OSA's Digital Library

Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 63, Iss. 12 — Dec. 1, 1973
  • pp: 1528–1543

Asymptotic theory of unstable resonator modes

Paul Horwitz  »View Author Affiliations


JOSA, Vol. 63, Issue 12, pp. 1528-1543 (1973)
http://dx.doi.org/10.1364/JOSA.63.001528


View Full Text Article

Acrobat PDF (1490 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The integral equation that describes the mode structure of unstable resonators of rectangular aperture is solved by a new method that makes use of an asymptotic expansion of the integral for large Fresnel number. This expansion is carried out to terms of order 1/F, where F is the Fresnel number, and includes effects of diffraction at the edges of the feedback mirror, as well as a term corresponding to the geometric-optics approximation. Results obtained by this method are in good agreement with those obtained by gaussian integration, even at an effective Fresnel number of unity. The method has the advantage that it involves finding the roots of a polynomial, rather than the eigenvalues of a matrix, and therefore requires less computer time and storage. The results of the present work are at variance with a class of theories based on geometric optics, as regards the higher-loss modes even for very large Fresnel number, although the lowest-loss symmetric mode is correctly given by geometric optics in this limit.

Citation
Paul Horwitz, "Asymptotic theory of unstable resonator modes," J. Opt. Soc. Am. 63, 1528-1543 (1973)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-63-12-1528


Sort:  Author  |  Year  |  Journal  |  Reset

References

  1. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
  2. A. E. Siegman and H. Y. Miller, Appl. Opt. 9, 2729 (1970).
  3. A. E. Siegman and R. Arrathoon, IEEE J. Quantum Electron. 3, 156 (1966).
  4. R. L. Sanderson and W. Streifer, Appl. Opt. 8, 2129 (1969).
  5. W. Streifer, IEEE J. Quantum Electron. 4, 229 (1967).
  6. L. Bergstein, Appl. Opt. 7, 495 (1968).
  7. S. R. Barone, Appl. Opt. 10, 935 (1970).
  8. A. E. Siegman, Proc. IEEE 53, 277 (1965).
  9. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  10. Mathematically, the reason for the continuous spectrum of eigenvalues is that taking only the first term in the expansion of I(x) corresponds to neglecting contributions from the end points of the integration, which contribute at the next order. The end-point contributions do not contribute if the integral is taken over the range (- ∞, ∞), provided it exists. Their neglect, therefore, is equivalent to considering the small mirror to be infinitely big, an approximation the dangers of which we have touched on above. It requires, of course, the introduction of a convergence factor to insure the finiteness of the integral if Reξ> 0. It also makes Eq. (8) a singular integral equation, in the sense that the range of integration is infinite, and the appearance of a continuum of eigenvalues is a common pathology of such equations. See, e.g. Ref. 11, p. 273.
  11. F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J. 1965).
  12. M. A. Jenkins and J. F. Traub, Numer. Math. 14, 252 (1970).
  13. A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  14. L. Sirovich, Techniques of Asymptotic Analysis (Springer, New York, 1971).
  15. N. Bleistein and A. Handelsman, SIAM J. Appl. Math. (Soc. Ind. Appl. Math.) 15, 422 (1967).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited