## Mode characteristics of twisted resonators composed of two cylindrical mirrors.

Optics Express, Vol. 2, Issue 5, pp. 184-190 (1998)

http://dx.doi.org/10.1364/OE.2.000184

Acrobat PDF (317 KB)

### Abstract

We examine the optical resonator composed of two astigmatic elements, in a twisted configuration. These cavities have mode cross-sections with principal axes that rotate on propagation. Explicit cavity mode equations are derived for the case of identical mirrors. Such a resonator is appropriate for a solid-state laser that is end-pumped with the output of a laser-diode array brought to a line focus. We present a simple analysis of the significance of rotational misalignment, which effects the pump-to-mode power coupling, beam quality, and cavity stability.

© Optical Society of America

## 1. Introduction

1. F. Krausz, J. Zehetner, T. Brabec, and E. Winter, “Elliptic-mode cavity for diode-pumped lasers,” Opt. Lett. **16**, 1496–1498 (1991). [CrossRef] [PubMed]

*x*and

*y*directions are separated and located at the end mirrors. When the mirrors are rotated obliquely, the resonator mode has a more general form of astigmatism. This type of beam has been considered by Arnaud and Kogelnik [8

8. J.A. Arnaud and H. Kogelnik, “Gaussian Light beams with General Astigmatism,” Appl. Opt. **8**, 1687 (1969). [CrossRef] [PubMed]

8. J.A. Arnaud and H. Kogelnik, “Gaussian Light beams with General Astigmatism,” Appl. Opt. **8**, 1687 (1969). [CrossRef] [PubMed]

9. A.W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE **679**, 129 (1986) [CrossRef]

13. J. Serna and G. Nemes, “Decoupling of coherent Gaussian beams with general astigmatism,” Opt. Lett. **18**, 1774 (1993). [CrossRef] [PubMed]

*et al*. [14

14. I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. **25**, 436 (1995). [CrossRef]

*et al*. [15,16

16. B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. **24**, 619 (1992). [CrossRef]

## 2. A generalised Gaussian beam

*z*=0 must include cross terms in the transverse variables:

*j*= √-1,

*M*are 2 × 2 real, symmetric matrices that contain transverse phase and amplitude information respectively. Cross terms in

*x*and

*y*enable us to have a beam with elliptical or hyperbolic contours of equal phase and amplitude. For physical beams, however, the amplitude contours must be elliptical.

*z*can be obtained by applying the Fresnel transform, ie

*I*is the 2 × 2 identity matrix. By using

## 3. Mode for resonator bounded by two identical cylindrical mirrors

*R*separated by a distance

*L*. Assuming identical end mirrors simplifies the algebra while still containing the essential features. The first mirror is fixed, being active in the

*x*direction. The second mirror is rotated some angle

*t*around the optical axis, being active in the

*y*direction when

*t*= 0. On solving the associated equations, we obtain the solution for the intensity distribution at the first mirror:

*t*and

*α*=

*L*/

*R*must satisfy

## 4. Resonator mode properties

17. K. Kubodera and K. Otsuka, “Single-transverse-mode LiNdP_{4}O_{12} slab waveguide laser,” J. Appl. Phys. **50**, 653 (1979). [CrossRef]

1. F. Krausz, J. Zehetner, T. Brabec, and E. Winter, “Elliptic-mode cavity for diode-pumped lasers,” Opt. Lett. **16**, 1496–1498 (1991). [CrossRef] [PubMed]

*π*/4 radians to each other. Note that the principal axes of the amplitude rotate by more than

*π*/4 radians during propagation, and that the principal axes of the phase and amplitude distributions are not aligned. The intensity distribution is always elliptical for the lowest order mode, but the principal axes of the intensity at each mirror do not coincide with those of the mirrors when the system is rotationally misaligned. We find that the rotation angle of the elliptical intensity foot print is a monotonically increasing function of the mirror rotation angle. The rotational angle of the intensity distribution is plotted against mirror rotation in Figure 4, for various values of

*α*=

*L*/

*R*. When

*α*falls in the range [1.0,2.0], the curves are plotted only for values of

*t*in which the resonator is stable. The non-linearity in these plots is reflected in the meandering of the principal axes of the beam footprint at the top mirror in the animation in Figure 1. The fact that

*s*reaches

*π*/2 when

*t*does is consistent with the fact that the average velocity of rotation of the beam footprint at the bottom mirror matches that of the top mirror.

*L*/

*R*. The curves for α ∈ (0.0,1.0) are reasonably well behaved, having a local maximum at

*t*=0, and a singularity at

*t*=

*π*/2. The curves for α ∈ [1.0,2.0] however, have singularities at some

*t*>0 and

*t*=

*π*/2. It can be seen from Figure 5, that an orthogonal resonator (

*t*= 0) with an aspect ratio of 10, can be created by choosing α=0.99. This resonator would not require critical rotational alignment of its elements, because the aspect ratio is insensitive to small changes in

*t*(due to the local maximum), as is the rotation of the intensity distribution (

*s*~0.54

*t*for small values of

*t*, when α~1.)

## 5. Conclusion

*L*/

*R*slightly less than 1.0. These resonators were found to be geometrically stable with respect to rotational misalignment for all angles whenever α is less than 1.0, and for a non-zero interval of mirror rotation angles for a between 1.0 and 2.0. Due to the highly elliptical cavity mode, the resonator composed of two cylindrical end-mirrors is appropriate for a solid-state laser that is end-pumped with a beam brought to a line focus.

## References

1. | F. Krausz, J. Zehetner, T. Brabec, and E. Winter, “Elliptic-mode cavity for diode-pumped lasers,” Opt. Lett. |

2. | J. Zehetner, “Highly efficient diode-pumped elliptical mode Nd:YLF laser,” Opt. Commun. |

3. | D. Kopf, U. Keller, M.A. Emanuel, R.J. Beach, and J.A. Skidmore, “1.1-W cw Cr:LiSAF laser pumped by a 1-cm diode array,” Opt. Lett. |

4. | J.L. Blows, J.M. Dawes, and J.A. Piper, “Highly astigmatic diode pumped laser cavities for intracavity nonlinear optical conversion,” in |

5. | J. Blows, J. Dawes, J. Piper, and G. Forbes, “A highly astigmatic diode end-pumped solid-state laser,” in |

6. | A.E. Siegman, |

7. | A.E. Siegman, |

8. | J.A. Arnaud and H. Kogelnik, “Gaussian Light beams with General Astigmatism,” Appl. Opt. |

9. | A.W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE |

10. | A.W. Greynolds, “Vector formulation of the ray-equivalent method for general Gaussian beam propagation,” Proc. SPIE |

11. | J.A. Arnaud, “Nonorthogonal Optical Waveguides and Resonators,” Bell Syst. Tech. J. |

12. | J.A. Arnaud, “Hamiltonian theory of beam mode propagation,” Prog. Opt. |

13. | J. Serna and G. Nemes, “Decoupling of coherent Gaussian beams with general astigmatism,” Opt. Lett. |

14. | I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. |

15. | B. Lü, S. Xu, G. Feng, and B. Cai, “Astigmatic resonator analysis using an eigenray vector concept,” Optik |

16. | B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. |

17. | K. Kubodera and K. Otsuka, “Single-transverse-mode LiNdP |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(140.3480) Lasers and laser optics : Lasers, diode-pumped

(140.4780) Lasers and laser optics : Optical resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 18, 1997

Revised Manuscript: November 25, 1997

Published: March 2, 1998

**Citation**

Justin Blows and Gregory Forbes, "Mode characteristics of twisted resonators composed of two cylindrical mirrors.," Opt. Express **2**, 184-190 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-5-184

Sort: Journal | Reset

### References

- F. Krausz, J. Zehetner, T. Brabec and E. Winter, "Elliptic-mode cavity for diode-pumped lasers," Opt. Lett. 16, 1496-1498 (1991).<br> [CrossRef] [PubMed]
- J. Zehetner, "Highly efficient diode-pumped elliptical mode Nd:YLF laser," Opt. Commun. 117, 273-267 (1995).<br> [CrossRef]
- D. Kopf, U. Keller, M.A. Emanuel, R.J. Beach and J.A. Skidmore, "1.1-W cw Cr:LiSAF laser pumped by a 1-cm diode array," Opt. Lett. 22, 99-101 (1997).<br> [CrossRef] [PubMed]
- J.L. Blows, J.M. Dawes and J.A. Piper, "Highly astigmatic diode pumped laser cavities for intracavity nonlinear optical conversion," in International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), MJ3.<br>
- J. Blows, J. Dawes, J. Piper and G. Forbes, "A highly astigmatic diode end-pumped solid-state laser," in Trends in Optics and Photonics, Vol. 10, Advanced Solid State Lasers, Clifford R. Pollock and Walter R. Bosenburg, eds. (Optical Society of America, Washington, DC, 1997), p. 376-379.<br>
- A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) 15.<br>
- A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) 19.1<br>
- J.A. Arnaud and H. Kogelnik, "Gaussian Light beams with General Astigmatism," Appl. Opt. 8, 1687 (1969).<br> [CrossRef] [PubMed]
- A.W. Greynolds, Propagation of generally astigmatic Gaussian beams along skew ray paths, Proc. SPIE 679, 129 (1986)<br> [CrossRef]
- A.W. Greynolds, Vector formulation of the ray-equivalent method for general Gaussian beam propagation, Proc. SPIE 560, 33 (1985)<br>
- J.A. Arnaud, "Nonorthogonal Optical Waveguides and Resonators," Bell Syst. Tech. J. 49, 2311 (1970).<br>
- J.A. Arnaud, "Hamiltonian theory of beam mode propagation," Prog. Opt. XI, 249 (1973).<br>
- J. Serna and G. Nemes, "Decoupling of coherent Gaussian beams with general astigmatism," Opt. Lett. 18, 1774 (1993).<br> [CrossRef] [PubMed]
- I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov and G.D. Laptev, "Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.," Quantum Electron. 25, 436 (1995).<br> [CrossRef]
- B. L, S. Xu, G. Feng and B. Cai, "Astigmatic resonator analysis using an eigenray vector concept," Optik 99, 158 (1992).<br>
- B. L, S. Xu, Y. Hu and B. Cai, "Matrix representation of three-dimensional astigmatic resonators," Opt. Quantum. Electron. 24, 619 (1992).<br> [CrossRef]
- K. Kubodera and K. Otsuka, "Single-transverse-mode LiNdP 4 O 12 slab waveguide laser," J. Appl. Phys. 50, 653 (1979). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.