## Mirrors movement-induced equivalent rotation effect in ring laser gyros |

Optics Express, Vol. 21, Issue 12, pp. 14458-14465 (2013)

http://dx.doi.org/10.1364/OE.21.014458

Acrobat PDF (1633 KB)

### 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

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]

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]

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

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

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

## 2. Theoretical analyses

*P*is pushed forward to

_{1}*P*, and the other spherical mirror

_{1}'*P*is pulled backward to

_{2}*P*. The change of the light path is elaborated in Fig. 2, in which the dotted and solid lines describe the optical path when

_{2}'*P*holding motionless and moving forward, respectively. Assume

_{1}*R*is common radius of

*P*and

_{1}*P*,

_{2}*A*is the incident angles and

_{i}*ε*is displacement of spherical mirror

_{1}*P*. According to the geometric relationship as shown in Fig. 2, the relationships are given as followingand

_{1}*r*| and |

_{o}*θ*| of the reflection ray can be obtained from Eq. (1) and Eq. (2) as where

_{o}*f*is the focal length of the spherical mirrors, which is equal to

*RcosA*in the meridian plane.

_{i}/2*r*,

_{i}*r*,

_{o}*θ*,

_{i}*θ*,

_{o}*ε*is positive or negative. According to the rules in [14,15],

_{1}*r*,

_{i}*r*are positive,

_{o}*θ*is positive,

_{i}*θ*is negative and

_{o}*ε*is positive in Eq. (3) and Eq. (4). So If the extended matrix of

_{1}*P*is expressed as

_{1}*M*(

*P*), the relationship of

_{1}*r*,

_{i}*r*,

_{o}*θ*,

_{i}*θ*can be written asSubstitute Eq. (5) and Eq. (6) into Eq. (7), the extended matrix

_{o}*M*(

*P*) of

_{1}*P*can be deduced

_{1}*M*(

*P*) of

_{2}*P*can be deduced. Assuming

_{2}*P*is pushed forward by the displacement of

_{1}*ε*and

*P*is pulled backward by -

_{2}*ε*, the following conclusions can be deduced

*M*(

*l*) as the ray matrix of the free-space ray propagating along the path

_{i}*l*(

_{i}*i*= 1,2,3 and 4), then

*M*(

*l*) can be expressed asThe ray matrix for round-trip propagation in a resonator is the product of each individual matrix in proper sequential orderwhere

_{i}*A*,

*B*,

*C*and

*D*are standard ray matrix elements;

*β*and

*δ*are disorder ray matrix elements.

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]

*r*and

_{1}*θ*can be obtained where

_{1}*l*is length of

*l*is defined as the side ratio of sum length of the total optical path except

_{1}, m*l*to

_{1}*l,*therefore

*l*+

_{2}*l*+

_{3}*l*. The result of the Eq. (14) is the angle we concerned, which represents the rotation of the ray between

_{4}= ml*P*and

_{4}*P*corresponding to the original optical axis after

_{1}*P*is pushed forward by the displacement of ε and

_{1}*P*is pulled backward by -

_{2}*ε*.

*θ*=

_{1}*θ*=

_{3}*θ*<

_{4}*θ*since

_{2}*m*>1. Thus it can be seen that

*θ*is the common component in the four optical paths in the laser loop. We define

_{1}*θ*as the equivalent rotation angle after

_{1}*P*is pushed and

_{1}*P*is pulled and the optical paths in the closed-loop rotate

_{2}*θ*. We can get the velocity of the equivalent rotation angle from the time derivation of Eq. (14)Here

_{1}*dε/dt*indicates the velocity of the movement of spherical mirrors

*P*and

_{1}*P*. 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

_{2}*P*and

_{1}*P*.

_{2}## 3. Discussions

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]

*l*/

*R*meetsFor typical RLG resonators,

*l*/

*R*<< cos

*A*/

_{i}*m*, so Eq. (18) can be simplified to

*l*,

*A*and

_{i}*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*= 15°, 30° and 45° and

_{i}*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*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

_{i}*m*increases from 2 to 4 the absolute value of the sensitivity decreases.

## 4. Experiments

*P*and

_{1}*P*are mounted on two piezoelectric transducers respectively, which can push or pull

_{2}*P*and

_{1}*P*with variable drive voltage

_{2}*V*. A triangle signal generated by an oscillator is differentially amplified and then a pair of differential triangle signal is generated to drive

*P*and

_{1}*P*in opposite directions. The velocities, defined as

_{2}*dɛ*/

*dt*, of

*P*and

_{1}*P*can be obtained from the voltage derivation of the time

_{2}*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.

*l*is 0.08m, the side ratio

_{1}*m*is 3, the incident angles

*A*is 45° and the common radius of

_{i}*P*and

_{1}*P*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.

_{2}*P*and

_{1}*P*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

_{2}*ε*of

*P*. 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

_{1}*ɛ*is triangle wave. This means the output of the gyro is modulated by the derivation of the displacement

*dɛ*/

*dt*.

*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

*R*and

*l*, the smaller

*A*and

_{i}*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

## 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. |

2. | M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. |

3. | F. Aronowitz, “Mode coupling due to backscattering in a He-Ne traveling-wave ring laser,” Appl. Phys. Lett. |

4. | F. Aronowitz, “Fundamentals of the ring laser gyro,” Gyroscopes Optiques Et Leurs Applications |

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. |

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. |

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 |

14. | A. E. Siegman, |

15. | O. Svelto, |

16. | J. Yuan, X. Long, B. Zhang, F. Wang, and H. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. |

**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

- 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]
- M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt.19(3), 101–115 (1988). [CrossRef]
- F. Aronowitz, “Mode coupling due to backscattering in a He-Ne traveling-wave ring laser,” Appl. Phys. Lett.9(1), 55–58 (1966). [CrossRef]
- F. Aronowitz, “Fundamentals of the ring laser gyro,” Gyroscopes Optiques Et Leurs Applications15, 339 (1999).
- D. Loukianov, R. Rodloff, H. Sorg, and B. Stieler, “Optical gyros and their application,” RTO-AGA-339(1999).
- R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron.23(4), 438–445 (1987). [CrossRef]
- J. E. Killpatrick, “Random bias for laser angular rate sensor,” U.S.A Patent No.3467472 (1969).
- J. E. Killpatrick, Dither control system for a ring laser gyro,” U.S.A Patent No.6476918B1 (2002).
- 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).
- W. H. Egli, and Minneapolis, “Ring laser angular rate sensor with modulated scattered waves,” U.S.A Patent No.4592656 (1986).
- W. L. Lim and F. H. Zeman, “Laser gyro system,” U.S.A Patent No.4824252 (1989).
- 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).
- 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).
- A. E. Siegman, Lasers, (University Science Books, Mill Valley, CA, 1986) Chap. 15.
- O. Svelto, Principles of Lasers, (Springer , 1998).
- 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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