INTERFEROMETER WITH KERR EFFECT COMPENSATION

Abstract: A fiber optic interferometer provides kerr-effect compensation by using a multimode source (10) to provide counter-propagating waves in a loop (14) of fiber optic material to reduce the difference between the average intensity weighted phase shifts of the waves after they have traversed the fiber loop (14).

Description: I TERFEROMETER WITH KERR EFFECT COMPENSATION Background of the Invention The present invention relates to fiber optic interferometers, and particularly, to fiber optic ring interferometers for rotation sensing. A fiber optic ring interf rometer typically comprises a loop of fiber optic material to which lightwaves are coupled for propagation around the loop in opposite directions. After traversing the loop, the counter- 10 propagating waves are combined so that they constructively or destructively interfere to form an optical output signal. The intensity of this optical output signal varies as. a function of the type and amount of interference, which, in turn, is dependent upon the ' relative phases of the counter-propagating waves. Ring interferometers have proven particularly useful for rotation sensing. Rotation of the loop creates a relative phase difference between the counter-propagating waves, in accordance with the well-known "Sagnac" effect, ^ υ with the amount of phase difference corresponding to the velocity of rotation. The optical output signal produced by the interference of the counter-propagating waves, when recombined, varies in intensity as a function of the rotation rate of the loop. Rotation sensing is 5 accomplished by detection of this optical output signal. While mathematical calculations indicate that ring interferometers should be capable of providing rotation sensing accuracies substantially better than are required for inertial navigation (e.g., 0.001 degrees per hour or 0 less), the results achieved in practice have generally not conformed to theoretical expectations. Some of the reasons for the disparity between theoretical and actual results have been identified as including spurious waves caused by Rayleigh backscattering, and non-rotationally- 5 induced phase differences caused by residual fiber birefringence. More recently, it was discovered that the accuracy of rotation sensing is also limited by the A.C. Kerr effect. The A.C. Kerr effect is a well-known optical effect in which the refractive index of a substance changes when the substance is placed in a varying electric field. In optical fibers, the electric fields of light¬ waves propagating therethrough can themselves change the refractive index of the fiber, in accordance with the Kerr effect. The amount of change is proportional to the square of the electric field, or the light intensity. Since the propagation constant of the fiber, for each of the waves, is a function of refractive index, the Kerr effect manifests itself as intensity dependent perturba¬ tions of the propogation constants. Unless such perturba¬ tions happen to be exactly the same for each of the counter-propagating waves, the A.C. or optical Kerr effect will cause the waves to propagate with different velocities, resulting in a non-rotationally induced phase difference between the waves, and thereby creating a spurious signal, which is indistinguishable from a rota- tionally induced signal. This spurious, Kerr-induced signal is a major source of long-term drift in current, state-of-the-art, fiber optic rotation sensing interfer¬ ometers. Thus, there exists a need to reduce or eliminate the Kerr-induced phase difference in fiber optic interfer¬ ometers, particularly in those requiring high sending accuracies, such as in inertial navigation grade rotation sensors. Summary of the Invention The present invention comprises a rotation sensing interferometer in which errors caused by the Kerr effect are reduced or eliminated by using a light source which OMPI provides an output comprising oscillatory modes having a number of different wavelengths to supply light to a gyroscope comprising a loop of single mode optical fiber. It has been discovered that the Kerr effect errors are eliminated or substantially reduced if each of the counter-propagating waves has a 50% duty cycle. The different wavelengths beat with one another to cause intensity variations in the counter-propagating waves which results in a 50% duty cycle for each of the waves. i the counter-propagating waves originate from a multimode source, the non-reciprocal, intensity weighted average of the Kerr-induced phase shift accumulated by each wave during traverse of the interferometer loop will vary inversely as the number of modes, thus providing perfect Kerr effect compensation. As used herein, the term "multimode source" refers to a light source which produces plural frequencies, as opposed to a single mode source which produces light at a substantially single frequency. A single mode optical fiber has only one transverse power distribution in a predetermined wavelength range. However, a single mode fiber will simultaneously guide a number of wavelengths with all guided wavelengths having the same transverse power distribution, such as transverse electric, or TE, in which the electric field vector is perpendicular to the direction of propagation. The rotation sensor of the invention includes a fiber optic coupler which introduces the counter-propagating waves into the loop. After traversing the loop the waves recombine in the coupler to form an interference pattern. The oscillatory modes recombine in the coupler such that corresponding counter-propagating modes interfere with one another to produce an interference pattern from which the Sagnac phase shift may be measured. A second embodiment of the invention includes a multimode source which results in a duty cycle greater than 50% and a modulator which modulates the multimode output to produce a light signal having a 50% duty cycle for input to the loop. By compensating for the Kerr effect according to the present invention, a major source of sensing error is substantially reduced or eliminated. It is believed that Kerr effect compensation according to the invention is a major breakthrough for the development of inertial naviga¬ tion grade fiber optic rotation sensors for use in gyroscopes. Brief Description of the Drawings These and other advantages of the present invention are best understood through reference to the drawings, in which: Figure 1 is a schematic drawing of the rotation sensor of the present invention showing the fiber optic compo¬ nents positioned along a continuous, uninterrupted strand of fiber optic material, and further showing the signal generator, photodetector, lock-in amplifier, and display associated with the detection system; Figure 2 is a sectional view of one embodiment of a fiber optic directional coupler for use in the rotation sensing interferometer of Figure 1; Figure 3 is a sectional view of one embodiment of a fiber optic polarizer for use in the rotation sensor of Figure 1; Figure 4 is a perspective view of one embodiment of a fiber optic polarization controller for use in the rotation sensor of Figure 1; Figure 5 is a schematic diagram of the rotation sensor of Figure 1 with the polarizer, polarization controllers, and phase modulator removed therefrom; QMPI Figure 6 is a graph of the intensity of the optical output signal, as measured by the photodetector as a function of the rotationally induced Sagnac phase difference, illustrating the effects of birefringence- induced phase differences and birefringence-induced amplitude fluctuations; Figure 7 is a graph of phase difference as a function of time showing the phase modulation of each of the counter-propagating waves and the phase difference between the counter-propagating waves; Figure 8 is a schematic drawing illustrating the effect of the phase modulation upon the intensity ofthe optical output signal, as measured by the detector, when the loop is at rest; Figure 9 is a schematic drawing showing the effect of the phase modulation upon the intensity of the optical output signal, as measured by the detector, when the loop is rotating; and Figure 10 is a graph of the amplifier output signal as a function of the rotationally-induced Sagnac phase difference, illustrating an operating range for the rotation sensor of Figure 1; Figure 11 is a simplified schematic drawing of a pair of square wave, intensity modulated lightwaves, counter- propagating through the loop of Eiber optic material, and having substantially dissimilar peak intensities, to illustrate intensity-independent Kerr effect compensation; and Figure 12 is a graph representing experimental results obtained with the apparatus and method of the invention. Detailed Description of the Preferred Embodiment In addition to Kerr effect compensating means, the preferred embodiment of the present invention also includes a synchronous detection system for detecting the intensity of the optical output signal to indicate rotation rate, and a polarization controlling system to maintain the polarization of the light in the fiber loop. These systems are described in Optics Letters, Vol. 15, No. 11 (Nov. 1980) pp. 488-490, and in International Patent Application No. PCT/US82/00400, filed March 31, 1982, which are hereby incorporated herein by reference. The polarization control and synchronous detection systems described therein are appropriate for use with the Kerr effect compensation of the present invention, and contribute to the overall performance of the rotation sensing interferometer described herein. The preferred embodiment will first be described in reference to these systems, and subsequently, a detailed description directed specifcally to Kerr effect compensation will be provided. However, it should be understood at the outset that the Kerr effect compensation of the present invention has general application, and may be utilized in ring interferometers other than the type described in reference to the preferred embodiment. As shown in Figure 1, the rotation sensing interfer¬ ometer of the preferred embodiment comprises a light source 10 for introducing light into a continuous length or strand of optical fiber 12, a portion of which is wound into a sensing loop 14. As used herein, the reference numeral 12 designates generally the entire continuous strand of optical fiber, while the numeral 12 with letter suffixes (A, B, C, etc.) designates portions of the optical fiber 12. In the embodiment shown, the light source 10 prefera¬ bly comprises a super-radiant diode or a super luminescent diode which operates according to the principles of stimu¬ lated emission in a plurality of modes. By way of specific example, the light source 10 may comprise a OMPI GaAlAs double-heterostructure laser as described by Wang et al., Appl. Phy. Lett. 41(7), 1 October 1982, pp. 587- 589. The fiber optic strand 12 may comprise a single mode fiber having, for example, an outer diameter of 80 microns and a core diameter of 4 microns. The loop 14 comprises a plurality of turns of the fiber 12, wrapped about a spool or other suitable support (not shown) . By way of specific example, the loop 14 may have approximately 1,000 turns of fiber wound on a form having a diameter of 14 centimeters. Preferably, the loop 14 is wound symmetrically, starting from the center, so that symmetrical points in the loop 14 are in proximity. Specifically, the fiber is wrapped about the spool so that the turns of the central portion of the loop 14 are positioned innermost adjacent to the spool and the turns toward the ends of the loop are positioned outermost away from the spool so that both end portions of the fiber loop 14 are positioned symmetrically about the central turns and are freely accessible at the outside of the loop 14. It is believed that this reduces the environmental sensitivity of the 'rotation sensor, since such symmetry causes time varying temperature and pressure gradients to have a similar effect on both of the counter-propagating waves. Light from the light source 10 is optically coupled to one end of the fiber 12 by a lens 15. The laser 10 and lens 15 will be referred to collectively as the light source 16. Various components for guiding and processing the light are positioned or formed at various locations along the continuous strand 12. For the purpose of describing the relative locations of these components, the continuous fiber 12 will be described as being divided into seven portions, labeled 12A through 12G, respectively, with the portions 12A through 12E being on the side of the loop 14 that is coupled to the source 16, and the portions 12F and 12G being on the opposite side of the loop 14. Adjacent to the light source 16, between the fiber portions 12A and 12B, is a polarization controller 24. A suitable type of polarization controller for use as the controller 24 is described in Electronics Letters, Vol. 16, No. 20 (Sept. 25, 1980) pp. 778-780, which is hereby incorporated by reference herein. A description of the polarization controller 24 will be provided subsequently; however, it should be presently understood that the controller 24 permits adjustment of both the state and direction of polarization of the applied light. The fiber 12 then passes through ports, labeled A and B, of a directional coupler 26, located between the fiber portions 12B and 12C, for coupling optical power to a second strand of optical fiber 28 which passes through the ports labeled C and D of the coupler 26, the port C being on the same side of the coupler as the port A, and the port D being on the same side of the coupler as the port B. The end of the fiber 28 extending ' from the port D terminates non-reflectively at the point labeled "NC" (for "not connected") while the end of the fiber 28 extending from the port C is optically coupled to a photodetector 30. By way of specific example, the photodetector 30 may comprise a standard, reverse-biased, silicon, pin-type photodiode. A coupler suitable for use in the present invention is described in detail in European patent application No. 81.102677.3, published in Bulletin No. 81/42 of Oct. 21, 1981, under publication No. 0038023, which is incorporated by reference herein. After passing through the polarizer 32, the fiber 12 passes through ports, labeled A and B, of a directional coupler 34, located between the fiber portions 12D and OMPI 12E. The coupler 34 is preferably of the same type as described above in reference to the coupler 26. The fiber 12 is then wound into the loop 14, with a polarization controller 36 located between the loop 14 and fiber portion 12F. The polarization controller 36 may be of the type discussed in reference to the controller 24, and is utilized to adjust the polarization of the waves counter- propagating through the loop 14 so that the optical output signal, formed by superposition of these waves, has a polarization which will be efficently passed, with minimal optical power loss, by the polarizer 32. Thus, by utilizing both the polarization controllers 24, 36, the polarization of the light propagating through the fiber 12 may be adjusted for maximum optical power. phase modulator 38, driven by an AC generator 40, and connected thereto by a line 41, is mounted on the fiber 12, between the loop 14 and the fiber portion 12F. This modulator 38 comprises a PZT cylinder, around which the fiber 12 is wrapped. The fiber 12 is bonded to the cylinder so that when it expands radially in response to the modulating signal from the generator 40 , it stretches the fiber 12. An alternative type of phase modulator (not shown), suitable for use with the present invention, com¬ prises a PZT cylinder which longitudinally stretches four segments of the fiber 12 bonded to short lengths of capil¬ lary tubing at the ends of the cylinder. Those skilled in the art will recognize that this alternative type of modu¬ lator may impart a lesser degree of polarization modula¬ tion to the propagating optical signal than the modulator 38 however, it will be seen subsequently that the phase modulator 38 may be operated at a frequency which eliminates the undesirable effects of phase modulator- induced polarization modulation. Thus, either type of phase modulator is suitable for use in the present inven- -lo¬ tion. The fiber 12 then passes through ports, labeled C and D of the coupler 34, with the fiber portion 12F extending from the port D and the fiber portion 12G extending from the port C. Fiber portion 12G terminates non-reflectively at a point labeled "NC" (for "not connected"). The output signal from the AC generator 40 is supplied on a line 44 to a lock-in amplifier 46, which also is connected to receive the output of the photodetector 30 by a line 48. This signal to the amplifier 46 provides a reference signal for enabling the amplifier 46 to synchronously detect the detector output signal at the modulation frequency of the phase modulator 38. Thus, the amplifier 46 effectively provides a bandpass filter at the fundamental frequency (i.e., first harmonic) of the phase modulator 33 , blocking all other harmonics of this frequency. The detected signal is integrated over a period of time, typically on the order of 1 second to 1 hour, to average out random noises. It will be seen below that the magnitude of this first harmonic' component of the detector output signal is proportional, through an operating range, to the rotation rate of the loop 14. The amplifier 46 outputs a signal, which is proportional to this first harmonic component, and thus proivides a direct indication of the rotation rate, which may be visually displayed on a display panel 47 by supplying the amplifier output signal to the display panel 47 on a line 49. The Couplers 26 and 34 A preferred fiber optic directional coupler for use as the couplers 26 and 34 in the rotation sensor or gyroscope of the present invention is illustrated in Figure 2. The coupler includes two strands 50A and 50B of a single mode fiber optic material mounted in longitudinal arcuate grooves 52A and 52B, respectively, formed in optically flat, confronting surfaces of rectangular bases or blocks 53A and 53B, respectively. The block 53A with the strand 50A mounted in the groove 52A will be referred to as the coupler half 51A, and the block 53B with the strand 50B mounted in the groove 52B will be referred to as the coupler half 51B. The arcuate grooves 52A and 52B have a radius of curvature which is very large compared to the diameter of the fibers 50, and have a width slightly larger than the fiber diameter to permit the fibers 50, when mounted therein, to conform to a path defined by the bottom walls of the grooves 52. The depth of the grooves 52A and 52B varies from a minimum at the center of the blocks 53A and 53B, respectively, to a maximum at the edges of the blocks 53A and 53B, respectively. This advantageously permits the fiber optic strands 50A and 50B, when mounted in the grooves 52A and 52B, respectively, to gradually converge toward the center and diverge toward the edges of the blocks 53A, 53B, thereby eliminating any sharp bends or abrupt changes in direction of the fibers 50 which may cause power loss through mode perturbation. In the embodiment shown, the grooves 52 are rectangular in cross- section; however, it will be understood that other suitable cross-sectional contours which will accommodate the fibers 50 may be used alternatively, such as a U- shaped cross-section or a V-shaped cross-section. At the centers of the blocks 53, in the embodiment shown, the depth of the grooves 52 which mount the strands 50 is less than the diameter of the strands 50, while at the edges of the blocks 53, the depth of the grooves 52 is preferably at least as great as the diameter of the strands 50. Fiber optic material was removed from each of the strands 50A and 50B, e.g., by lapping to form respective oval-shaped planar surfaces, which are coplanar with the confronting surfaces of the blocks 53A, 53B. These oval surfaces, where the fiber optic material has -1la- been removed, will be referred to herein as the fiber "facing surfaces." Thus, the amount of fiber optic material removed increases gradually from zero towards the edges of the blocks 53 to a maximum towards the center ofthe blocks 53. This tapered removal of the fiber optic material enables the fibers to converge and diverge gradually, which is advantageous for avoiding backward -12- reflection and excess loss of light energy. In the embodiment shown, the coupler halves 51A and 51B are identical, and are assembled by placing -the confronting surfaces of the blocks 53A and 53B together, so that the facing surfaces of the strands 50A and 50B are in facing relationship. An index matching substance (not shown), such as index matching oil, is provided between the confronting surfaces of the blocks 53. This substance has a refractive index approximately equal to the refractive index of the cladding, and also functions to prevent the optically flat surfaces from becoming permanently locked together. The oil is introduced between the blocks 53 by capillary action. An interaction region 54 is formed at the junction of the strands 50, in which light is transferred between the strands by evanescent field coupling. It has been found that, to ensure proper evanescent field coupling, the amount of material removed from the fibers 50 must be carefully controlled so that the spacing between the core portions of the strands 50 is within a predetermined "critical zone". The evanescent fields extend into the cladding and decrease rapidly with distance outside their respective cores. Thus, sufficient material should be removed to permit each core to be positioned substantially within the evanescent field of the other. If too little material is removed, the cores will not be sufficiently close to permit the evanescent fields to cause the desired interaction of the guided modes, and thus, insufficient coupling will resu.t Conversely, if too much material is removed, the propagation characteristics of the fibers will be altered, resulting in loss of light energy due to mode perturbation. However, when the spacing between the cores of the strands 50 is within the critical zone, each strand receives a significant portion of the evanescent field energy from the other strand, and good coupling is OMPI -13- achieved without significant energy loss. The critical zone includes that area in which the evanescent fields of the fibers 50A and 50B overlap with sufficient strangth to provide coupling, i.e., each core is within the evanescent field of the other. However, as previously indicated, mode perturbation occurs when the cores are brought too close together. For example, it is believed that, for weakly guided modes, such as the E^ mode in single mode fibers, such mode perturbation begins to occur when sufficient material is removed from the fibers 50 to expose their cores. Thus, the critical zone is defined as that area in which the evanescent fields overlap with sufficient strength to cause coupling without substantial mode perturbation induced power loss. The extent of the critical zone for a particular coupler is dependent upon a number of interrelated factors such as the parameters of the fiber itself and the geometry of the coupler. Further, for a single mode fiber having a step-index profile, the critical zone can be quite narrow. In a single mode fiber coupler of the type shown, the required center-to-center spacing between the strands 50 at the center of the coupler is typically less than a few (e.g., 2-3) core diameters. Preferably, the strands 50A and 50B (1) are identical to each other; (2) have the same radius of curvature at the interaction region 54; and (3) have an equal amount of fiber optic material removed therefrom to form their respective facing surfaces. Thus, the fibers 50 are symmetrical, through the interaction region 54, in the plane of their facing surfaces, so that their facing surfaces are coestensive if superimposed, which ensures that the two fibers 50A and 50B will have the same propagation characteristics at the interaction region 54, and thereby avoids coupling attenuation associated with dissimilar propagation characteristics. OMPI -14- The blocks or bases 53 may be fabricated of any suitable rigic material. In one presently preferred embodiment, the bases 53 comprise generally rectangular blocks of fused quartz glass approximately 1 inch long, 1 inch wide, and 0.4 inch thick. In this embodiment, the fiber optic strands 50 are secured in the slots 52 by suitable cement, such as epoxy glue. One advantage of the fused quartz blocks 53 is that they have a coefficient of thermal expansion similar to that of glass fibers, and this advantage is particularly important if the blocks 53 and fibers 50 are subjected to any heat treatment during the manufacturing process. Another suitable material for the block 53 is silicon, which also has excellent thermal properties for this application. The coupler includes four ports, labeled A, B, C, and D, in Figure 2. When viewed from the perspective of Figure 2, ports A and C, which correspond to strands 50A and 50B, respectively, are on the left-hand side of the coupler, while the ports B and D, which correspond to the strands 50A and 50B, respectively, are on / the right-hand side of the coupler. For the purposes of discussion, it will be assumed that input light is applied to port A. This light passes through the coupler and is output at port B and/or port D, depending upon the amount of power that is coupled between the strands 50. In this regard, the term "normalized coupled power" is defined as the ratio of the coupled power to the total output power. In the above example, the normalized coupled power would be equal to the ratio of the power at port D of the sum of the power output at ports B and D. This ratio is also referred to as the "coupling efficiency", and when so used, is typically expresses as a percent. Thus, when the term "normalized coupled power" is used herein, it should be understood that the corresponding coupling efficiency is equal to the normalized coupled power times 100. In this regard, tests have shown that the coupler of the type OMPI -15- shown in Figure 2 has a coupling efficiency of up to 100%. However, the coupler may be "tuned" to adjust the coupling efficiency to any desired value between zero and the maximum, by offsetting the facing surfaces of the blocks 53. Such tuning is preferably accomplished by sliding the blocks 53 laterally relative to each other. The coupler is highly directional, with substantially all of the power applied at one side of the coupler being delivered to the other side of the coupler. That is, substantially all of the light applied to input Port A is delivered to the output Ports B and D, without contra- directional coupling to Port C. Likewise, substantially all of the light applied to input Port C is delivered to the output Ports B and D. Further, this directivity is symmetrical. Thus, light supplied to either input Port B or input Port D is delivered to the output Ports A and C. Moreover, the coupler is essentially non- discriminatory with respect to polarizations, and thus, preserves the polarization of the coupled light. Thus, for example, if a light beam having a vertical polarization is input to Port A, the light coupled from Port A to Port D, as well as the light passing straight through from Port A to Port B, will remain vertically polarized. From the foregoing, it can be seen that the coupler may function as a beam-splitter to divide the applied light into two counter-propagating waves W1 ,W2 (Figure 1). Further, the coupler may additionally function to recombine the counter-propagating waves after they have traversed the loop 14 (Figure 1). In the embodiment shown, each of the couplers 26,34 has a coupling efficiency of about 50%, which proviees maximum optical power at the photodetector 30 (Figure 1 ) . As used herein, the term "coupling efficiency" is defined as the power ratio of the coupled power to the total output power, expressed as a percent. For example. -16- referring to Figure 2, if light is applied to Port A, the coupling efficiency would be equal to the ratio of the power at Port D to the sum of the power output at Ports B and D. The terms "coupling ratio" or "splitting ratio" are defined as the coupling efficiency divided by 100. Thus, a coupling efficiency of 50% could be equivalent to a coupling ratio or splitting ratio of 0.5. The Polarizer 32 A preferred polarizer 32 for use in the rotation sensor of Figure 1, is illustrated in Figure 3 and described on Optics Letters, Vol 5, No. 11 (Nov. 1980) pp. 479-481, which is hereby incorporated by reference herein.. The polarizer 32 includes a birefringent cristal 60, positioned within the evanescent field of light transmitted by the fiber 12. The fiber 12 is mounted in a slot 62 which opens to the upper face 63 of a generally rectangular quartz block 64. The slot 62 has an arcuately curved bottom wall, and the fiber is mounted in the slot 62 so that it follows the contour of this bottom wall. The upper surface 63 of the block 64 is lapped to remove a portion of the cladding from the fiber 1,2 is a region 67. The cyrstal 60 is mounted on the block 64, with the lower surface 68 of the crystal facing the upper surface 63 of the block 64, to position the crystal 60 within the evanescent field of the fiber 12. The relative indices of refraction of the fiber 12 and the birefringent material 60 are selected to that the wave velocity of the desire polarization mode is greater in the birefringent crystal 60 than in the fiber 12, while the wave velocity of an undesired polarization mode is greater in the fiber 12 than in the birefringent crystal 60. The light of the desire polarization mode remains guided by the core portion of the fiber 12, whereas light of the undesired polarization mode is coupled from the fiber 12 to the birefringent crystal 60. Thus, the polarizer 32 permits passage of light in one polarization mode, while OMPI -17- preventing passage of light in the other polarization mode. As previously indicated, the polarization controllers 24,36 (Figure 1) may be utilized to adjust the polarizations of the applied light and optical output signal, respectively, so that optical power loss through the polarizer 32 is minimized. The Polarization Controllers 24,36 One type of polarization controller suitalbe for use in the rotation sensor of Figure 1 is illustrated in Figure 4. The controller includes a base 70 on which a plurality of upright blocks 72A through 72D are mounted. Between adjacent ones of the blocks 72, spools 74A through 74C are tengentialy mounted on shafts 76A through 76C, respectively. The shafts 76 are axially aligned with each other, and are rotatably mounted between the blocks 72. The spools 74 are generally cylindrical and are positioned tangentially to the shafts 76, with the axes of the spools 74 perpendicular to the axes of the shafts 76. The strand 12 extends through axial bores in the shafts 76 and is wrapped about each of the spools 74 to form three coils 78A through 78C. The radii of the coils 78 are such that the fiber 12 is stressed to form a birefringent medium in each of the coils 78. The three coils 78A through 78C may be rotated independently of each other about the axes of the shafts 74A through 74C, respectively, to adjust the birefringence of the fiber 12 and, thus, control the polarization of the light passing through the fiber 12. The diameter and number of turns in the coils 78 are such that the outer coils 78A and C provide a spatial delay of one-quarter wavelength, while the central coil 783 provides a spatial delay of one-half wavelength. The quarter wavelength coils 78A and C control the ellipticity of the polarization, and the half wavelength coil 78B controls the direction of polarization. This provides a full range of adjustment of the polarization of the light propagating through the fiber 12. It will be understood. -18- however, that the polarization controller may be modified to provide only the two quarter wave coils 78A and C, since the direction of polarization (otherwise provided by the central, coil 78B) may be controlled indirectly through proper adjustment of the ellipticity of polarization by means of the two quarter wave coils 78A and C. Accordingly, the polarization controllers 24 and 36 are shown inFigure 1 as including only the two quarter wave coils 78A and C. Since this configuration reduces the overall size of the controllers 24-36, it may be advantageous for certain applications of the present invention involving space limitations. Thus, the polarization controllers 24 and 36 provide means for establishing, maintaining, and controlling the polarization of both the applied light and the counter- propagating waves. Operation Without Phase Modulation or Polarization Control In order to fully understand the function and importance of the polarizer 32 (Figure 1) and phase modulator 38, the operation of the rotation sensor will first be described as if these components had been removed from the system. Accordingly, Figure 5 shows the rotation sensor of Figure 1, in schematic block diagram form, with the modulator 38, polarizer 32, and associated components removed therefrom. Light is coupled from the source 16 to the fiber 12 for propagation therethrough. The light enters Port A of the coupler 26, where a portion of the light is lost through Port D. The remaining portion of the light propagates from Port B of the coupler 26 to Port A of the coupler 34, where it is split into two counter-propagating waves W1,W2, the wave W1 propagates from the Port B in a clockwise direction about the loop 14, while the wave W2 propagates from Port D in a counter-clockwise direction around the loop 14. After the waves W1 ,W2 have traversed OMPI -19- the loop 14, they are recombined by the coupler 34 to form an optical output signal, which propagates from Port A of the coupler 34 to Port B of the coupler 26. A portion of the optical output signal is coupled from Port B to Port C of the coupler 26 for propagation along the fiber 28 to the photodetector 30. This photodetector 30 outputs an electrical signal which is proportional to theintensity of the light impressed thereon by the optical output signal. The intensity of the optical output signal will vary in accordance with the amount and type (i.e., constructive or destructive) of interference between the waves W1,W2 when they are recomgined or superposed at the coupler path around the loop 14. Thus, assuming the loop 14 is at rest, when the waves Wl,W2 are recombined at the coupler 34, they will interfere constructively, with no phase difference therebetween, and the intensity of the optical output signal will be at a maximum. However, when the loop 14 is rotated, the counter-propagating waves Wl,W2 will be shifted in phase, in accordance with the Sagnac effect, so that when they are superposed at the coupler 34, they destructively intefere to reduce the intensity of the optical output signal. The Sagnac phase difference between the waves W1,W2, caused by rotation of the loop 14, is defined by the following relationship: Λ _ 8ηNA 0 ,, . Φ ws λc " Ω (1) Where A is the area bounded by the loop 14 of optical fiber, N is the number of turns of the optical fiber about the area A, Ω is the angular velocity of the loop about an axis which is perpendicular to the plane of the loop, and λ and c are the free space values of the wavelength and the velocity, respectively, of the light applied to the loop. OMPI 20 The intensity of the optical output signal (I τ ) is a function of the Sagnac phase difference (φws) between the waves W1,W2, and is defined by the following equation: T T = τ i + T 2 + 2 ϊ Λ COsφ ws (2) where I. and I_ are the individual intensities of the waves Wl,W2, respectively. From Equations (1) and (2) , it may be seen that the intensity of the optical output signal is a function of the rotation rate (Ω) . Thus, an indication of such rotation rate may be obtained by measuring the intensity of the optical output signal, utilizing the detector 30. Figure 6 shows a curve 80, which illustrates this relationship between the intensity of the optical output signal (I ) and the Sagnac phase difference (φ ) X s between the counter-propagating waves Wl,W2. The curve 80 has the shape of a cosine curve, and the intensity of the optical output signal is at a maximum when the Sagnac p r hase difference (φws) is zero*. If it is assumed that the only source of phase difference between the counter-propagating waves W1.W2 is the rotation of the loop 14, the curve 80 will vary symmetrically about the vertical axis. However, in practice, a phase difference between the counter- propagating waves W1,W2 may be caused not only by rotation of the loop 14, but also by the residual birefringence of the optical fiber 12. Birefringence- induced phase differences occur because fiber birefringence tends to cause each of the two polarization r.odes of the single mode fiber 12 to propagate light at a different velocity. This creates a non-reciprocal, non-rotationally induced phase difference between the waves V?1,W2, which causes the waves W1,W2 to interfere in a manner that distorts or shifts the curve 80 of Figure 6, for eκarrple, as illustrated by the curve 82, shown in phantom lines. OMPI 21 Such birefringence-induced, non-reciprocal phase difference is indistinguishable from a rotationally-induced Sagnac phase difference, and is dependent on environmental factors which vary fiber birefringence, such as temperature and pressure. Thus, fiber birefringence may cause errors in rotation sensing. Operation With the Polarizer 32 The problem of non-reciprocal operation due to fiber birefringence is solved by means of the polarizer 32 (Figure 1) which, as discussed above, permits utilization of only a single polarization mode. Thus, when the polarizer 32 is introduced into the system, at the point designated by the reference numeral 84 in Figure 5, light input through the polarizer 32 propagates into the loop 14 in the desired polarization mode. Further, when the counter-propagating waves are recombined to form the optical output signal, any light that is not of the same polarization as the light applied to the loop is prevented from reaching the photodetector 30, since the optical output signal, as it travels from Port A of coupler 34 to Port B of coupler 26, also passes through the polarizer 32. Thus, the optical output signal, when it reaches the detector 30, will have precisely the same polarization as the light applied to the loop. Therefore, by passing the input light and optical output signal through the same polarizer 32, only a single optical path is utilized, thereby eliminating the problem of birefringence-induced phase difference. Further, it should be noted that the polarization controllers 24,36 (Figure 1) may be used to adjust the polarization of the applied light, and optical output signal, respectively, to reduce optical power loss at the polarizer 32, and thus, maximize the signal intensity at the detector 30. 22 Operation With the Phase Modulator 38 Referring again to Figure 6, it will be seen that, because the curve 80 is a cosine function, the intensity of the optical output signal is nonlinear for small phase differences (Φ W.o_) between the waves Wl,W2. Further, the optical output signal intensity is relatively insensitive to changes in phase difference, for small values of φ . Such nonlinearity and insensitivity makes it difficult to transform the optical intensity (I ) measured by detector 30 into a signal indicative of the rate of rotation Ω (Equation 1) of the loop 14. Further, although birefringence-induced phase differences between the waves W1,W2 are eliminated, as discussed above, by use of the polarizer 32, fiber birefringence may also cause a reduction in the optical intensity of the optical output signal, since light may be prevented from reaching the photodetector 30 by the polarizer 32. Thus, changes in fiber birefringence may cause the amplitude of the curve 80 of Figure 6 to vary, for example, as illustrated by the curve 84. The foregoing problems are solved by means of a synchronous detection system utilizing the phase modulator 38, signal generator 40, and lock-in amplifier 46, shown in Figure 1. Referring to Figure 7, the phase modulator 38 modulates the phase of each of the propagating waves W1,W2 at the frequency of the signal ' generator 40. However, as may be seen from Figure 1, the phase modulator 38 is located at one end of the loop 14. Thus, the modulation of the wave Wl is not necessarily in phase with the modulation of the wave W2. Indeed, it is necessary for proper operation of this synchronous detection system that the modulation of the waves Wl,W2 be out of phase. Referring to Figure 7, it is preferable that the modulation of the wave Wl, represented by the sinusoidal curve 90, be 180 degrees out of phase with - - modulation of the wave W2, represented by the curve Q' 23 Use of a modulation frequency which provides such 180-degree phase difference between the modulation of the wave Wl relative to that of W2 is particularly advantageous in that it eliminates modulator-induced amplitude modulation in the optical output signal measured by the detector 30. This modulation frequency (f ) may be calculated using the following equation: where L is the differential fiber length, between the coupler 34 and modulator 38, for the counter-propagating waves W1,W2 (i.e., the distance, measured along the fiber, between the modulator 38 and a symmetrical point on the other side of the loop 14) ; neq is the equivalent refractive index for the single mode fiber 12, and c is the free space velocity of the light applied to the loop 14. At this modulation frequency (f ) , the phase difference (Φ ) between the counter-propagating waves W1,W2, due to phase modulation of these waves in accordance with the curves 90 and 92, is illustrated by the sinusoidal curve 94 in Figure 7. This modulation of the phase difference between the waves W1,W2 will modulate the intensity (I-,) of the optical output signal in accordance with the curve 80 of Figure 6, since such phase modulation i is indistinguishable from rotationally- induced Sagnac phase differences The foregoing may be understood more fully through reference to Figures 8 and 9 which graphically illustrate the effect of (a) the phase modulation ς , defined by the curve 94 of Figure 7, and (b) the Sagnac phase difference cws, u-oon the intensity (I_T) of the ootical output signal, represented by the curve 80 of Figure 6. However, before proceeding with a discussion of Figures 7 and 8, it should first be understood that the intensity OMPI 24 (I_) of the modulated optical output signal is a function of the total phase difference between the waves W1,W2. Further, such total phase difference is comprised of both the rotationally-induced Sagnac phase difference φ and the time varying modulation-induced phase difference φwm. Thus, the total phase difference „w between the waves 1,W2 may be expressed as follows: Accordingly, since the effects of the modulation-induced ' phase difference <•!> , as well as the rotationally-induced phase difference Φ τ Vv r S_, will be considered in reference to Figures 8 and 9, the horizontal-axis for the curve 80 has been relabeled as φw to indicate that the total p * • hase difference is being considered, rather than only the rotationally-induced phase difference, as in Figure 6. Referring now to Figure 8, the effect of the phase modulation φ τ w„m (curve 94) ■ up e-on the intensity J. I_f l ¬ of the optical output signal (curve 80) will be discussed. In Figure 8,. it is assumed that the loop 14 is at rest, and thus, the optical signal is not affecte'd by the Sagnac effect. Specifically, it may be seen that the modulation- induced phase difference curve 94 varies the optical output signal in accordance with the curve 80, symmetrically about its vertical axis, so that the optical intensity measured by the detector 30 varies periodically at a frequency equal to the second harmonic of the modulating frequency, as shown by the curve 96. Since, as discussed above, the lock-in amplifier 46 is enabled by the signal generator 40 (Figure 1) to synchronously detect the detector output signal at the modulation frequency (i.e., first harmonic) of the modulator 38, and since the detector output signal is at the second harmonic of the modulation frequency, as shown by the curve 96, the amplifier output signal will be zero and the display 47 will indicate a rotation rate of zero. It should be 25 noted that, even if birefringence-induced amplitude fluctuations occur in the optical output signal, as discussed in reference to the curve 84 ofFigure 6, the curve 96 of Figure 8 will remain at a second harmonic frequency. Thus, such birefringence-induced amplitude fluctuations will not affect the amplifier 46 output signal. The detection system, therefore, provides a substantially stable operating point that is insensitive to changes in birefringence, particularly when the loop. 14 is at rest. When the loop 14 is rotated, the counter-propagating waves W1,W2 are shifted in phase, as discussed above, in accordance with the Sagnac effect. The Sagnac phase shift provides a phase difference φ ¥»S which adds to the phase difference φw created by t-he modulator 38, so that the entire curve 94 is translated in phase from the position shown in Figure 8, by an amount equal to φ , to the position shown in Figure 9. This causes the optical output signal to vary non-symmetrically in accordance with the curve 80, thereby harmonically distorting this signal, as shown by the curve 96 of Figure 9, so that it includes a component at the fundamental (i.e., first harmonic) frequency of the modulator 38, as illustrated in phantom lines by the sinusoidal curve 98. It will be seen subsequently that the RMS value of this sinusoidal curve 98 is proportional to the sine of the rotationally-induced, Sagnac phase difference φ WS. Since the amplifier 46 synchronously detects signals having the fundamental frequency of the modulator 38, the amplifier 46 will output a signal to the display 47 that is proportional to the RMS value of the curve 98 to indicate the rotation rate of the loop. The drawings of Figure 9 illustrate the intensity waveform of the optical output signal for one direction of rotation (e.g., clockwise) of the loop 14. However, it will be understood that, if the loop 14 is rotated in the opposite direction (e.g., counter-clockwise) at an equal velocity, the intensity waveform 96 of the optical output signal will be exactly the same as illustrated in Figure 9, except that it will be translated so that the curve 98 is shifted 180 degrees from the position shown in Figure 9. The lock-in amplifier 46 detects this 180-degree phase difference for the curve 98, by comparing its phase with the phase of the reference signal from the signal generator 40, to determine whether the rotation of the loop is clockwise or counter-clockwise. Depending on the direction of rotation, the amplifier 46 outputs either a positive or negative signal to the display 47. However, regardless of the direction of rotation, the magnitude of the signal is the same for equal rates of rotation of the loop 14. The waveform of the amplifier output signal is shown in Figure 10 as the curve 100. It will be seen that this curve 100 is sinusoidal and varies positively or negatively from zero rotation rate depending on whether the rotation of the loop 14 is clockwise or counter-clockwise. Further, the curve 100 has a substantially linear portion 102 which varies symmetrically about the origin and provides a relatively wide operating rate for measuring rotation. Moreover, the slope of the curve 100 provides excellent sensitivity throughout its linear operating range 102. Thus, by utilizing the synchronous detection system, the above-described problems of non-linearity, insensitivity, and birefringence-induced amplitude flunctuations are reduced or eliminated. A further advantage of this detection system relates to the fact that state-of-the-art phase modulators, such as the modulator 38, induce amplitude modulation in the optical output signal, either directly, or indirectly through polarization modulation. However, 27 it will be recalled from the discussion in reference to Equation 3 that, by operating at a specific frequency at which the phase difference between the modulation of. the waves Wl and W2 is 180 degrees, the odd harmonic frequency components of the amplitude modulation, that are induced in each of the counter-propagating waves W1,W2 by the modulator 38, cancel each other when the waves are superposed to form the optical output signal. Thus, since the above-described detection system detects only an odd harmonic (i.e., the fundamental frequency) of the optical output signal, the effects of amplitude modulation are eliminated. Therefore, by operating at the specific frequency, defined by Equation 3, and detecting only an odd harmonic of the optical output signal, the rotation sensor of the present invention may operate independently of modulator-induced amplitude and polarization modulation. A further benefit of operating at the specific frequency is that even harmonics of the phase modulation, induced by the modulator 38 in each of the <counter- propagating phases W1,W2, cancel when these waves are superposed to form the optical output signal. Since these even harmonics may produce spurious odd harmonics • in the optical output signal which might otherwise be detected by the detection " system, their elimination improves the accuracy of rotation sensing. -. In addition to operating the phase modulator 38 at the frequency, defined by Equation 3, it is also preferable to adjust the magnitude of the phase modulation so that the amplitude of the detected first harmonic of the optical output signal intensity is maximized, since this provides improved rotation sensing sensitivity and accuracy. It has been found that the first harmonic of the optical output signal intensity is at the maximum, for a given rotation rate, when the amplitude of the modulator-induced phase difference between the waves W1,W2, 28 indicated by the dimension labeled z in Figures 7, 8, and 9, is 1.84 radians. This may be understood more full through reference to the following equation for the total intensity (I_) of two superposed waves having individual intensities .of I. and I 2 , respectively, with a phase difference φ therebetween. X T = Z ϊ + Z 2 + 2 ϊ Λ COSφ w (5) where: φ ψ w = φ τ ws + - φ ψ wm (6) and Φ w. m = z sin (2τrf t) (7) Thus, « w = Φw wos + Z Sil1 ^^ mm*) (8 ) the Fourier expansion of cosine φ is : no cos φ w = cos Φ ws ^J 0 (z ) + Σ J 2n (z ) cos [2τr (2nf πι t) ] } n=l CO -sin t]} (9) where Jn(z) is the n Bessel function of the variable z , and z is the peak amplitude. of the modulator-induced phase difference between the waves W1,W2. Therefore, detecting only the first harmonic of I yields: ^ ( i ) = 4l *i*2 J ι< z)sin Φ ws sin ( 2τrf m t ) ( 10 ) Thus, the amplitude of the first harmonic of the optical output signal intensity is dependent upon the value of the first Bessel function J, (z). Since J.(z) is a maximum when z equals 1.84 radians, the amplitude of the phase modulation should preferably be selected so that the magnitude of the modulator-induced phase difference (z) between the waves W1,W2 is 1.84 radians. Reducing the Effects of Backscatter As is well known, present state-of-the-art optical fibers are not optically perfect, but have imperfections 29 which cause scattering of small amounts of light. This phenomena is commonly referred to as rayleigh scattering. Although such scattering causes some light, to be lost from the fiber, the amount of such loss is relatively small, and therefore, is not a major concern. The principal problem associated with rayleigh scattering • relates not to scattered light which is lost, but rather, to light which is reflected so that it propagates through the fiber in a direction opposite to its original direction of propagation. This is commonly referred" to as "backscattered" light. Since such backscattered light is coherent with the light comprising the counter- propagating waves W1,W2, it can constructively or • destructively interfere with-such propagating waves, and thereby cause "noise" in the system, i.e., cause spurious variations in the intensity of the optical output signal, as measured by the detector 30. Destructive or constructive interference between the backscattered waves and the propagating waves may be reduced by means of the additional phase modulator 39 at the center of the fiber loop 14. This phase modulator is driven by a signal generator (not shown) , which is not synchronized with the modulator 38. The propagating waves will pass through this additional phase modulator 39 one time only, on their travel around the loop. For backscatter which occurs from a propagating wave before the wave reaches the additional modulator, the backscatter will not be phase modulated by this additional modulator, since neither its source propagating wave nor the backscatter itself has passed through the additional modulator. On the other hand, for backscatter which occurs from a propagating wave after the wave passes through this additional phase modulator, the backscatter will be effectively twice phase modulated, once when the -30- propagating wave passed through the additional phase modu¬ lator, and once the backscatter passed through the addi¬ tional modulator. Thus, if the additional phase modulator introduces a phase shift of φ (t), the backscattered wave originating at any point except at the center of the loop 14 will have a phase shift of either zero, or 2 φ (t), either of which is time varying with respect to the φ (t) phase shift for the propagating wave. This time varying interference will average out over time, effectively eliminating the effects of the backscattered light. Kerr Effect Compensation As previously indicated, the Kerr effect refers to a phenomenon in which the refractive index of a substance changes in the presence of a varying electric field. In a non-linear medium the optical Kerr effect may induce an unequal phase delay for counter-propagating waves. Fused silica optical fibers such as the fiber 12 are sufficient¬ ly non-linear that the Kerr-induced phase difference in a fiber optic rotation sensor may be much' larger than the phase difference due to the Sagnac effect. The optical Kerr effect manifests itself as intensity- dependent pertrubations of the propagation constants of the fiber 12. The perturbations of the propagation constants are functions of the states of polarization of the two counter-propagating waves. Therefore, in a fiber rotation sensor it is preferable that the states of polarization of the counter-propagating waves be identical. The polarizer 32 permits utilization of only a single polarization mode in the loop 14. Therefore, in the following analysis of the Kerr it is assumed that the counter-propagating waves have identical, linear polariza¬ tions. The perturbations of the propagating constants of the counter-propagating waves are: 4πηn -31- βk χ (z,t) = δ [I χ (z,t) + 2I 2 (z,t)] (11) 4πηn βk 2 (z,t) = [I 2 (z,t) 2I ] _(z,t)] (12) where η is the impedance of the medium, λ is the vacuum wavelength of the waves, n is the Kerr coefficient of the 10 medium, and δ is a factor of order unity that depends on the transverse distribution of the light mode in the fiber 12. I* j _ (z,t) and I (z,t) are the wave transverse peak intensities, which depend upon the displacement around the loop and upon time. ^5 An important feature of equations 11 and 12 is the factor of 2 between I-. and I 2 that shows, that the inten¬ sity of wave 2 has twice the effect on the propagation constant of wave 1 as does the intensity of wave 1 on itself. The term "self effect" is used herein to refer to ° the effect which the electric field of f a lightwave has upon the propagation constant seen by the same lightwave. The term "cross effect" is used herein to refer to the change in propagation constant of the lightwave caused by the electric field of another lightwave which propagates in the opposite direction. Referring to Equation 12, the cross effect of wave 1 on the propagation constant of wave 2 is twice the self-effect of wave 2. The cross effects being twice the self effects " gives rise to a non- reciprocity by causing the perturbations of the 0 propagation constants to be different if the intensities I-_ and I 2 are not the same. No non-reciprocity would occur if the perturbations were to depend simply on the sum of the two intensities and no non-reciprocity would occur if the intensities I-, and I 2 were the same because 5 then I- j _ + 2I 2 = I 2 + 2I- j _. The non-reciprocity causes the propagation velocities -32- of the 2 waves, Wl and W2, to be different because propa¬ gation velocity is a function of the propagation constant. Therefore, one of the waves, Wl, W2 will traverse the loop 14 more rapidly than the other, thereby creating a phase difference between the waves Wl, W2 at the coupler 34. The Kerr effect-induced phase difference is indistinguish¬ able from a rotationally-induced phase difference, thereby resulting in a spurious rotation signal. As explained in Optics Letters, Vol. 7, No. 6 (June 1982) pp. 282-284, and hereby incorporated herein by reference, modulation of the waves, Wl and W2, in accordance with a specific waveform, serves to reduce or eliminate the relative influence of the cross effect. According to that article, as the waves, Wl and W2, propagate, the cross effect is particularly important in giving rise to Kerr effect-induced phase differences. The use of modulation to control the phase shift caused by the Kerr effect may be more easily understood through reference to Figure 11, which illustrates a pair of square wave intensity modulated counter-propagating lightwaves, which each have a duty cycle of 50%. The cross effect is present when the intensities of the two waves, Wl and W2, have coincident peaks; otherwise the cross effect is absent; and each wave Wl, W2, experiences only the self effect. Each portion of the waves, Wl, W2, experiences the self effect all the time and the cross effect half of the time because of the 50% duty cycle. Therefore, the factor of 2 in equations 11 and 12 is reduced to an average value of unity, thereby effectively cancelling the non-reciprocity. Non-reciprocal phase accumulated over half of the cycle is compensated for by the non-reciprocal phase of the opposite sign accumulated by an adjustment in the duty cycle of the pulse train in order to obtain full compensation. Still referring to Figure 11, it will be assumed for purposes of illustration that wave Wl has a peak intensity of 3 in some arbitrary units while the wave -33- W2 has a peak intensity of 1 in the same units. Part of the square wave corresponding to peak intensity is referred to as the crest portion and the part of the square wave that is at zero intensity is referred herein as the trough portion. Because of the Kerr effect, the propagation constant seen by the trough portion of a given wave will be different than the propagation constant seen by the crest portion of that same wave. In this particular example, the intensity of the wave at the trough portions is negli¬ gible and it will not contribute to the rotation rate error, so it may be ignored. Thus, in this example, only the propagation constant seen by the crest portions need be examined to determine the intensity weighted average phase shift of the waves. Because of the Kerr effect, the propagation constant of the crest portions of either of the waves Wl,W2 will change as the wave travels through successive crest and trough portions of the oncoming wave. For example, if one imagines an observer at an arbitrary reference point on the crest portion of the wave Wl, as at point A in Figure 11, traveling with the wave Ul, the propagation constant seen by that observer will be at a first value when point A is within a crest portion of the oncoming wave W2, and will be at a second value when point A is within a trough portion of the oncoming wave W2. Since the duty cycle of the wave W2 is 50%, and the observer sees trough portions and crest portions of this wave an equal amount of time, the average propagation constant of the wave Wl (e.g., seen by the observer at point A) will simply be the average of the sum of these first and second values. The situation is similar for an observer traveling with the wave W2 on its crest portion, as at the reference point B in Figure 11. The propagation constant of the wave W2 (e.g., seen by the point B observer) will change between first and second values as it travels through successive -34- crest and trough portions of the oncoming wave Wl, i.e., the propagation constant will be at a first value when point B is within a crest portion of the wave Wl and at the second value when point B is within a trough portion of the wave Wl. Since the wave Wl also has a 50% duty cycle, the average propagation constant of the wave W2 (e.g., the point B observer) is the average of the sum of these first and second values. It should be noted that the first and second values for the wave Wl may be different than those for the wave W2, however, if the Kerr effect is fully compensated, the average propagation constants for the waves Wl and W2 will be the same. The average, Kerr-induced change in propagation constants for each of the waves Wl,W2 will now be calcula¬ ted for the example described above in reference to Figure 11. For the wave Wl, the Kerr-induced change in propaga¬ tion constant (Δβ-^) seen, e.g., at point A, when point A is within a crest of the oncoming wave W2, may be defined as: Δβ-. = k(I- 2I 2 ) (11a) However, when point A is within a trough of the oncoming wave W2, the Kerr-induced change in propagation constant (β-t) seen by the wave Wl at point A is: Δβ χ = k(I χ ) (12a) where I- j _ and I 2 are the intensities of the waves W1,W2, respectively. The constant k is included as a proportion¬ ality constant. From Equations 11a and 12a, it may be seen that, as expected, when point A of the wave Wl is within a crest of the oncoming wave W2, there is both a self-effect (due to OMPI -35- Wl) and a cross-effect (due to W2) , while, when point A is within a trough of the oncoming wave, there is only a self-effect. Substituting the appropriate values into Equations 11a and 12a, it may be seen that when point A is within a crest, the change in propagation constant is equal to 5k, and when it is within a trough, the change in refractive index is 3k. Thus, the average propagation constant of the wave Wl is equal to 4k. For the wave W2, the change in propagation constant when point B is within a crest of the oncoming wave Wl, may be defined as: Δβ., = k(I. 2I χ ) (13) However, when point B of the wave W2 is within a trough of the oncoming wave Wl, the change in propagation constant is: Δβ. = k(I 2 ) (14) Inserting the appropriate intensity values into Equation 13, the change in propagation constant, when point B is within a crest portion of the oncoming wave, is equal to 6K. Similarly, using Equation 14, it may be found that the change in propagation constant, when point B is within a trough of the oncoming wave, is equal to Ik. Since the duty cycle is 50%, and the wave W2 has each propagation constant an equal amount of time, the average propagation constant is simply the average of Ik + 7k, which is, again, 4k. Therefore, even though the respective intensities of the waves W1,W2, in the above example, were quite differ¬ ent, the average change in the propagation constant for OMPI -36- each of the waves was the same (e.g., 4k) over a complete period of square wave modulation. Using other intensities for the waves W1,W2 should yield the. same results. The Kerr-induced rotation rate error Ω k , for the rotation sensor of the preferred embodiment, is a function of the intensity weighted average of the phase differences between the waves. Also, the detection system described above provides a signal proportional to the intensity weighted average of the non-reciprocal (Kerr-induced) phase shift. Accordingly, the Kerr-induced rotation rate error may be expressed mathematically as follows: <I 0 2 (t)> - 2 <I 0 (t)> 2 Q. = - ηnδ(l-2K) (15) R <3: o (t)> where c is the velocity of light in a vacuum, R is the radius of the fiber coil, η is the impedence of the medium, n is the Kerr coefficient of the medium, δ is a factor on the order of unity that depends upon the trans- verse distribution of the mode, K is the coupling ratio, Ig(t) is the intensity of the modulated source wave as a function of time. The angled brackets indicate an average over time. The Kerr-induced rotation rate error Ω k , for the rota- tion sensor of the preferred embodiment, is a function of the intensity weighted average of the phase differences between the waves. Also, the detection system described above provides a signal proportional to the intensity weighted average of the non-reciprocal (Kerr-induced) phase shift. Referring back to Equation 15, it may be seen that non-reciprocal operation caused by the Kerr effect may be eliminated, at least theoretically, by adjusting the coup¬ ler 34 so that the splitting ratio is 0.5 and the waves W1,W2 are equal in intensity. However, to achieve suffi¬ cient Kerr compensation for inertial navigation applica- -37- tions, requiring an accuracy of .001 degrees/hours, it is estimated that the splitting ratio of the coupler 34 would need to be adjusted within a tolerance on the order of magnitude of 0.5 x 10""'*, assuming a cw souce. In prac- tice, it appears that this is not possible, or is at least impractical, even under laboratory conditions. It is believed that the best achievable tolerance, under labora¬ tory conditions, would be no more than about 0.5 > 10~3, which would not provide sufficient Kerr compensation for many applications. Moreover, maintaining such a tolerance would be very difficult, particularly in commercial appli¬ cations where the interferometer is subjected to vibration or other physical disturbances, as in aircraft gyroscopes. Kerr Compensation With Thermal Sources When used with the apparatus described herein and according to the method of the present invention, a large class of light sources 10 provide the intensity variations required to compensate for Kerr effect-induced errors in rotation sensors. The relation between the rotation rate error Ω-^ and the source intensity, I(t) is: Ω χ α <I (t)> τ - 2 <I(t) <I(t)> 2τ > T (17) where the angled brackets indicate time averaging over time intervals T or 2 τ, as indicated by the subscripts, where T is the detection system integration time and τ is the transit time of light around the fiber loop 14. Assuming that the average intensity is established over a time 2τ, Equation 17 becomes: Ω α <I 2 (t)> - 2 <I(t)> 2 (18) In the following it is assumed that I(t) is a stochastic process varying randomly with time; that the -38- average value of I(t) is constant with respect to time; and that I(t) is ergodic so that the occurrence of any particular value is equally likely. By defining I = I(t) and identifying the variance as: 5 permits Equation 18 to be written as: 2 — 2 a τ ~ - = I ~~ - I (19) ^ where the bar over a quantity indicates an ensemble average or mean value. 2 - 2 Ω α σ τ - I (20) 15 which shows that the rotation rate error induced by the Kerr effect vanishes if the standard deviation, σ- j -, is equal to the mean value, I, of the intensity. Polarized thermal light, such as is output by a superluminescent 0 diode, satisfies the condition that σ-**- = I. Therefore, polarized thermal light may be used to provide a rotation sensor which is insensitive to the Sagnac effect. By virtue of the central limit theorem, which is a well-known mathematical relationship, the amplitude of ^ polarized thermal light is a circular complex Gaussian random process. The corresponding intensity is a random process with an exponential probability density defined by: 0 exp - I > 0 I P(D = Z (21) 0 I < 0 5 The second moment of the intensity of polarized thermal light is: OMPI -39- which after integration by parts becomes: (23) = 21 Therefore, referring to Equation 18, when polarized thermal light is used in the rotation-sensing device described herein, the Kerr effect produces no error in the sensed rotation rate. Kerr Compensation With Multimode Source Laser light comprising a number of oscillatory modes, or frequencies, i.e., multimode lasers, 'may also be used in the rotation sensing device described herein to provide a lower rotation rate error .than is possible with unmodulated light of a single frequency. It is assumed that the laser cavity (not shown) is dispersive such that the laser oscillatory modes are not evenly spaced in frequency as is the case with diode lasers. The light amplitude in the dispersive case is the sum of the contributions of the individual modes and may be expressed as: A (t) « A e if(n) ωt (24) n=l π __OMPI -40- Where A_ and f ( n ) ω are the complex ampl itude and frequency, respectively of the n oscillating mode; and N is the number of modes or frequencies at which the cavity oscillates. A similar result is obtained without dispersion if the phases of each of the modes are random and statistically independent. Neglecting the finite line width of the individual modes and expanding f(n) in a Taylor series gives: N A(t) I A r i (ω + nΔω + n 2δsω )t (25) n=l The corresponding intensity is I(t) = A(t)A*(t) = A Am e it(n"m)Δω + ( n 2 -m 2)δ ω+... ] t n,m=l n m The average intensity is the sum of the averages of the individual terms. The averages of the exponentials are zero unless n=m, which yields: N < I(t) > = I |A r (27) n=l The square of the intensity is; I 2 ( t ) = I A A A-A e i[(π-m+l-k)Δω+(n 2 -m 2 +l 2 -k 2 )δω+...]t ^ n m 1 k n,m,l,k=l OMPI -41- The time average of the exponentials is zero unless n-m+1- k = 0 and n 2-m2-12-k2 = 0. For the possibility that n=m, it is required that l=k in order that I(t) ≠O. If n≠m, then it is required that n=k and m=l in order that I(t) ≠Q - The foregoing restrictions on n, m, 1 and k cause all the terms in the exponent of Equation 28 to vanish, giving: N N N <ι 2 (t]> = I IA 2 ^ ! 2 + I |A n | 2 |A m | 2 - I |A n | (29) n,l»l n,m=l n=l which may be expressed as N <I ώ (t)> = 2 <I(t)>' n (30) n=l The foregoing assumes amplitudes which are constant with respect to time. In order to account for the finite times over which the time averaged values were determined in applying Equation 30, it is sufficient that 2Φ{τ>>l and φ{T >>1, which are not overly restrictive since τ is on the order of microseconds, and T is on the order of seconds. Consideration of the case of N equal amplitude modes such that yields: -42- which shows that the Kerr-induced rotation rate error is inversely proportional to the number of oscillating modes of the laser. Experimental Results A working model of the rotation sensor described herein employed a GaAs laser diode as the light source 10. This particular laser diode has more than 50 oscillating modes and 10 oscillating modes within the full width of the half maximum power points of the laser output spectrum. The light source 10 inserted an average of about 120 microwatts of power into the fiber 12 and about 10 microwatts were received by the detector 30. The relative amounts of power of the counterpropagating waves in the loop 14 were adjusted by tuning the coupling ratio K of the coupler 34. If the intensity of the counterpropagating waves were constant in time, the rotation sensor would suffer a rotation rate error due to the optical Kerr effect of up to approximately 10 degrees per hour. No such error was observed within the limits of accuracy of the measurements when the multimode laser described hereinabove was used as the light source 10. Within the experimental accuracy, the fluctuating intensity light source 10 produced a Kerr induced rotation rate error which was less than 1% of the predicted error for a constant intensity source. Adjustment of the polarization controller 36 to vary the polarization of the counter propagating waves did not produce a Kerr induced rotation rate error, which is in agreement with theoretical predictions that Kerr effect compensation using the apparatus and methods of the OMPI -43- present invention is independent of polarization. Figure 12 illustrates the rotation signal from the rotation sensing device of the invention for a five hour period. Figure 12 illustrates that the present invention provides a substantial improvement in the long-term stability of fiber optics rotation sensing systems. The offset of just over 0.1 deg/h, which is about 1% of the earth's rotation rate, as measured by turning off the light source 10 may reflect inaccuracy in the oreientation of the axis of the coil 14, which should be perpendicular to the earth's axis of rotation. The curve exhibits an RMS short-term noise value of 0.02 deg/h, which is due to the electronics of the system. If the light source 10 does not provide an output having a sufficient number of oscillatory modes to produce satisfactory Kerr effect compensation, a modulator 13 may be included between the light source 10 and the lens 15. Ordinarily the modulator 13 will modulate the output of the light source 10 so that counter propagating light waves having 50% duty cycles are coupled into the loop 14. For example, if the light source' 10 outputs two frequencies which beat together to have a duty cycle of 67%, the modulator 13 provides modulation to reduce the duty cycle to 50%. Such modulation may be provided by square wave modulation of the output of the light source 10. ~

Claims: -44- WHAT IS CLAIMED IS: 1. A fiber optic interferometer with reduced Kerr- effeet-induced error comprising: a loop (14) of fiber optic material, said fiber 5 optic material having a predetermined index of refraction; means for coupling (34) a first light wave and a second light wave into said loop, said first and second light waves propagating in opposite directions 10 in said loop, each of said first and second light waves having an electric field associated therewith which produces a variation in the index of refraction of said fiber optic material; and a light source (10) connected to said coupling 15 means for generating said first and second light waves, said light source providing light having a plurality of oscillatory modes to said coupler means, said plurality of oscillatory modes combining to produce a light intensity which varies with time for 20 reducing Kerr-effeet-induced error. 2. A fiber optic interferometer according to Claim 1 wherein said plurality of oscillatory modes combine to produce a light intensity having a duty cycle of 50 percent. ' 25 3. A fiber optic interferometer as defined in Claim 1, additionally comprising: means for modulating (13) the light intensity from said light source, said modulating means cooperating with the variations in said light intensity produced by said plurality of oscillatory modes 30 to reduce Kerr-effect-induced errors. 4. A fiber optic interferometer as defined in Claim 3 wherein said modulating means (13) cooperates with said variation in light intensity produced by said plurality of oscillatory modes to yield an effective amplitude duty 35 cycle for said light source of 50%. 5. A fiber optic interferometer as defined in Claim OMPI -45- 1 wherein said fiber optic material is single mode for said plurality of oscillatory modes of said light source. 6. A method for reducing the Kerr-effeet-induced error in a fiber optic ring interferometer, comprising the steps of: selecting a light source (10) having a sufficient number of oscillatory modes to cause interference between said modes which will yield a duty cycle selected to compensate for Kerr-effeet-induced errors; coupling light from said source (10) into the interferometer (14) such that two counter-propagating waves travel therethrough, said counter-propagating waves experiencing a phase shift in response to rotation of said interferometer; combining (34) the counter-propagating waves to produce an interference pattern; and detecting (30) intensity variations in the interference pattern to measure the rate of rotation of said interferometer. 7. An interferometer having reduced Kerr-effect- induced error, comprising: a light source (10) selected to produce plural oscillatory modes which interfere with one another to modulate the intensity of light from said source; a loop (14) of optical fiber; a coupler (34) connecting said light source (10) to said loop (14), said coupler (34) coupling a first portion of the light from said source to circulate in said loop (14) in a first direction and a second portion of said light to circulate in the other direction; said coupler (34) further combining said first and second portions, subsequent to their circulation in said loop (14), to provide an interference pattern, said modulation reducing the Kerr-effect-induced error in said interference pattern; and OMPI -46- means for detecting (30) said interference pattern. 8. An interferometer as defined in Claim 7 wherein said light source (10) is a thermal source. 9. An interferometer as defined in Claim 7 wherein said light source (10) comprises a multi-mode source having a deterministic phase relationship between said modes. 10. An interferometer as defined in Claim 6 wherein said light source (10) comprises a multi-mode light source having a random phase relationship between said modes.

Inventors: BERGH RALPH A, CULSHAW BRIAN, CUTLER CASSIUS C, LEFEVRE HERVE C, SHAW HERBERT JOHN, BERGH, RALPH, A., CULSHAW, BRIAN, CUTLER, CASSIUS, C., LEFEVRE, HERVE, C., SHAW, HERBERT, JOHN
Applicants: UNIV LELAND STANFORD JUNIOR, THE BOARD OF TRUSTEES OF THE LELAND STANFORD JUNIO
Application year: 1982
Main classification: G01C19
IPC codes: G01C19, G01P9, G02B6, G02B26, G02F1