## Kaleidoscope modes in large aperture Porro prism resonators

Optics Express, Vol. 16, Issue 17, pp. 12707-12714 (2008)

http://dx.doi.org/10.1364/OE.16.012707

Acrobat PDF (456 KB)

### Abstract

We apply a new method of modeling Porro prism resonators, using the concept of rotating loss screens, to study stable and unstable Porro prism resonator. We show that the previously observed petal–like modal output is in fact only the lowest order mode, and reveal that a variety of kaleidoscope beam modes will be produced by these resonators when the intra–cavity apertures are sufficiently large to allow higher order modes to oscillate. We also show that only stable resonators will produce these modes.

© 2008 Optical Society of America

## 1. Introduction

2. M. Henriksson, L. Sjöqvista, and T. Uhrwing, “Numerical simulation of a battlefield Nd:YAG laser,” Proc. SPIE **5989**, 59890I (2005). [CrossRef]

5. A. Rapaport and L. Weichman, “Laser Resonator Design Using Optical Ray Tracing Software: Comparisons with Simple Analytical Models and Experimental Results,” IEEE J. Quantum Electron. **37**1401–1408 (2001). [CrossRef]

6. I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express **15**, 14065–14077 (2007). [CrossRef] [PubMed]

6. I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express **15**, 14065–14077 (2007). [CrossRef] [PubMed]

10. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. **53**, 537–624 (2003). [CrossRef]

11. M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Different field distributions obtained with an axicon and an amplitude mask,” Opt. Commun. **281**, 401–407 (2008). [CrossRef]

10. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. **53**, 537–624 (2003). [CrossRef]

## 2. Porro prism resonator model

6. I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express **15**, 14065–14077 (2007). [CrossRef] [PubMed]

*α*, the oscillating field is sub–divided into a finite number (

*N*) of petals, given by:

*i*,

*j*, and

*m*are integers, with certain constraints placed on

*j*[6

**15**, 14065–14077 (2007). [CrossRef] [PubMed]

*L*=10 cm, and lasing at

*λ*=1064 nm. Two Porro prisms at either end of the laser formed the resonator, replacing traditional mirrors. The stability of the resonators was determined by the two identical intra–cavity lenses (of focal length

*f*) each placed adjacent to a prism. The resonator was confined in the transverse direction by circular apertures placed immediately in front of each prism, and with radius

*a*.

## 3. Generalized modal patterns

*f*, and the aperture radius

*a*in order to investigate the impact of resonator stability and effective Fresnel number on the oscillating modes. Because of the symmetry of the lens-aperture configuration, any chosen stable resonator can be described in terms of just two parameters:

*G*is the equivalent G–parameter of the resonator, and

*N*is the effective Fresnel number;

_{F}*λ*denotes the wavelength of the laser light in vacuum and

*L*is the total optical path length inside the resonator.

*G*=0.75 and with Porro angles (

*α*) of 60°, 45° and 30° respectively. When the intracavity aperture is very small (

*N*~1.5), no mode is able to resonate. At intermediate aperture sizes (

_{F}*N*~3.5) the conventional petal–like modes are observed, with 6, 8 and 12 petals for

_{F}*α*=60°, 45° and 30° respectively. At large aperture sizes (

*N*>6) the petal–like modes give way to more complex mode patterns. This increase in mode complexity as the aperture size increases suggests that the petal–like modes are in fact the lowest order modes of Porro prism resonators, while previously unreported higher order modes also exist, and can be made to resonate if given sufficiently large transverse freedom. These results are shown in Fig. 3.

_{F}*G*also influences the oscillating mode, as one might expect.

*α*=45°, similar results are found at other Porro angles.

## 4. Mode periodicity

*α*but rather of

*G*, and is the result of the resonator’s complex eigenvalues. One can understand this periodicity if one considers the similarities to the well–known Herriot cell resonator [7] and by following the path of a ray through the resonator. Such resonators result in a periodicity that is not a double pass through the resonator, as is the case in a standard Fabry–Perot system, but rather is based on a uni–directional analysis, where the number of passes can be made very large for a complete “round trip” – in this case “round trip” refers to the condition that the beam repeats a previous path through the resonator. The number of reflections and the orientation of the beam, and hence the periodicity of the resonator, can be controlled by judicious choice of the resonator parameters. This concept is illustrated in Fig. 5 where a standard resonator is operated as a non–planar ring laser [8

8. W. Liu, Y. Huo, X. Yin, and D. Zhao, “Modes of Multi-End-Pumped Nonplanar Ring Laser,” IEEE Photonics Technol. Lett. **17**, 1776–1778 (2005). [CrossRef]

*M*is given by:

*ν*

_{0}describing both the position and angular deviation of the ray. After

*p*round trips through the resonator,

*ν*

_{0}will be transformed into a new vector

*ν*according to:

_{p}*λ*and

_{i}*ν⃗*are the eigenvalues and eigenvectors of the matrix

_{i}*M*respectively and

*α*are the coefficients required for the expansion of

_{i}*ν*

_{0}in terms of the eigenvectors. For repeatability of the mode we require

*ν*=

_{p}*ν*

_{0}, found from the solution to the simultaneous equations (for each

*i*)

*λ*=1.

^{p}_{i}**15**, 14065–14077 (2007). [CrossRef] [PubMed]

8. W. Liu, Y. Huo, X. Yin, and D. Zhao, “Modes of Multi-End-Pumped Nonplanar Ring Laser,” IEEE Photonics Technol. Lett. **17**, 1776–1778 (2005). [CrossRef]

9. C. Bollig, W. A. Clarkson, D. C. Hanna, D. S. Lovering, and G. C. W. Jones, “Single-frequency operation of a monolithic Nd:glass ring laser via the acousto-optics effect,” Opt. Commun. **133**, 221–224 (1997). [CrossRef]

## 5. Kaleidoscope modes

10. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. **53**, 537–624 (2003). [CrossRef]

11. M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Different field distributions obtained with an axicon and an amplitude mask,” Opt. Commun. **281**, 401–407 (2008). [CrossRef]

**53**, 537–624 (2003). [CrossRef]

*ψ*=

*π*/

*n*. The similarity between this and a rotating loss on the field at angles

*α*=

*π/n*(Eq. (1) with

*i*=1) in Porro prism resonators probably accounts for the likeness in output modes. In [11

11. M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Different field distributions obtained with an axicon and an amplitude mask,” Opt. Commun. **281**, 401–407 (2008). [CrossRef]

**15**, 14065–14077 (2007). [CrossRef] [PubMed]

**53**, 537–624 (2003). [CrossRef]

**281**, 401–407 (2008). [CrossRef]

## 6. Conclusion

*N*is sufficiently large, and that these higher order modes closely resemble recently reported kaleidoscope modes due to the fundamental property of field sub–division in Porro prism resonators. The appearance of first the petal mode and then increasingly complex kaleidoscope modes with increasing aperture size leads to the conclusion that the petal–like modes are the lowest order modes of Porro prism resonators, while higher order modes exist in the form of kaleidoscope–like fields. We also predict that the standard petal mode is only observable from stable Porro prism resonators, and indicate how the stability criteria (

_{F}*G*parameter) impacts on the cyclical nature of the higher order modes. We believe it is possible to observe these modes experimentally, but acknowledge that there are some technical challenges to overcome before doing so.

## Acknowledgment

## References and Links

1. | B. A. See, K. Fuelop, and R. Seymour, “An Assessment of the Crossed Porro Prism Resonator,” Technical Memorandum ERL-0162-TM, Electronics Research Lab Adelaide (Australia) (1980). |

2. | M. Henriksson, L. Sjöqvista, and T. Uhrwing, “Numerical simulation of a battlefield Nd:YAG laser,” Proc. SPIE |

3. | M. Henriksson and L. Sjöqvist, “Numerical simulation of a flashlamp pumped Nd:YAG laser,” Report # ISRN FOI-R-1710-SE, Swedish Defence Research Agency, Linkoeping, Sensor Technology (2007). |

4. | M. Ishizu, “Laser Oscillator,” US Patent 6816533 (2004). |

5. | A. Rapaport and L. Weichman, “Laser Resonator Design Using Optical Ray Tracing Software: Comparisons with Simple Analytical Models and Experimental Results,” IEEE J. Quantum Electron. |

6. | I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express |

7. | N. Hodgson and H. Weber, |

8. | W. Liu, Y. Huo, X. Yin, and D. Zhao, “Modes of Multi-End-Pumped Nonplanar Ring Laser,” IEEE Photonics Technol. Lett. |

9. | C. Bollig, W. A. Clarkson, D. C. Hanna, D. S. Lovering, and G. C. W. Jones, “Single-frequency operation of a monolithic Nd:glass ring laser via the acousto-optics effect,” Opt. Commun. |

10. | Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. |

11. | M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Different field distributions obtained with an axicon and an amplitude mask,” Opt. Commun. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.3410) Lasers and laser optics : Laser resonators

(140.4780) Lasers and laser optics : Optical resonators

(230.5480) Optical devices : Prisms

(260.0260) Physical optics : Physical optics

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: April 28, 2008

Revised Manuscript: June 28, 2008

Manuscript Accepted: August 1, 2008

Published: August 7, 2008

**Citation**

Liesl Burger and Andrew Forbes, "Kaleidoscope modes in large aperture Porro prism resonators," Opt. Express **16**, 12707-12714 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12707

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### References

- B. A. See, K. Fuelop, and R. Seymour, "An Assessment of the Crossed Porro Prism Resonator," Technical Memorandum ERL-0162-TM, Electronics Research Lab Adelaide (Australia) (1980).
- M. Henriksson, L. Sjöqvista, and T. Uhrwing, "Numerical simulation of a battlefield Nd:YAG laser," Proc. SPIE 5989, 59890I (2005). [CrossRef]
- M. Henriksson and L. Sjöqvist, "Numerical simulation of a flashlamp pumped Nd:YAG laser," Report # ISRN FOI-R-1710-SE, Swedish Defence Research Agency, Linkoeping, Sensor Technology (2007).
- M. Ishizu, "Laser Oscillator," US Patent 6816533 (2004).
- A. Rapaport and L. Weichman, "Laser Resonator Design Using Optical Ray Tracing Software: Comparisons with Simple Analytical Models and Experimental Results," IEEE J. Quantum Electron. 37, 1401-1408 (2001). [CrossRef]
- I. A. Litvin, L. Burger, and A. Forbes, "Petal-like modes in Porro prism resonators," Opt. Express 15, 14065-14077 (2007). [CrossRef] [PubMed]
- N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005), Chap. 20.
- W. Liu, Y. Huo, X. Yin, and D. Zhao, "Modes of Multi-End-Pumped Nonplanar Ring Laser," IEEE Photon. Technol. Lett. 17, 1776-1778 (2005). [CrossRef]
- C. Bollig, W. A. Clarkson, D. C. Hanna, D. S. Lovering, and G. C. W. Jones, "Single-frequency operation of a monolithic Nd:glass ring laser via the acousto-optics effect," Opt. Commun. 133, 221-224 (1997). [CrossRef]
- Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-624 (2003). [CrossRef]
- M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, "Different field distributions obtained with an axicon and an amplitude mask," Opt. Commun. 281, 401-407 (2008). [CrossRef]

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