Abstract

We present an analytic variational scheme which predicts the modes and losses for empty resonators with large Fresnel number <i>N</i>. The method is most accurate for stable cavities, with decreasing effectiveness as the cavity becomes unstable. We consider mainly the boundary area between the stable and unstable regimes described by neither Gaussian beams nor approximate asymptotic solutions. Numerical results in this region for circular mirrors are presented. The method may be extended to stable cavities.

© 1980 Optical Society of America

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  1. See references in the following reviews: H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); A. E. Siegman, "Unstable optical resonators," Appl. Opt. 13, 353 (1974).
  2. A. G. Fox and T. Li, "Resonant Modes in a Maser Interferometer," Bell Syst. Tech. J. 40, 453–488 (1961).
  3. R. L. Sanderson and W. Streifer, "Comparison of laser mode calculations," Appl. Opt. 8, 131–140 (1969).
  4. A. E. Siegman and H. Y. Miller, "Unstable optical resonator loss calculations using the Prony method," Appl. Opt. 9, 2729–2736 (1970).
  5. D. Rensch and A. Chester, "Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators," Appl. Opt. 12, 997–1010 (1972).
  6. W. Murphy and M. L. Bernabe, "Numerical procedures for solving nonsymmetric eigenvalue problems associated with optical resonators," Appl. Opt. 15, 2358–2365 (1978).
  7. M. Lax, G. Agrawal, and W. Louisell, "Continuous Fourier transform spline solution of unstable resonator field distribution," Opt. Lett. 4, 303–305 (1979).
  8. P. Horwitz, "Asymptotic theory of unstable resonator modes," J. Opt. Soc. Am. 63, 1528–1543 (1973).
  9. R. R. Butts and P. V. Avizonis, "Asymptotic analysis of unstable laser resonators with circular mirrors," J. Opt. Soc. Am. 68, 1072–1078 (1978).
  10. G. C. Dente, "Polarization effects in resonators," Appl. Opt. 18, 2911 (1979).
  11. W. H. Louisell, in Physics of Quantum Electronics, edited by S. F. Jacobs, M. O. Scully, M. Sargent, and C. D. Cantrell (Addison-Wesley, Reading, Massachusetts, 1976), Vol. 3, p. 369.
  12. P. Morse and H. Feshbach, Methods in Mathematical Physics (McGraw-Hill, New York, 1953).
  13. G. Boyd and H. Kogelnik, "Generalized confocal resonator theory," Bell Syst. Tech. J. 41, 1347–1369 (1962).
  14. L. Weinstein, Open Resonators and Waveguides (Golem, Boulder, 1969).
  15. A. E. Siegman, "Unstable Optical Resonators for Laser Applications," Proc. IEEE 53, 277–287 (1965).
  16. W. Kahn, "Unstable optical resonators," Appl. Opt. 4, 407–417 (1966).
  17. L. Bergstein, "Modes of stable and unstable optical resonators," Appl. Opt. 7, 495–504 (1968).
  18. L. Chen and L. Felsen, "Coupled Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. QE9, 1102–1113 (1973).
  19. C. Santana and L. Felsen, "Unstable open resonators: Two dimensional and three-dimensional losses by a waveguide analysis," Appl. Opt. 15, 1470–1478 (1976).
  20. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York 1964).
  21. G. Watson, Theory of Bessel Functions (Cambridge University, Cambridge, 1966).
  22. G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the Selective Properties of Open Unstable Cavities," Opt. Spectrosc. 34, 437–432 (1973).
  23. L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Claredon, Oxford, 1974).
  24. L. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).
  25. S. Morgan, "On the Integral Equations of Laser Theory," IEEE Trans. Microwave Theory Tech. 8, 191–193 (1963).
  26. D. Newman and S. Morgan, "Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory," Bell Syst. Tech. J. 43, 113–126 (1964).
  27. J. Cochran, "The Existence of Eigenvalues for the Integral Equations in Laser Theory," Bell Syst. Tech. J. 44, 77–88 (1965).
  28. H. Hochstadt, "Integral Equations in Laser Theory," SAIM Rev. 8, 62–65 (1966).
  29. J. Nagel, D. Rogovin, P. Avizonis, and R. Butts, "Asymptotic Approaches to Marginally Stable Resonators," Opt. Lett. 4, 300–302 (1979).
  30. The condition is M > 1 + N-1. and not M > 1 + N. as stated in Ref. 29.
  31. A. Siegman and R. Arrathoon, "Modes in Unstable Optical Resonators and Lens Waveguides," IEEE J. Quantum Electron. QE3, 156–163 (1967).
  32. R. Sanderson and W. Streifer, "Unstable laser resonator modes," Appl. Opt. 8, 2129–2136 (1969).
  33. R. Butts and A. Paxton (private communication).

1979 (3)

1978 (2)

1976 (1)

1973 (3)

P. Horwitz, "Asymptotic theory of unstable resonator modes," J. Opt. Soc. Am. 63, 1528–1543 (1973).

L. Chen and L. Felsen, "Coupled Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. QE9, 1102–1113 (1973).

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the Selective Properties of Open Unstable Cavities," Opt. Spectrosc. 34, 437–432 (1973).

1972 (1)

1970 (1)

1969 (2)

1968 (1)

1967 (1)

A. Siegman and R. Arrathoon, "Modes in Unstable Optical Resonators and Lens Waveguides," IEEE J. Quantum Electron. QE3, 156–163 (1967).

1966 (3)

H. Hochstadt, "Integral Equations in Laser Theory," SAIM Rev. 8, 62–65 (1966).

W. Kahn, "Unstable optical resonators," Appl. Opt. 4, 407–417 (1966).

See references in the following reviews: H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); A. E. Siegman, "Unstable optical resonators," Appl. Opt. 13, 353 (1974).

1965 (2)

A. E. Siegman, "Unstable Optical Resonators for Laser Applications," Proc. IEEE 53, 277–287 (1965).

J. Cochran, "The Existence of Eigenvalues for the Integral Equations in Laser Theory," Bell Syst. Tech. J. 44, 77–88 (1965).

1964 (1)

D. Newman and S. Morgan, "Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory," Bell Syst. Tech. J. 43, 113–126 (1964).

1963 (1)

S. Morgan, "On the Integral Equations of Laser Theory," IEEE Trans. Microwave Theory Tech. 8, 191–193 (1963).

1962 (1)

G. Boyd and H. Kogelnik, "Generalized confocal resonator theory," Bell Syst. Tech. J. 41, 1347–1369 (1962).

1961 (1)

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

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York 1964).

Agrawal, G.

Arrathoon, R.

A. Siegman and R. Arrathoon, "Modes in Unstable Optical Resonators and Lens Waveguides," IEEE J. Quantum Electron. QE3, 156–163 (1967).

Avizonis, P.

Avizonis, P. V.

Bergstein, L.

Bernabe, M. L.

Boyd, G.

G. Boyd and H. Kogelnik, "Generalized confocal resonator theory," Bell Syst. Tech. J. 41, 1347–1369 (1962).

Butts, R.

Butts, R. R.

Chen, L.

L. Chen and L. Felsen, "Coupled Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. QE9, 1102–1113 (1973).

Chester, A.

Cochran, J.

J. Cochran, "The Existence of Eigenvalues for the Integral Equations in Laser Theory," Bell Syst. Tech. J. 44, 77–88 (1965).

Delves, L. M.

L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Claredon, Oxford, 1974).

Dente, G. C.

Felsen, L.

C. Santana and L. Felsen, "Unstable open resonators: Two dimensional and three-dimensional losses by a waveguide analysis," Appl. Opt. 15, 1470–1478 (1976).

L. Chen and L. Felsen, "Coupled Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. QE9, 1102–1113 (1973).

Feshbach, H.

P. Morse and H. Feshbach, Methods in Mathematical Physics (McGraw-Hill, New York, 1953).

Fox, A. G.

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

Hochstadt, H.

H. Hochstadt, "Integral Equations in Laser Theory," SAIM Rev. 8, 62–65 (1966).

Horwitz, P.

Kahn, W.

W. Kahn, "Unstable optical resonators," Appl. Opt. 4, 407–417 (1966).

Kogelnik, H.

See references in the following reviews: H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); A. E. Siegman, "Unstable optical resonators," Appl. Opt. 13, 353 (1974).

G. Boyd and H. Kogelnik, "Generalized confocal resonator theory," Bell Syst. Tech. J. 41, 1347–1369 (1962).

Lax, M.

Li, T.

See references in the following reviews: H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); A. E. Siegman, "Unstable optical resonators," Appl. Opt. 13, 353 (1974).

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

Louisell, W.

Louisell, W. H.

W. H. Louisell, in Physics of Quantum Electronics, edited by S. F. Jacobs, M. O. Scully, M. Sargent, and C. D. Cantrell (Addison-Wesley, Reading, Massachusetts, 1976), Vol. 3, p. 369.

Lyubimov, V. V.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the Selective Properties of Open Unstable Cavities," Opt. Spectrosc. 34, 437–432 (1973).

Miller, H. Y.

Morgan, S.

D. Newman and S. Morgan, "Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory," Bell Syst. Tech. J. 43, 113–126 (1964).

S. Morgan, "On the Integral Equations of Laser Theory," IEEE Trans. Microwave Theory Tech. 8, 191–193 (1963).

Morse, P.

P. Morse and H. Feshbach, Methods in Mathematical Physics (McGraw-Hill, New York, 1953).

Murphy, W.

Nagel, J.

Newman, D.

D. Newman and S. Morgan, "Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory," Bell Syst. Tech. J. 43, 113–126 (1964).

Orlova, I. B.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the Selective Properties of Open Unstable Cavities," Opt. Spectrosc. 34, 437–432 (1973).

Paxton, A.

R. Butts and A. Paxton (private communication).

Rall, L.

L. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).

Rensch, D.

Rogovin, D.

Sanderson, R.

Sanderson, R. L.

Santana, C.

Siegman, A.

A. Siegman and R. Arrathoon, "Modes in Unstable Optical Resonators and Lens Waveguides," IEEE J. Quantum Electron. QE3, 156–163 (1967).

Siegman, A. E.

A. E. Siegman and H. Y. Miller, "Unstable optical resonator loss calculations using the Prony method," Appl. Opt. 9, 2729–2736 (1970).

A. E. Siegman, "Unstable Optical Resonators for Laser Applications," Proc. IEEE 53, 277–287 (1965).

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York 1964).

Streifer, W.

Vinokurov, G. N.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the Selective Properties of Open Unstable Cavities," Opt. Spectrosc. 34, 437–432 (1973).

Walsh, J.

L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Claredon, Oxford, 1974).

Watson, G.

G. Watson, Theory of Bessel Functions (Cambridge University, Cambridge, 1966).

Weinstein, L.

L. Weinstein, Open Resonators and Waveguides (Golem, Boulder, 1969).

Appl. Opt. (9)

Bell Syst. Tech. J. (4)

G. Boyd and H. Kogelnik, "Generalized confocal resonator theory," Bell Syst. Tech. J. 41, 1347–1369 (1962).

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

D. Newman and S. Morgan, "Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory," Bell Syst. Tech. J. 43, 113–126 (1964).

J. Cochran, "The Existence of Eigenvalues for the Integral Equations in Laser Theory," Bell Syst. Tech. J. 44, 77–88 (1965).

IEEE J. Quantum Electron. (2)

L. Chen and L. Felsen, "Coupled Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. QE9, 1102–1113 (1973).

A. Siegman and R. Arrathoon, "Modes in Unstable Optical Resonators and Lens Waveguides," IEEE J. Quantum Electron. QE3, 156–163 (1967).

IEEE Trans. Microwave Theory Tech. (1)

S. Morgan, "On the Integral Equations of Laser Theory," IEEE Trans. Microwave Theory Tech. 8, 191–193 (1963).

J. Opt. Soc. Am. (2)

Opt. Lett. (2)

Opt. Spectrosc. (1)

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the Selective Properties of Open Unstable Cavities," Opt. Spectrosc. 34, 437–432 (1973).

SAIM Rev. (1)

H. Hochstadt, "Integral Equations in Laser Theory," SAIM Rev. 8, 62–65 (1966).

Other (11)

W. H. Louisell, in Physics of Quantum Electronics, edited by S. F. Jacobs, M. O. Scully, M. Sargent, and C. D. Cantrell (Addison-Wesley, Reading, Massachusetts, 1976), Vol. 3, p. 369.

P. Morse and H. Feshbach, Methods in Mathematical Physics (McGraw-Hill, New York, 1953).

L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Claredon, Oxford, 1974).

L. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York 1964).

G. Watson, Theory of Bessel Functions (Cambridge University, Cambridge, 1966).

See references in the following reviews: H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); A. E. Siegman, "Unstable optical resonators," Appl. Opt. 13, 353 (1974).

L. Weinstein, Open Resonators and Waveguides (Golem, Boulder, 1969).

A. E. Siegman, "Unstable Optical Resonators for Laser Applications," Proc. IEEE 53, 277–287 (1965).

R. Butts and A. Paxton (private communication).

The condition is M > 1 + N-1. and not M > 1 + N. as stated in Ref. 29.

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