## Generalized sensitivity factors for optical-axis perturbation in nonplanar ring resonators |

Optics Express, Vol. 19, Issue 20, pp. 19752-19757 (2011)

http://dx.doi.org/10.1364/OE.19.019752

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### Abstract

By utilizing the generalized ray matrix for spherical mirror reflection, two new sensitivity factors are introduced considering the perturbation of the optical-axis caused by the radial and axial displacements of a spherical mirror in a nonplanar ring resonator. Based on this, a novel way for finding the location of the singular points of the sensitivity factors is presented. It is found that some nonplanar ring resonators with the effective modes may have the singular points of the sensitivity factors. The unsuitable regions for nonplanar ring resonators are also obtained from the perspective of the sensitivity factors.

© 2011 OSA

## 1. Introduction

5. Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. **282**(15), 3108–3112 (2009). [CrossRef]

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

7. J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. **281**(5), 1204–1210 (2008). [CrossRef]

8. G. B. Al’tshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. **7**(7), 857–859 (1977). [CrossRef]

10. A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectros. (USSR) **40**(6), 657–660 (1984). [CrossRef]

*et al*. [7

7. J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. **281**(5), 1204–1210 (2008). [CrossRef]

11. S.-C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett. **19**(10), 683–685 (1994). [CrossRef] [PubMed]

## 2. Analysis method and generalized sensitivity factors

*L*(

_{j}*j*= 1, 2, 3, 4), a reflection on one spherical mirror with radius of curvature

*R*(infinite for the plane mirror) and incident angle

_{j}*A*, and a coordinate rotation angle

_{j}*φ*.

_{j}12. A. H. Paxton and W. P. Latham Jr., “Unstable resonators with 90 ° beam rotation,” Appl. Opt. **25**(17), 2939–2946 (1986). [CrossRef] [PubMed]

*L*as T(

_{j}*L*) and

_{j}*R*(

*φ*), respectively. The generalized ray matrix for spherical mirror reflection can be written as [15

_{j}15. J. Yuan, X. W. Long, and M. X. Chen, “Generalized ray matrix for spherical mirror reflection and its application in square ring resonators and monolithic triaxial ring resonators,” Opt. Express **19**(7), 6762–6776 (2011). [CrossRef] [PubMed]

*θ*and

_{jx}*θ*are the misalignment angles of the spherical mirror P

_{jy}*in its local tangential and sagittal planes, respectively;*

_{j}*δ*and

_{jx}*δ*are the radial displacements, and

_{jy}*δ*is the axial displacement [16

_{jz}16. X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett. **8**(12), 1135–1138 (2010). [CrossRef]

*M*operates on a five-component ray vector

*V*given bywhere,

*r*,

_{x}*r*are the optical-axis heights from the reference axis along the

_{y}*x*,

*y*axes, and

*r*

^{'}

*,*

_{x}*r*

^{'}

*are the angles between the optical-axis and the reference axis in the*

_{y}*x*,

*y*planes, respectively. On the basis that the optical-axis has to reappear itself in the ring resonator,

*V*can be obtained by solving the following equation

7. J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. **281**(5), 1204–1210 (2008). [CrossRef]

**281**(5), 1204–1210 (2008). [CrossRef]

*/L*(

*L*is the total length of the resonator) before

*∂r*'

*/∂θ*is contained. SD1 and ST1 present the axis decentration and tilt sensitivity of the optical-axis on the

*i*th mirror as the result of the misalignment angles, respectively.

*j*th mirror (

*θ*,

_{jx}*θ*,

_{jy}*δ*,

_{jx}*δ*,

_{jy}*δ*) possibly exist, it is needed to analyze the effect of each source has on the movement of the optical-axis. That is, the effects of the radial and axial displacements of the

_{jz}*j*th mirror on the movement of the mirror optical-axis in a nonplanar ring resonator should be considered by introducing two new sensitivity factors defined as followsWhere, SD2 and ST2 represent the axis decentration and tilt sensitivity of the mirror optical-axis as a result of the radial and axial displacement of the

*j*th mirror, respectively. SD1, ST1, SD2 and ST2 constitute the generalized sensitivity factors with which one can take all the perturbation sources of a mirror into consideration. Considering that SD1 and ST1 have been studied in Refs [7

**281**(5), 1204–1210 (2008). [CrossRef]

11. S.-C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett. **19**(10), 683–685 (1994). [CrossRef] [PubMed]

_{1}, P

_{2}are spherical mirrors with the identical radius of curvature

*R*, while P

_{3}, P

_{4}are planar mirrors (Fig. 1), the numerical results presented here are for the case that the incident angles on all four mirrors are identical [11

11. S.-C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett. **19**(10), 683–685 (1994). [CrossRef] [PubMed]

*A*

_{1}=

*A*

_{2}=

*A*

_{3}=

*A*

_{4}=

*A*. The beam travels along the direction P

_{1}→P

_{2}→P

_{3}→P

_{4}→P

_{1}and the perturbation originates from the misalignment of mirror P

_{1}. Figure 2 shows SD2 and ST2 versus

*L/R*with

*A*= 43.87° (corresponding image rotation angle

*ρ*=90°).

_{1}have two common singular points at

*L/R*=3.98 and

*L/R*=7.26, where the optical-axis movements diverge. The values of the sensitivity factors approach infinite at these singular points and the absolute value of the sensitivity factors increase sharply when

*L/R*approaches the singular points. We have also studied the behavior of the optical-axis on other mirrors and found that the SD2 and ST2 have the same singular points as discussed above. Similarly, beginning from the definition of the sensitivity factors, we can identify that the singular points locate at

*L/R=*4.51 and

*L/R=*6.54 when

*A*=40.06° (

*ρ*=180°), while the singular points locate at

*L/R=*5.07 and

*L/R=*5.73 when

*A=*31.74° (

*ρ*=270°).

*M'*, we can obtain the determinant of

*M'*(det

*M'*) versus

*L/R*as shown in Fig. 3 . Comparing with the singular points mentioned above, it can be seen that the location of the singular points overlap with the zero value point of det

*M'*. As we know that det

*M'*can be expressed by

*L/R*and

*A*, so we can solve the equation of det

*M'*=0 and

*L/R*can be expressed in terms of

*A*(here the expression is named as

*f*(

*A*) and

*f*(

*A*) is the function of the zero points in det

*M'*). By expressing the sensitivity factors in terms of

*L/R*and

*A*, it is found that the left (right) limit of the sensitivity factors approach plus (minus) or minus (plus) infinity when

*L/R*are close to

*f*(

*A*). So that the zero points in det

*M'*is just the singular points of the sensitivity factors. It is an easier way to find the accurate location of the singular points by calculating the determinant of

*M'*rather than the traditional way.

## 3. Analysis of the generalized sensitivity factors

17. J. Yuan, X. W. Long, L. M. Liang, B. Zhang, F. Wang, and H. C. Zhao, “Nonplanar ring resonator modes: generalized Gaussian beams,” Appl. Opt. **46**(15), 2980–2989 (2007). [CrossRef] [PubMed]

**281**(5), 1204–1210 (2008). [CrossRef]

*ρ*are 270°, 90° or 0°. However, the cases for other image rotation angles haven’t been studied yet. We have numerically calculated all the singular points of SD1, ST1, SD2 and ST2 when

*ρ*varies from 0° to 360° (0°<

*A*<45°). As we know that there is only one singular point for some sensitivity factors in a given resonator while there are two singular points for other sensitivity factors in the same resonator [7

**281**(5), 1204–1210 (2008). [CrossRef]

*A*<10°. However, it split into two branches with the augment of

*A*. Although the branch II spreads to the unstable area, the branch I crosses the 2nd stable area when 5.095<

*L/R*<5.335, which means the corresponding resonators with effective modes have the singular points of the sensitivity factors.

## 4. Conclusion

*M'*is just the singular points of the sensitivity factors, so we can find the singular points by analyzing the determinant of the coefficient matrix of the linear equations. It provides an easier way to find the accurate location of the singular points. By calculating all the singular points with

*A*ranging from 0° to 45°, it can be found that some nonplanar ring resonators with effective modes may suffer the singular points of sensitivity factors. An example of unsuitable areas regarding the large sensitivity factors is also presented. The analysis in this paper could be helpful to avoid the violent movement of the optical-axis to small misalignment of the mirrors in nonplanar ring resonators.

## Acknowledgements

## References and links

1. | H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, “The multioscillator ring laser gyroscope,” in |

2. | W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. |

3. | A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. |

4. | F. Zomer, Y. Fedala, N. Pavloff, V. Soskov, and A. Variola, “Polarization induced instabilities in external four-mirror Fabry-Perot cavities,” Appl. Opt. |

5. | Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. |

6. | R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. |

7. | J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. |

8. | G. B. Al’tshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. |

9. | I. W. Smith, “Optical resonator axis stability and instability from first principles,” Proc. SPIE |

10. | A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectros. (USSR) |

11. | S.-C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett. |

12. | A. H. Paxton and W. P. Latham Jr., “Unstable resonators with 90 ° beam rotation,” Appl. Opt. |

13. | A. H. Paxton and W. P. Latham Jr., “Ray matrix method for the analysis of optical resonators with image rotation,” Proc. SPIE |

14. | A. E. Siegman, |

15. | J. Yuan, X. W. Long, and M. X. Chen, “Generalized ray matrix for spherical mirror reflection and its application in square ring resonators and monolithic triaxial ring resonators,” Opt. Express |

16. | X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett. |

17. | J. Yuan, X. W. Long, L. M. Liang, B. Zhang, F. Wang, and H. C. Zhao, “Nonplanar ring resonator modes: generalized Gaussian beams,” Appl. Opt. |

**OCIS Codes**

(140.3370) Lasers and laser optics : Laser gyroscopes

(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: July 26, 2011

Revised Manuscript: September 7, 2011

Manuscript Accepted: September 7, 2011

Published: September 23, 2011

**Citation**

Dandan Wen, Dong Li, and Jianlin Zhao, "Generalized sensitivity factors for optical-axis perturbation in nonplanar ring resonators," Opt. Express **19**, 19752-19757 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19752

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### References

- H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, “The multioscillator ring laser gyroscope,” in Laser Handbook, M. I. Stitch, and M. Bass, eds. (North Holland, 1985), pp. 229–327.
- W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985). [CrossRef]
- A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron.6(6), 1389–1399 (2000). [CrossRef]
- F. Zomer, Y. Fedala, N. Pavloff, V. Soskov, and A. Variola, “Polarization induced instabilities in external four-mirror Fabry-Perot cavities,” Appl. Opt.48(35), 6651–6661 (2009). [CrossRef] [PubMed]
- Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun.282(15), 3108–3112 (2009). [CrossRef]
- R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron.23(4), 438–445 (1987). [CrossRef]
- J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun.281(5), 1204–1210 (2008). [CrossRef]
- G. B. Al’tshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron.7(7), 857–859 (1977). [CrossRef]
- I. W. Smith, “Optical resonator axis stability and instability from ﬁrst principles,” Proc. SPIE412, 203–206 (1983).
- A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectros. (USSR)40(6), 657–660 (1984). [CrossRef]
- S.-C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett.19(10), 683–685 (1994). [CrossRef] [PubMed]
- A. H. Paxton and W. P. Latham., “Unstable resonators with 90 ° beam rotation,” Appl. Opt.25(17), 2939–2946 (1986). [CrossRef] [PubMed]
- A. H. Paxton and W. P. Latham., “Ray matrix method for the analysis of optical resonators with image rotation,” Proc. SPIE554, 159–163 (1985).
- A. E. Siegman, Lasers (University Science, 1986).
- J. Yuan, X. W. Long, and M. X. Chen, “Generalized ray matrix for spherical mirror reflection and its application in square ring resonators and monolithic triaxial ring resonators,” Opt. Express19(7), 6762–6776 (2011). [CrossRef] [PubMed]
- X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett.8(12), 1135–1138 (2010). [CrossRef]
- J. Yuan, X. W. Long, L. M. Liang, B. Zhang, F. Wang, and H. C. Zhao, “Nonplanar ring resonator modes: generalized Gaussian beams,” Appl. Opt.46(15), 2980–2989 (2007). [CrossRef] [PubMed]

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