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.

© 1965 Optical Society of America

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References

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  1. L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964). Equations are numbered as in this reference.
    [CrossRef]
  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).
    [CrossRef]
  4. I. Fredholm, Acta Math. 27, 365 (1903).
    [CrossRef]
  5. J. A. Cochran, Bell System Tech. J. 44, 77 (1965).
    [CrossRef]
  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.
    [CrossRef]
  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).
    [CrossRef]
  13. C. L. Tang, Appl. Opt. 1, 768 (1962).
    [CrossRef]
  14. S. P. Morgan, I.E.E.E. Trans. Microwave Theory Tech. MTT-11, 191 (1963).
    [CrossRef]
  15. A. G. Fox, T. Li, and S. P. Morgan, Appl. Opt. 2, 544 (1963).
    [CrossRef]
  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.
    [CrossRef]
  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.

1965 (1)

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

1964 (4)

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

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

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

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

1963 (3)

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

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

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

1962 (2)

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

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

1961 (1)

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

1903 (1)

I. Fredholm, Acta Math. 27, 365 (1903).
[CrossRef]

Bergstein, L.

Christian, J. R.

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

Cochran, J. A.

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

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).
[CrossRef]

Fox, A. G.

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

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

Fredholm, I.

I. Fredholm, Acta Math. 27, 365 (1903).
[CrossRef]

Goubau, G.

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

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).
[CrossRef]

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

Morgan, S. P.

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

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).
[CrossRef]

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

Newman, D. J.

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

Schachter, H.

Tang, C. L.

Vainshtein, L. A.

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

Acta Math. (1)

I. Fredholm, Acta Math. 27, 365 (1903).
[CrossRef]

Appl. Opt. (2)

Bell System Tech. J. (3)

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

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

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

I.E.E.E. Trans. Microwave Theory Tech. (3)

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).
[CrossRef]

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

I.R.E. Trans. Microwave Theory Tech. (1)

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

J. Opt. Soc. Am. (1)

Zh. Eksperim. i Teor. Fiz. (1)

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

Other (7)

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

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.

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

Ref. 6, p. 42.

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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Figures (4)

F. 1
F. 1

Relative power loss for two lowest modes between plane infinite-strip reflectors, as a function of Fresnel number. ——Bergstein and Schachter. – – –Fox and Li.

F. 2
F. 2

Relative power loss for 01 and 11 modes between plane circular reflectors, as a function of Fresnel number. —— Bergstein and Schachter. – – – Fox and Li.

F. 3
F. 3

Eigenfunction of 01 mode between plane circular reflectors for H= 2. (a) Amplitude; (b) phase. —— Bergstein and Schachter. – – – Fox and Li.

F. 4
F. 4

Eigenfunction of 11 mode between plane circular reflectors for H=1. (a) Amplitude; (b) phase. —— Bergstein and Schachter. – – – Fox and Li.

Tables (1)

Tables Icon

Table I Calculated values of relative power loss at H = 10.

Equations (20)

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γ ¯ f ( ξ ) = ( H 2 i ) 1 2 1 1 e i ( π / 2 ) H ( ξ ξ 1 ) 2 f ( ξ 1 ) d ξ 1 .
K ( ξ , ξ 1 ) = j = 1 n a j ( ξ ) b j ( ξ 1 ) .
1 1 K ( ξ , ξ ) d ξ 0 .
k = 0 ( β j k γ ¯ δ j k ) α k = 0 , j = 0 , 1 , 2 , ,
[ β j k γ ¯ δ j k ] = 0
[ B j k γ ¯ δ j k ] = 0
| 1 1 ( sin [ π ( H ξ k ) ] π ( H ξ k ) ) sin cos π j ξ d ξ | | 1 1 e i π H ξ 2 ( sin [ π ( H ξ k ) ] π ( H ξ k ) ) sin cos π j ξ d ξ | ,
P 2 ( 20 ) = { 1.361 % ( 100 intervals ) , 1.345 % ( 200 intervals ) .
P 01 ( 20 ) = { 0.879 % ( 50 intervals ) , 0.843 % ( 100 intervals ) .
γ ¯ n A n n ,
A n n = ( H 2 i ) 1 2 1 1 1 1 e i ( π / 2 ) H ( ξ ξ 1 ) 2 sin [ 1 2 n π ( ξ + 1 ) ] × sin [ 1 2 n π ( ξ 1 + 1 ) ] d ξ d ξ 1 .
γ ¯ n ( H 2 i ) 1 2 1 1 1 1 e i ( π / 2 ) H ( ξ ξ 1 ) 2 φ n ( ξ ) φ n ( ξ 1 ) d ξ d ξ 1 1 1 φ n 2 ( ξ ) d ξ ,
φ n ( ξ ) = sin [ 1 2 n π ( ξ + 1 ) ]
P n = 2 π 2 n 2 [ β ( M + β ) ] / { [ ( M + β ) 2 + β 2 ] 2 } ,
M = ( 4 π H ) 1 2 , β = 0.824 .
P m n = 8 p m n 2 [ β ( M + β ) ] / { [ ( M + β ) 2 + β 2 ] 2 } ,
P n 0.365 n 2 / H 3 2 ,
P n 0.167 n 2 / H 3 2 ,
P m n 0.148 p m n 2 / H 3 2 ,
P m n 0.0675 p m n 2 / H 3 2 .