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(*u*^{2}+ν^{2}) 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) ≡ [β_{jk}Γ-δ_{j,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 *C*_{r}^{(M)} exist for all *r*, and (2) an *N* < ∞ exist such that |*C*_{r}|≤ *A*^{-r}, for all *r*>*N*. This can be shown to be the case if the elements β_{jk} decrease at least as fast 1/*jk* when *j* and *k* approach infinit, i.e., if beyond a certain *J*≤*N* and *K*≤*N*, β_{jk},≤ (*B*/*jk*) for all *j*>*J* and *k*>*K*, *B* being a finite constant. For example, it is readily found that | *C*_{M}^{(M)} | ≤ (*B*^{M}/*M*|) and, therefore [equation] for all *A*<*R*<∞ (since we can always take *M*≥ *eBA*). Similarly, |*C*_{M-1}^{(M)}| ≤[⅓*MB*^{M-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 *p*_{mk}≍(½*m*+*k*-¼)π.

The only exception is ƒ_{01}′(*r*,Ø,*z*) = (π)^{-½} exp {*i*β*z*}, 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.