterclockwise beam of coherent light, such that the counterclockwise beam is parallel but oppositely directed with respect to the clockwise beam. At least three reflectors constrain the clockwise and counterclockwise beams to propagate in a closed path. A means for detecting the frequency difference between the clockwise and counterclockwise beams is provided. An acentric photorefractive plate is positioned within the closed path such that the c-axis of the plate is parallel to the clockwise and counterclockwise beams in the plate.

BACKGROUND OF THE INVENTION 
This invention is concerned with eliminating lock-in problems in ring laser 
gyroscopes. 
A mechanical gyroscope utilizes the inertia of a spinning mass to provide a 
reference direction useful in various applications, such as the navigation 
of an airplane or a spacecraft. The moving parts required in a mechanical 
gyroscope, however, introduce some undesirable attributes, such as high 
drift rates resulting from friction, into the device. The ring laser 
gyroscope was developed to avoid some of these difficulties. 
The ring laser gyroscope maintains a constant frame of reference by 
circulating massless light waves in a closed path. A typical ring laser 
gyroscope, for example, consists of a resonant cavity defined by three or 
four corner mirrors. A gas laser generates a monochromatic light beam 
which is split into two beams. These beams are made to propagate in 
clockwise and counterclockwise directions in the cavity. If the gyroscope 
is rotated about an axis which has a component normal to the plane of the 
optical path, the frequency of one of the beams will be increased, while 
the frequency of the other beam will decrease, because of the Doppler 
effect. The beams can then be extracted from the cavity and combined to 
produce a beat frequency which can be related to the magnitude and 
direction of the rotation. 
In an inertial frame, the optical path lengths for the clockwise and 
counterclockwise beams of a conventional ring laser cavity are exactly the 
same. Thus, when the ring gyroscope lases, the two beams will exhibit 
exactly the same frequency. If the gyroscope is rotating, the effective 
optical path lengths for the clockwise and the counterclockwise beams are 
different. In practice, the two beams tend to oscillate at the same 
frequency for small rotation rates. This is known as the "lock-in" problem 
and is due to the coupling of the two beams which results from 
backscattering of the laser beams by the mirrors in the beam path. 
Backscattering causes a small amount of light from each laser beam to be 
transferred to the oppositely traveling wave. At slow rates of rotation, 
this coupling causes the frequencies of the two beams to lock together at 
a single frequency, thereby preventing the measurement of slow rotation 
rates. 
The lock-in problem is conventionally avoided by mechanically vibrating the 
ring cavity or using the magneto-optic effect to cause the two beams to 
oscillate at different frequencies in an inertial frame. If this frequency 
difference is made large enough, lock-in will not occur. The art of ring 
laser gyroscopes would be considerably advanced, however, if the 
oscillation frequencies could be split with a simpler, more compact, and 
more reliable technique. 
Another technique for reducing the lock-in frequency requires the addition 
of a wavefront conjugating element to the ring cavity (Diels, U.S. Pat. 
No. 4,525,843; Diels, et al., Influence of wave-front-conjugated coupling 
on the operation of a laser gyro, Optics Letters, Volume 6, Page 219 
(1981)). The intracavity conjugating element causes a fraction of the 
energy in each beam to be phase shifted and added to the 
oppositely-directed beam, thereby introducing a coupling between the 
counter-rotating beams similar to the coupling caused by backscattering. 
The phase shifts introduced by the wave front conjugation are cumulative 
with the phase shifts due to rotation, so that the frequency at which 
lock-in of the gyro occurs is reduced. Because this wave front conjugation 
process is symmetrical, however, the phase shift introduced in the 
clockwise wave by the conjugated coupling is equal in magnitude to the 
phase shift introduced in the counterclockwise wave and the two 
frequencies of oscillation remain identical when there is no rotation or 
rotation at a rate below the threshold value. Consequently, lock-in will 
still occur at frequencies below the lowered threshold frequency and this 
technique is thus only a partial solution because it does not completely 
eliminate the lock-in problem. 
SUMMARY OF THE INVENTION 
The photorefractive effect in an acentric photorefractive crystal is used 
to couple the two oppositely directed traveling waves in a ring gyroscope. 
The nonlocal response of the crystal, which gives rise to the asymmetrical 
coupling of the waves, results in unequal transmissivities as well as 
unequal phase shifts when the two waves traverse the photorefractive 
material. This inequality in phase shifts and transmissivities removes the 
degeneracy of the modes and leads to a splitting in both oscillation 
frequency and intensity. 
An asymmetric ring laser gyroscope constructed according to this invention 
includes a laser gain medium for generating a clockwise beam of coherent 
light and a counterclockwise beam of coherent light. The counterclockwise 
beam is parallel but oppositely directed with respect to the clockwise 
beam. At least three reflectors constrain the beams to propagate in a 
closed path. A means is provided for detecting the frequency difference 
between the clockwise and counterclockwise beams, and an acentric 
photorefractive plate is positioned within the closed path such that the 
c-axis of the plate is parallel to the clockwise and counterclockwise 
beams. 
As long as the c-axis of the plate is not perpendicular to the clockwise 
and counterclockwise beams in the plate, the plate will nonreciprocally 
couple the beams, causing a first phase shift in the clockwise beam and a 
second phase shift different from the first phase shift in the 
counterclockwise beam.

DESCRIPTION OF THE INVENTION 
The photorefractive effect in electrooptic crystals has been widely studied 
for many applications, including real-time holography, optical data 
storage, and phase-conjugate wave-front generation. Increasing attention 
has been focused on utilizing the nonreciprocal energy transfer in 
two-wave mixing. These new applications include image amplification, 
vibration analysis, and self-oscillation. 
It is known that hologram formation in photorefractive media allows a phase 
shift between the interference fringes of light and the refractive-index 
modulation, provided the materials are acentric. This phase shift permits 
a nonreciprocal steady-state transfer of energy between the light beams. 
This problem has been formulated and solved by many workers. Most of this 
work, however, was focused on codirectional coupling, in which the 
amplitudes of the mixing waves increase or decrease along the direction of 
their bisector (+z). Contradirectional two-wave mixing in the same 
photorefractive medium has recently been studied. See Yeh, 
"Contra-Directional Two-Wave Mixing in Photorefractive Media", Optical 
Communications, Volume 45, Page 323 (1983). In this coupling, the 
amplitudes of the waves decrease or increase along a direction which is 
perpendicular to the bisector of their wave vectors. It is an outstanding 
feature of this invention to place a thin plate of acentric 
photorefractive material (e.g., LiNbO.sub.3, LiTaO.sub.3, BaTiO.sub.3, 
BSO, BGO, KTN, etc.) in the cavity of a ring laser gyroscope with the 
c-axis of the crystal aligned with the counterrotating beams. This causes 
the oscillation frequencies as well as intensities of the two beams to be 
predictably different. By solving the coupled equation for 
counterpropagating two-wave mixing in a photorefractive crystal and 
obtaining expressions for the transmissivities as well as the relative 
phase shifts, the split in the oscillation frequency and the intensity of 
the ring laser can be calculated. 
Although in the preferred embodiment of the invention the c-axis is aligned 
with the counterrotating beams, this is not essential for the operation of 
the invention. It is only necessary that the c-axis not be perpendicular 
to the beams, i.e., the c-axis direction vector must have a component 
parallel to the beam direction. 
FIG. 1 is a schematic diagram illustrating a preferred embodiment of an 
asymmetric ring laser gyroscope constructed according to the present 
invention. A laser gain medium 10 generates a clockwise beam of coherent 
light 12 and a counterclockwise beam of coherent light 14, such that the 
counterclockwise beam is parallel but oppositely directed with respect to 
the clockwise beam. Three mirrors 16, 18, and 20 constrain the beams 12 
and 14 to propagate in a closed, triangular path. The mirrors 16 and 18 
are highly reflective, while the mirror 20 is made partially reflective, 
so that a portion of the energy in each beam is transmitted through the 
latter mirror. The transmitted portion of the clockwise beam 12 also 
propagates through another partially reflecting mirror 22, while the 
transmitted portion of the counterclockwise beam 14 is reflected by a 
mirror 24 and is reflected by the mirror 22. Thus both transmitted 
portions are directed together into a detector 26, which measures the 
frequency difference between the clockwise and counterclockwise beams. An 
acentric photorefractive plate 28 is positioned within the closed path 
such that the c-axis of the plate is parallel to the clockwise and 
counterclockwise beams in the plate. 
In the absence of the photorefractive plate, both beams will oscillate at 
the same frequency .nu..sub.o and with the same intensity I.sub.0 
(w/cm.sup.2). When the crystal plate of photorefractive material is 
present in the ring laser cavity, however, an interference pattern is 
formed inside the plate due to the counterpropagating beams. This 
interference pattern will generate and redistribute photocarriers. As a 
result, a periodic space charge field is created inside the crystal. This 
field will produce an index grating via the Pockels effect. Consequently, 
a frequency split will be introduced between the two counterpropagating 
waves, allowing even slow rates of rotation to be measured by the 
gyroscope without the onset of lock-in between the beams. 
The third-order nonlinearity which gives rise to phase conjugation can, 
because of symmetry, be nonvanishing in any medium, including isotropic 
materials (gases, liquids, and glasses) as well as cubic crystals. Yariv, 
et al., Optical Waves in Crystals, Page 553 (John Wiley & Sons, 1984). Not 
all materials in which phase conjugation can occur, however, are 
satisfactory for the operation of the present invention. In this 
invention, the photorefractive crystal must be an acentric or 
non-centrosymmetric crystal, i.e., a crystal lacking inversion symmetry. 
Thus the crystal can be selected from the triclinic, monoclinic, 
orthorhombic, tetragonal, trigonal, hexagonal, or cubic symmetry classes. 
Yariv, Pages 227-229. This requirement ensures, as explained further 
below, that the frequency split between the counterpropagating waves will 
not be symmetric, which in turn will cause lock-in to be eliminated even 
at a zero rotation rate. 
The manner in which this invention eliminates lock-in can be explained by 
considering the propagation of electromagnetic waves in a ring resonator. 
In the region occupied by the photorefractive crystal plate, the electric 
field of the two waves can be written: 
EQU E.sub.j =A.sub.j (z) exp[i(k.sub.j z-.omega..sub.j t)]+c.c. j=1,2 1) 
where z is measured along the beam path and k.sub.1 =-k.sub.2 =k=n.sub.0 
2.pi./.lambda., where n.sub.0 is the ordinary index of refraction of the 
crystal. In Equation (1), it is assumed for simplicity that both waves 
have the same linear state of polarization and that the beam path is 
parallel to the c-axis (optic axis) of the crystal. It is further assumed 
that no optical rotation is present in the material. A.sub.1 and A.sub.2 
are the wave amplitudes and are taken as functions of z only for the 
steady-state situation. 
In the photorefractive medium (from z=0 to z=1), these two waves generate 
an interference pattern (which is traveling if .omega..sub.1 
.noteq..omega..sub.2). This pattern may generate and redistribute 
photocarriers. As a result, a spatial charge field (which is also 
traveling if .omega..sub.1 .noteq..omega..sub.2) is created in the medium. 
This field induces an index volume grating via the Pockels effect. In 
general, the index grating will have a finite spatial phase shift relative 
to the interference pattern, which will make the index grating shift 
toward one of the beams. This asymmetry leads to a nonreciprocal exchange 
of power, causing one beam to donate and the other beam to accept energy. 
The net result is that one beam will suffer a loss and the other will 
experience a gain, leading to a split in the oscillation intensity. 
Following the notation of Fischer, et al., "Amplified Reflection, 
Transmission, and Self-Oscillation in Real-Time Holography", Optics 
Letters, Volume 6, Page 519 (1981), the fundamental component of the 
intensity-induced grating can be written as: 
##EQU1## 
where 
EQU I.sub.0 =I.sub.1 +I.sub.2 .vertline.A.sub.1 .vertline..sup.2 
+.vertline.A.sub.2 .vertline..sup.2 (3) 
.phi. is real, n.sub.1 is a real and positive number, K=2k, and 
.OMEGA.=.omega..sub.1 -.omega..sub.2. Here again, for the sake of 
simplicity, a scalar grating is assumed. The phase .phi. indicates the 
degree to which the index grating is shifted spatially with respect to the 
light interference pattern. According to Huignard, et al., "Coherent 
Signal Beam Amplification in Two Wave Mixing Experiments with 
Photorefractive B.S.O Crystals", Optics Communications, Volume 38, Page 
249 (1981), .phi. and n.sub.1 can be written, respectively: 
EQU .phi.=.phi..sub.0 +tan.sup.-1 (.OMEGA..tau.) (4) 
and 
##EQU2## 
where .tau. is the time constant required to build up the holograph 
grating, .DELTA.n.sub.s is the saturation value of the photoinduced index 
change, and .phi..sub.0 is a constant phase shift related to the nonlocal 
response of the crystal under fringe illumination. Both .DELTA.n.sub.s and 
.phi..sub.0 depend on the grating spacing (2.pi./K) and its direction, as 
well as on the material properties of the crystal, such as its 
electro-optic coefficient. Expressions for .DELTA.n.sub.s and .phi..sub.0 
can be found in Kukhtarev, et al., "Holographic Storage in Electrooptic 
Crystals. Beam Coupling and Light Amplification", Ferroelectrics, Volume 
22, Page 961 (1979) and Feinberg, et al., "Photorefractive Effects and 
Light Induced Charge Migration in Barium Titanate", Journal of Applied 
Physics, Volume 51, Page 1297 (1980). In photorefractive media which 
operate by diffusion only (i.e., no external static field is required), 
such as, for example, BaTiO.sub.3, the magnitude of .phi..sub.0 is .pi./2, 
with its sign depending on the direction of the c-axis. 
Using Equation (2) for n and the scalar-wave equation, the following 
coupled equations can be derived by using the parabolic approximation 
(i.e., assuming slowly varying amplitudes): 
##EQU3## 
where .psi..sub.1 and .psi..sub.2 are phases of the amplitudes A.sub.1 and 
A.sub.2, respectively. Using Equations (7) and (3), the coupled equations 
(6) can be written as: 
##EQU4## 
Note that the Poynting power flow along +z is conserved, i.e.: 
##EQU5## 
The solution of Equation (8) is: 
##EQU6## 
where B and C are constants and relate to the boundary conditions. B and C 
can be expressed in terms of any two of the four boundary values I.sub.1 
(O), I.sub.2 (O), I.sub.1 (l), and I.sub.2 (l), where l is the length of 
interaction. In terms of I.sub.1 (O) and I.sub.2 (O), B and C are given 
by: 
##EQU7## 
In practice, it is convenient to express B and C in terms of the incident 
intensities I.sub.1 (O) and I.sub.2 (l). In this case, B and C become: 
##EQU8## 
According to Equation (8), both I.sub.1 (z) and I.sub.2 (z) are increasing 
functions of z, provided .gamma. is positive. The transmissivities for the 
two waves, according to Equations (14) and (15), are: 
##EQU9## 
where m is the incident intensity ratio m.tbd.I.sub.2 (l)/I.sub.1 (O). 
Note that T.sub.1 &gt;1 and T.sub.2 &lt;1 for positive .gamma.. The sign of 
.gamma. depends on the direction of the c-axis. These expressions for 
transmissivity are formally identical to those describing codirectional 
coupling even though the spatial variations of I.sub.1 (z) and I.sub.2 (z) 
are very different. 
With I.sub.1 (z) and I.sub.2 (z) known, the phases .psi..sub.1 and 
.psi..sub.2 can be integrated directly from Equations (9). The phase 
shifts in traversing through the medium are kl+.psi..sub.1 (l)-.psi..sub.1 
(O) and kl+.psi..sub.2 (O)-.psi..sub.2 (l) for waves E.sub.1 and E.sub.2, 
respectively. These two phase shifts differ by an amount 
.DELTA.=.psi..sub.2 (O)-.psi..sub.2 (l)-[.psi..sub.1 (l)-.psi..sub.1 (O)] 
which, according to Equation (9), is given by: 
##EQU10## 
This difference in phase shift is zero when I.sub.2 (z)=I.sub.1 (z) 
between z=O and z=l, which corresponds to C=O in Equation (13). Using 
Equation (13) and carrying out the integration in Equation (17), the 
following expression for this phase shift difference is obtained: 
##EQU11## 
where T.sub.1 is the beam intensity transmissivity given by Equation (16). 
Note that .DELTA. can also be written as 
.DELTA.=(2.beta./.gamma.)logT.sub.2 +.beta.l. For small couplings, i.e., 
.gamma.l&lt;&lt;1, this difference in phase shift can be written approximately 
as: 
##EQU12## 
The nonreciprocal property of the transmissivity and phase shift in 
photorefractive media may have important applications in many optical 
systems. It is known that in linear optical media, the transmissivity as 
well as the phase shift of beams traversing through a layered structure 
(including absorbing media) are independent of the side of incidence (the 
so-called left-and-right incidence theorem). The nonlocal response of the 
photorefractive crystal, which is responsible for the nonreciprocity, 
makes it possible to provide asymmetrical mode coupling in the ring laser 
gyro of this invention. 
In a conventional ring laser gyroscope, the oscillation frequency as well 
as the intensity are the same for the two beams in an inertial frame. The 
oscillation occurs at those frequencies f: 
##EQU13## 
which lie within the gain curve of the laser medium (e.g., He-Ne). Here L 
is the effective length of a complete loop, and N is a large integer. For 
L.ltoreq.30 cm, these frequencies are separated by the mode spacing 
c/L.gtoreq.1 GHz. Since the width of the gain curve is typically 1.5 GHz 
due to principally Doppler broadening, the gyro usually oscillates at a 
single longitudinal mode. When the gyroscope rotates, the effective 
optical path lengths are different, leading to a difference .DELTA.f 
between the frequencies of the laser oscillation for the two beams. The 
difference is: 
##EQU14## 
where f is the frequency of oscillation in an inertial frame, A is the 
area of the loop, the .OMEGA. is the angular velocity of the rotation. 
The oscillation intensity inside the laser cavity is determined by the gain 
as well as the loss and is given by: 
EQU I.sub.0 =K(g.sub.O -g.sub.t) (22) 
where K is a constant which depends on the laser medium, g.sub.O is the 
unsaturated gain factor per pass, and g.sub.t is the threshold gain 
factor. Note that both g.sub.O and g.sub.t are dimensionless. In a 
conventional ring resonator, the threshold gain for both traveling waves 
is given by: 
EQU g.sub.t .varies.L-log R (23) 
where .varies. is the loss constant (including bulk absorption and 
scattering) and R is the product of the three-mirror reflectivities. 
In the presence of photorefractive coupling, the unequal transmissivities 
make the threshold gains different for the two waves: 
##EQU15## 
where T.sub.1 and T.sub.2 are the beam transmissivities given by Equations 
(16). The difference in the threshold gains leads to a split in the 
oscillation intensity. The fractional difference in the oscillation 
intensity is given approximately by: 
##EQU16## 
If it is now assumed that the beam intensities are nearly uniform in the 
photorefractive material (i.e., .gamma.l&lt;&lt;1), the difference in phase 
shift .DELTA. can be written, according to Equations (17) and (25): 
##EQU17## 
This expression agrees with Equation (19), provided (g.sub.O -g.sub.t)&lt;&lt;2, 
which is a legitimate limitation because (g.sub.O -g.sub.t) is typically 
of the order of 10.sup.-2. 
The unequal phase shift for the oppositely directed traveling waves 
corresponds to different effective optical path lengths for the waves. 
This results in a difference .OMEGA. between the angular frequencies of the 
laser oscillation of the two beams. The difference is .sub..OMEGA..tbd. 
.omega..sub.1 -.omega..sub.2 c.DELTA./L, which can be written, according 
to Equations 26), 11), and 10): 
##EQU18## 
Note that .OMEGA. is not zero, provided sin.phi. cos.phi.=0. This 
frequency difference is proportional to .beta..gamma.l.sup.2. Thus it is 
possible to choose the thickness l such that 1/.DELTA..upsilon. is greater 
than the response time (e.g., approximately 1 msec for LiNbO.sub.3) of the 
photorefractive material. 
Now consider the angular frequency split .OMEGA. for various cases. For the 
pure diffusion case (i.e., no external electric field) in a 
photorefractive material, the phase shift .phi. is given by .phi.=.pi./2 
+tan.sup.-1 .OMEGA..tau., according to Equation 4) and Huignard, et al., 
supra. Thus Equation 27) becomes: 
##EQU19## 
which has three solutions. The trivial one is .OMEGA.=0, which corresponds 
to an unsplit oscillation. The other roots are given by: 
##EQU20## 
Taking .tau.=100 msec, L=30 cm, g.sub.O -g.sub.t =0.01, l=1 mm, 
.DELTA.n.sub.s =10.sup.-5, and .lambda.=-0.6328 .mu.m, Equation 29) yields 
.OMEGA..sub.0 =10.sup.3 sec.sup.-1, which corresponds to a frequency split 
of 160 Hz. Whether the ring gyro will oscillate at the same frequency 
(.OMEGA.=0) or with a split .OMEGA..sub.0, or both, depends upon the mode 
stability, which is discussed below. 
In the general case when the external field is present (i.e., the so-called 
drift case), the phase .phi..sub.0 is given by: 
##EQU21## 
where E.sub.O is the applied electric field along the grating momentum K, 
and E.sub.d and E.sub.p are electric fields characteristic of diffusion 
and maximum space charge, respectively. E.sub.d =k.sub.B TK/e and E.sub.p 
=N.sub.T e/K.epsilon., where K is the grating momentum, k.sub.B is the 
Boltzmann constant, T is the temperature, e is the electron charge, 
N.sub.T is the trap density in the photorefractive material, and .epsilon. 
is the dielectric permittivity of the medium. Using Equations 30) and 4), 
the factor sin.phi. cos.phi. in Equation 27) can be written: 
##EQU22## 
Consider, for example, a trapping density of N.sub.T =10.sup.15 cm.sup.-3 
in a BSO crystal at T=300 K. Using .lambda.=0.6328 .mu.m, n=2.54, 
.epsilon./.epsilon..sub.0 =6.5, and K=4.pi.n/.lambda., E.sub.d =13 kV/cm 
and E.sub.p =550 V/cm. Note that E.sub.d &gt;&gt;E.sub.p in the range of fringe 
spacing (.LAMBDA..about.0.1 .mu.m) considered. Assuming an externally 
applied field of E.sub.O =1 kV/cm, Equation 30) leads to a stationary 
phase of .phi..sub.0 =.pi./2-0.0031 (tan.phi..sub.0 =322). Substituting 
Equations 31) and 4) for sin.phi. cos.phi. and n.sub.1, respectively, in 
Equation 27) gives: 
##EQU23## 
In the practical region of interest (where tan.phi..sub.0 &gt;&gt;1), Equation 
32) has three roots. The small root, which is unstable and thus is of no 
interest in this invention, is given by .OMEGA..tau.=cot.phi..sub.0, which 
is virtually zero. The other two roots 
(.vertline..OMEGA..tau..vertline.&gt;&gt;1) are given approximately by: 
##EQU24## 
provided .vertline..OMEGA..tau..vertline.&lt;&lt;tan.phi..sub.0 where 
.OMEGA..sub.0 is given by Equation 29). Note that a small applied field 
(i.e., E.sub.O &lt;&lt;E.sub.d so that 
.vertline..OMEGA..tau..vertline.&lt;&lt;tan.phi..sub.0) has little effect on the 
frequency split. When the applied field is large enough so that 
tan.phi..sub.0 .about..OMEGA..tau., Equation 32) must be solved 
numerically for .OMEGA..tau.. FIG. 2 shows the frequency split due to 
photorefractive coupling in a BSO crystal as a function of the applied 
electric field. The ordinate scale on the right is for the dashed curve, 
while the ordinate scale on the left is for the solid curves. 
The stability of these three oscillation modes will determine the actual 
mode of oscillation in the steady state. To investigate this issue, the 
effect of small perturbations on the oscillation frequencies needs to be 
examined. Using Equation 27) and .phi.=.pi./2+tan.sup.-1 .OMEGA..tau., 
consider that the frequency difference .OMEGA. is slightly deviated from 
the solution by .delta..OMEGA.. This .delta..OMEGA. will change the 
holographic grating phase shift by .delta..phi.. Equation 27) will then 
yield the resulting frequency difference .OMEGA.+d.OMEGA. after 
substituting .phi.+.delta..phi. for .phi. on the right hand side. The 
criterion for stable oscillation is: 
##EQU25## 
Using Equations 4) and 27), the right hand side of Equation 27) can be 
plotted as a function of .OMEGA..tau.. The solution of Equation 27) can 
then be obtained by drawing a straight line through the origin with a 
slope of 1/.tau.. This plot is illustrated in FIG. 3. The intersections of 
the straight line with the curve give the solutions. The ratio 
(d.OMEGA./.delta..OMEGA.) is proportional to the slope at the 
intersections. Note that the solution at .OMEGA.=0 has a positive slope 
which indicates that this mode of oscillation is unstable according to the 
criterion of Equation 34). The other two solutions of Equation 29) are 
stable because they have a negative slope. A negative slope indicates that 
any deviation .phi..OMEGA. caused by perturbation will eventually damp 
out. The two roots of Equation 33) for the general case can also be shown 
to be stable in a similar way. 
In conclusion, photorefractive coupling of oppositely directed traveling 
waves in a ring laser resonator may be utilized to bias a laser gyroscope 
away from its lock-in region. If the photorefractive crystal is acentric, 
the nonlocal response of the crystal will lead to unequal transmissivity 
and phase shifts of the two waves. These, in turn, lead to a split in the 
oscillation intensity as well as the oscillation frequency. It is this 
frequency split which may be used to bias the gyro away from lock-in. 
Those skilled in the art will undoubtedly find additional embodiments and 
modifications apparent. In the derivation of this phenomenon, for example, 
the bulk absorption in the photorefractive material is neglected. This is 
legitimate provided .varies.&lt;&lt;.gamma., which is generally true in most 
photorefractive crystals. Furthermore, attenuation in the crystal may 
affect the difference in phase shift according to Equation 17) because 
I.sub.2 I.sub.1 will no longer be a constant. In the latter case, 
numerical analysis will be required to account for the attenuation and 
obtain a more accurate result. In addition, techniques for detecting the 
frequency difference between the beams are well known to those skilled in 
the art and thus do not need to be presented in any detail here. 
Similarly, those skilled in the art will appreciate that many arrangements 
for introducing counterpropagating beams to the closed path and for 
extracting the beams from the path for frequency measurements are known in 
the art. Consequently, the exemplary embodiments should be considered as 
illustrative, rather than inclusive, and the appended claims are intended 
to define the full scope of the invention. 
The teachings of the following documents referred to herein are 
incorporated by reference: 
Diels, U.S. Pat. No. 4,525,843. 
Diels, et al., Influence of wave-front-conjugated coupling on the operation 
of a laser gyro, Optics Letters, Volume 6, Page 219 (1981). 
Fischer, et al., "Amplified Reflection, Transmission, and Self-Oscillation 
in Real-Time Holography", Optics Letters, Volume 6, Page 519 (1981) 
Huignard, et al., "Coherent Signal Beam Amplification in Two Wave Mixing 
Experiments with Photorefractive B.S.O. Crystals", Optics Communications, 
Volume 38, Page 249 (1981) 
Kukhtarev, et al., "Holographic Storage in Electrooptic Crystals. Beam 
Coupling and Light Amplification", Ferroelectrics, Volume 22, Page 961 
(1979) 
Feinberg, et al., "Photorefractive Effects and Light Induced Charge 
Migration in Barium Titanate", Journal of Applied Physics, Volume 51, Page 
1297 (1980) 
Yeh, "Contra-Directional Two-Wave Mixing in Photorefractive Media", Optical 
Communications, Volume 45, Page 323 (1983). 
Yariv, et al., Optical Waves in Crystals (John Wiley & Sons, 1984).