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 namely0(1z/R1)(1z/R2)1.

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References

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  1. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express17(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. Express17(14), 12057–12069 (2009).
    [CrossRef] [PubMed]
  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.28(5), 976–983 (1989).
    [CrossRef] [PubMed]
  8. E. Acosta and S. Bará, “Variable aberration generators using rotated Zernike plates,” J. Opt. Soc. Am. A22(9), 1993–1996 (2005).
    [CrossRef] [PubMed]
  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. A27(10), 2344–2346 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. A. E. Siegman, Lasers (University Science Books, 1986).
  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]
  14. A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J.40, 453–488 (1961).
  15. O. Svelto, Principles of Lasers, 3rd edition (Plenum Press, 1989), pp. 189–190.

2012

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

2011

2010

2009

2006

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]

2005

1989

1961

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J.40, 453–488 (1961).

Acosta, E.

Akturk, S.

Bará, S.

Bonnefois, A. M.

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]

Buralli, D. A.

Burger, L.

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

Burns, D.

Forbes, A.

Fox, A. G.

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J.40, 453–488 (1961).

Gilbert, M.

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]

Griffith, M.

Laycock, L.

Li, T.

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J.40, 453–488 (1961).

Litvin, I. A.

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

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 and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express17(18), 15891–15903 (2009).
[CrossRef] [PubMed]

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

Lubeigt, W.

Morris, G. M.

Ngcobo, S.

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

Rogers, J. R.

Sasián, J.

Soylu, B.

Thro, P. Y.

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]

Weulersse, J. M.

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]

Yalizay, B.

Appl. Opt.

Bell Syst. Tech. J.

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J.40, 453–488 (1961).

J. Mod. Opt.

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

J. Opt. Soc. Am. A

Nat. Photonics

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

Opt. Commun.

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]

Opt. Express

Opt. Lett.

Other

O. Svelto, Principles of Lasers, 3rd edition (Plenum Press, 1989), pp. 189–190.

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).

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (3)

Fig. 1
Fig. 1

(a) A schematic representation of the laser cavity with non-conventional mirrors and (b) the stationary phase approximation.

Fig. 2
Fig. 2

(a) The dependence of the cavity stability on a spherical aberration value (β) for a constant focal length intra–cavity lens f = 1m (solid), f = 0.3m (dashed) and constant cavity length z = 0.4 m.(b, c, d) The Monte-Carlo method of propagation of rays in the particular laser cavity namely (b) the dependence of initial coordinate of rays and the number of rays which did not leave the particular laser cavity and (d) the radial coordinate dependence on the number of intra–cavity passes for the randomly generated rays in the stability region (red points) (see Rmax point of Fig. 2(a)) and outside the stability region (blue points) for similar parameters of laser cavity. Following parameters of the laser cavity were used namely the intra–cavity lens with a focal length of f = 1m and a spherical aberration parameter β = 7.2 105 on one side of the cavity and the plane mirror on another side of the cavity. For the generation Fig. 2(b) and 2(d) 400 rays were generated at every radial position with step of 0.01 mm and with a random initial angle (see Fig. 2(c)), 200 passes were completed to generate Fig. 2(b).

Fig. 3
Fig. 3

(a) The behavior of the laser cavity stability with a lens–axicon doublet and the corresponding intensity of the TEM00 mode for the following fixed parameters of the laser cavity, f = −0.9 m, z = 0.4 m, n = 1.5. Where the dashed lines present the stability graph and the solid lines present the corresponding intensity distribution of the TEM00 mode. (b) A schematic representation of the laser cavity with a lens-axicon doublet where M, are the laser mirrors (output couplers), A, is an axicon and L, is a lens.

Tables (2)

Tables Icon

Table 1 The dependence of the eigenvalues of the first three radial modes on the spherical aberration coefficient for a constant focal length of the intra–cavity lens f = 1m (solid curves of Fig. 2(a)) with a cavity length z = 0.4 m. The eigenvalues were calculated by Fox–Li method [1, 14].

Tables Icon

Table 2 The behavior of the eigenvalue of the fundamental TEM00 mode of the laser cavity with a lens–axicon doublet that depends on the axicon cone angle.

Equations (3)

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0 1 4 ( 2 + C 1 ( r ) z ) ( 2 + C 2 ( r ) z ) 1,
C 1 ( r )=0, C 2 ( r )=4β r 2 1 f ,
C 1 ( r )=0, C 2 ( r )=(n1) γ r 1 f ,

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