Abstract

A theoretical framework is presented for calculating three-dimensional resonator modes of both stable and unstable laser resonators. The resonant modes of an optical resonator are computed using a kernel formulation of the resonator round-trip Huygens–Fresnel diffraction integral. To substantiate the validity of this method, both stable and unstable resonator mode results are presented. The predicted lowest loss and higher order modes of a semi-confocal stable resonator are in agreement with the analytic formulation. Higher order modes are determined for an asymmetrically aberrated confocal unstable resonator, whose lowest loss unaberrated mode is consistent with published results. The three-dimensional kernel method provides a means to evaluate multi-mode configurations with two-dimensional aberrations that cannot be decomposed into one-dimensional representations.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53(3), 277–287 (1965).
    [CrossRef]
  2. A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. 3(4), 156–163 (1967).
    [CrossRef]
  3. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13(2), 353–367 (1974).
    [CrossRef] [PubMed]
  4. A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the prony method,” Appl. Opt. 9(12), 2729–2736 (1970).
    [CrossRef] [PubMed]
  5. R. L. Sanderson and W. Streifer, “Unstable laser resonator modes,” Appl. Opt. 8(10), 2129–2136 (1969).
    [CrossRef] [PubMed]
  6. P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63(12), 1528–1543 (1973).
    [CrossRef]
  7. L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. 9(11), 1102–1113 (1973).
    [CrossRef]
  8. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
  9. P. J. Davis and I. Polonsky, “Numerical Interpolation, Differentiation, and Integration,” in Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, M. Abramowitz and I. A. Stegun, eds. (Dover, 1972) pp. 875–924.
  10. V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. 23(9), 1105–1134 (1991).
    [CrossRef]
  11. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
    [CrossRef] [PubMed]
  12. W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. 5(12), 575–586 (1969).
    [CrossRef]
  13. D. B. Rensch and A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. 12(5), 997–1010 (1973).
    [CrossRef] [PubMed]

1991

V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. 23(9), 1105–1134 (1991).
[CrossRef]

1974

1973

1970

1969

R. L. Sanderson and W. Streifer, “Unstable laser resonator modes,” Appl. Opt. 8(10), 2129–2136 (1969).
[CrossRef] [PubMed]

W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. 5(12), 575–586 (1969).
[CrossRef]

1967

A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. 3(4), 156–163 (1967).
[CrossRef]

1966

1965

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53(3), 277–287 (1965).
[CrossRef]

1961

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

Arrathoon, R. W.

A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. 3(4), 156–163 (1967).
[CrossRef]

Chen, L. W.

L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. 9(11), 1102–1113 (1973).
[CrossRef]

Chester, A. N.

De Silvestri, S.

V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. 23(9), 1105–1134 (1991).
[CrossRef]

Felsen, L. B.

L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. 9(11), 1102–1113 (1973).
[CrossRef]

Fox, A. G.

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

Horwitz, P.

Kogelnik, H.

Krupke, W. F.

W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. 5(12), 575–586 (1969).
[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, 453–488 (1961).

Magni, V.

V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. 23(9), 1105–1134 (1991).
[CrossRef]

Miller, H. Y.

Rensch, D. B.

Sanderson, R. L.

Siegman, A. E.

A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13(2), 353–367 (1974).
[CrossRef] [PubMed]

A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the prony method,” Appl. Opt. 9(12), 2729–2736 (1970).
[CrossRef] [PubMed]

A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. 3(4), 156–163 (1967).
[CrossRef]

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53(3), 277–287 (1965).
[CrossRef]

Sooy, W. R.

W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. 5(12), 575–586 (1969).
[CrossRef]

Streifer, W.

Valentini, G.

V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. 23(9), 1105–1134 (1991).
[CrossRef]

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).

IEEE J. Quantum Electron.

A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. 3(4), 156–163 (1967).
[CrossRef]

W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. 5(12), 575–586 (1969).
[CrossRef]

L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. 9(11), 1102–1113 (1973).
[CrossRef]

J. Opt. Soc. Am.

Opt. Quantum Electron.

V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. 23(9), 1105–1134 (1991).
[CrossRef]

Proc. IEEE

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53(3), 277–287 (1965).
[CrossRef]

Other

P. J. Davis and I. Polonsky, “Numerical Interpolation, Differentiation, and Integration,” in Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, M. Abramowitz and I. A. Stegun, eds. (Dover, 1972) pp. 875–924.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Semi-confocal stable resonator modes determined using the three-dimensional Huygens–Fresnel kernel formulation. The nine images detail the mode progression (left to right, top to bottom, or inferred from null number) sustained by the working aperture. The axial intensity plots are presented adjacent to the modes they represent. The intensity is normalized to unity and the length scale is normalized to the fundamental mode beam waist.

Fig. 2
Fig. 2

Phase aberrations investigated: (a) tilt in xy, (b) cylindrical focus in xy, (c) astigmatism, and (d) coma. The magnitude is normalized from −1 to 1.

Fig. 3
Fig. 3

Confocal unstable resonator lowest loss mode determined using the Fox–Li iterative method with aberrations: (a) unaberrated, (b) 1/100 wave tilt in xy, (c) 1/20 wave cylindrical focus in xy, (d) 1/20 wave astigmatism, and (e) 1/20 wave of coma.

Fig. 4
Fig. 4

Effect of aberrations on the transverse mode structure of a confocal unstable resonator determined by the Huygens–Fresnel kernel formulation: column (i) shows the lowest loss mode followed by the next lowest loss modes (ii)-(v). Each row details the effect of a different resonator aberration as follows: (a) unaberrated, (b) 1/100 wave tilt in xy, (c) 1/20 wave cylindrical focus in xy, (d) 1/20 wave astigmatism, and (e) 1/20 wave of coma.

Fig. 5
Fig. 5

The transverse modes of a high Fresnel number rectangular aperture confocal unstable resonator. Lowest loss mode determined using the (a) Fox–Li iterative method and (b) Huygen–Fresnel kernel method for comparison. The high order mode progression is shown (c) to (e).

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

γ U ( x 2 , y 2 ) = i λ B x B y ρ ( x 1 , y 1 ) U ( x 1 , y 1 ) exp [ i π λ ( A x x 1 2 + D x x 2 2 2 x 1 x 2 B x + A y y 1 2 + D y y 2 2 2 y 1 y 2 B y ) ] d x 1 d y 1 ,
M A B C D = ( 2 g 1 g 2 1 2 g 2 L 2 g 1 L 2 g 1 g 2 1 ) ,
γ U ( ξ 2 , η 2 ) = i N x N y exp [ i π A ( N x ξ 2 2 + N y η 2 2 ) ] U ( ξ 1 , η 1 ) exp [ i π A ( N x ξ 1 2 + N y η 1 2 ) ] exp [ 2 i π ( N x ξ 1 ξ 2 + N y η 1 η 2 ) ] d ξ 1 d η 1 ,
γ u , v Φ u , v = i = 1 N j = 1 M K i , u K j , v Φ i , j ,
γ α Φ α = β = 1 S K β , α Φ β ,

Metrics