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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 12 — Jun. 17, 2013
  • pp: 14458–14465
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Mirrors movement-induced equivalent rotation effect in ring laser gyros

Guangfeng Lu, Zhenfang Fan, Shaomin Hu, and Hui Luo  »View Author Affiliations


Optics Express, Vol. 21, Issue 12, pp. 14458-14465 (2013)
http://dx.doi.org/10.1364/OE.21.014458


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Abstract

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.

© 2013 OSA

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

1. W. Chow, J. Gea-Banacloche, L. Pedrotti, V. Sanders, W. Schleich, and M. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985). [CrossRef]

5

5. D. Loukianov, R. Rodloff, H. Sorg, and B. Stieler, “Optical gyros and their application,” RTO-AGA-339(1999).

]. To avoid the lock-in phenomenon or reduce the lock-in threshold, variable technologies have been proposed [1

1. W. Chow, J. Gea-Banacloche, L. Pedrotti, V. Sanders, W. Schleich, and M. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985). [CrossRef]

,2

2. M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988). [CrossRef]

,6

6. R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987). [CrossRef]

13

13. X. G. Kong, T. S. Feng, and G. F. Jin, “Lock-in variation in the frequency-stabilized ring laser gyroscope,” Chin. J. Lasers 19, 744–748 (1992).

]. In [6

6. R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987). [CrossRef]

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

7. J. E. Killpatrick, “Random bias for laser angular rate sensor,” U.S.A Patent No.3467472 (1969).

], 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

2. M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988). [CrossRef]

,9

9. C. Guo and J. L. Wang, and H. G. Lv, “Test System of frequency stabilization and lock stabilization control parameter for ring laser gyroscope,” Opt. Technol. 32, 448–451 (2006).

13

13. X. G. Kong, T. S. Feng, and G. F. Jin, “Lock-in variation in the frequency-stabilized ring laser gyroscope,” Chin. J. Lasers 19, 744–748 (1992).

].

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
Fig. 1 Light path in the square ring resonator before and after the positions of P1 and P2 are changed.
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
Fig. 2 Schematic diagram of the light path before and after changing the position of P1.
, 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
{AC=ricosAiCB=2|ε1|tgAi|r0|=(AC+CB)cosAi
(1)
and

{β+Ai=|θo|+Ai|θi|+Ai=β+Aiβ=AC+|ε1|tgAiR
(2)

So, the parameters |ro| and |θo| of the reflection ray can be obtained from Eq. (1) and Eq. (2) as
|ro|=|ri|+2|ε1|sinAi,
(3)
|θo|=|θi|+|ri|f+|ε1|sinAif.
(4)
where f is the focal length of the spherical mirrors, which is equal to RcosAi/2 in the meridian plane.

It is important to decide whether ri, ro, θi, θo, ε1 is positive or negative. According to the rules in [14

14. A. E. Siegman, Lasers, (University Science Books, Mill Valley, CA, 1986) Chap. 15.

,15

15. O. Svelto, Principles of Lasers, (Springer , 1998).

], ri, ro are positive, θi is positive, θo is negative and ε1 is positive in Eq. (3) and Eq. (4). So
ro=ri+2ε1sinAi,
(5)
θo=θirifε1sinAif.
(6)
If the extended matrix of P1 is expressed as M(P1), the relationship of ri, ro, θi, θo can be written as
[roθo1]=M(P1)[riθi1].
(7)
Substitute Eq. (5) and Eq. (6) into Eq. (7), the extended matrix M(P1) of P1 can be deduced

M(P1)=[102ε1sinAi1f1ε1sinAif001]
(8)

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(P2)=M(P1)=[102εsinAi1f1εsinAif001].
(9)

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 as
M(li)=[1li0010001],(i=1,2,3,4).
(10)
The ray matrix for round-trip propagation in a resonator is the product of each individual matrix in proper sequential order
M=[ABβCDδ001]=M(l4)M(l3)M(l2)M(P2)M(l1)M(P1),
(11)
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

14. A. E. Siegman, Lasers, (University Science Books, Mill Valley, CA, 1986) Chap. 15.

16

16. J. Yuan, X. Long, B. Zhang, F. Wang, and H. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. 46(25), 6314–6322 (2007). [CrossRef] [PubMed]

]
[r1θ11]=M[r1θ11].
(12)
Substitute Eq. (9), Eq. (10) and Eq. (11) to Eq. (12), r1 and θ1 can be obtained
r1=m(lR2cosAi)(m+1)cosAimlRεsinAi,
(13)
θ1=2lR4cosAi((m+1)cosAimlR)lεsinAi.
(14)
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
θ2=θ1+2(m1)lRi((m+1)cosAimlR)lεsinAi,
(15)
θ3=θ1,
(16)
θ4=θ1.
(17)
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=2lR4cosAil[(m+1)cosAimlR]sinAidεdt.
(18)
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 Ai = 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.
.

3. Discussions

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

14. A. E. Siegman, Lasers, (University Science Books, Mill Valley, CA, 1986) Chap. 15.

16

16. J. Yuan, X. Long, B. Zhang, F. Wang, and H. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. 46(25), 6314–6322 (2007). [CrossRef] [PubMed]

]
1<A+D2<1.
(19)
Substitute Eq. (11) to Eq. (19), l/R meets
lR<cosAim.
(20)
For typical RLG resonators, l/R << cosAi/m, so Eq. (18) can be simplified to

Ω=4l(m+1)sinAidεdt.
(21)

The sensitivity of resonator parameters l, Ai and m on the equivalent rotation are analyzed according Eq. (21). Figure 4
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.
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.

4. Experiments

Schematic diagram of the experimental system is shown as Fig. 5
Fig. 5 Schematic diagram of the experimental system
. 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 /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
Fig. 6 Real time output of the gyro when spherical mirrors P1 and P2 are vibrated in opposite directions.
and Fig. 7
Fig. 7 Equivalent rotation rate vs. velocity of mirrors’ movement
.

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 /dt.

By changing the vibration frequency from 0.01Hz to 0.25Hz, different velocities /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 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.

Acknowledgments

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

References and links

1.

W. Chow, J. Gea-Banacloche, L. Pedrotti, V. Sanders, W. Schleich, and M. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985). [CrossRef]

2.

M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988). [CrossRef]

3.

F. Aronowitz, “Mode coupling due to backscattering in a He-Ne traveling-wave ring laser,” Appl. Phys. Lett. 9(1), 55–58 (1966). [CrossRef]

4.

F. Aronowitz, “Fundamentals of the ring laser gyro,” Gyroscopes Optiques Et Leurs Applications 15, 339 (1999).

5.

D. Loukianov, R. Rodloff, H. Sorg, and B. Stieler, “Optical gyros and their application,” RTO-AGA-339(1999).

6.

R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987). [CrossRef]

7.

J. E. Killpatrick, “Random bias for laser angular rate sensor,” U.S.A Patent No.3467472 (1969).

8.

J. E. Killpatrick, Dither control system for a ring laser gyro,” U.S.A Patent No.6476918B1 (2002).

9.

C. Guo and J. L. Wang, and H. G. Lv, “Test System of frequency stabilization and lock stabilization control parameter for ring laser gyroscope,” Opt. Technol. 32, 448–451 (2006).

10.

W. H. Egli, and Minneapolis, “Ring laser angular rate sensor with modulated scattered waves,” U.S.A Patent No.4592656 (1986).

11.

W. L. Lim and F. H. Zeman, “Laser gyro system,” U.S.A Patent No.4824252 (1989).

12.

J. H. Simpson and J. G. Koper, “Ring laser gyroscope utilizing phase detector for minimizing beam lock-in,” U.S.A Patent No.4473297 (1984).

13.

X. G. Kong, T. S. Feng, and G. F. Jin, “Lock-in variation in the frequency-stabilized ring laser gyroscope,” Chin. J. Lasers 19, 744–748 (1992).

14.

A. E. Siegman, Lasers, (University Science Books, Mill Valley, CA, 1986) Chap. 15.

15.

O. Svelto, Principles of Lasers, (Springer , 1998).

16.

J. Yuan, X. Long, B. Zhang, F. Wang, and H. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. 46(25), 6314–6322 (2007). [CrossRef] [PubMed]

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.3410) Lasers and laser optics : Laser resonators
(140.3560) Lasers and laser optics : Lasers, ring
(140.4780) Lasers and laser optics : Optical resonators

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 18, 2013
Revised Manuscript: June 1, 2013
Manuscript Accepted: June 4, 2013
Published: June 10, 2013

Citation
Guangfeng Lu, Zhenfang Fan, Shaomin Hu, and Hui Luo, "Mirrors movement-induced equivalent rotation effect in ring laser gyros," Opt. Express 21, 14458-14465 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-14458


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References

  1. W. Chow, J. Gea-Banacloche, L. Pedrotti, V. Sanders, W. Schleich, and M. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985). [CrossRef]
  2. M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt.19(3), 101–115 (1988). [CrossRef]
  3. F. Aronowitz, “Mode coupling due to backscattering in a He-Ne traveling-wave ring laser,” Appl. Phys. Lett.9(1), 55–58 (1966). [CrossRef]
  4. F. Aronowitz, “Fundamentals of the ring laser gyro,” Gyroscopes Optiques Et Leurs Applications15, 339 (1999).
  5. D. Loukianov, R. Rodloff, H. Sorg, and B. Stieler, “Optical gyros and their application,” RTO-AGA-339(1999).
  6. R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron.23(4), 438–445 (1987). [CrossRef]
  7. J. E. Killpatrick, “Random bias for laser angular rate sensor,” U.S.A Patent No.3467472 (1969).
  8. J. E. Killpatrick, Dither control system for a ring laser gyro,” U.S.A Patent No.6476918B1 (2002).
  9. C. Guo and J. L. Wang, and H. G. Lv, “Test System of frequency stabilization and lock stabilization control parameter for ring laser gyroscope,” Opt. Technol.32, 448–451 (2006).
  10. W. H. Egli, and Minneapolis, “Ring laser angular rate sensor with modulated scattered waves,” U.S.A Patent No.4592656 (1986).
  11. W. L. Lim and F. H. Zeman, “Laser gyro system,” U.S.A Patent No.4824252 (1989).
  12. J. H. Simpson and J. G. Koper, “Ring laser gyroscope utilizing phase detector for minimizing beam lock-in,” U.S.A Patent No.4473297 (1984).
  13. X. G. Kong, T. S. Feng, and G. F. Jin, “Lock-in variation in the frequency-stabilized ring laser gyroscope,” Chin. J. Lasers19, 744–748 (1992).
  14. A. E. Siegman, Lasers, (University Science Books, Mill Valley, CA, 1986) Chap. 15.
  15. O. Svelto, Principles of Lasers, (Springer , 1998).
  16. J. Yuan, X. Long, B. Zhang, F. Wang, and H. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt.46(25), 6314–6322 (2007). [CrossRef] [PubMed]

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