Abstract

The resonant modes of optic interferometer cavities are investigated in the angular spectrum domain. The investigation is based on an integral equation (governing the relation between the normal modes and cavity geometry) which is derived by using the self-consistent Rayleigh formulation for solving diffraction problems. For plane-parallel cavities this integral equation can be solved by means of a series expansion of orthogonal functions characteristic of the cavity geometry without making any assumption about the relative magnitudes of end reflector dimensions and separation. The solution also provides, in addition to a comparison with the solutions of the approximated Huygens’ integral equation as found by other investigators, a direct way for obtaining the angular plane-wave spectrum (or the radiation pattern) of the beam emerging from the cavity. The particular cases of plane-parallel cavities with infinite-strip and circular end reflectors are considered in this paper.

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  1. A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
  2. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
  3. G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
  4. A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).
  5. S. R. Barone, J. Appl. Phys. 34, 831 (1963).
  6. H. Schachter and L. Bergstein, in MRI Symposium Proceedings XIII, Optical Masers (Polytechnic Institute of Brooklyn, New York, 1963), pp. 173–198.
  7. L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).
  8. H. Schachter, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June 1964).
  9. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.
  10. An alternative way of obtaining Eq. (11) is to set w≈1-½(u22) in the integrand of Eq. (6) and then integrate over u and ν.
  11. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962).
  12. The proof of this statement is somewhat lengthy and is only outlined here. Let γ˜ = 1/Γ. Equation (27) then becomes, D(Γ) ≡ [βjkΓ-δj,k] = 0. To show that Eq. (27.2) is true for all γ˜ < ∞ (with the possible exception of γ˜ = 0) it is obviously sufficient to show that [equation] converges uniformly to D (Γ) for all |Γ| <R<∞. To show this we use a comparison test. The real series [equation] with finite but arbitrary C¨r, and C¨, converges uniformly (as M → ∞) to [equation] for all |x| <A<R<∞. D(M)(Γ) therefore converges uniformly to D (Γ) for all |Γ| <A<R<∞ if: (1) the limit of Cr(M) exist for all r, and (2) an N < ∞ exist such that |Cr|≤ A-r, for all r>N. This can be shown to be the case if the elements βjk decrease at least as fast 1/jk when j and k approach infinit, i.e., if beyond a certain JN and KN, βjk,≤ (B/jk) for all j>J and k>K, B being a finite constant. For example, it is readily found that | CM(M) | ≤ (BM/M|) and, therefore [equation] for all A<R<∞ (since we can always take MeBA). Similarly, |CM-1(M)| ≤[⅓MBM-1/(M-1)!], and lim [equation] etc. [equation].
  13. For the determination of the eigenvalues and eigenfunctions of the three low-order even-symmetric modes, (1), (3), (5), and the three low-order odd-symmetric modes, (2), (4), (6), it was found in each case sufficient to use only 5×5 terms of the determinant of Eq. (27.1). Truncating the determinant after 6×6 or more terms led to essentially the same results.
  14. L. Bergstein and H. Schachter, J. Opt. Soc. Am. 55, 1226 (1965).
  15. E. Wolf and E. W. Marchand, J. Opt. Soc. Am. 54, 587 (1964)
  16. L. A. Vainshtein, Soviet Physics—JETP 17, 714 (1963).
  17. H. Ogura and Y. Yoshida, Japan, J. Appl. Phys. 3, 546 (1964).
  18. H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).
  19. E. Marom, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June 1965).
  20. To a very good approximation pmk≍(½m+k-¼)π.
  21. The only exception is ƒ01′(r,Ø,z) = (π) exp {iβz}, which is a uniform plane wave.
  22. For the determination of the eigenvalues and eigenfunctions of the six low-order modes (01), (02), (03), (11), (12), (13), it was found sufficient to use only 5×5 terms of the determinant of Eq. (46).
  23. W. Grobner and H. Hofreiter, Integraltafel, Part II, Bestimmte Integrale (Springer-Verlag, Vienna, Austria, 1958), 2nd ed.

Barone, S. R.

S. R. Barone, J. Appl. Phys. 34, 831 (1963).

Bergstein, L.

H. Schachter and L. Bergstein, in MRI Symposium Proceedings XIII, Optical Masers (Polytechnic Institute of Brooklyn, New York, 1963), pp. 173–198.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 55, 1226 (1965).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

Boyd, G. D.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

Fox, A. G.

A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).

Gordon, J. P.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Grobner, W.

W. Grobner and H. Hofreiter, Integraltafel, Part II, Bestimmte Integrale (Springer-Verlag, Vienna, Austria, 1958), 2nd ed.

Hofreiter, H.

W. Grobner and H. Hofreiter, Integraltafel, Part II, Bestimmte Integrale (Springer-Verlag, Vienna, Austria, 1958), 2nd ed.

Ikenoue, J.

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).

Kogelnik, H.

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

Li, T.

A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).

Marchand, E. W.

E. Wolf and E. W. Marchand, J. Opt. Soc. Am. 54, 587 (1964)

Marom, E.

E. Marom, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June 1965).

Ogura, H.

H. Ogura and Y. Yoshida, Japan, J. Appl. Phys. 3, 546 (1964).

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962).

Schachter, H.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 55, 1226 (1965).

H. Schachter and L. Bergstein, in MRI Symposium Proceedings XIII, Optical Masers (Polytechnic Institute of Brooklyn, New York, 1963), pp. 173–198.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).

H. Schachter, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June 1964).

Vainshtein, L. A.

L. A. Vainshtein, Soviet Physics—JETP 17, 714 (1963).

Wolf, E.

E. Wolf and E. W. Marchand, J. Opt. Soc. Am. 54, 587 (1964)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

Yoshida, Y.

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).

H. Ogura and Y. Yoshida, Japan, J. Appl. Phys. 3, 546 (1964).

Other (23)

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

G. D. Boyd and H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).

S. R. Barone, J. Appl. Phys. 34, 831 (1963).

H. Schachter and L. Bergstein, in MRI Symposium Proceedings XIII, Optical Masers (Polytechnic Institute of Brooklyn, New York, 1963), pp. 173–198.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).

H. Schachter, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June 1964).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

An alternative way of obtaining Eq. (11) is to set w≈1-½(u22) in the integrand of Eq. (6) and then integrate over u and ν.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962).

The proof of this statement is somewhat lengthy and is only outlined here. Let γ˜ = 1/Γ. Equation (27) then becomes, D(Γ) ≡ [βjkΓ-δj,k] = 0. To show that Eq. (27.2) is true for all γ˜ < ∞ (with the possible exception of γ˜ = 0) it is obviously sufficient to show that [equation] converges uniformly to D (Γ) for all |Γ| <R<∞. To show this we use a comparison test. The real series [equation] with finite but arbitrary C¨r, and C¨, converges uniformly (as M → ∞) to [equation] for all |x| <A<R<∞. D(M)(Γ) therefore converges uniformly to D (Γ) for all |Γ| <A<R<∞ if: (1) the limit of Cr(M) exist for all r, and (2) an N < ∞ exist such that |Cr|≤ A-r, for all r>N. This can be shown to be the case if the elements βjk decrease at least as fast 1/jk when j and k approach infinit, i.e., if beyond a certain JN and KN, βjk,≤ (B/jk) for all j>J and k>K, B being a finite constant. For example, it is readily found that | CM(M) | ≤ (BM/M|) and, therefore [equation] for all A<R<∞ (since we can always take MeBA). Similarly, |CM-1(M)| ≤[⅓MBM-1/(M-1)!], and lim [equation] etc. [equation].

For the determination of the eigenvalues and eigenfunctions of the three low-order even-symmetric modes, (1), (3), (5), and the three low-order odd-symmetric modes, (2), (4), (6), it was found in each case sufficient to use only 5×5 terms of the determinant of Eq. (27.1). Truncating the determinant after 6×6 or more terms led to essentially the same results.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 55, 1226 (1965).

E. Wolf and E. W. Marchand, J. Opt. Soc. Am. 54, 587 (1964)

L. A. Vainshtein, Soviet Physics—JETP 17, 714 (1963).

H. Ogura and Y. Yoshida, Japan, J. Appl. Phys. 3, 546 (1964).

H. Ogura, Y. Yoshida, and J. Ikenoue, Japan, J. Appl. Phys. 4, 598 (1965).

E. Marom, Ph.D. dissertation, Polytechnic Institute of Brooklyn (June 1965).

To a very good approximation pmk≍(½m+k-¼)π.

The only exception is ƒ01′(r,Ø,z) = (π) exp {iβz}, which is a uniform plane wave.

For the determination of the eigenvalues and eigenfunctions of the six low-order modes (01), (02), (03), (11), (12), (13), it was found sufficient to use only 5×5 terms of the determinant of Eq. (46).

W. Grobner and H. Hofreiter, Integraltafel, Part II, Bestimmte Integrale (Springer-Verlag, Vienna, Austria, 1958), 2nd ed.

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