P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 116.

R. B. Adler, "Waves on inhomogeneous cylindrical structures," Proc. IRE 40, 339–348 (1952).

A. W. Snyder and J. D. Love, Optical waveguide theory [Wiley and Chapman and Hall, London (in press)].

Recall that the potential of an electrostatic dipole can be expressed by two independent parameters, the dipole moment *p* = *Qd* and the separation distance *d* between a plus and minus charge, each of strength *Q*. If *d* is sufficiently small, then the potential ø at an arbitrary positions is ø (*p,d*)≃ø (*p,0*) where ø (*p,0*) is the potential of a point (*d* = 0) dipole. A point dipole is unphysical because (a) charges that occupy the same position cancel and (b) *p* is arbitrary because *Q* = ∞. Nevertheless, the fields of a physical (*d* ≠ 0) dipole are well approximated by those of the point dipole. Note that *p, d*, and *Q* of the dipole play the roles of *V*, θ_{c}and λ^{-1}, respectively, of the waveguide.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1963), p. 285.

It may appear that the ▽_{t} as a large effect for rapidly varying ∊ profiles; however, this term is bounded by θ^{2}_{c}. Taking the extreme situation of a step profile, where ∊ = ∊_{co} + (∊_{cl} - ∊_{co})*S*(*r* - ρ) and S (r - ρ) is a step function, we see that ▽_{t} ln∊ = θ^{2}_{c}δ(*r* - ρ). In other words, ▽_{t} ln∊ is zero except at *r* = ρ, but the delta function δ(*r* - ρ) has a minute strength of θ^{2}_{c}, where θ_{c} is given by Eq. (8). While it is true that the ▽_{t} ln∊ term is most significant for step profiles, it is usually incorpqrated into the mathematics via the boundary conditions. We show that our procedure leads to highly accurate results even for this extreme case (Sec. IV).

A circularly symmetric waveguide is unchanged if it is rotated through an arbitrary angle or reflected in an axis. Hence if a mode of the waveguide is rotated through an arbitrary angle it must remain a mode (not necessarily the same mode) with the same β. Now, if the pattern e¯_{xe} of Fig. 2(a), for example, is rotated through an arbitrary angle it is then represented by a linear combination of all four patterns in Fig. 2(a). Thus if the individual patterns are modes of the waveguide, all four must have the same β. But if the fields e¯_{xe}, e¯_{xo}, e¯_{yo} are substituted into Eq. (17) we find that the four corrected β's are not all equal. Therefore *n*_{co} = *n*_{cl} modes are not *n*_{co} ≅ *n*_{cl} modes.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, 1972).

The patterns el and e_{4} of Fig. 2(b) are unchanged by reflection in an arbitrary axis and by rotation through an arbitrary angle, consistent with their being nondegenerate modes of a circularly symmetric waveguide. However, under arbitrary rotation and reflections e_{2} changes into a pattern which is a linear combination of e_{2} and e_{3}. Symmetry demands that this new combination is also a mode, which in turn requires that e_{2} and e_{3} have identical β's. Indeed, by substituting e_{2} and e_{3} into Eq. (17) we verify that they have identical β's. Thus the linear combinations given by Eq. (23) are consistent with the requirements of symmetry and the results of Eq. (17). Analogous arguments show that these consistencies remain when *l* ≠ 1.

M. Born and E. Wolf, Principles of Optics, 3rd edition (Pergamon, New York, 1964).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 159, 164.

This result is found by substituting the fields e¯_{x} = ψ ◯ and e¯_{y} = ψ ŷ into Eq. (17) to determine β_{x} and β_{y}, where ψ is the solution to the scalar wave equation in elliptical geometry. It is not sufficient to approximate ψ by the solutions of the circular symmetric fiber as done for β_{e} - β_{0} in Appendix D. Instead, higher-order terms are necessary. Alternatively,^{9} one can perturb about Maxwell's equations in circular symmetry as formulated in Ref. 21; however, because et must include terms of order θ^{2}_{c}, the previously reported result of Marcuse (Ref. 22) is inaccurate. We have included θ^{2}_{c}, terms using the expansion presented in Ref. 1.

A. W. Snyder, "Coupled mode theory for optical fibers," J. Opt. Soc. Am. 62, 1268–1277 (1972). Note that the coupling coefficient is *C* = (β¯_{+} - β¯_{-})/2, where *C* is given by Eq. (26a) of the 1972 paper.