Abstract

A study is made of the resonant or normal modes of optic and quasioptic interferometer cavities with plane-parallel end reflectors. The solution of the integral equation governing the relation between the normal modes and the geometry of the cavity is found by means of a series expansion of orthogonal functions. The terms of the series for the normal modes can be interpreted as Fraunhofer diffraction patterns characteristic of the geometry of the end reflectors. Various geometries, such as the infinite-strip, rectangular, and circular end reflector cavities, are considered and the results plotted and interpreted.

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  1. A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
  2. A. G. Fox and TinGye Li, Bell System Tech. J. 40, 453 (1961).
  3. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).
  4. R. F. Sooho, Proc. IEEE 51, 70 (1963).
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 744–751.
  6. J. Focke, Ber. Sachs. Ges. (Akad.) Wiss. 101, No. 3 (1954).
  7. It should be pointed out that the longitudinal component of the field is small for both the TE and TM modes so that the field is predominantly transverse and the modes are nearly TEM.
  8. Kantorovich and Krylov, A pproximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).
  9. C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).
  10. When cos(πH) =0, higher order terms in the expansion of Si(α) and Ci(α) must be taken into consideration. However, these terms lead to an expression for Bjk′ that tends to zero as (1/j4) and (1/k4) with increasing j and k. Similar considerations apply to the expression for Bjk″ when sin(πH) =0.
  11. As in Sec. 2, we say that Eqs. (41) and (43) exist if the sum Σ |βm,jk | over all j and k is finite. Now, we recognize immediately that[equation](42*)is a majorant of βm,jk. Since the terms Bmjk are obtained by solving the homogeneous integral equation γ¯mRm(ρ) = πHʃ10Rm(ρ1)JmHρρ1)ρ1dρ1 (36*)which is symmetrical in Hermitian sense, the algebraic equation [Bm,jk-γ¯mδj,k] =0 exists and so do Eqs. (43) and (41).
  12. An expansion of Eq. (52) gives [equation]
  13. An expansion of Eq. (69) gives [equation]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 744–751.

Boyd, G. D.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).

Focke, J.

J. Focke, Ber. Sachs. Ges. (Akad.) Wiss. 101, No. 3 (1954).

Fox, A. G.

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

Gordon, J. P.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).

Li, TinGye

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

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).

Sooho, R. F.

R. F. Sooho, Proc. IEEE 51, 70 (1963).

Townes, C. H.

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 744–751.

Other (13)

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).

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

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 10, 489 (1961).

R. F. Sooho, Proc. IEEE 51, 70 (1963).

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), pp. 744–751.

J. Focke, Ber. Sachs. Ges. (Akad.) Wiss. 101, No. 3 (1954).

It should be pointed out that the longitudinal component of the field is small for both the TE and TM modes so that the field is predominantly transverse and the modes are nearly TEM.

Kantorovich and Krylov, A pproximate Methods of Higher Analysis (Interscience Publishers, Inc., New York, 1958).

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, California, 1957).

When cos(πH) =0, higher order terms in the expansion of Si(α) and Ci(α) must be taken into consideration. However, these terms lead to an expression for Bjk′ that tends to zero as (1/j4) and (1/k4) with increasing j and k. Similar considerations apply to the expression for Bjk″ when sin(πH) =0.

As in Sec. 2, we say that Eqs. (41) and (43) exist if the sum Σ |βm,jk | over all j and k is finite. Now, we recognize immediately that[equation](42*)is a majorant of βm,jk. Since the terms Bmjk are obtained by solving the homogeneous integral equation γ¯mRm(ρ) = πHʃ10Rm(ρ1)JmHρρ1)ρ1dρ1 (36*)which is symmetrical in Hermitian sense, the algebraic equation [Bm,jk-γ¯mδj,k] =0 exists and so do Eqs. (43) and (41).

An expansion of Eq. (52) gives [equation]

An expansion of Eq. (69) gives [equation]

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