## Stability of a laser cavity with non-parabolic phase transformation elements |

Optics Express, Vol. 21, Issue 9, pp. 10706-10711 (2013)

http://dx.doi.org/10.1364/OE.21.010706

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### Abstract

In this paper we present a general approach to determine the stability of a laser cavity which can include non-conventional phase transformation elements. We consider two pertinent examples of the detailed investigation of the stability of a laser cavity firstly with a lens with spherical aberration and thereafter a lens axicon doublet to illustrate the implementation of the given approach. In the particular case of the intra–cavity elements having parabolic surfaces, the approach comes to the well–known stability condition for conventional laser resonators namely

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## 1. Introduction

1. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express **17**(18), 15891–15903 (2009). [CrossRef] [PubMed]

2. I. A. Litvin and A. Forbes, “Gaussian mode selection with intra-cavity diffractive optics,” Opt. Lett. **34**(19), 2991–2993 (2009). [CrossRef] [PubMed]

3. I. A. Litvin, “Implementation of intra-cavity beam shaping technique to enhance pump efficiency,” J. Mod. Opt. **59**(3), 241–244 (2012). [CrossRef]

4. W. Lubeigt, M. Griffith, L. Laycock, and D. Burns, “Reduction of the time-to-full-brightness in solid-state lasers using intra-cavity adaptive optics,” Opt. Express **17**(14), 12057–12069 (2009). [CrossRef] [PubMed]

7. D. A. Buralli, G. M. Morris, and J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. **28**(5), 976–983 (1989). [CrossRef] [PubMed]

8. E. Acosta and S. Bará, “Variable aberration generators using rotated Zernike plates,” J. Opt. Soc. Am. A **22**(9), 1993–1996 (2005). [CrossRef] [PubMed]

4. W. Lubeigt, M. Griffith, L. Laycock, and D. Burns, “Reduction of the time-to-full-brightness in solid-state lasers using intra-cavity adaptive optics,” Opt. Express **17**(14), 12057–12069 (2009). [CrossRef] [PubMed]

10. B. Yalizay, B. Soylu, and S. Akturk, “Optical element for generation of accelerating Airy beams,” J. Opt. Soc. Am. A **27**(10), 2344–2346 (2010). [CrossRef] [PubMed]

11. E. Acosta and J. Sasián, “Phase plates for generation of variable amounts of primary spherical aberration,” Opt. Express **19**(14), 13171–13178 (2011). [CrossRef] [PubMed]

13. A. M. Bonnefois, M. Gilbert, P. Y. Thro, and J. M. Weulersse, “Thermal lensing and spherical aberration in high–power transversally pumped laser rods,” Opt. Commun. **259**(1), 223–235 (2006). [CrossRef]

1. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express **17**(18), 15891–15903 (2009). [CrossRef] [PubMed]

## 2. Stability of a laser cavity concept

*A*and

*D*we implement a well-known condition for a stationary phase approximation and apply this to the Fresnel diffraction integral. This leads to a relationship of the radial coordinate of a ray at some initial plane (r

_{0}) and its radial coordinate upon intersecting some interest plane (

*r*

_{i}) (see Fig. 1(b)) separated by a distance z namely

*k*

_{0}

*f*(

*r*

_{0}) is the phase distribution of an electromagnetic field at the initial plane and

*k*

_{0}= 2π/

*λ*.

_{0 }– the radial coordinate of the ray at the plane of the phase transformation element. Now we can find the ABCD matrix of propagation in the cavity and extract the required coefficients A and D. We arrive at the following formula for the stability of a laser cavity with intra–cavity elements of the arbitrary shapes:where

*f*

_{1}(

*r*) and

*f*

_{2}(

*r*) present the phase transformation of the phase of the incident beam on the first and second mirrors respectively (see Fig. 1(a)).

*C*

_{1}(

*r*) and

*C*

_{2}(

*r*) are now the new coefficients for the stability of the laser cavity (

### 2.1. Spherical aberration

*f*with an induced spherical aberration and plane second mirror. The spherical aberration can be a result of the mirror itself or a thermal effect of the laser crystal positioned close to the second mirror [13

13. A. M. Bonnefois, M. Gilbert, P. Y. Thro, and J. M. Weulersse, “Thermal lensing and spherical aberration in high–power transversally pumped laser rods,” Opt. Commun. **259**(1), 223–235 (2006). [CrossRef]

*f*(defocus) and β (spherical) (see Fig. 2 (a)).

_{max}of Fig. 2(a)) close to the central region of the cavity. As a result we identify the dramatically decreasing eigenvalues (the round trip losses) of the higher order modes as presented in Table 1. The given behavior of the dependence of the cavity stability on the radial coordinate can be implemented as one of the techniques of higher order mode discrimination.

_{max}of Fig. 2(a))) have the stable oscillations in the cavity (independent of the initial angle) and conversely, the rays which are initially positioned outside the region of R

_{max}are unstable and will disperse out of this particular laser cavity.

### 2.2. Intra–cavity axicon

_{00}) depends strongly on the first and second order aberrations which include defocus (lens) and an axicon (the general view of a phase transformation equation of the axicon in cylindrical coordinates is –

*k*

_{0}(

*n*–1)γ

*r*where γ is the axicon base angle). This is primarily due to the value of the central part of the higher order aberrations which are negligible in this area (see Fig. 3(a)). Consequently the central part of an unstable cavity in terms of the conventional stability condition

## 3. Conclusion

## References and links

1. | I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express |

2. | I. A. Litvin and A. Forbes, “Gaussian mode selection with intra-cavity diffractive optics,” Opt. Lett. |

3. | I. A. Litvin, “Implementation of intra-cavity beam shaping technique to enhance pump efficiency,” J. Mod. Opt. |

4. | W. Lubeigt, M. Griffith, L. Laycock, and D. Burns, “Reduction of the time-to-full-brightness in solid-state lasers using intra-cavity adaptive optics,” Opt. Express |

5. | H. Harry, “Aspheric optical elements.” US Philips Sep, 14 1976: US patent 3980399 (1976) |

6. | G. J. Swanson and W. B. Veldkamp, “High-efficiency, multilevel, diffractive optical elements,” US patent 4895790 (1990). |

7. | D. A. Buralli, G. M. Morris, and J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. |

8. | E. Acosta and S. Bará, “Variable aberration generators using rotated Zernike plates,” J. Opt. Soc. Am. A |

9. | S. Ngcobo, I. A. Litvin, L. Burger, and A. Forbes, “The digital laser,” Nat. Photonics (submitted to). |

10. | B. Yalizay, B. Soylu, and S. Akturk, “Optical element for generation of accelerating Airy beams,” J. Opt. Soc. Am. A |

11. | E. Acosta and J. Sasián, “Phase plates for generation of variable amounts of primary spherical aberration,” Opt. Express |

12. | A. E. Siegman, |

13. | A. M. Bonnefois, M. Gilbert, P. Y. Thro, and J. M. Weulersse, “Thermal lensing and spherical aberration in high–power transversally pumped laser rods,” Opt. Commun. |

14. | A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. |

15. | O. Svelto, |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(140.3410) Lasers and laser optics : Laser resonators

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 11, 2013

Revised Manuscript: March 30, 2013

Manuscript Accepted: March 31, 2013

Published: April 24, 2013

**Citation**

Igor A. Litvin, "Stability of a laser cavity with non-parabolic phase transformation elements," Opt. Express **21**, 10706-10711 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10706

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### References

- I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express17(18), 15891–15903 (2009). [CrossRef] [PubMed]
- I. A. Litvin and A. Forbes, “Gaussian mode selection with intra-cavity diffractive optics,” Opt. Lett.34(19), 2991–2993 (2009). [CrossRef] [PubMed]
- I. A. Litvin, “Implementation of intra-cavity beam shaping technique to enhance pump efficiency,” J. Mod. Opt.59(3), 241–244 (2012). [CrossRef]
- W. Lubeigt, M. Griffith, L. Laycock, and D. Burns, “Reduction of the time-to-full-brightness in solid-state lasers using intra-cavity adaptive optics,” Opt. Express17(14), 12057–12069 (2009). [CrossRef] [PubMed]
- H. Harry, “Aspheric optical elements.” US Philips Sep, 14 1976: US patent 3980399 (1976)
- G. J. Swanson and W. B. Veldkamp, “High-efficiency, multilevel, diffractive optical elements,” US patent 4895790 (1990).
- D. A. Buralli, G. M. Morris, and J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt.28(5), 976–983 (1989). [CrossRef] [PubMed]
- E. Acosta and S. Bará, “Variable aberration generators using rotated Zernike plates,” J. Opt. Soc. Am. A22(9), 1993–1996 (2005). [CrossRef] [PubMed]
- S. Ngcobo, I. A. Litvin, L. Burger, and A. Forbes, “The digital laser,” Nat. Photonics (submitted to).
- B. Yalizay, B. Soylu, and S. Akturk, “Optical element for generation of accelerating Airy beams,” J. Opt. Soc. Am. A27(10), 2344–2346 (2010). [CrossRef] [PubMed]
- E. Acosta and J. Sasián, “Phase plates for generation of variable amounts of primary spherical aberration,” Opt. Express19(14), 13171–13178 (2011). [CrossRef] [PubMed]
- A. E. Siegman, Lasers (University Science Books, 1986).
- A. M. Bonnefois, M. Gilbert, P. Y. Thro, and J. M. Weulersse, “Thermal lensing and spherical aberration in high–power transversally pumped laser rods,” Opt. Commun.259(1), 223–235 (2006). [CrossRef]
- A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J.40, 453–488 (1961).
- O. Svelto, Principles of Lasers, 3rd edition (Plenum Press, 1989), pp. 189–190.

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