Abstract

The original numerical wave model of the open resonator has been employed in the investigation of conditions for multipass mode generation. It is shown that for Fresnel numbers larger than unity, multiple reflections of radiation from the stable resonator mirrors lead to sustained quasi-stationary oscillations which are indicative of multipass mode generation. Various types of ray trajectories have been considered at the paraxial resonance conditions. Trajectory selecting techniques are suggested to provide the high quality output beams at large Fresnel numbers. The results of numerical experiments on amplitude-phase distribution of output radiation are presented for the suggested schemes.

© 2013 Optical Society of America

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References

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  1. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
    [Crossref]
  2. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
    [Crossref] [PubMed]
  3. A. E. Siegman, Lasers (University Science, 1986).
  4. N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer Verlag, Berlin, 1997).
  5. Y.A. Ananiev, Resonateurs Optiques et Probleme de Divergence du Rayonnement Laser (Mir, 1982).
  6. L. A. Weinstein, Open resonators and open waveguides (Golem Press, Boulder, 1969).
  7. S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic Press, 1986).
  8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970).
    [Crossref] [PubMed]
  9. C.-H. Chen, P.-T. Tai, W.-F. Hsieh, and M.-D. Wei, “Multibeam-waist modes in an end-pumped Nd-YVO4 laser,” J. Opt. Soc. Am. B 20(6), 1220–1226 (2003).
    [Crossref]
  10. H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid-state laser with different fractionally degenerate resonator configurations,” J. Opt. Soc. Am. B 18(1), 7–12 (2001).
    [Crossref]
  11. C. H. Chen and C. F. Chiu, “Generating a geometric mode for clarifying differences between an operator method and SU(2) wave representation,” Opt. Express 15(20), 12692–12698 (2007).
    [Crossref] [PubMed]
  12. C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12(1), 015708 (2010).
    [Crossref]
  13. J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188(5-6), 345–351 (2001).
    [Crossref]
  14. B. Sterman, A. Gabay, S. Yatsiv, and E. Dagan, “Off-axis folded laser beam trajectories in a strip-line CO2 laser,” Opt. Lett. 14(23), 1309–1311 (1989).
    [Crossref] [PubMed]
  15. D. Dick and F. Hanson, “M modes in a diode side-pumped Nd:glass slab laser,” Opt. Lett. 16(7), 476–477 (1991).
    [Crossref] [PubMed]
  16. V. G. Niziev and R. V. Grishaev, “Dynamics of Mode Formation in an Open Resonator,” Appl. Opt. 49(34), 6582–6590 (2010).
    [Crossref] [PubMed]
  17. A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 046608 (2005).
    [Crossref] [PubMed]
  18. V. I. Voronov, “Spatial characteristics of multipass modes in lasers with ring cross-section of active medium,” J. Tech. Phys. 65, 98–107 (1995).
  19. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000).
    [Crossref]
  20. A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32(22), 2871–2875 (1999).
    [Crossref]
  21. M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3 kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
    [Crossref] [PubMed]
  22. V. G. Niziev and D. Toebaert, “Formation of transverse mode in axially symmetric lasers,” Appl. Opt. 51(7), 954–962 (2012).
    [Crossref] [PubMed]

2012 (1)

2010 (2)

V. G. Niziev and R. V. Grishaev, “Dynamics of Mode Formation in an Open Resonator,” Appl. Opt. 49(34), 6582–6590 (2010).
[Crossref] [PubMed]

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12(1), 015708 (2010).
[Crossref]

2007 (2)

2005 (1)

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 046608 (2005).
[Crossref] [PubMed]

2003 (1)

2001 (2)

2000 (1)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000).
[Crossref]

1999 (1)

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32(22), 2871–2875 (1999).
[Crossref]

1995 (1)

V. I. Voronov, “Spatial characteristics of multipass modes in lasers with ring cross-section of active medium,” J. Tech. Phys. 65, 98–107 (1995).

1991 (1)

1989 (1)

1970 (1)

1966 (1)

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Ahmed, M. A.

Chen, C. H.

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12(1), 015708 (2010).
[Crossref]

C. H. Chen and C. F. Chiu, “Generating a geometric mode for clarifying differences between an operator method and SU(2) wave representation,” Opt. Express 15(20), 12692–12698 (2007).
[Crossref] [PubMed]

Chen, C.-H.

Chiu, C. F.

Dagan, E.

Degnan, J. J.

Dick, D.

Dingjan, J.

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188(5-6), 345–351 (2001).
[Crossref]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Gabay, A.

Graf, T.

Grishaev, R. V.

Hanson, F.

Hsieh, W. F.

Hsieh, W.-F.

Huang, P. Y.

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12(1), 015708 (2010).
[Crossref]

Kogelnik, H.

Kuo, C. W.

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12(1), 015708 (2010).
[Crossref]

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
[Crossref] [PubMed]

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 046608 (2005).
[Crossref] [PubMed]

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000).
[Crossref]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32(22), 2871–2875 (1999).
[Crossref]

Niziev, V. G.

V. G. Niziev and D. Toebaert, “Formation of transverse mode in axially symmetric lasers,” Appl. Opt. 51(7), 954–962 (2012).
[Crossref] [PubMed]

V. G. Niziev and R. V. Grishaev, “Dynamics of Mode Formation in an Open Resonator,” Appl. Opt. 49(34), 6582–6590 (2010).
[Crossref] [PubMed]

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 046608 (2005).
[Crossref] [PubMed]

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000).
[Crossref]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32(22), 2871–2875 (1999).
[Crossref]

Parriaux, O.

Pommier, J. C.

Ramsay, I. A.

Schulz, J.

Sterman, B.

Tai, P.-T.

Toebaert, D.

van Exter, M. P.

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188(5-6), 345–351 (2001).
[Crossref]

Voronov, V. I.

V. I. Voronov, “Spatial characteristics of multipass modes in lasers with ring cross-section of active medium,” J. Tech. Phys. 65, 98–107 (1995).

Voss, A.

Wei, M.-D.

Woerdman, J. P.

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188(5-6), 345–351 (2001).
[Crossref]

Wu, H. H.

Yakunin, V. P.

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32(22), 2871–2875 (1999).
[Crossref]

Yatsiv, S.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

J. Opt. (1)

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12(1), 015708 (2010).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. D Appl. Phys. (2)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000).
[Crossref]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32(22), 2871–2875 (1999).
[Crossref]

J. Tech. Phys. (1)

V. I. Voronov, “Spatial characteristics of multipass modes in lasers with ring cross-section of active medium,” J. Tech. Phys. 65, 98–107 (1995).

Opt. Commun. (1)

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188(5-6), 345–351 (2001).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 046608 (2005).
[Crossref] [PubMed]

Other (5)

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

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer Verlag, Berlin, 1997).

Y.A. Ananiev, Resonateurs Optiques et Probleme de Divergence du Rayonnement Laser (Mir, 1982).

L. A. Weinstein, Open resonators and open waveguides (Golem Press, Boulder, 1969).

S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic Press, 1986).

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Figures (8)

Fig. 1
Fig. 1 The illustration of stationary (or quasi-stationary) generation established after multiple bounces in the stable resonators at different stability parameters g1 = g2. |E| is electric field amplitude, the Fresnel number is 4.5.

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Fig. 2
Fig. 2 The typical ray trajectories at different N of paraxial resonance. The parameters of resonator stability are gi = 1-L/Ri.

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Fig. 3
Fig. 3 A six-pass closed ray trajectory with respect to the azimuthal angle.

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Fig. 4
Fig. 4 Main elements of the resonator construction for generation of a multipass mode and selection of ray trajectories. Reflecting diffraction mirror – 1. Reflecting mirror – 2. Non-reflecting ring zone – 3. Partially reflecting output zone – 4. The filled zone depicts the occupation of the resonator volume with three-pass radiation (g1 = g2 = 0.5) from the viewpoint of optical geometry. The non-reflecting ring zone introduces loss for ecliptic trajectories (blue lines).

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Fig. 5
Fig. 5 Radial distribution of the field amplitude (upper curve) and phase (lower curve) on mirror 2 (Fig. 4) in the steady-state mode of quasi-stationary generation. The calculation parameters: g1 = g2 = 0.5, N = 3, non-ecliptic, azimuthal polarization. Two identical non-reflecting zones are located on the both mirrors. The outer ring radius is r/rm = 0.5, while the annular width Δr/rm = 0.03. The output zone radius (filled in the Fig.) is r/rm = 0.33, with a reflectance of 50% and a Fresnel number F = 4.5. The phase distribution is shown on the selected spherical surface; its radius is R = 1.43L.

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Fig. 6
Fig. 6 Radial distribution of the field amplitude (upper curve) and phase (lower curve) on the flat mirror in the steady-state mode of quasi-stationary generation. The calculation parameters: g1 = 1, g2 = 0.5, N = 4, ecliptic, azimuthal polarization. The resonator mirrors have two non-reflecting zones. On the flat mirror it has the form of a ring with a mean radius r/rm = 0.3 and a width Δr/rm = 0.1. On the concave mirror, the non-reflecting zone at the mirror centre has the radius r/rm = 0.25. The output zone radius (filled in the Fig.) is r/rm = 0.25, and has a reflectance of 50%, and a Fresnel number, F = 4.5. The phase distribution is shown on the selected spherical surface; its curvature radius is R = 3.3L.

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Fig. 7
Fig. 7 Radial distribution of the field amplitude (upper curve) and phase (lower curve) on the right mirror in the steady-state mode of quasi-stationary generation. The calculation parameters: g1 = g2 = 0.383, N = 8, ecliptic, azimuthal polarization. The radius of the non-reflecting circle zone on the left mirror is r/rm = 0.153, the radius of the output zone on the right mirror (filled in the Fig.) is r/rm = 0.21, the output zone reflectance is 50%, and the Fresnel number is F = 10.1. The phase distribution is shown on the mirror surface.

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Fig. 8
Fig. 8 Radial distribution of the field amplitude on the mirror in the steady-state mode of quasi-stationary generation. The radius of the outlet hole (filled in the Fig.) is r/rm = 0.1. The Fresnel number F = 4.5. The polarization is plane. The calculation parameters are g1 = g2 = 0.5 (a) and g1 = g2 = 0.53 (b).

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Equations (1)

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g 1 ⋅ g 2 = 1+cosθ 2 ;   θ=2π K N  ;    0≤K≤N/2

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