## Generalized ray matrix for spherical mirror reflection and its application in square ring resonators and monolithic triaxial ring resonators |

Optics Express, Vol. 19, Issue 7, pp. 6762-6776 (2011)

http://dx.doi.org/10.1364/OE.19.006762

Acrobat PDF (1302 KB)

### Abstract

To the best of our knowledge, the generalized ray matrix, an augmented 5×5 ray matrix for a spherical mirror reflection with all the possible perturbation sources including three kinds of displacements and its detailed deducing process have been proposed in this paper for the first time. Square ring resonators and monolithic triaxial ring resonators have been chosen as examples to show its application, and some novel results of the optical-axis perturbation have been obtained. A novel method to eliminate the diaphragm mismatching error and the gain capillary mismatching error in monolithic triaxial ring resonators more effectively has also been proposed. Both those results and method have been confirmed by related experiments and the experimental results have been described with diagrammatic representation. This generalized ray matrix is valuable for ray analysis of various kinds of resonators. These results are important for the cavity design, cavity improvement and alignment of high accuracy and super high accuracy ring laser gyroscopes.

© 2011 OSA

## 1. Introduction

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

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

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

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]

*C*of the MTRR, which has been proposed in our previous paper [15

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. **47**(5), 628–631 (2008). [CrossRef] [PubMed]

*C*and

*C*and

## 2. Analysis method

*x*and

*y*axes respectively and we call them optical-axis decentration.

*x*and

*y*plane respectively and we call them optical-axis tilt.

*x*and

*y*axes.

_{i}with radius R

_{i}has been chosen as an example to show the perturbation sources in Fig. 1 and Fig. 2 . The incident angle is A

_{i}. As shown in Fig. 1(a), generally the mirror M

_{i}has 3 kinds of translational displacements and 3 kinds of angular misalignments.

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]

_{i1}and M

_{i2}represent the mirror M

_{i}before and after the axial displacement

_{1}and to point P

_{2}. For a linear resonator, the ray is incident vertically and

_{1}and P

_{2}is zero and the reflection point has not been changed under the axial displacement

_{i}and L2

_{i}are chosen for examples. L1

_{o1}is the reflection ray of L1

_{i}via the reflection at the point P

_{1}of M

_{i1}. L2

_{o1}is the reflection ray of L2

_{i}via the reflection at the point P

_{3}of M

_{i1}. L2

_{o2}is the reflection ray of L2

_{i}via the reflection at the point P

_{2}of M

_{i2}. The coordinate axes of the incident ray are

*x*and

*y*axes are analyzed. For the incident ray L2

_{i}, the coordinates of the point P

_{2}are

_{i}(1, 5) should be modified into

_{i}in the coordinate axis of reflection ray, the exit angle of L2

_{o1}is

_{o2}is 0. This angle is under the direction of

*x*axis and the angle modification under the direction of

*y*axis is 0, so the standard ray-matrix elements M

_{i}(2, 5) should be modified into

_{i}(1, 5) and M

_{i}(2, 5) should be modified into

*x*and

*y*axes are analyzed. For the incident ray L1

_{i}, the coordinates of the point P

_{4}are

_{i}(1, 5) should be modified into

_{o2}is

_{o1}is 0. This angle is under the direction of

*x*axis and the angle modification under the direction of

*y*axis is 0, so the standard ray-matrix elements M

_{i}(2, 5) should be modified into

_{i}(4, 5) should be modified into

_{i}behind the rotation axis

*x*and

*y*axes are analyzed. For the incident ray L1

_{i}, the coordinates of the point P

_{1}are both

_{i}(1, 5) should not be modified. The exit angle of L1

_{o3}is

_{o1}is 0. This angle is under the direction of

*x*axis and the angle modification under the direction of

*y*axis is 0, so the standard ray-matrix elements M

_{i}(2, 5) should be modified into

_{i}(4, 5) should be modified into

_{i}with all kinds of possible perturbation sources including

## 3. Analysis of square ring resonators

*x*and

*y*axes respectively, and the center of the longest gain capillary is also the center of the gain medium. The positive orientation of

*e*) and the center of the gain capillary (point

*g*) simultaneously.

*e*) and the center of the gain capillary (point

*g*) along the

*x*and

*y*axes respectively, and

17. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. **23**(17), 2944–2949 (1984). [CrossRef] [PubMed]

18. N. M. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. **29**(3), 394–403 (1990). [CrossRef] [PubMed]

_{a}and M

_{b}) and two planar mirrors (M

_{c}and M

_{d}). A light bulb (LB) is used to illuminate the diaphragm and the longest gain capillary. He-Ne laser (HNLP) with wavelength

19. J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. **74**(3), 1362–1365 (2003). [CrossRef]

20. J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. **76**(12), 125106 (2005). [CrossRef]

19. J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. **74**(3), 1362–1365 (2003). [CrossRef]

20. J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. **76**(12), 125106 (2005). [CrossRef]

_{a}is removed from the cavity first, then the light bulb (LB) is switched off and the laser (HNLP) is switched on. The facular image of incidental laser at different locations can be obtained by adjusting the focusing lens (CFLAD). The centers of the facular image of the incidental laser at different locations can be made located at the center of the CCD area by adjusting the focusing lens and the adjusting device of two reflectors (RAD1 and RAD2). As a consequence, the optical-axis of the incidental laser has been made aligned with the optical-axis of the CCD area and focusing lens and the ideal optical-axis of the passive ring cavity too.

_{a}is mounted on the cavity first, then the path length control device of laser (HNLP) is adjusted and the frequency of the input laser is modified, when the input frequency has a resonance with the passive ring cavity, the centers of the beam frequency facular images at point

*e*and

*g*can be obtained by adjusting the focusing lens (CFLAD). The optical-axis of the beam transmitted by the resonator is called the real optical-axis in this paper and it passes through the centers of the beam frequency facular images at point

*e*and

*g*. When the perturbation sources of

## 4. Analysis of monolithic triaxial ring resonators

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. **47**(5), 628–631 (2008). [CrossRef] [PubMed]

_{1}, M

_{2}, M

_{3}, M

_{4}, M

_{5}and M

_{6}are respectively positioned in the center of each cube body face. The cube is machined such that a small diameter bore connects adjacent mirrors. A closed optical cavity is defined between four coplanar mirrors, which are interconnected by bores. There are three mutually orthogonal closed beam paths, each of which is used to detect angular rotation about its normal axis. The planar ring resonator which is defined by the optical cavity between the mirrors M

_{2}, M

_{3}, M

_{4}and M

_{6}is called cavity I, the resonator defined by M

_{1}, M

_{3}, M

_{5}and M

_{6}is called cavity II, and the resonator defined by M

_{1}, M

_{2}, M

_{5}and M

_{4}is called cavity III.

*C*of MTRR has also been found out in previous papers [8

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

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. **47**(5), 628–631 (2008). [CrossRef] [PubMed]

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]

*C*cannot be decreased by modifying the angles of the terminal surfaces and the terminal mirrors. In other words, the three monoaxial ring resonators cannot be aligned to pass through the center of their diaphragms simultaneously by angular misalignments. In this paper, the perturbation sources of all 6 mirror’s axial displacements and all 3 spherical mirror’s radial displacements will be considered. The points P

_{A}, P

_{B}and P

_{C}, which are the diaphragms of the cavities I, II and III, and their symmetrical points P

_{D}, P

_{E}and P

_{F}, which are the center of the gain capillary for each cavity, will be chosen for analysis. For MTRR, it would be better that the optical-axes of all the three monoaxial ring resonators pass through the center of their diaphragms (P

_{A}, P

_{B}and P

_{C}) simultaneously, and this mean the diaphragm mismatching error

*C*should be eliminated [15

**47**(5), 628–631 (2008). [CrossRef] [PubMed]

_{D}, P

_{E}and P

_{F}) simultaneously too, and this mean the gain capillary mismatching error

**47**(5), 628–631 (2008). [CrossRef] [PubMed]

_{A}, P

_{B}, P

_{C}, P

_{D}, P

_{E}and P

_{F}, caused by all the 3 spherical mirror’s translations displacements and all the 6 mirror’s axial displacements can be written asand

**47**(5), 628–631 (2008). [CrossRef] [PubMed]

**8**(12), 1135–1138 (2010). [CrossRef]

**47**(5), 628–631 (2008). [CrossRef] [PubMed]

*t*.

*t*can be written as:

*C*and

_{A}, P

_{B}, P

_{C}and their symmetrical point P

_{D}, P

_{E}and P

_{F}at the time of t can be written as

**47**(5), 628–631 (2008). [CrossRef] [PubMed]

**47**(5), 628–631 (2008). [CrossRef] [PubMed]

*C*and

*C*and

*t*can be decreased and even eliminated to 0 by choosing appropriate

_{A}, P

_{B}, P

_{C}have been made smallest. At the same time,

_{D}, P

_{E}, P

_{F}have been made smallest too. The total diffraction loss of the monolithic triaxial ring resonator has been made lowest.

**8**(12), 1135–1138 (2010). [CrossRef]

**8**(12), 1135–1138 (2010). [CrossRef]

## 5. Conclusion

*C*of the MTRR has been found out that it does not has any relation with the planar mirror’s axial displacement and it has the relation with the spherical mirror’s axial displacement. The gain capillary mismatching error C2 of the MTRR has been defined in this paper. A novel method to eliminate the diaphragm mismatching error

*C*and the gain capillary mismatching error

*C*and gain capillary mismatching error

## Acknowledgments

## References and links

1. | 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. |

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

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

4. | J. A. Arnaud, “Degenerate Optical Cavities. II: Effect of Misalignments,” Appl. Opt. |

5. | G. B. Altshuler, 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. |

6. | H. R. Bilger and G. E. Stedman, “Stability of planar ring lasers with mirror misalignment,” Appl. Opt. |

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

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

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. Spectrosc. (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. | B. E. Currie, G. E. Stedman, and R. W. Dunn, “Laser stability and beam steering in a nonregular polygonal cavity,” Appl. Opt. |

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

15. | J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. |

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

17. | D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. |

18. | N. M. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. |

19. | J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. |

20. | J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. |

**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: January 26, 2011

Revised Manuscript: March 13, 2011

Manuscript Accepted: March 15, 2011

Published: March 24, 2011

**Citation**

Jie Yuan, Xingwu Long, and Meixiong Chen, "Generalized ray matrix for spherical mirror reflection and its application in square ring resonators and monolithic triaxial ring resonators," Opt. Express **19**, 6762-6776 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6762

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### References

- 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), 61–86 (1985). [CrossRef]
- M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988). [CrossRef]
- A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000). [CrossRef]
- J. A. Arnaud, “Degenerate Optical Cavities. II: Effect of Misalignments,” Appl. Opt. 8(9), 1909–1917 (1969). [CrossRef] [PubMed]
- G. B. Altshuler, 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, 857–859 (1977). [CrossRef]
- H. R. Bilger and G. E. Stedman, “Stability of planar ring lasers with mirror misalignment,” Appl. Opt. 26(17), 3710–3716 (1987). [CrossRef] [PubMed]
- R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987). [CrossRef]
- J. Yuan, X. W. Long, B. Zhang, F. Wang, and H. C. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. 46(25), 6314–6322 (2007). [CrossRef] [PubMed]
- I. W. Smith, “Optical resonator axis stability and instability from first principles,” Proc. SPIE 412, 203–206 (1983).
- A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectrosc. (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]
- B. E. Currie, G. E. Stedman, and R. W. Dunn, “Laser stability and beam steering in a nonregular polygonal cavity,” Appl. Opt. 41(9), 1689–1697 (2002). [CrossRef] [PubMed]
- J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. 281(5), 1204–1210 (2008). [CrossRef]
- J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [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]
- D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23(17), 2944–2949 (1984). [CrossRef] [PubMed]
- N. M. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. 29(3), 394–403 (1990). [CrossRef] [PubMed]
- J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003). [CrossRef]
- J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005). [CrossRef]

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