Scale factor characteristics of laser gyroscopes of different sizes Zhenfang Fan, Guangfeng Lu, Shomin Hu, Zhiguo Wang and Hui Luo a) National University of Defense Technology, Changsha, Hunan 410073, China (Received XXXXX; accepted XXXXX; published online XXXXX) The scale factor correction characteristics of two ring laser gyroscopes of different sizes are investigated systematically in this paper. The variation in the scale factor can reach 144 or 70 ppm for square gyroscopes with arm lengths of 8.4 cm or 15.6 cm, respectively, during frequency tuning. A dip in the scale factor is observed at the line center of the gain characteristic for both gyroscope sizes. When a different longitudinal mode is excited, the scale factor behavior remains the same, but the scale factor values differ slightly from those derived from geometric prediction. The scale factor tends to decrease with increasing discharge current, but the sensitivity of the scale factor to variations in the excitation decreases with increasing discharge current. I. INTRODUCTION The ring laser gyroscope is a high-accuracy angular movement sensor that is widely used in navigation systems and many other applications. 1 4 The scale factor error is an important error term for these gyroscopes because it determines the performance of the system directly. Lamb and Aronowitz made major contributions to the development of the basic theory of the ring laser gyroscope, 5 7 which is called semi-classical theory. Semi-classical theory can explain many phenomena that occur in ring laser gyroscopes, but the theory is based on weak signal conditions and third-order polarization approximations, and can thus only explain some of the problems of the device qualitatively. Experimental data is highly important to the development of ring laser gyroscopes. The theory can successfully predict the line shape of the scale factor, but many detailed scale factor correction characteristics are not disclosed. Therefore, these characteristics must be investigated experimentally. The lock-in phenomenon also leads to scale factor errors, but this problem has been discussed by many previous researchers, 8-10 and this paper is therefore focused on other aspects affecting the scale factor. To provide a full overview of the ring laser gyroscope s scale factor characteristics, two types of ring laser gyroscope with different path lengths ----- a) Author to whom correspondence should be addressed. Electronic mail: lasergyro@nudt.edu.cn. are used to perform the experiments. The gain medium is a mixture of He and Ne. The ratio of Ne 20 and Ne 22 is about 1:1, and this ratio can effectively suppress competition between opposing traveling light beams. 12 Through the experiments, many scale factor characteristics, which cannot be found in the theory, is revealed. The experimental data disclosed in this paper will aid the electrical servo system design of the gyroscope and improve the accuracy of the scale factor in the application. II. CORRECTION THEORY A set of equations has been derived that describe the intensities and the frequencies of the ring laser gyroscope. By initially neglecting the backscattering coupling term, the equations can be written as 5, 12 I c / L I I I, 1 1 1 1 12 2 1 I c / L I I I, 2 2 2 2 21 1 2 I I, 1 1 1 1 1 12 2 2 2 2 2I2 21I1. The first two equations describe the intensities of the opposing traveling beams, where I is the dimensionless intensity, α is the single pass gain minus the cavity loss, and β and θ are the self and cross saturation coefficients, respectively. In the phase equations, ω i is the cavity frequency, σ is the dispersion correction due to the gain medium, and ρ and τ are the self and cross gain saturations, respectively. By substracting the latter two equations, the beat frequency can be obtained as: (1) 1
2-1 (2) 0 2-1 2-12 I2 1-21 I1, where ω 0 is the empty cavity beat note, and it can be written as 1 4A 0=2 in. (3) L In Eq. (3), Ω in is the input angular velocity, A is the covered area of the resonant cavity, and L is the perimeter of the gyro. The slopes of the coefficients σ, ρ and τ lead to the correction of the scale factor. In addition, the radiation trapping effect also contributes to the correction of the scale factor, and thus must also be taken into account. 12 Therefore, the scale factor correction formula can be written as 1+C, (4) where 0 C A S A S S S S (5) 0 0 R. In Eq. (5), A 0 is a constant that is related to the gain of the discharge, S σ, S ρ and S τ represent the corrections based on coefficients σ, ρ and τ, respectively, and S R is the correction based on the radiation trapping effect. By combining Eq. (3) and Eq. (4), the beat frequency can be written as 4A =2 1+ C in. (6) L For a gyroscope with a square resonant cavity, Eq. (6) can be written as = L 1+ C in. 2 (7) From Eq. (7), it can be concluded that the scale factor is related to both the geometric parameter L and the dispersion parameters. relatively small, and they can be neglected in this case. The radiation trapping term S R opens downwards, and thus results in a dip in the total scale factor correction curve. III. EXPERIMENTS AND DISCUSSION The semi-classical theory determines the basic line shape of the scale factor, but many of the finer details cannot be determined on that basis. A complete image of the scale factor characteristics can only be obtained experimentally. The experiments are carried out using laser gyroscopes with an operating wavelength of 0.6328 μm. Two types of ring laser gyroscopes with different path lengths of 8.4 cm and 15.6 cm are chosen. The relationship between the longitudinal mode distance and the path length can be expressed as c v. (8) L Therefore, the longitudinal mode distances of the two gyroscopes are approximately 3571 MHz and 1923 MHz. Each gyroscope is mounted on a high-accuracy rotary table to obtain the scale factor. To minimize the effects of the lock-in threshold, the rotation speed is chosen to be as high as 800 /s. The rotary table is rotated clockwise 10 rounds and then counterclockwise 10 rounds to balance the Earth rotation and the associated bias. A. Scale factor dispersion characteristic To obtain the frequency detuning characteristic, a voltage sweep is applied to the piezoelectric ceramics, which were assembled on the reflective mirrors. Figure 2 shows the scale factor of the 8.4 cm path length gyroscope, where the light intensity is drawn on the right axis. FIG. 1. Scale factor correction distributions for every correction term. Figure 1, which is drawn based on numerical simulation, shows the contributions of every scale factor correction term in Eq.(5). The purple curve is the sum of all. The mode pulling term S σ is in the domain position, so the entire curve opens upwards. The contributions of S ρ and S τ are 2 FIG. 2. Frequency tuning of gyroscope with 8.4 cm path length. In Figure 2, it can be seen that a dip exists in the middle of the scale factor curve. This is consistent with the simulation result in Figure 1. By comparing the scale factor curve with that of
the light intensity, it was found that the maximum point of the intensity curve corresponds to the dip point of the scale factor curve. Over the whole detuning range, the scale factor variation can be as much as 144 ppm. The two minimum points, which are located 300 MHz (left) and 330 MHz (right) away from the center dip point, have scale factor deviations of 10 ppm and 15 ppm, respectively. The laser beam only appears when the laser gain exceeds the cavity loss. As the detuning frequency increases, the gain weakens. Because the path length is relatively short, the longitudinal mode distance is so large that when the detuning frequency is far away from the gain center frequency, the gain cannot exceed the cavity loss; then, the laser beam vanishes and the scale factor cannot be measured. Figure 3 shows the scale factor and the intensity of the gyroscope with the perimeter of 15.6 cm. between the gain centers of NE 20 and NE 22 is 875 MHz 12, which is approximately half of the longitudinal mode span of 1923 MHz at the detuning frequency boundary. Both mode N and mode N+1 acquire almost the same gain, and the gain is not so weak as to cause the laser to be extinguished, and thus these modes can exist simultaneously. At point (b), multiple longitudinal modes appear, and thus the scale curve changes rapidly. Beyond point (b), mode N becomes weak and mode N+1 becomes stronger, so the light intensity curve becomes flat. The situation is identical at point (a). With increasing perimeter length, the longitudinal distance becomes narrower, and only the downward opening area near the dip point can be observed; this is verified in Ref. [14]. B. Scale factors for different modes In different modes, the gyroscope has different perimeters, and this induces a geometric scale factor difference. Assuming that the perimeter of mode N is L, mode N+1 will then have the perimeter L+λ, and the relative scale factor error is SFN 1 SFN SF (9) SFN L For the gyroscope with the 8.4 cm perimeter, the neighboring modes will have a geometric scale factor difference of 7.5 ppm. FIG. 3. Frequency tuning of gyroscope with 15.6 cm path length. The dip in the scale factor curve also exists in Figure 3. The scale factor variation over the entire frequency detuning range is approximately 70 ppm, which is half of the range of the previous gyroscope. The sideband minimum points, which are both located at a distance to the center of 350 MHz (the left point and the right point are located at almost identical distances), have scale factor deviations of 10 ppm and 14 ppm from the dip point, which are almost the same as the figures for the smaller sized gyroscope. When the detuning frequency is located far away from the center frequency, the 15.6 cm gyroscope exhibits different characteristics to those of the smaller device. Beyond point (a) and point (b), the light intensity curve becomes flat. However, this does not indicate that the laser beam is extinguished, because the beat frequency also exists at this time. The scale factor changes sharply at both point (a) and point (b), after which the curve becomes flat. This can be explained by its specific longitudinal mode span. The distance FIG. 4. Piezoelectric scan of gyroscope with 8.4 cm path length. Scanning of the path length control (PLC) voltage over a wide range allows the scale factor curves for the different modes to be obtained. Figure 4 shows the scale factor curve and light intensity curve of the 8.4 cm perimeter gyroscope obtained from the path length scan. Clearly, at junctions between neighboring modes, the laser is extinguished and the scale factor cannot be measured; this results in the discontinuous scale factor curve shown in the figure. However, considering that the gyroscope is under path length control, the gyroscope is always working in the dip points, and thus the list of scale factor dip points 3
that is shown in Table I can be obtained. From Table I, it can be seen that the scale factor divergence characteristics of the different modes are not in accordance with those of the geometric prediction obtained using Eq. (9). This is perhaps because the cavity gains and losses are different in the different modes, which will then induce differences in the scale factor correction. However, the overall trend is the same as that of the geometric prediction, i.e., as the cavity expands, the scale factor also increases. TABLE I. Scale factors of different modes of gyroscope with Mode 8.4 cm perimeter. Scale factor (ppm) 1-85.07 Scale factor change relative to previous mode (ppm) 2-73.80 11.27 3-67.96 5.84 4-57.21 10.75 5-51.28 5.93 Average (ppm) 8.45 Closed loop path length control(plc) must be applied to the gyroscope to ensure that the light intensity is always maximized, and that the scale factor is locked to the dip point. Under circumstances of temperature change and self-heating, the path length control voltage of the gyroscope must be modulated to track the path length changes, and PLC errors are inevitable. By zooming in on the scale factor curve around the third mode, we obtain Fig. 5, in which the slope is 4.85 ppm/v. To achieve scale factor accuracy of 1 ppm, the accuracy of the path length control voltage must be 0.2 V, and this level of accuracy can be achieved easily. In fact, the slope near the dip point is flat, and the PLC demands can thus be further reduced. FIG. 5. Slope of scale factor with PLC voltage. Figure 6 shows the scale factor curve and the light intensity curve of the 15.6 cm perimeter gyroscope obtained from the path length scan. When compared with the results for the other gyroscope, it is clear that the laser is never extinguished during the entire scan trip and that the scale factor curve is continuous, but there are obvious jumps when multi-mode conditions appear in the middle of two modes. FIG. 6. Piezoelectric scan of gyroscope with 15.6 cm path length. Table II shows the scale factor for the different modes of the 15.6 cm perimeter gyroscope. According to Eq. (9), The geometric scale factor difference of neighboring modes is calculated as 4.1 ppm. The experimental value also differs from the expedition. When compared with the 8.4 cm gyroscope, the errors under the different mode conditions are obviously smaller. TABLE II. Scale factors of different modes of gyroscope with 15.6 cm perimeter. Mode Scale factor Scale factor change 4
(ppm) 1-25.96 relative to previous mode (ppm) 2-19.88 6.08 3-15.35 4.53 Average (ppm) 5.31 C. Scale factors under different discharge currents FIG. 7. Current sensitivity of gyroscope with 8.4 cm path length. Figure 7(a) shows the scale factor curves around the dip point for the 8.4 cm path length gyroscope under various discharge current conditions. The scale factor is shown to decrease with increasing discharge current. Figure 7(b) shows the scale factor around the dip point; the slope can be calculated to be 21 ppm/ma, which is the current sensitivity of the scale factor. However, in Fig. 7(b), it is also found that as the discharge current increases, the slope becomes flatter. various sizes. IV. CONCLUSIONS Scale factor accuracy is highly significant for high-performance ring laser gyroscopes. Scale factor correction is related to the dispersion of the gain medium, and a set of equations has been developed to describe this correction. However, the correction theory has been developed on the basis of weak signal conditions and a third-order polarization approximation. Geometrical theory can predict the scale differences for every mode, but the effect of the gain medium is not taken into account. Therefore, the theory can only describe the scale factor characteristics in a general fashion. Because of the limitations of the theory, experiments were carried out to research the scale factor characteristics in detail, and gyroscopes with different perimeters were used for comparison. Results for scale factor changes produced by frequency tuning are given, along with those predicted using dispersion theory; both types of characteristics have a dip at the point at which the light intensity reaches a maximum, and the depths and widths of the characteristics are similar for both. The scale factor was also investigated for different modes, and was found to increase with the expansion of the resonant cavity, but the value change obtained is different to that obtained using the geometrical prediction. The scale factor varies inversely with discharge current, and it is beneficial to select a high discharge current to reduce current sensitivity. The experimental data disclosed in this paper will aid the electrical servo system design of the gyroscope and improve the accuracy of the scale factor. ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (Grant No. 61308059). FIG. 8. Current sensitivity of gyroscope with 15.6 cm perimeter. Figure 8(a) shows the scale factors for different discharge currents for the 15.6 cm perimeter gyroscope. The figure also shows that the scale factor decreases with increasing discharge current. The scale factor current sensitivity can be calculated to be 25 ppm/ma from Fig. 8(b). It is again found that the curve becomes flatter as the discharge current increases. This indicates that it is beneficial to choose a slightly higher current to reduce the scale factor current sensitivity, and this conclusion has been applied to gyroscopes of 5 1 W. W. Chow, Rev. Mod. Phys. 57, 61 (1985). 2 E. J. Post, Rev. Mod. Phys. 39, 475 (1967). 3 A. D. King, GEC Rev. 13, 140 (1998). 4 G. E. Stedman, Rep. Prog. Phys. 60, 615(1997). 5 F. Aronowitz, Phys. Rev. 139, 635 (1965). 6 L. N. Mengozzi and W. E. Lamb, Phys. Rev. A. 8, 2103 (1973). 7 W. E. Lamb, Phys. Rev. A. 134, 1429 (1964). 8 J. E. Killpatrick, IEEE Spect. 4, 44 (1967). 9 H. A. Haus, J. Quantum Elect. 21, 78(1985). 10 Z. Fan, H. Luo, G. Lu and S. Hu, Chin. Opt. Lett. 10, 61403(2012). 11 S. Song, J. Lee, S. Hong and D. Chwa, J. Optics. 12, 115501(2010). 12 Y. N. Jiang, Ring Laser Gyro (Tsinghua, Beijing, 1984).
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