INTERFEROMETRIC SENSOR USING TIME DOMAIN MEASUREMENTS

Abstract: An interferometer used as a rotation sensor is constructed using a strand of optical fiber (112), a portion of which is formed into a sensing loop (114). A pair of lightwaves (W₁, W₂) are caused to counterpropagate in the sensing loop (114) and are combined to form an optical output signal that has an intensity that varies in accordance with the difference in the phases of the two counterpropagating lightwaves (W₁, W₂). A phase modulator (192) is positioned on the optical fiber (112) in the sensing loop (114) at a location such that the two counterpropagating lightwaves (W₁, W₂) are modulated approximately 180° out of phase. The time-varying phase modulation causes a time-varying phase difference (ΔΦ M (t)) that is combined with a rotationally-induced Sagnac effect phase to provide a total phase difference that is detected by a photodetector (160). The photodetector (160) provides an electrical output signal this is differentiated to provide a differentiated output signal (520, 450, 460) that has zero-crossings (H′, L′) that correspond to maxima and minima of the electrical output signal. The time intervals (Δt₁′) between selected zero-crossings (H′, L′) are measured and the rotationally-induced phase difference (ΔΦ R ) caused by the Sagnac effect is calculated from the measured time intervals (Δt₁′) The rotation rate (Q) is then calculated from the Sagnac phase difference (ΔΦ R ).

Description: Background of the Invention Field of the Invention The present invention relates to optical interferometers and, more particularly, to interferometric sensors such as fiber optic gyroscopes. Description of the Related Art Fiber optic rotation sensors typically include a loop of fiber optic material to which lightwaves are coupled such that a pair of lightwaves propagate around the loop in opposite directions (i.e., the two lightwaves are counterpropagating). When the loop is rotated, a relative phase difference is induced between the counterpropagating lightwaves in accordance with the well-known "Sagnac effect." The amount of phase difference corresponds to the rotational velocity of the loop. The counterpropagating lightwaves, when recombined, interfere constructively or destructively to produce an optical output signal that varies in intensity in accordance with the rotation rate of the loop. Rotation sensing is commonly accomplished by detection of this optical output signal. A number of devices and techniques have been developed to process the detected optical output signal to provide an electrical output signal that represents the velocity and direction of rotation of the loop. Known devices and techniques monitor the intensity of the optical output signal to measure the phase difference between the counterpropagating lightwaves to determine the rotational velocity and direction in accordance with the Sagnac equation: where: ΔΦ R is the Sagnac phase difference caused by rotation of the optical loop; A is the area bounded by the optical loop in which lightwaves counterpropagate; N is the number of times that the lightwaves propagate around the loop before being recombined; Q is the angular velocity of the loop about an axis that is perpendicular to the plane of the loop; and λ and c are the free space values of the wavelength and velocity, respectively, of the lightwaves applied to the optical loop. The intensity of the optical output signal is a function of the Sagnac phase difference ΔΦ R between the two counterpropagating lightwaves as follows: I T = I₁ + I₂ + 2√I₁I₂ cos(ΔΦ R ) (2) where I T is the intensity of the optical output signal, and I₁ and I₂ are the individual intensities of the two counterpropagating lightwaves. It has been found that simple measurement of the intensity of the optical output signal will not provide sufficient information from which the direction and rate of rotation can be determined. For example, the sensitivity of the phase difference measurement is effectively zero for phase differences that are integral multiples of π (i.e., ΔΦ R = Nπ for N = ...-2,-1,0,1,2,...) because the interference intensity is an even, periodic function of the phase difference (i.e., the interference intensity varies as a cosine function as set forth in Equation (2) above). The sensitivity of the phase difference measurement can only be measured module π and is negligible when ΔΦ R is an integral multiple of π because the cosine function in Equation (2) varies as 1 - ΔΦ R2 /2 (module π) near a maximum and thus the intensity of the output signal does not vary linearly with ΔΦ R . Small phase differences cannot therefore be directly measured near a zero rotation rate. Typically, this difficulty in measuring small phase differences is overcome by dynamic biasing wherein an additional phase modulation is introduced into the closed path around which the lightwaves are propagating. A time-varying modulation with a zero mean amplitude is generally used rather than a static modulation. The dynamic biasing causes the phase signal to reach values where the sensitivity is suitably large so that the phase difference is readily measurable. A gyroscope with dynamic biasing can be readily used to measure small rotation rates. When the phase modulation is a harmonic signal of a predetermined frequency, a small phase shift caused by the rotation of the optical loop causes the intensity of the optical output signal to include a time-varying component at the predetermined frequency. This time-varying component can be demodulated to provide a measure of the Sagnac phase shift and thus the rotation rate. Although the dynamic biasing allows phase differences to be measured with high sensitivity at low rotation rates, the rotation-induced interference intensity is periodic and thus cannot be used directly to measure high rotation rates. In order to provide an extended dynamic range for measuring rotation rates, two basic approaches have been used. One approach is to provide a feedback signal from the demodulated optical output signal to a second device in the optical loop. The second device provides an additional, controllable, nonreciprocal phase difference and the feedback signal is varied so as to null the demodulated optical output signal. The feedback signal required to null the demodulated optical output signal is monitored to measure the Sagnac phase shift and thus the rotation rate. The performance of this type of closed-loop gyroscope depends in part upon the stability, linearity and phase range of the nonreciprocal phase shifter used for the second device and upon the characteristics of the electronics that provides the feedback signal. An alternative approach to extending the dynamic range of an optical fiber gyroscope is an open-loop approach wherein no feedback is provided to null the demodulated output signal. All the information necessary to reconstruct the Sagnac phase difference is included in the interference intensity caused by the combination of the rotation and the dynamic modulation. Typically, extensive signal processing is required to extract the phase difference information from the optical output signal when the phase difference is not limited to values near zero. When an extended dynamic range is to be attained, the signal processing can be quite demanding. In many designs of open-loop optical gyroscopes and other interferometers, the quality of the electronic signal processing circuitry, rather than the quality of the optical circuit, determines the dynamic range and accuracy of the sensing system. Both analog and digital processing may be used to process the optical output signal, and it is preferable that most of the processing be performed in the digital domain. However, typical commercially available analog-to-digital convertors needed to convert the analog electrical representation of the optical output signal to digital data do not have sufficient dynamic range to provide quality digital processing. Thus, complex analog circuitry is typically required as part of the electronics that processes the optical output signal. Other electronic circuitry has been used that relies less heavily upon digital processing. However, generally, the electronic components must be carefully selected and stabilized in order to achieve the high accuracy that is required to provide a sensitive interferometer with an extended dynamic range. Summary of the Invention According to the present invention in a first aspect, there is disclosed an interferometric sensor for sensing an ambient effect comprising a light source for producing a lightwave, an interferometer comprising an optical waveguide optically coupled to the light source, the optical waveguide defining first and second optical paths, at least one of the optical paths being exposed to the ambient effect, a coupler for introducing the lightwave into the optical paths in the form of first and second counterpropagating lightwaves (W₁, W₂), the first and second counterpropagating waves (W₁, W₂) recombining after the traverse in opposite directions of the optical paths so as to produce an output lightwave having an output waveform, the phase difference (ΔΦ R ) between the first and the second counterpropagating waves (W₁, W₂) after the traverse varying with the ambient effect and a modulator coupled to at least one of the optical paths for modulating the phase of light propagating therethrough to cause a time varying modulating phase difference (ΔΦ M ) between the first and second counterpropagating lightwaves (W₁, W₂), whereby the modulator changes the output waveform into a modulated output waveform. The interferometric sensor of the present invention is characterized by a signal processing circuit for detecting a first pair of points (H', L') on the modulated output waveform, the signal processing circuit further measuring the time domain separation (Δt₁') between the points of the first pair of points (H', L') and computing the phase difference (ΔΦ R ) as a function of the time domain separation (Δt₁') and selected characteristics of the modulating phase difference (ΔΦ M ) so as to measure the ambient effect. The selected characteristics of the modulating phase difference (ΔΦ M ) preferably comprise the maximal amplitude (ΔΦ Mmax ) and the period (T) of the modulating phase difference (ΔΦ M ). In a particular embodiment of the present invention, the first pair of points (H', L') on the modulated output waveform substantially corresponds to points of the unmodulated output waveform having a same first amplitude. The first pair of points (H', L') corresponds to extrema on the unmodulated output waveform. More particularly, one of the first pair of points (H') corresponds to a first maximum on the unmodulated output waveform and the other of the first pair of points (L') corresponds to a second consecutive maximum on the unmodulated output waveform. In an alternate embodiment, the signal processing circuit further detects a second pair of points (I', K') on the modulated output waveform, the signal processing circuit further measuring the time domain separation (Δt₂') between the points of the second pair of points (I', K') and computing the phase difference (ΔΦ R ) as a function of the time domain separations (Δt₁', Δt₂') and the period (T) of the modulating phase difference (ΔΦ M ). Preferably, the second pair of points (I', K') on the modulated output waveform substantially corresponds to points of the unmodulated output waveform having a same second amplitude. Advantageously, the second pair of points (I', K') corresponds to extrema on the unmodulated output waveform. Preferably, one of the second pair of points (I') corresponds to a first minimum on the unmodulated output waveform and the other of the first pair of points (K') corresponds to a second consecutive minimum on the unmodulated output waveform. A preferred embodiment of the processing circuit advantageously comprises a differentiator for differentiating the modulated output waveform so as to provide a differentiated waveform, a zero-crossing detector for detecting the zero crossings on the differentiated waveform corresponding to the first pair of points (H', L') and a circuit that measures the time domain separation (Δt₁') between the zero crossings corresponding to the first pair of points (H', L'). Preferably, the zero-crossing detector further detects the zero crossings corresponding to the second pair of points (I', K') and wherein the circuit measures the time domain separation (Δt₂') between the zero crossings corresponding to the second pair of points (I', K'), the circuit further comprising a processor for computing the phase difference (ΔΦ R ) as a function of the time domain separations (Δt₁', Δt₂') the period (T) of the modulating phase difference (ΔΦ M ). The modulating phase difference (ΔΦ M ) may have a maximal amplitude (ΔΦ Mmax ) of at least π radians or of no more than 2π radians. Advantageously, the modulator has a phase excursion which is from a value between π and 2π radians to a value between -π and -2π radians. The modulating phase difference (ΔΦ M ) may be a sine function or a triangular function. In the particular embodiment of the present invention, the ambient effect is induced by rotation and is detected by the rotationally induced Sagnac effect produced in one optical path submitted to rotation, whereas the signal processing circuit comprises digital electronics. The optical waveguide forms a loop optically closed by the coupler. According to a second aspect of the present invention, there is disclosed a method of sensing an ambient effect, comprising the steps of counterpropagating first and second lightwaves (W₁, W₂) in first and second optical paths, submitting at least one of the optical paths to the ambient effect, combining the counterpropagating lightwaves (W₁, W₂) after propagation through the optical paths to form an optical output signal having an output waveform, the phase difference (ΔΦ R ) between the first and the second counterpropagating waves (W₁, W₂) after the traverse varying with the ambient effect, modulating the phase difference (ΔΦ R ) between the counterpropagating lightwaves (W₁, W₂) to cause a time varying modulating phase difference (ΔΦ M ) between the first and second counterpropagating lightwaves (W₁, W₂) and changing the output waveform into a modulated output waveform by means of the modulating phase difference (ΔΦ M ). The method is characterized by the steps of detecting a first pair of points (H', L') on the modulated output waveform, measuring the time domain separation (Δt₁') between the points of the first pair of points (H', L'), computing the phase difference (ΔΦ R ) as a function of the time domain separation (Δt₁') selected characteristics of the modulating phase difference (ΔΦ M ) and measuring the ambient effect by means of the phase difference (ΔΦ R ). Preferably, the phase difference (ΔΦ R ) is computed as a function of the maximal amplitude (ΔΦ Mmax ) and the period (T) of the modulating phase difference (ΔΦ M ). The method disclosed in the present invention preferably comprises the steps of selecting one point (H') of the first pair of points (H', L') so that the point (H') corresponds to a first maximum of the unmodulated output waveform and selecting the other (L') of the first pair of points (H', L') so that the point (L') corresponds to a second consecutive maximum of the unmodulated output waveform. It may also comprise the steps of detecting a second pair of points (I', K') on the modulated output waveform, measuring the time domain separation (Δt₂') between the points of the second pair of points (I', K') and computing the phase difference (ΔΦ R ) as a function of the time domain separations (Δt₁', Δt₂') and the period (T) of the modulating phase difference (ΔΦ M ). Advantageously, it also comprises the steps of selecting one of the second pair of points (I') so that the point (I') corresponds to a first minimum of the unmodulated output waveform and selecting the other of the first pair of points (K') so that the point (K') corresponds to a second consecutive minimum of the unmodulated output waveform. The method ideally comprises the steps of differentiating the modulated output waveform so as to provide a differentiated waveform, detecting the zero crossings on the differentiated waveform corresponding to the first pair of points (H', L') and measuring the time domain separation (Δt₁') between the zero crossings corresponding to the first pair of points (H', L'). It is advantageous that the method comprises the steps of detecting the zero crossings corresponding to the second pair of points (I', K'), measuring the time domain separation (Δt₂') between the zero crossings corresponding to the second pair of points (I', K') and computing the phase difference (ΔΦ R ) as a function of the time domain separations (Δt₁', Δt₂') and the period (T) of the modulating phase difference (ΔΦ M ). Preferably, the step of submitting at least one of the optical paths comprises the step of rotating the one optical path and wherein the step of measuring the ambient effect is performed by detecting the rotationally induced Sagnac effect produced in the one optical path submitted to rotation. Brief Description of the Drawings Figure 1 is a schematic drawing of a Sagnac interferometer of the prior art, showing fiber optic components positioned along a continuous, uninterrupted length of fiber optic material, and further showing a photodetector that detects the optical output signal and processing circuitry to process the electrical signal generated by the photodetector. Figure 2 is a schematic drawing of the interferometer of Figure 1 with the phase modulator removed therefrom. Figure 3 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. Figure 4 is a graph of phase difference as a function of time, showing the phase modulation of each of the counterpropagating lightwaves and the phase difference between the counterpropagating lightwaves. Figure 5 is a schematic drawing illustrating the effect of the phase modulation upon the intensity of the optical output signal, as measured by the photodetector, when the sensing loop is not rotating. Figure 6 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 sensing loop is rotating in a first direction. Figure 7 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 sensing loop is rotating in a second direction, opposite the first direction. Figure 8 is a schematic drawing similar to Figure 5 illustrating the effect of a triangular phase modulation upon the intensity of the optical output signal, as measured by the photodetector, when the sensing loop is not rotating, and further illustrating an electrical waveform produced by differentiating the electrical output of the photodetector. Figure 9 is a schematic drawing similar to Figure 8 illustrating the effect of a triangular phase modulation upon the intensity of the optical output signal, as measured by the photodetector, when the sensing loop is rotating, and further illustrating an electrical waveform produced by differentiating the electrical output of the photodetector. Figure 10 is a schematic drawing of an enlarged portion of the triangular modulation waveform of Figure 8. Figure 11 is a schematic drawing of an enlarged portion of the triangular modulation waveform of Figure 9. Figure 12 is a block diagram of an exemplary embodiment of the apparatus of the present invention showing the rotation sensor, the photodetector, the electrical differentiating circuit connected to the electrical output of the photodetector, the zero-crossing detector and the counter/timer circuit. Figure 13 is a block diagram of an exemplary embodiment of the counter/timer circuit of Figure 12. Figure 14 is a schematic drawing similar to Figure 9 illustrating the effect of a sinusoidal phase modulation upon the intensity of the optical output signal, as measured by the photodetector, when the sensing loop is rotating, and further illustrating an electrical waveform produced by differentiating the electrical output of the photodetector. Figure 15 is a schematic drawing of an enlarged portion of the sinusoidal modulation waveform of Figure 14. Detailed Description of the Preferred Embodiment Prior Art As shown in Figure 1, an open loop rotation sensor includes a light source 110 that introduces light into a continuous length of a first optical fiber 112. A portion of the first optical fiber 112 is wound into a sensing loop 114. The first optical fiber 112 is preferably a single-mode optical fiber having, for example, an outer cladding diameter of about 80 microns and an inner core diameter of about 4 microns. The single-mode fiber 112 is preferably a high birefringence fiber, although non-birefringent fiber may also be utilized. The sensing loop 114 comprises a plurality of turns of the first optical fiber 112 wrapped around a spool or other support (not shown). By way of specific example, the sensing loop 114 may comprise approximately 1000 turns of the first optical fiber 112 wound on a support having a diameter of approximately 14 centimeters. Preferably, the sensing loop 114 is wound symmetrically, starting from the center of the length of fiber in the loop 114, so that symmetrical points in the loop are in proximity. Symmetrical winding of the sensing loop 114 reduces the environmental sensitivity of the rotation sensor 100 because the symmetry causes time-varying temperature and pressure gradients to have a similar effect on both counterpropagating waves in the sensing loop 114. Light from the light source 110 is optically coupled to one end of the first optical fiber 112 by, for example, butting a first end 120 of the first optical fiber 112 against the light source 110. An input end portion (or first end portion) 122 of the first optical fiber 112 provides an optical path from the first end 120 to a first directional coupler 124 that is positioned on the first optical fiber 112. The first directional coupler 124 is advantageously constructed in accordance with U.S. Patent Nos. 4,493,528 and 4,536,058, which are incorporated herein by reference. The first directional coupler 124 has a first port 130, a second port 132, a third port 134 and a fourth port 136. The first optical fiber 112 passes through the first directional coupler 124 between the first port 130 and the second port 132, and continues as an intermediate portion 140 of the first optical fiber 112. Within the first directional coupler 124, the first optical fiber 112 is juxtaposed with a second optical fiber 150 which may be of the same type as the first fiber 112. The second fiber 150 has a first end portion 152 and a second end portion 154. The first end portion 152 of the second optical fiber extends from the third port 134 of the first directional coupler 124 and is optically coupled to a photodetector 160. The second end portion 154 extends from the fourth port 136 of the first directional coupler 124, and is terminated non-reflectively by a light-absorbing terminator 162. The directional coupler 124 is preferably constructed to couple approximately 50 percent of the light in one of the optical fibers to the other optical fiber. Accordingly, approximately 50 percent of the light propagating in the input end portion 120 of the first optical fiber 112 is coupled to the end portion 154 of the second optical fiber 150. The other 50 percent of the light exits the second port 132 of the first directional coupler 124 and propagates in the intermediate portion 140. The intermediate portion 140 of the first optical fiber 112 provides an optical propagation path from the first directional coupler 124 to a second directional coupler 170 that may be constructed in a similar manner to the first directional coupler 124. The second directional coupler 170 has a first port 172, a second port 174, a third port 176 and a fourth port 178. The intermediate portion 140 of the first optical fiber 112 enters the second directional coupler 170 at the first port 172. After passing through the second directional coupler 170 between the first port 172 and the second port 174, the first optical fiber 112 forms the sensing loop 114, as discussed above. The first optical fiber 112 then passes back through the second directional coupler 170 from the fourth port 178 and the third port 176. A second end portion 180 of the first optical fiber 112 extends from the third port 176 and is terminated by a terminator 182 that comprises a light-absorbing material. The second directional coupler 170 is constructed to have substantially 50 percent coupling efficiency. When light is input to the second directional coupler 170 via the first port 172, the light will be split so that the light output from the second directional coupler 170 will be divided substantially equally between the two output ports 174 and 178. The light output from the second directional coupler 170 forms a first lightwave, represented by an arrow W₁, that propagates around the sensing loop 114 in a clockwise direction, as viewed in Figure 1, and a second lightwave, represented by an arrow W₂, that propagates around the sensing loop 114 in a counterclockwise direction. Thus, the second directional coupler 170 forms a pair of counterpropagating lightwaves that traverse the sensing loop 114. Since the coupling efficiency of the second directional coupler 170 is 50 percent, the intensities of the two lightwaves W₁ and W₂ are substantially equal. After traversing the entire length of the first optical fiber 112 that forms the sensing loop 114, the first lightwave W₁, enters the second directional coupler 170 via the fourth port 178, and the second lightwave W₂ enters the second directional coupler 170 via the second port 174, such that the waves W₁ and W₂ are recombined. Approximately 50 percent of the recombined light exits the first port 172 for propagation to the intermediate portion 140 of the first optical fiber 112. The remaining 50 percent of the recombined light exits via the third port 176 to be absorbed by the light-absorbing terminator 182. The recombined lightwaves propagate via the intermediate portion 140 of the first optical fiber 112 to the second port 132 of the first directional coupler 124. Approximately 50 percent of the recombined lightwaves that enter the second port 132 are coupled to the first end portion 152 of the second optical fiber 150 via the third port 134 and propagate to the photodetector 160. The photodetector 160 outputs an electrical signal that is proportional to the intensity of the optical output signal, and the electrical signal is processed by a processing circuit to produce a signal indicative of the rotation rate of the loop. A polarizer 190 is advantageously positioned on the intermediate portion 140 of the first optical fiber 112 between the first directional coupler 124 and the second directional coupler 170. The polarizer 190 may be constructed in accordance with U.S. Patent No. 4,386,822, which is incorporated herein by reference. The polarizer 190 is positioned on the intermediate portion 140 of the first optical fiber 112 so that light of one polarization mode is permitted to pass through the intermediate portion 140 while the passage of light of an orthogonal polarization mode is prevented. As set forth in U.S. Patent 4,410,275, which is incorporated by reference herein, the purpose of the polarizer 190 is to ensure reciprocal operation of the rotation sensor of Figure 1. Spurious nonreciprocal phase differences may occur in the sensor because the light travelling in each of the two polarization modes of a single mode fiber, such as the first optical fiber 112, travels at a different propagation velocity. Such a nonreciprocal phase difference is indistinguishable from a rotationally induced Sagnac phase difference and is dependent on environmental factors, such as temperature and pressure. When the polarizer 190 is included, light passing through the polarizer 190 and into the sensing loop 114 via the second directional coupler 170 propagates in only a selected one of the two polarization modes. Furthermore, when the two counterpropagating lightwaves W₁ and W₂ are recombined to form the recombined optical signal, any light in the recombined optical signal that does not have the same polarization as the light applied to the sensing loop 114 is prevented from reaching the first directional coupler 124 and thus does not form part of the optical output signal that is detected by the photodetector 160. Accordingly, by passing the input light (i.e., the light input to the sensing loop 114) and the output light (i.e., the light output from the sensing loop 114) through the same polarizer 190, only a single optical path is utilized for propagating the lightwaves. Thus, nonreciprocity caused by the different propagation velocities of the two polarization modes is eliminated. The single mode optical fiber 112 is preferably a highly birefringent optical fiber having two well-defined polarization modes, and the polarizer is adjusted to select light propagating in one of the two polarization modes. If non-birefringent fiber is used, it may be preferable to include a pair of polarization controllers (not shown) to adjust the polarization of the input optical signal and the polarization of the light propagating in the sensing loop so that the polarization is aligned with the polarization axis of the polarizer 190, thereby reducing optical power loss in the polarizer 190. As set forth in U.S. Patent No. 4,529,312, which is incorporated by reference herein, reciprocity may also be achieved through the use of unpolarized light. It has been found that unpolarized light causes the nonreciprocal phase errors to cancel. As further illustrated in Figure 1, a phase modulator 192 is formed on the optical fiber 112 proximate to the fourth port 178 of the second directional coupler 170 between the fourth port 178 and the sensing portion 114. The phase modulator 192 is electrically connected to and driven by an electrical signal generator 194. The phase modulator 192 may comprise a PZT cylinder, around which a portion of the first optical fiber 112 is wrapped. The first optical fiber 112 is bonded to the PZT cylinder so that when the PZT cylinder expands radially in response to an electrical modulating signal from the signal generator 194, the portion of the first optical fiber bonded to the PZT cylinder stretches. Other modulators, such as electro-optic modulators, can also be used. In order to understand the function of the phase modulator 192, the operation of the rotation sensor 100 will first be described as if the phase modulator 192 were not present. For the purposes of this discussion, the rotation sensor 100 is illustrated in Figure 2 without the phase modulator 192. As set forth above, light from the source 110 traverses the sensing loop 114 and is combined to form an optical output signal that is detected by the photodetector 160. The intensity of the optical output signal received by the photodetector 160 will vary in accordance with the amount and type (i.e., constructive or destructive) interference between the two counterpropagating lightwaves W₁ and W₂ when they are recombined at the second directional coupler 170. Assuming the sensing loop 114 is at rest (i.e., is not rotating), the two waves W₁ and W₂ travel the same optical path around the sensing loop 114. Thus, when the two waves W₁ and W₂ are recombined at the second directional coupler 170, they will interfere constructively, with no phase difference therebetween, and the intensity of the optical output signal detected by the photodetector 160 will be at a maximum. However, when the sensing loop 114 is rotated, the counterpropagating lightwaves W₁ and W₂ will be shifted in phase in accordance with the Sagnac effect, so that when they are superimposed at the second directional coupler 170, they destructively interfere, thereby causing a reduction in the intensity of the optical output signal and thus reducing the magnitude of the electrical output signal generated by the photodetector 160. The Sagnac phase difference between the lightwaves W₁ and W₂ is defined by the following relationship: where: ΔΦ R is the Sagnac phase difference caused by rotation of the optical loop; A is the area bounded by the optical loop in which the lightwaves counterpropagate; N is the number of times that the lightwaves propagate around the loop before being recombined; Q is the angular velocity of the loop about an axis that is perpendicular to the plane of the loop; and λ and c are the free space values of the wavelength and velocity, respectively, of the lightwaves applied to the optical loop. The intensity of the optical output signal is a function of the Sagnac phase difference ΔΦ R between the two counterpropagating lightwaves as follows: I T = I₁ + I₂ + 2√ I₁ I₂ cos(ΔΦ R ) (2) where I T is the intensity of the optical output signal, and I₁ and I₂ are the individual intensities of the two counterpropagating lightwaves. From Equations (1) and (2), it may be seen that the intensity of the optical output signal is a function of the rotation rate Q. Thus, an indication of the rotation rate Q may be obtained by measuring the intensity of the optical output signal by using the photodetector 160. Figure 3 shows a curve 200 that illustrates the relationship between the intensity I T of the optical output signal and the Sagnac phase difference ΔΦ R between the two counterpropagating lightwaves W₁ and W₂. The curve 200 has the shape of a cosine curve in accordance with Equation (2), and the intensity of the optical output signal is at a maximum when the Sagnac phase difference ΔΦ R is zero or an even multiple of π (i.e., ...-4π, -2π, 0, 2π, 4π ...) and at a minimum when the Sagnac phase difference ΔΦ R is a odd multiple of π (i.e., ...-3π, -π, π, 3π, 5π ...). When the phase difference between the counterpropagating waves is caused entirely by rotation of the sensing loop 114, the curve 200 will vary symmetrically about a vertical axis 202. Referring to Figure 3, it can be seen that, because the curve 200 is a cosine function, the intensity of the optical output signal detected by the photodetector 160 is highly nonlinear for small Sagnac phase differences ΔΦ R between the two counterpropagating lightwaves W₁ and W₂. Furthermore, the optical output signal intensity is relatively insensitive to changes in phase difference for small changes in the value of ΔΦ R . Such nonlinearity and insensitivity makes it very difficult to transform the intensity I T of the optical output signal, as measured by the photodetector 160, into a signal that is indicative of the rate of rotation of the sensing loop 114 in accordance with Equations (1) and (2). For example, assuming that the horizontal axis of the Figure 3 to the right of the vertical axis 202 represents Sagnac phase differences ΔΦ R caused by clockwise rotation of the sensing loop 114 and that the horizontal axis to the left of the vertical axis 202 represents Sagnac phase differences ΔΦ R caused by counterclockwise rotation of the sensing loop 114, it can be seen that the direction of rotation cannot be determined by simply measuring the intensity of the optical output signal since the intensity is symmetrical about the vertical axis 202 and is the same for equal rotation rates in either direction. A number of devices and techniques have been used to measure the rotation rate by modulating the phases of the two counterpropagating lightwaves W₁ and W₂ such that the intensity of the optical output signal detected by the photodetector 160 includes sufficient information so that small rotation rates can be measured and so that rotation in the clockwise direction can be distinguished from rotation in the counterclockwise direction. For example, U.S. Patent Nos. 4,410,275; 4,456,377; 4,529,312; 4,634,282; and 4,637,722; and U.S. Patent Application Serial No. 488,111, entitled "Fiber Optic Rotation Sensor With Extended Dynamic Range," filed on April 25, 1983 (now U.S. Patent No. 4,687,330, issued on August 18, 1987); U.S. Patent Application Serial No. 581,304, entitled "Gated Fiber Optic Rotation Sensor With Extended Dynamic Range," filed on February 17, 1984 (now U.S. Patent No. 4,728,192, issued on March 1, 1988); and U.S. Patent Application Serial No. 581,303, entitled "Gated Fiber Optic Rotation Sensor With Linearized Scale Factor," filed on February 17, 1984 (now U.S. Patent No. 4,707,136, issued on November 17, 1987), which are incorporated herein by reference, disclose devices and techniques for modulating the phase of the counterpropagating lightwaves in a sensing loop and for determining the rotation rate from the intensity of the optical output signal. Referring to Figure 4, the phase modulator 192 modulates the phase of each of the counterpropagating lightwaves W₁ and W₂ at the frequency of the signal generator 194. As set forth above, the phase modulator 192 is preferably located proximate to the second directional coupler 170, near one end of the sensing loop 114. Thus, the modulation of the clockwise propagating lightwave W₁ is not necessarily in phase with the modulation of the counterclockwise propagating lightwave W₂. The modulation of the lightwave W₁ is typically 180° out of phase with the modulation of the lightwave W₂. This is illustrated in Figure 4, wherein the modulation of the phase of the lightwave W₁ is represented by a sinusoidal curve 210 that is 180° out of phase with a sinusoidal curve 212 that represents the modulation of the phase of the lightwave W₂. Use of a modulation frequency that provides such 180° phase difference between the modulation of the lightwave W₁ relative to that of the lightwave W₂ is particularly advantageous because it eliminates the effects of modulator induced amplitude modulation in the optical output signal measured by the photodetector 160. The preferred modulation frequency f m that provides such 180° phase difference between the modulations of the two lightwaves may be calculated in accordance with the following equation: where: L is the differential fiber length between the second directional coupler 170 and the phase modulator 192 for the counterpropagating lightwaves W₁ and W₂, i.e., the distance, measured along the fiber, between the phase modulator 192 and a symmetrical point on the opposite end of the sensing loop 114; n eq is the equivalent refractive index for the optical fiber 112; and c is the free space velocity of the light applied to the sensing loop 114. At the selected modulation frequency f m , which is often called the "proper frequency", the phase difference ΔΦ M between the two counterpropagating lightwaves W₁ and W₂ caused by the phase modulation of the two lightwaves is illustrated by a sinusoidal curve 214 in Figure 4. The curve 214 is obtained by subtracting the curve 212 from the curve 210 to obtain the phase difference between the lightwave W₁ and the lightwave W₂. The modulation of the phase difference between the lightwave W₁ and the lightwave W₂ also modulates the intensity I T of the optical output signal in accordance with the curve 200 of Figure 3. Such phase modulation ΔΦ M is indistinguishable from rotationally-induced Sagnac phase difference ΔΦ R . The foregoing may be more fully understood by referring to Figures 5 and 6 which graphically illustrate the combined effect of the phase modulation ΔΦ M , defined by the curve 214 of Figure 4, and the rotationally-induced Sagnac phase difference ΔΦ R , upon the intensity I T of the optical output signal. It should be understood that the intensity I T is a function of the total phase difference between the lightwaves W₁ and W₂, and that the total phase difference is comprised of both the rotationally-induced Sagnac phase difference ΔΦ R and the time-varying modulation-induced phase difference ΔΦ M . The total phase difference ΔΦ T between the two lightwaves W₁ and W₂ may be expressed as follows: ΔΦ T = ΔΦ R + ΔΦ M (4) As indicated above, the effects of the modulation-induced phase difference ΔΦ M , as well as the rotationally-induced phase difference ΔΦ R , will be considered in reference to Figures 5 and 6. Accordingly, the horizontal axis for the graph of the curve 200 in Figures 5 and 6 has been labelled as ΔΦ T to indicate that the total phase difference is being considered, rather than only the rotationally-induced phase difference ΔΦ R , as in Figure 3. Referring now to Figure 5, the effect of the phase modulation ΔΦ M (represented by the curve 214) upon the intensity I T will be first discussed. The curve 200 represents the relationship between (1) the intensity of the optical output signal resulting from interference of the two waves W₁ and W₂ at the coupler 170 and (2) to the phase difference ΔΦ T between the two lightwaves at the coupler 170. When the modulation-induced phase difference ΔΦ M between the two lightwaves is zero, as illustrated at a location 220 on the curve 200 in Figure 5, the resultant intensity I T of the combined lightwaves is a maximum, as illustrated at the location 222 on the curve 200. When the relative phase angle between the two lightwaves W₁ and W₂ is non-zero, the combined optical signal will have a lower intensity, depending upon the magnitude of the phase difference ΔΦ M . The intensity continues to decrease with increasing ΔΦ M until the relative phase difference ΔΦ M is either plus or minus 180°, as illustrated at a location 224 and at a location 226, respectively, in Figure 5. At a phase difference of plus or minus 180° (i.e., +π or -π), the two counterpropagating lightwaves completely destructively interfere, and the resultant intensity is zero, as illustrated at the location 224 and the location 226. Since the curve 200 is a periodic function of the total phase difference ΔΦ T , if the total phase difference ΔΦ T is increased beyond +π or -π, the intensity will increase, as further illustrated in Figure 5. In Figure 5, it is assumed that the sensing loop 114 is at rest, and thus, the optical output signal is not affected by the Sagnac effect. Specifically, it may be seen that the modulation-induced phase difference curve 214 causes the intensity I T of the optical output signal to vary in time, as illustrated by an intensity curve 230 in Figure 5. The intensity curve 230 is obtained by translating the points on the curve 214 (representing the instantaneous modulation phase difference ΔΦ M between the two lightwaves W₁ and W₂) onto the curve 200 (representing the resultant optical intensity for a phase difference of that magnitude). When all the points on the curve 214 are translated onto the curve 200, and the corresponding intensities are plotted, the curve 230 results. At zero rotation, the translation of the curve 214 through the curve 200 is symmetrical about the vertical axis 202 of the curve 200. Thus, the optical intensity represented by the curve 230 and measured by the photodetector 160 varies periodically at a frequency equal to the second harmonic of the modulating frequency f m . When the sensing loop 114 is rotated, the counterpropagating lightwaves W₁ and W₂ are shifted in phase in accordance with the Sagnac effect, as discussed above. The Sagnac phase shift provides a constant phase difference ΔΦ R for a constant rotational velocity. The Sagnac phase shift adds to the phase difference ΔΦ M caused by the phase modulator 192 so that the entire curve 214 is translated in phase from the position shown in Figure 5 by an amount equal to ΔΦ R , as illustrated in Figure 6. The addition of the rotationally-induced phase difference to the modulator-induced phase difference causes the intensity I T of the optical output signal to vary nonsymmetrically along the curve 200 between a point 240 and a point 242, and thus causes the optical output signal to have a time-varying intensity as illustrated by a curve 250 in Figure 6. The curve 250 can be obtained by translating points on the curve 214 onto the curve 200, as discussed above in connection with Figure 5. For example, the combined phase difference ΔΦ T , illustrated at a location 260 on the curve 214 translates through a location 262 on the curve 200 to a location 264 on the curve 250. It should be understood that the location 264 corresponds in time to the location 260 and corresponds in amplitude to the location 262. Similarly, a location 266 on the curve 214 translates through the location 242 on the curve 200 to a location 268 on the curve 250; a location 270 on the curve 214 translates through the location 262 on the curve 200 to a location 272 on the curve 250; a location 274 on the curve 214 translates through a location 276 on the curve 200 to a location 278 on the curve 250; a location 280 on the curve 214 translates through the location 240 on the curves 250 to a location 282 on the curve 250; a location 284 on the curve 214 translates through the location 276 on the curve 200 to a location 286 on the curve 250; a location 288 on the curve 214 translates through the location 262 on the curve 200 to a location 290 on the curve 250; and a location 292 on the curve 214 translates through the location 242 on the curve 200 to a location 294 on the curve 250. It can be seen that after the curve 214 completes one cycle, for example, from the location 260 to the location 288, and begins repeating, the corresponding locations on the curve 250 also repeat. Figure 7 illustrates a curve 300 for the optical output signal when the phase difference ΔΦ R results from rotation in the opposite direction to the rotation in Figure 6 and thus causes the curve 214 to be shifted in the opposite direction along the ΔΦ T axis. It can be seen that for substantially the same magnitude of Sagnac phase shift ΔΦ R , the curve 300, representing the intensity of the optical output signal, is substantially similar to the curve 250 in Figure 6; however, the curve 300 is shifted in phase with respect to the curve 250. The optical output signal has a first harmonic component as illustrated by sinusoidal curve 310 (shown in phantom) in Figure 7. The amplitude of this first harmonic component is indicative of the rotationally-induced Sagnac phase shift. Thus, by detecting this first harmonic, an indication of the rotation rate of the sensing loop 114 may be obtained. An example of a previous rotation sensor that utilizes this technique is disclosed in U.S. Patent No. 4,410,275. Such rotation sensors typically used a lock-in amplifier to synchronously detect the component of the electrical representation of the optical output signal to provide an amplified output signal having a value proportional to the RMS value of the first harmonic component of the optical output signal. The amplified output signal is used in such rotation sensors to indicate the rotation rate of the sensing loop. Description of the Preferred Embodiment The present invention provides an apparatus and method for determining the direction and rate of rotation of the sensing loop 114 that does not require the use of a lock-in amplifier or other such device. The present invention is particularly advantageous in that it lends itself to digital processing of the optical output signal. Figure 8 illustrates a set of four curves. Three of the curves, 400, 410 and 420, are similar to the three curves 200, 214 and 300 discussed above in connection with Figure 7. The curve 400 illustrates the output intensity I T as a function of the total differential phase shift ΔΦ T between the two counterpropagating lightwaves W₁ and W₂. As before, the curve 400 is cosinusoidal such that the maximum output intensity I T corresponds to zero phase shift and to even multiples of π phase shift, and the minimum output intensity corresponds to odd multiples of π phase shift. The curve 410 represents the time-varying differential modulation ΔΦ M (t) of the phase caused by the operation of the phase modulator 192 (Figure 1). In Figure 8, the curve 410 is illustrated as a triangular waveform; however, as will be shown below, other waveforms for the phase modulation (e.g., sinusoidal) can be alternatively used. In preferred embodiments, the modulation is symmetrical such that the increasing portion of the modulation on one side of a maximum (or a minimum) is mirrored by a corresponding decreasing portion on the other side of a maximum (or a minimum) in each half cycle and such that the slopes of the increasing and decreasing portions are finite (i.e., have a well-defined, non-vertical slope). In Figure 8, it is assumed that the sensing loop 114 is not rotating so that the time-varying total phase difference ΔΦ T (t) between the two lightwaves is the same as the differential phase modulation ΔΦ M (t). The time-varying phase difference ΔΦ T (t) translates through the curve 400 to a curve 420 that represents the intensity I T (t) of the optical output signal as a function of time. In Figure 8, the amplitude of the phase modulation is selected so that the total phase difference ΔΦ T has an absolute magnitude that is greater than π and less than 2π. For example, in Figure 8, the total phase difference ΔΦ T varies from a magnitude that is between -2π and -π to a magnitude that is between +π and +2π. This selection of the amplitude of the phase modulation assures that the phase modulation curve 410 will have an excursion that passes the minima on the curve 400 at -π and at +π. The curve 420 has a number of distinct maximum and minimum that correspond in time to the maxima and minima of the curve 410, as translated through the curve 400, and a number of distinct maxima and minima that correspond to the maxima and minima of the curve 400. Each of the maxima and minima on the curve 410 and each of the locations on the curve 410 that corresponds to a maxima or a minima of the curve 400 has been labelled with a letter A, B, C, D..., O, P. The corresponding locations on the curve 420 have been labelled with the same letter so that the correspondence can be clearly understood. For example, at a time t A , the differential phase modulation ΔΦ M has a magnitude that corresponds to a differential phase shift of +π and thus causes the optical output signal to have an intensity substantially equal to zero, as illustrated by the zero magnitude of the curve 420 at the time t A (i.e., the location A on the curve 420). Similarly, at a time t B , the differential phase modulation ΔΦ M has a maximum positive magnitude. This magnitude translates through the curve 400 to a location B on the curve 420 at the time t B . It can be seen that the locations A, C, E, G, I, K, M and O correspond to the translation of the like-identified locations on the curve 410 through the minima of the curve 400 at phase differences of -π and +π; and the locations D, H, L and P on the curve 420 correspond to the translation of the like-identified locations on the curve 410 through the maximum of the curve 400 at a phase difference of zero. The peaks of the curve 420 at the locations B, F, J and N correspond to the translation of the like-identified peaks of the curve 410 through the curve 400. One can see that the peaks of the curve 420 at the locations B, F, J and N will be always spaced apart in time by intervals that are equal to one-half the period of the selected modulation. For example, in Figure 8, the period of the phase modulation is shown as T, and one-half the period of the modulation is shown as T/2. So long as the frequency of the phase modulation is held constant, the spacing between the peaks B, F, J and N will be constant. On the other hand, as will be described below in connection with Figure 9, the times at which the maxima and minima of the curve 420 at the locations A, C, D, E, G, H, I, K, L, M, O and P occur depends upon whether or not the modulation curve 410 is offset by a rotationally-induced phase difference ΔΦ R caused by the Sagnac effect. As set forth above, the amplitude ΔΦ Mmax of the curve 410 is selected to be at least π so that the phase excursion of the modulation passes through the minima and maxima of the curve 400. Preferably, the amplitude ΔΦ Mmax is selected to be greater than π so that the locations of zero slope on the curve 420 (also referred to as extrema) caused by the minima and maxima of the curve 400 (e.g., the locations A, C, D, E, G, H, I, K, L, M, O) are distinguishable from the locations of zero slope caused by the peaks of the curve 410 (e.g., the locations B, F, J, N). Figure 9 illustrates the curve 400 that corresponds to the curve 400 in Figure 8, and also illustrates a curve 410' that generally corresponds to the curve 410 in Figure 8. However, in Figure 9, a rotationally-induced phase difference ΔΦ R has been added to the modulated phase difference curve 410 of Figure 8 so that the modulated phase difference curve 410' is offset from the zero phase difference axis in the positive direction (i.e., towards +π). As in Figure 8, the locations on the curve 410' in Figure 9 are translated through the curve 400 to produce a curve 430 that represents the optical output intensity I T (t) as a function of time. For example, the relative peaks at the locations B', F', J' and N' on the curve 430 correspond to the maximum and minimum excursion of the curve 410' with respect to the curve 400. The magnitudes of the peaks at the locations B', F', J' and N' of the optical output intensity I T (t) curve 430 in Figure 9 have changed with respect to the corresponding locations B, F, J and N on the curve 420 in Figure 8 since the maxima and minima of the curve 410' have shifted with respect to the differential phase shift axis ΔΦ T of the curve 400. However, the times at which the peaks at the locations B', F', J' and N' are fixed since they directly correspond to the times at which the like-identified maxima and minima of the curve 410 occur. In contrast, the magnitudes of the maxima and minima A and A', C and C', D and D', E and E', G and G', H and H', I and I', K and K', L and L', M and M', O and O', and P and P' are fixed, but the times at which they occur vary in accordance with the amount of Sagnac phase shift ΔΦ R that has been added to the phase modulation ΔΦ M . For example, when Figure 9 is compared with Figure 8, it can be seen that in Figure 9 the two locations H' and L' on the curve 410' that translate through the maximum of the curve 400 at 0 have shifted in time with respect to the locations H and L in Figure 8. In particular, the locations H' and L' in Figure 9 are spaced further apart in time by a time interval Δt₁' that is greater than a corresponding time interval Δt₁ between the locations H and L in Figure 8. Similarly, the locations I' and K' in Figure 9 are spaced apart by a time interval Δt₂′ that is greater than a corresponding time interval Δt₂ between the locations I and K in Figure 8. The change in the time interval between the locations H and L and the locations H' and L' and the change in the time interval between the locations I and K and the locations I' and K' are directly related to the offset in the curve 410' relative to the curve 410 in Figure 8 caused by the rotationally-induced Sagnac phase difference ΔΦ R . This can be more fully understood by referring to Figure 10 wherein the portion of the curve 410 in Figure 8 between the locations H and L is expanded for clarity and to Figure 11 wherein the portion of the curve 410' between the locations H' and L' is expanded for clarity. Referring first to Figure 10, which represents the curve 410 when there is no additional phase shift ΔΦ R caused by rotation, it can be seen that the magnitudes of the time intervals Δt₁ and Δt₂ can be determined by geometric calculations based upon the maximum amplitude of the phase modulation ΔΦ M (t), shown as ΔΦ Mmax . As was illustrated in Figure 8, and as shown again in Figure 10, the magnitude of the time interval between the location H and the location L is Δt₁ and the time interval between the location I and the location K is Δt₂. In Figure 10, the locations H and L represent a phase difference of zero and correspond to the maxima on the optical output intensity curve 420 in Figure 8. The locations I and K represent a phase difference of π and correspond to the minima on the optical output intensity curve 420 in Figure 8. Thus, the locations H and L are always separated from the locations I and K by a phase difference of π radians. This is further illustrated in Figure 11 wherein the locations I' and K' are shown as being separated from the locations H' and L' by phase angle difference of π radians. However, in Figure 11, the time interval Δt₁' between the location H' and the location L' is larger than the time interval Δt₁ between the location H and the location L in Figure 10. Similarly, the time interval Δt₂' between the location I' and the location K' in Figure 11 is greater than the time interval Δt₂ between the location I and the location K in Figure 10. The change in the time difference is directly related to the offset in the phase difference caused by the rotationally induced phase difference ΔΦ R . The relationship between the rotation-induced phase difference ΔΦ R and the points H', L', I' and K' in Figure 11 may be analyzed using geometry. A first triangle is formed by points X, Y and Z of the curve 410. The points X and Y are the points of the curve 410 which are displaced from zero phase difference by the ΔΦ R effect. A second triangle is formed by points H', L' and Z. Finally, a third triangle is formed by points I', K' and Z. The first triangle has a base of T/2 and has a height of ΔΦ Mmax , which is one-half of the peak-to-peak phase excursion caused by the phase modulator. The second triangle has a base of Δt₁' and a height of (ΔΦ Mmax + ΔΦ R ) . The third triangle has a base of Δt₂' and a height of (ΔΦ Mmax + ΔΦ R -π) . Thus, it follows that: (T/2)/ΔΦ Mmax = Δt₁'/(ΔΦ Mmax + ΔΦ R ) (5) and (T/2)/ΔΦ Mmax = Δt₂'/(ΔΦ Mmax + ΔΦ R -π) (6) Both Equations (5) and (6) can be rearranged to obtain the following: (ΔΦ R /ΔΦ Mmax ) = (2Δt₁'-T)/T = δ₁ (7) and ((ΔΦ R -πΔΦ Mmax ) = (2Δt₂'-T)/T = δ₂ (8) where δ₁ and δ₂ are selected to represent the two equalities in Equations (7) and (8), respectively. When the maximum modulation-induced phase difference ΔΦ Mmax is held constant, then it follows that the difference between δ₁ and δ₂ is a constant as follows: δ₁ - δ₂ = (ΔΦ R /ΔΦ Mmax ) -((ΔΦ R -π)/ΔΦ Mmax )= π/ΔΦ Mmax = constant (9) If the plane excursion ΔΦ Mmax caused by the phase modulator 192 is preselected such that it is a known value, the value of the constant can be calculated directly from Equation (9). If this plane excursion ΔΦ Mmax is not known in advance, δ₁ and δ₂ can be determined by measuring the time intervals Δt₁' and Δt₂', and the constant can be calculated using Equations (7), (8) and (9). Furthermore, once the constant is determined, then the maximum modulation-induced phase difference ΔΦ Mmax can be determined by dividing the constant by π. After the maximum modulation-induced phase difference ΔΦ Mmax is calculated, then the rotationally-induced phase difference ΔΦ R can be determined by using either of Equations (7) and (8) as follows: ΔΦ R = δ₁xΔΦ Mmax = ΔΦ Mmax x(2Δt₁'-T)/T (10) and ΔΦ R = (δ₂xΔΦ Mmax ) + π = (ΔΦ Mmax x(2Δt₂'-T)/T) + π (11) ΔΦ R can be expressed in terms of δ₁ and δ₂ as follows: ΔΦ R = π/(1 - δ₂/δ₁) (11') Referring back to Figure 9, it can be seen that the time interval Δt₁' can be measured by detecting the two peaks at the locations H' and L' and measuring the time interval between the occurrences of the two peaks. The two peaks are distinguishable from the peak at the location J' both because the two peaks H' and L' are larger than the peak at J' and also because they do not coincide with a maximum of the time-varying modulation ΔΦ M (t). Similarly, the time interval Δt₂' can be measured by detecting the two minima at the locations I' and K'. The two minima at I' and K' are distinguishable from the minima at the location G' because the minima at the locations I' and K' are adjacent in time to the maximum in the time-varying modulation ΔΦ M (t) at the location J'. Thus, the electrical signal from the electrical signal generator 194 can be advantageously used to synchronize circuitry, to be described below, that detects the maxima at the locations H' and L' and the minima at the locations I' and K'. The above measurements can also be accomplished using a pair of points corresponding to a preselected location on the curve 400, not necessarily a maximum or a minimum but also, for example, a point where the slope is maximal (at π/2, 3π/2,...). The measurement could also be performed at locations corresponding to two successive minima (ΔΦ R + π, ΔΦ R - π) . Before describing the preferred embodiment of the detection system, reference is again made to Figures 8 and 9. As set forth above, the time intervals Δt₁ and Δt₂ (and Δt₁' and Δt₂') are measured by detecting the maxima and minima in the optical output signal represented by the curve 420 (Figure 8) and the curve 430 (Figure 9). Although the maxima and minima can be detected by direct measurement, such direct measurements may be difficult to accomplish using digital techniques. Thus, in preferred embodiments of the present invention, the electrical representation of the time-varying optical output signal is not measured directly. Rather, the electrical representation of the optical output signal is differentiated to provide a differentiated output signal d[I T (t)]/dt that is illustrated by a curve 450 in Figure 8 and a curve 460 in Figure 9. As illustrated by the curves 450 and 460, the relative maxima and minima of curves 420 and 430 correspond to zero-crossings of the curves, 450 and 460, respectively. For example, the maximum at the location H of the curve 420 of Figure 8 corresponds to a zero-crossing at a corresponding location H on the curve 450. Since the location H on the curve 420 is a maximum, the zero crossing at the corresponding location H on the curve 450 is shown as a transition from a positive value to a negative value. Similarly, a negative-to-positive going zero-crossing at a location I on the curve 450 in Figure 8 corresponds to a minimum at the location I on the curve I T (t) 420 in Figure 8. In many cases, it is easier to construct a circuit that precisely detects the zero-crossings of the differentiated signal d[I T (t)]/dt than it is to construct a circuit that precisely detects the relative maxima and minima of the signal I T (t). The preferred embodiments of the present invention measure the time intervals Δt₁' and Δt₂' by measuring the time between the zero-crossings of the differentiated output signal d[I T (t)]/dt. A preferred embodiment of a rotation sensor constructed in accordance with the present invention is illustrated in Figure 12. The elements of the optical elements of the rotation sensor correspond to the elements described above in connection with Figure 1. The optical output signal from the first end portion 152 of the second optical fiber 150 is detected by the photodetector 160, and the photodetector 160 generates an electrical output signal on a line 500 that has a magnitude that is proportional to the time-varying intensity of the optical output signal detected by the photodetector 160. Thus, the electrical output signal has a time-varying magnitude that corresponds to the I T (t) curve 420 of Figure 8, the I T (t) curve 430 of Figure 9, or a similar curve having maxima and minima that occur at times that vary in accordance with the magnitude and direction of the rotation of the sensing loop 114. The electrical output signal on the line 500 is provided as an input to a differentiating circuit 510 that receives the electrical output signal and generates a differentiated output signal d[I T (t)]/dt on a line 520 that corresponds to the d[I T (t)]/dt curve 450 of Figure 8, the d[I T (t)]/dt curve 460 of Figure 9, or another similar curve having zero-crossings that occur in time in accordance with the magnitude and direction of the rotation of the sensing loop 114. The differentiated output signal on the line 520 is provided as an input to a zero-crossing detection circuit 530. The zero-crossing detection circuit provides a positive zero-crossing output +ZC on a line 532 that is activated when the differentiated output signal crosses through zero magnitude from a negative value to a positive value; and provides a negative zero-crossing output -ZC on a line 534 that is activated when the differentiated output signal crosses through zero magnitude from a positive value to a negative value. The positive zero-crossing signal +ZC and the negative zero-crossing signal -ZC on the lines 532 and 534, respectively, are provided as inputs to a counter/timer circuit 540. The counter/timer circuit 540 also receives a pair of synchronization input signals on a line 542 and a line 544 from the signal generator 194. The synchronization input signal on the line 542 synchronizes the counter/timer circuit 540 with the positive peaks of the electrical modulation signal output of the signal generator 194 corresponding to the locations B and J on the ΔΦ M (t) curve 410 in Figure 8 and the locations B' and J' on the ΔΦ M (t) curve 410' in Figure 9. The synchronization input signal on the line 544 synchronizes the counter/timer circuit 540 with the negative peaks of the electrical modulation signal output of the signal generator 194 corresponding to the locations F and N on the ΔΦ M (t) curve 410 in Figure 8 and the locations F' and N' on the ΔΦ M (t) curve 410, in Figure 9. The counter/timer circuit 540 is responsive to the synchronization signals on the lines 542 and 544 so that the counter/timer circuit 540 measures the time intervals between the zero-crossing signals corresponding to the locations H and H', I and I', K and K', and L and L', on the curves 450 and 460 in Figures 8 and 9, respectively, rather than measuring the time intervals that start or stop at the zero-crossings at the locations B and B', F and F', J and J', and N and N'. An exemplary counter/timer circuit 540 is illustrated in Figure 13. The counter/timer circuit 540 includes a counter 560, a timer/oscillator 562 and a data storage memory 564. The counter 560 is preferably a precision counter that is reset by one of the synchronization signals on the line 542 or the line 544 (e.g., the line 542, as illustrated in Figure 12). The counter 560 counts in response to the output of the timer/oscillator 562. The counter 560 provides a digital output on a bus 570 that represents the number of cycles of the output of the timer/oscillator 562 that have occurred since the counter 560 was most recently reset by the synchronization signal on the line 542 or 544. The digital output of the counter 560 is provided as a data input to the data storage memory 564. The data storage memory 564 has a plurality of data locations into which the data from the counter 560 are stored. The storage of data into the data storage memory 564 is controlled by a write control circuit 580 that is synchronized with the two synchronization signals on the lines 542 and 544 and with the zero-crossing signals +ZC and -ZC on the lines 532 and 534, respectively. Whenever the write control circuit 580 receives a zero-crossing signal +ZC or -ZC, it initiates a memory write cycle to the data storage memory 564 and stores the current digital output of the counter 560 on the bus 570 into a selected location of the data storage memory 564 along with an indicator that indicates whether the data corresponds to a positive-going zero-crossing signal from the +ZC line 532 or a negative-going zero-crossing signal from the -ZC line 534. Thus, the time of occurrence of each of the zero-crossings is stored in the data storage memory 564. Preferably, the write control circuit 580 also stores the current digital output of the counter 560 in the data storage memory 564 whenever each of the synchronization signals occurs along with an indicator that indicates that the count corresponds to the particular synchronization signal. The data storage memory 564 is also addressable by a processor 590 which is advantageously a microprocessor. The processor 590 reads the data stored in the data storage memory 564, calculates the time intervals between selected zero-crossing signals and calculates the magnitude of the rotationally-induced phase difference ΔΦ R in accordance with Equations (10), (11) or (11'), above. Thereafter, the angular velocity of rotation Q of the sensing loop 114 is calculated in accordance with Equation (1), above. It should be understood that the present invention is able to provide the above-described calculations without requiring a predetermined amplitude or frequency for the modulation signal provided by the signal generator 194. As was illustrated above in connection with Equation (9), the maximum amplitude ΔΦ Mmax of the phase modulation ΔΦ T (t) is calculated from the measured time intervals. Thus, it is not necessary to provide this maximum amplitude as an input to the processor 590 for the calculations. Furthermore, in the event that the maximum amplitude ΔΦ Mmax drifts from steady-state as the system operates, such as may occur if the signal generator varies in voltage or if the modulation is subject to temperature variations, the changes in the amplitude ΔΦ Mmax are automatically incorporated into the calculations described above. The calculated amplitude ΔΦ Mmax can also be provided as an output by the processor 590 to provide feedback to the signal generator 194 so that the amplitude of the signal generator output 194 is adjusted to maintain the amplitude within a selected range. By way of example, the amplitude ΔΦ Mmax may be maintained within the range of π to +2π. This feedback is illustrated by a phantom line 594 from the counter/timer circuit 540 to the signal generator 194. The signal generator 194 can advantageously be a conventional signal generator that includes an error comparator circuit that automatically adjusts the amplitude of the signal generator output to the phase modulator 192 in response to variations in the calculated amplitude of ΔΦ Mmax . As set forth above, the write control circuit 580 preferably causes the counts associated with the zero-crossings caused by the maxima and minima of the phase modulation signal to be stored in the data storage memory 564. Thus, the time intervals T and T/2 can also advantageously be calculated by the processor 590. Therefore, it is not necessary to provide the time interval as an input to the processor 590. Once the processor 590 has calculated the rotationally-induced phase difference ΔΦ R , the rotation rate Q is calculated using Equation (1), above, and the rotation rate provided as an output signal on a line 592 to be displayed or to be used to control other equipment (not shown). It can thus be seen that the present invention provides an open loop rotation sensing system that is substantially independent of the system parameters (e.g., the modulation frequency and amplitude) that may change with time. Although described above in connection with a triangular wave, it should be understood that other waveforms may be advantageously used. For example, Figure 14 is similar to Figure 9, but with a sinusoidal waveform, illustrated by a curve 600, substituted for the triangular waveform represented by the curve 410' of Figure 9. As illustrated in Figure 14, the sinusoidal curve 600 is offset by a rotationally-induced phase difference ΔΦ R . The curve 600 is translated through a cosinusoidal curve 602 that illustrates the optical output intensity I T as a function of total phase difference ΔΦ T and that corresponds to the curve 400 in Figure 9. As before, the optical output signal as a function of time (i.e., I T (t)) is provided and is represented by a curve 604. The optical output signal is differentiated to provide a differentiated output signal d[I T (t)]/dt, represented by a curve 606. The curve 606 has zero-crossings that correspond to the maxima and minima of the curve 604. As in Figures 8 and 9, the time intervals between the maxima and minima of the optical output signal intensity curve 604 and the zero-crossings of the differentiated output signal curve 606 can be measured by electronic circuitry, as discussed in reference to Figure 13. The measured time intervals are used to calculate the rotationally-induced phase difference ΔΦ R . This is illustrated in Figure 15, which is an expanded illustration of a portion of the phase modulation curve 600 of Figure 14. In Figure 15, a solid line 700 represents the solid line in Figure 14 corresponding to zero total phase difference (i.e., ΔΦ T (t) = 0), and a dashed line 702 represents the offset of the time-varying modulation ΔΦ M (t) by the rotationally-induced phase difference ΔΦ R . A dashed line 704 represents a total phase difference ΔΦ T (t) of when the optical output intensity I T (t) is zero. The distance from the line 702 to the line 704 is equal to π - ΔΦ R and is directly related to the measurable time interval Δt₂ as follows: (π - ΔΦ R ) = ΔΦ Mmax {sin[(π/2)(T-2Δt₂)/T]} (12) Similarly, the distance from the line 700 to the line 702 is equal to ΔΦ R and is directly related to the measurable time interval Δt₁ as follows: ΔΦ R = ΔΦ Mmax {sin[(π/2)(2Δt₁-T)/T]} (13) The two constants δ₁ and δ₂ can be defined for sinusoidal modulation as follows: δ₁ = (2Δt₁-T)/T = (2/π)sin⁻¹(ΔΦ R /ΔΦ Mmax ) (14) δ₂ = (2Δt₂-T)/T = (2/π)sin⁻¹[(ΔΦ R -π)/ΔΦ Mmax (15) Thus: sin((π/2)δ₁) = ΔΦ R /ΔΦ Mmax (16) and sin((π/2)δ2) = (ΔΦ R -π)/ΔΦ Mmax (17) The modulation amplitude can be determined as follows: sin((π/2)δ₁)-sin((π/2)δ₂) = π/ΔΦ Mmax (18) It can therefore be seen that the rotationally-induced phase difference ΔΦ R can be determined by measuring the time intervals between zero crossings and performing the calculations in accordance with Equations (14)-(18), above. The amplitude of the time-varying phase modulation can also be calculated in accordance with Equations (14)-(18). The rotationally-induced phase difference can also be directly calculated using the following equation: ΔΦ R = π/(1 - sin ((π/2)δ₂)/sin((π/2)δ₁)) (19) Similarly to the triangular modulation waveform, another pair of points can be chosen to determine ΔΦ R . In particular, a preselected pair of points on the curve 600 corresponding to a preselected location on the curve 602 can be used to perform the time measurements. In particular, two pairs of points corresponding respectively to two successive minima on the curve 602 (-π, +π) allow a direct calculation of ΔΦ R . As mentioned above for the triangular waveform, other points can be selected such as points of maximal slope on the curve 602 (π/2, 3π/2,...). Although the invention has been described in the context of a Sagnac interferometer for use as a rotation sensor, those skilled in the art will understand that the invention may be utilized in other types of interferometers for sensing ambient effects other than rotation. For example, the invention may be implemented in Mach-Zehnder interferometers, Michelson interferometers, Fabray-Perot interferometers (see, e.g., U.S. Patent No. 4,469,397), or the like. Further, while the invention has been described in terms of a digital implementation, those skilled in the art will recognize that the invention may be implemented alternatively utilizing analog components.

Claims: An interferometric sensor for sensing an ambient effect comprising: a light source (110) for producing a lightwave; an interferometer comprising an optical waveguide (112) optically coupled to said light source (110), said optical waveguide defining first and second optical paths, at least one of said optical paths being exposed to said ambient effect; a coupler (170) for introducing said lightwave into said optical paths in the form of first and second counterpropagating lightwaves (W₁, W₂), said first and second counterpropagating waves (W₁, W₂) recombining after the traverse in opposite directions of said optical paths so as to produce an output lightwave having an output waveform (400), the phase difference (ΔΦ R ) between said first and said second counterpropagating waves (W₁, W₂) after said traverse varying with said ambient effect; and a modulator (192) coupled to at least one of said optical paths for modulating the phase of light propagating therethrough to cause a time varying modulating phase difference (ΔΦ M ) between said first and second counterpropagating lightwaves (W₁, W₂), whereby said modulator changes said output waveform (400) into a modulated output waveform (430), characterized by: a signal processing circuit (510, 530, 540) for detecting a first pair of points (H', L') on said modulated output waveform (430), said signal processing circuit (510, 530, 540) further measuring the time domain separation (Δt₁') between said points of said first pair of points (H', L') and computing said phase difference (ΔΦ R ) as a function of the time domain separation (Δt₁') and selected characteristics of said modulating phase difference (ΔΦ M ) so as to measure said ambient effect. The interferometric sensor as defined in Claim 1, wherein said selected characteristics of said modulating phase difference (ΔΦ M ) comprise the maximal amplitude (ΔΦ Mmax ) and the period (T) of said modulating phase difference (ΔΦ M ). The interferometric sensor as defined in Claim 1 or 2, wherein said first pair of points (H', L') on said modulated output waveform (430) substantially corresponds to points of the unmodulated output waveform (400) having a same first amplitude. The interferometric sensor as defined in Claim 3, wherein said first pair of points (H', L') corresponds to extrema on said unmodulated output waveform (400). The interferometric sensor as defined in Claim 4, wherein one of said first pair of points (H') corresponds to a first maximum on said unmodulated output waveform (400) and the other of said first pair of points (L') corresponds to a second consecutive maximum on said unmodulated output waveform (400). The interferometric sensor as defined in any one of Claims 1 to 5, wherein said signal processing circuit (510, 530, 540) further detects a second pair of points (I', K') on said modulated output waveform (430), said signal processing circuit (510, 530, 540) further measuring the time domain separation (Δt₂') between said points of said second pair of points (I', K') and computing said phase difference (ΔΦ R ) as a function of the time domain separations (Δt₁', Δt₂') and the period (T) of said modulating phase difference (ΔΦ M ). The interferometric sensor as defined in Claim 6, wherein said second pair of points (I', K') on said modulated output waveform (430) substantially corresponds to points of the unmodulated output waveform (400) having a same second amplitude. The interferometric sensor as defined in Claim 7, wherein said second pair of points (I', K') corresponds to extrema on said unmodulated output waveform (400). The interferometric sensor as defined in Claim 8, wherein one of said second pair of points (I') corresponds to a first minimum on said unmodulated output waveform (400) and the other of said first pair of points (K') corresponds to a second consecutive minimum on said unmodulated output waveform (400). The interferometric sensor as defined in any one of Claims 1 to 9, wherein said processing circuit (510, 530, 540) comprises: a differentiator (510) for differentiating said modulated output waveform (430) so as to provide a differentiated waveform (460); a zero-crossing detector (530) for detecting the zero crossings on said differentiated waveform (460) corresponding to said first pair of points (H', L'); and a circuit (540) that measures the time domain separation (Δt₁') between said zero crossings corresponding to said first pair of points (H', L'). The interferometric sensor as defined in Claim 10, wherein said zero-crossing detector (530) further detects the zero crossings corresponding to said second pair of points (I', K') and wherein said circuit (540) measures the time domain separation (Δt₂') between said zero crossings corresponding to said second pair of points (I', K'), said circuit (540) further comprising a processor (590) for computing said phase difference (ΔΦ R ) as a function of the time domain separations (Δt₁', Δt₂') and the period (T) of said modulating phase difference (ΔΦ M ). The interferometric sensor as defined in any one of Claims 1 to 11, wherein said modulating phase difference (ΔΦ M ) has a maximal amplitude (ΔΦ Mmax ) of at least π radians. The interferometric sensor as defined in any one of Claims 1 to 12, wherein said modulating phase difference (ΔΦ M ) has a maximal amplitude (ΔΦ Mmax ) of no more than 2π radians. The interferometric sensor as defined in any one of Claims 1 to 13, wherein the modulator (192) has a phase excursion which is from a value between π and 2π radians to a value between -π and -2π radians. The interferometric sensor as defined in any one of Claims 1 to 14, wherein the modulating phase difference (ΔΦ M ) is a sine function. The interferometric sensor as defined in any one of Claims 1 to 14, wherein the modulating phase difference (ΔΦ M ) is a triangular function. The interferometric sensor as defined in any one of Claims 1 to 16, wherein said ambient effect is induced by rotation and is detected by the rotationally induced Sagnac effect produced in said one optical path submitted to rotation. The interferometric sensor as defined in any one of Claims 1 to 17, wherein said signal processing circuit (510, 530, 540) comprises digital electronics. The interferometric sensor as defined in any one of Claims 1 to 18, wherein said optical waveguide (112) forms a loop (114) optically closed by said coupler (170). A method of sensing an ambient effect by using an interferometric sensor as defined in claim 1, comprising the steps of: counterpropagating first and second lightwaves (W₁, W₂) in first and second optical paths (114); submitting at least one of said optical paths (114) to said ambient effect; combining said counterpropagating lightwaves (W₁, W₂) after propagation through said optical paths (114) to form an optical output signal having an output waveform (400), the phase difference (ΔΦ R ) between said first and said second counterpropagating waves (W₁, W₂) after said traverse varying with said ambient effect; modulating the phase difference (ΔΦ R ) between said counterpropagating lightwaves (W₁, W₂) to cause a time varying modulating phase difference (ΔΦ M ) between said first and second counterpropagating lightwaves (W₁, W₂); and changing said output waveform (400) into a modulated output waveform (430) by means of said modulating phase difference (ΔΦ M ), characterized by the steps of: detecting a first pair of points (H', L') on said modulated output waveform (430); measuring the time domain separation (Δt₁') between said points of said first pair of points (H', L'); computing said phase difference (ΔΦ R ) as a function of the time domain separation (Δt₁') and selected characteristics of said modulating phase difference (ΔΦ M ); and measuring said ambient effect by means of said phase difference (ΔΦ R ). The method of sensing an ambient effect as defined in Claim 20, wherein said phase difference (ΔΦ R ) is computed as a function of the maximal amplitude (ΔΦ Mmax ) and the period (T) of said modulating phase difference (ΔΦ M ). The method of sensing an ambient effect as defined in Claim 20 or 21, further comprising the steps of selecting one point (H') of said first pair of points (H', L') so that said point (H') corresponds to a first maximum of said unmodulated output waveform (400); and selecting the other (L') of said first pair of points (H', L') so that said point (L') corresponds to a second consecutive maximum of said unmodulated output waveform (400). The method of sensing an ambient effect as defined in any one of Claims 20 to 22, further comprising the steps of detecting a second pair of points (I', K') on said modulated output waveform (430); measuring the time domain separation (Δt₂') between said points of said second pair of points (I', K'); and computing said phase difference (ΔΦ R ) as a function of the time domain separations (Δt₁', Δt₂') and the period (T) of said modulating phase difference (ΔΦ M ). The method of sensing an ambient effect as defined in Claim 23, further comprising the steps of: selecting one of said second pair of points (I') so that said point (I') corresponds to a first minimum of said unmodulated output waveform (400); and selecting the other of said first pair of points (K') so that said point (K') corresponds to a second consecutive minimum of said unmodulated output waveform (400). The method of sensing an ambient effect as defined in any one of Claims 20 to 24, further comprising the steps of: differentiating said modulated output waveform (430) so as to provide a differentiated waveform (460); detecting the zero crossings on said differentiated waveform (460) corresponding to said first pair of points (H', L'); and measuring the time domain separation (Δt₁') between said zero crossings corresponding to said first pair of points (H', L'). The method of sensing an ambient effect as defined in Claim 25, further comprising the steps of: detecting the zero crossings corresponding to said second pair of points (I', K'); measuring the time domain separation (Δt₂') between said zero crossings corresponding to said second pair of points (I', K'); and computing said phase difference (ΔΦ R ) as a function of the time domain separations (Δt₁', Δt₂') and the period (T) of said modulating phase difference (ΔΦ M ). The method of sensing an ambient effect as defined in any one of Claims 20 to 26, wherein said step of submitting at least one of said optical paths (114) comprises the step of rotating said one optical path and wherein the step of measuring said ambient effect is performed by detecting the rotationally induced Sagnac effect produced in said one optical path submitted to rotation.

Inventors: KIM, BYOUNG YOON
Applicants: THE BOARD OF TRUSTEES OF THE LELAND STANFORD JUNIOR UNIVERSITY
Application year: 1988
Main classification: G01C19
IPC codes: G01C19