Mirrors movement-induced equivalent rotation effect in ring laser gyros

Guangfeng Lu; Zhenfang Fan; Shaomin Hu; Hui Luo

doi:10.1364/OE.21.014458

1. Introduction

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 [6], Rodloff presented a theory which is helpful to minimize the lock-in threshold by optimizing of the resonator geometry. In [7], 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, A_i 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

{\begin{matrix} A^{'} C^{'} = \frac{r_{i}}{\cos A_{i}} \\ C^{'} B^{'} = 2 | ε_{1} | t g A_{i} \\ | r_{0} | = (A^{'} C^{'} + C^{'} B^{'}) \cos A_{i} \end{matrix}

and

Fig. 1 Light path in the square ring resonator before and after the positions of P₁ and P₂ are changed.

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Fig. 2 Schematic diagram of the light path before and after changing the position of P₁.

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{\begin{matrix} β + A_{i}^{'} = | θ_{o} | + A_{i} \\ | θ_{i} | + A_{i}^{'} = β + A_{i} \\ β = \frac{A^{'} C^{'} + | ε_{1} | t g A_{i}}{R} \end{matrix}

So, the parameters |r_o| and |θ_o| of the reflection ray can be obtained from Eq. (1) and Eq. (2) as

| r_{o} | = | r_{i} | + 2 | ε_{1} | \sin A_{i},

| θ_{o} | = - | θ_{i} | + \frac{| r_{i} |}{f} + \frac{| ε_{1} | \sin A_{i}}{f} .

where f is the focal length of the spherical mirrors, which is equal to RcosA_i/2 in the meridian plane.

It is important to decide whether r_i, r_o, θ_i, θ_o, ε₁ is positive or negative. According to the rules in [14,15], r_i, r_o are positive, θ_i is positive, θ_o is negative and ε₁ is positive in Eq. (3) and Eq. (4). So

r_{o} = r_{i} + 2 ε_{1} \sin A_{i},

θ_{o} = θ_{i} - \frac{r_{i}}{f} - \frac{ε_{1} \sin A_{i}}{f} .

If the extended matrix of P₁ is expressed as M(P₁), the relationship of r_i, r_o, θ_i, θ_o can be written as

[\begin{matrix} r {}_{o} \\ θ_{o} \\ 1 \end{matrix}] = M (P_{1}) [\begin{matrix} r_{i} \\ θ_{i} \\ 1 \end{matrix}] .

Substitute Eq. (5) and Eq. (6) into Eq. (7), the extended matrix M(P₁) of P₁ can be deduced

M (P_{1}) = [\begin{matrix} 1 & 0 & 2 ε_{1} \sin A_{i} \\ - \frac{1}{f} & 1 & - \frac{ε_{1} \sin A_{i}}{f} \\ 0 & 0 & 1 \end{matrix}]

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

M (P_{2}) = M (P_{1}) = [\begin{matrix} 1 & 0 & 2 ε \sin A_{i} \\ - \frac{1}{f} & 1 & - \frac{ε \sin A_{i}}{f} \\ 0 & 0 & 1 \end{matrix}] .

Then the ray matrix for round-trip propagation in RLG can be analyzed. We define M(l_i) as the ray matrix of the free-space ray propagating along the path l_i (i = 1,2,3 and 4), then M(l_i) can be expressed as

M (l_{i}) = [\begin{matrix} 1 & l_{i} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], (i = 1, 2, 3, 4) .

The ray matrix for round-trip propagation in a resonator is the product of each individual matrix in proper sequential order

M = [\begin{matrix} A & B & β \\ C & D & δ \\ 0 & 0 & 1 \end{matrix}] = M (l_{4}) M (l_{3}) M (l_{2}) M (P_{2}) M (l_{1}) M (P_{1}),

where A, B, C and D are standard ray matrix elements; β and δ are disorder ray matrix elements.

The resonator optical axis is invariant under the round-trip propagation. Also in accordance with the principles of the self-consistent, the following relationship exists [14–16]

[\begin{matrix} r_{1} \\ θ_{1} \\ 1 \end{matrix}] = M [\begin{matrix} r_{1} \\ θ_{1} \\ 1 \end{matrix}] .

Substitute Eq. (9), Eq. (10) and Eq. (11) to Eq. (12), r₁ and θ₁ can be obtained

r_{1} = \frac{m (\frac{l}{R} - 2 \cos A_{i})}{(m + 1) \cos A_{i} - m \frac{l}{R}} ε \sin A_{i},

θ_{1} = \frac{2 \frac{l}{R} - 4 \cos A_{i}}{((m + 1) \cos A_{i} - m \frac{l}{R}) l} ε \sin A_{i} .

where l is length of l₁, m is defined as the side ratio of sum length of the total optical path except l₁ to l, therefore l₂ + l₃ + l₄ = ml. The result of the Eq. (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 as

θ_{2} = θ_{1} + \frac{2 (m - 1) {\frac{l}{R}}_{i}}{((m + 1) \cos A_{i} - m \frac{l}{R}) l} ε \sin A_{i},

θ_{3} = θ_{1},

θ_{4} = θ_{1} .

In Eq. (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)

Ω = \frac{\frac{2 l}{R} - 4 \cos A_{i}}{l [(m + 1) \cos A_{i} - \frac{m l}{R}]} \sin A_{i} \cdot \frac{d ε}{d t} .

Here dε/dt indicates the velocity of the movement of spherical mirrors P₁ and P₂. From the 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 A_i = 45°, the equivalent rotation effect exists in other two types of RLG with different cavity structures as shown in Fig. 3.

Fig. 3 The cavity structures of the two types of RLG.

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3. Discussions

For a planar ring resonator, the stability condition is [14–16]

- 1 < \frac{A + D}{2} < 1.

Substitute Eq. (11) to Eq. (19), l/R meets

\frac{l}{R} < \frac{\cos A_{i}}{m} .

For typical RLG resonators, l/R << cosA_i/m, so Eq. (18) can be simplified to

Ω = \frac{- 4}{l (m + 1)} \sin A_{i} \cdot \frac{d ε}{d t} .

The sensitivity of resonator parameters l, A_i 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 A_i = 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 A_i 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.

Fig. 4 Sensitivity of equivalent rotation with incident angles of 15°, 30° and 45° versus l, with R = 8m and different values for m: (a) m = 2, (b) m = 3, (c) m = 4.

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4. Experiments

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.

Fig. 5 Schematic diagram of the experimental system

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A square RLG resonator, in which the length of l₁ is 0.08m, the side ratio m is 3, the incident angles A_i 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.

Fig. 6 Real time output of the gyro when spherical mirrors P₁ and P₂ are vibrated in opposite directions.

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Fig. 7 Equivalent rotation rate vs. velocity of mirrors’ movement

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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.

5. Conclusions

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 A_i 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.

Acknowledgments

This work is supported by Program for New Century Excellent Talents in University 2010.

References and links

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Mirrors movement-induced equivalent rotation effect in ring laser gyros

Abstract

1. Introduction

2. Theoretical analyses

3. Discussions

4. Experiments

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (21)

Optics Express