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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 70, Iss. 12 — Dec. 1, 1980
  • pp: 1525–1538

Analytic variational method for determining the modes of marginally stable resonators

J. Nagel and D. Rogovin  »View Author Affiliations

JOSA, Vol. 70, Issue 12, pp. 1525-1538 (1980)

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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

J. Nagel and D. Rogovin, "Analytic variational method for determining the modes of marginally stable resonators," J. Opt. Soc. Am. 70, 1525-1538 (1980)

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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).

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