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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 56, Iss. 1 — Jan. 1, 1966
  • pp: 16–31

Angular Spectra of Optic Cavities

LEONARD BERGSTEIN and EMANUEL MAROM  »View Author Affiliations

JOSA, Vol. 56, Issue 1, pp. 16-31 (1966)

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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.

LEONARD BERGSTEIN and EMANUEL MAROM, "Angular Spectra of Optic Cavities," J. Opt. Soc. Am. 56, 16-31 (1966)

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  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].
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  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).
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