In this letter, the relationship between the change of the closed-loop optical path and the movement of two adjacent spherical mirrors in ring laser gyros is investigated by matrix optical approach. When one spherical mirror is pushed forward and the other is pulled backward to maintain the total length of the closed-loop optical path constant, an equivalent rotation of the closed-loop optical path is found for the first time. Both numerical simulations and experimental results show the equivalent rotation rate is proportional to the velocities of the mirrors’ movement.
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In the field of He-Ne ring laser gyros (RLG), it is well known that lock-in phenomenon, which is primarily attributed to back scattering, is a major source of angular rate information error [1–5]. To avoid the lock-in phenomenon or reduce the lock-in threshold, variable technologies have been proposed [1,2,6–13]. In , Rodloff presented a theory which is helpful to minimize the lock-in threshold by optimizing of the resonator geometry. In , Killpatrick eliminates the lock-in effect by additionally randomizing the bias so that errors caused by the lock-in effect are no longer cumulative. Another typical method of reducing the lock-in effect is to modulate scattered waves reflected from the mirrors by driving two mirrors in opposite directions, so that the coupling of counter-propagating beams are minimized, thus the lock-in threshold is reduced [2,9–13].
However, when the two mirrors are driven in opposite directions to reduce the lock-in threshold, an equivalent rotation effect is observed in our experiments. Because RLG is used to measure the angle motion of RLG relative to inertial space, the equivalent rotation effect considered as an error source will also appear in the test result of RLG. Although the induced rotation averages out to zero for a long test time, this error cannot be neglected for short time application. To the best of our knowledge, the equivalent rotation effect mentioned in this paper, which is important for design and improvement of RLG, has never been reported..
In this paper, based on the method of optical propagation matrix, the disorder matrix of spherical mirror in RLG is analyzed and deduced firstly. Then the propagation matrix of laser after a round-trip in RLG is analyzed. The expression of angular rate in the equivalent rotation effect is deduced and the effect of cavity parameters on the equivalent rotation is analyzed and discussed by numerical simulations. Finally, the equivalent rotation effect is experimentally demonstrated.
2. Theoretical analyses
Figure 1 shows the original light path (dotted line) in the square ring resonator and the changed light path (solid line) when the spherical mirror P1 is pushed forward to P1', and the other spherical mirror P2 is pulled backward to P2'. The change of the light path is elaborated in Fig. 2, in which the dotted and solid lines describe the optical path when P1 holding motionless and moving forward, respectively. Assume R is common radius of P1 and P2, Ai is the incident angles and ε1 is displacement of spherical mirror P1. According to the geometric relationship as shown in Fig. 2, the relationships are given as following
It is important to decide whether ri, ro, θi, θo, ε1 is positive or negative. According to the rules in [14,15], ri, ro are positive, θi is positive, θo is negative and ε1 is positive in Eq. (3) and Eq. (4). SoEq. (5) and Eq. (6) into Eq. (7), the extended matrix M(P1) of P1 can be deduced
In the same way, the disorder matrix M(P2) of P2 can be deduced. Assuming P1 is pushed forward by the displacement of ε and P2 is pulled backward by -ε, the following conclusions can be deduced
Then the ray matrix for round-trip propagation in RLG can be analyzed. We define M(li) as the ray matrix of the free-space ray propagating along the path li (i = 1,2,3 and 4), then M(li) can be expressed asEq. (9), Eq. (10) and Eq. (11) to Eq. (12), r1 and θ1 can be obtainedEq. (14) is the angle we concerned, which represents the rotation of the ray between P4 and P1 corresponding to the original optical axis after P1 is pushed forward by the displacement of ε and P2 is pulled backward by -ε.
In the same way, the tilt angle of the other three optical paths can be deduced asEq. (15) to Eq. (17), θ1 = θ3 = θ4 <θ2 since m>1. Thus it can be seen that θ1 is the common component in the four optical paths in the laser loop. We define θ1 as the equivalent rotation angle after P1 is pushed and P2 is pulled and the optical paths in the closed-loop rotate θ1. We can get the velocity of the equivalent rotation angle from the time derivation of Eq. (14)Eq. (18), we can draw a simple but important conclusion: the equivalent rotating angular velocity is proportional to the velocities of the spherical mirrors P1 and P2.
It should be noticed that although the result above is deduced for the square RLG shown in Fig. 1, in which Ai = 45°, the equivalent rotation effect exists in other two types of RLG with different cavity structures as shown in Fig. 3.
3. DiscussionsEq. (11) to Eq. (19), l/R meetsEq. (18) can be simplified to
The sensitivity of resonator parameters l, Ai and m on the equivalent rotation are analyzed according Eq. (21). Figure 4 shows the sensitivity of the equivalent rotation as a function of l when Ai = 15°, 30° and 45° and m = 2, 3 and 4. It can be seen that the absolute value of the sensitivity of the equivalent rotation decreases with the increase of l and tends to be stable. In addition, when Ai increases from 15° to 45° the absolute value of the sensitivity of the equivalent rotation increases as shown in Fig. 4(a) to 4(c), but when m increases from 2 to 4 the absolute value of the sensitivity decreases.
Schematic diagram of the experimental system is shown as Fig. 5. The spherical mirrors P1 and P2 are mounted on two piezoelectric transducers respectively, which can push or pull P1 and P2 with variable drive voltage V. A triangle signal generated by an oscillator is differentially amplified and then a pair of differential triangle signal is generated to drive P1 and P2 in opposite directions. The velocities, defined as dɛ/dt, of P1 and P2 can be obtained from the voltage derivation of the time t (dV/dt). The output signal of RLG, which is measured by a counter, is Fourier transformed and the component whose frequency is caused by the triangle signal can be calculated.
A square RLG resonator, in which the length of l1 is 0.08m, the side ratio m is 3, the incident angles Ai is 45° and the common radius of P1 and P2 is 8m, is used in our experiment. The gyro is operated under mechanical dither and path length controller is enabled. The experimental results are shown in Fig. 6 and Fig. 7.
The real-time output of the gyro when spherical mirrors P1 and P2 are vibrated in opposite directions are shown in Fig. 6, in which the red line denotes the output of the gyro and the blue line denotes the displacement ε of P1. The vibration frequency is 0.25Hz and the sample frequency is 500Hz. It can be seen that the output of the gyro is square wave when the displacement ɛ is triangle wave. This means the output of the gyro is modulated by the derivation of the displacement dɛ/dt.
By changing the vibration frequency from 0.01Hz to 0.25Hz, different velocities dɛ/dt can be gotten. In Fig. 7, the continuous line with ‘ + ’ denotes the amplitudes of the equivalent rotation effect in experiments, and the continuous line denotes theory data. It can be seen that the equivalent rotation rate is proportional to the velocities of mirrors’ movement. The experiment results are in good agreement with theoretical analysis.
When two adjacent spherical mirrors are driven in opposite directions for planar ring resonators including triangular and square RLG, an equivalent rotation is induced and the equivalent rotation rate is proportional to the velocities of the mirrors’ movement. The relationship between equivalent rotation effect and resonator parameters is studied. The theoretical analyses show that the shorter R and l, the smaller Ai and m, the higher the sensitivity of the equivalent rotation. For the resonator in our experiment, 0.1μm/s the velocities of the mirrors’ movement approximately induces 0.178°/h the equivalent rotation rate, which is significant error for a high performance RLG. Therefore, the velocities of the mirrors should be carefully controlled to when the means of driving two adjacent mirrors in opposite directions is implemented to reduce lock-in threshold.
This work is supported by Program for New Century Excellent Talents in University 2010.
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