Abstract

Critical comments are made on a recent paper by Bergstein and Schachter [J. Opt. Soc. Am. 54, 887 (1964)], which deals with the integral equations satisfied by the normal modes of laser resonators with plane-parallel end reflectors. It is pointed out that the paper lacks rigor, and that the numerical results do not agree very well with other published computations. The present state of knowledge relating to these integral equations is summarized.

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  1. L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964). Equations are numbered as in this reference.
  2. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, Inc., New York, 1953), Vol. 1, p. 114.
  3. D. J. Newman and S. P. Morgan, Bell System Tech. J. 43, 113 (1964).
  4. I. Fredholm, Acta Math. 27, 365 (1903).
  5. J. A. Cochran, Bell System Tech. J. 44, 77 (1965).
  6. L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).
  7. Ref. 6, p. 42.
  8. A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961). In the present notation, H = 2N.
  9. Ref. 1, Fig. 5, and Ref. 8, Fig. 8.
  10. Ref. 1, Fig. 11, and Ref. 8, Fig. 8.
  11. F. B. Hildebrand, Methods of Applied Mathematics (Prentice–Hall, Inc., Englewood Cliffs, New Jersey, 1952), pp. 494–495.
  12. W. Culshaw, I.R.E. Trans. Microwave Theory Tech. MTT-10, 331 (1962).
  13. C. L. Tang, Appl. Opt. 1, 768 (1962).
  14. S. P. Morgan, I.E.E.E. Trans. Microwave Theory Tech. MTT-11, 191 (1963).
  15. A. G. Fox, T. Li, and S. P. Morgan, Appl. Opt. 2, 544 (1963).
  16. S. Kaplan and S. P. Morgan, I.E.E.E. Trans. Microwave Theory Tech. MTT-12, 254 (1964).
  17. L. A. Vainshtein, Zh. Eksperim. i Teor. Fiz. 44, 1050 (1963) [English transl.: Soviet Phys.—JETP 17, 709 (1963)].
  18. G. Goubau and J. R. Christian, I.E.E.E. Trans. Microwave Theory Tech. MTT-12, 212 (1964); Fig. 2.
  19. H. Hochstadt, SIAM Rev. 7 (to be published) has proved that Eq. (12.1) has an infinite number of eigenvalues, except possibly for a countable set of values of H. We wish to thank Professor Hochstadt for sending us an advance copy of the proof.

Bergstein, L.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964). Equations are numbered as in this reference.

Christian, J. R.

G. Goubau and J. R. Christian, I.E.E.E. Trans. Microwave Theory Tech. MTT-12, 212 (1964); Fig. 2.

Cochran, J. A.

J. A. Cochran, Bell System Tech. J. 44, 77 (1965).

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, Inc., New York, 1953), Vol. 1, p. 114.

Culshaw, W.

W. Culshaw, I.R.E. Trans. Microwave Theory Tech. MTT-10, 331 (1962).

Fox, A. G.

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961). In the present notation, H = 2N.

A. G. Fox, T. Li, and S. P. Morgan, Appl. Opt. 2, 544 (1963).

Fredholm, I.

I. Fredholm, Acta Math. 27, 365 (1903).

Goubau, G.

G. Goubau and J. R. Christian, I.E.E.E. Trans. Microwave Theory Tech. MTT-12, 212 (1964); Fig. 2.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, Inc., New York, 1953), Vol. 1, p. 114.

Hildebrand, F. B.

F. B. Hildebrand, Methods of Applied Mathematics (Prentice–Hall, Inc., Englewood Cliffs, New Jersey, 1952), pp. 494–495.

Hochstadt, H.

H. Hochstadt, SIAM Rev. 7 (to be published) has proved that Eq. (12.1) has an infinite number of eigenvalues, except possibly for a countable set of values of H. We wish to thank Professor Hochstadt for sending us an advance copy of the proof.

Kantorovich, L. V.

L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).

Kaplan, S.

S. Kaplan and S. P. Morgan, I.E.E.E. Trans. Microwave Theory Tech. MTT-12, 254 (1964).

Krylov, V. I.

L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).

Li, T.

A. G. Fox, T. Li, and S. P. Morgan, Appl. Opt. 2, 544 (1963).

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961). In the present notation, H = 2N.

Morgan, S. P.

S. Kaplan and S. P. Morgan, I.E.E.E. Trans. Microwave Theory Tech. MTT-12, 254 (1964).

S. P. Morgan, I.E.E.E. Trans. Microwave Theory Tech. MTT-11, 191 (1963).

D. J. Newman and S. P. Morgan, Bell System Tech. J. 43, 113 (1964).

A. G. Fox, T. Li, and S. P. Morgan, Appl. Opt. 2, 544 (1963).

Newman, D. J.

D. J. Newman and S. P. Morgan, Bell System Tech. J. 43, 113 (1964).

Schachter, H.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964). Equations are numbered as in this reference.

Tang, C. L.

C. L. Tang, Appl. Opt. 1, 768 (1962).

Vainshtein, L. A.

L. A. Vainshtein, Zh. Eksperim. i Teor. Fiz. 44, 1050 (1963) [English transl.: Soviet Phys.—JETP 17, 709 (1963)].

Other (19)

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964). Equations are numbered as in this reference.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, Inc., New York, 1953), Vol. 1, p. 114.

D. J. Newman and S. P. Morgan, Bell System Tech. J. 43, 113 (1964).

I. Fredholm, Acta Math. 27, 365 (1903).

J. A. Cochran, Bell System Tech. J. 44, 77 (1965).

L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).

Ref. 6, p. 42.

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961). In the present notation, H = 2N.

Ref. 1, Fig. 5, and Ref. 8, Fig. 8.

Ref. 1, Fig. 11, and Ref. 8, Fig. 8.

F. B. Hildebrand, Methods of Applied Mathematics (Prentice–Hall, Inc., Englewood Cliffs, New Jersey, 1952), pp. 494–495.

W. Culshaw, I.R.E. Trans. Microwave Theory Tech. MTT-10, 331 (1962).

C. L. Tang, Appl. Opt. 1, 768 (1962).

S. P. Morgan, I.E.E.E. Trans. Microwave Theory Tech. MTT-11, 191 (1963).

A. G. Fox, T. Li, and S. P. Morgan, Appl. Opt. 2, 544 (1963).

S. Kaplan and S. P. Morgan, I.E.E.E. Trans. Microwave Theory Tech. MTT-12, 254 (1964).

L. A. Vainshtein, Zh. Eksperim. i Teor. Fiz. 44, 1050 (1963) [English transl.: Soviet Phys.—JETP 17, 709 (1963)].

G. Goubau and J. R. Christian, I.E.E.E. Trans. Microwave Theory Tech. MTT-12, 212 (1964); Fig. 2.

H. Hochstadt, SIAM Rev. 7 (to be published) has proved that Eq. (12.1) has an infinite number of eigenvalues, except possibly for a countable set of values of H. We wish to thank Professor Hochstadt for sending us an advance copy of the proof.

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