(a) A. W. Snyder and R. A. Sammut, "Radiation modes of optical waveguides," Electron. Lett. 15, 4–5 (1979). (b) A. W. Snyder, "Continuous mode spectrum of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

A. W. Snyder and R. A. Sammut, "Radiation from optical waveguides: leaky mode interpretation," Electron. Lett. 15, 58–60 (1979).

E. G. Rawson, "Analysis of scattering from fiber waveguides with irregular core surfaces," Appl. Opt. 13, 2270–2275 (1974).

A. W. Snyder, "Leaky-ray theory of optical waveguides of circular cross section," Appl. Phys. 4, 273–298 (1974).

A. W. Snyder, "Radiation losses due to variations of radius on dielectric or optical fibres," IEEE Trans. Microwave Theory Tech. MTT-18, 608–615(1970).

G. L. Yip, "Launching efficiency of the HE_{11} surface wave mode on a dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-18, 1033–1041 (1970).

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides," IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

P. J. B. Clarricoats, "Optical fibre waveguides—a review" in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1976).

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

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965).

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).

E. G. Rawson, "Analysis of scattering from fiber waveguides with irregular core surfaces," Appl. Opt. 13, 2270–2275 (1974).

(a) A. W. Snyder and R. A. Sammut, "Radiation modes of optical waveguides," Electron. Lett. 15, 4–5 (1979). (b) A. W. Snyder, "Continuous mode spectrum of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

A. W. Snyder and R. A. Sammut, "Radiation from optical waveguides: leaky mode interpretation," Electron. Lett. 15, 58–60 (1979).

A. W. Snyder and R. A. Sammut, "Radiation from optical waveguides: leaky mode interpretation," Electron. Lett. 15, 58–60 (1979).

(a) A. W. Snyder and R. A. Sammut, "Radiation modes of optical waveguides," Electron. Lett. 15, 4–5 (1979). (b) A. W. Snyder, "Continuous mode spectrum of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

A. W. Snyder and W. R. Young, "Modes of optical waveguides," J. Opt. Soc. Am. 68, 297–309 (1978).

I. A. White and A. W. Snyder, "Radiation from dielectric optical waveguides: a comparison of techniques," Appl. Opt. 16, 1470–1472 (1977).

A. W. Snyder, "Leaky-ray theory of optical waveguides of circular cross section," Appl. Phys. 4, 273–298 (1974).

A. W. Snyder, "Radiation losses due to variations of radius on dielectric or optical fibres," IEEE Trans. Microwave Theory Tech. MTT-18, 608–615(1970).

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides," IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).

H. G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977).

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).

G. L. Yip, "Launching efficiency of the HE_{11} surface wave mode on a dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-18, 1033–1041 (1970).

A. W. Snyder, "Leaky-ray theory of optical waveguides of circular cross section," Appl. Phys. 4, 273–298 (1974).

(a) A. W. Snyder and R. A. Sammut, "Radiation modes of optical waveguides," Electron. Lett. 15, 4–5 (1979). (b) A. W. Snyder, "Continuous mode spectrum of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

A. W. Snyder and R. A. Sammut, "Radiation from optical waveguides: leaky mode interpretation," Electron. Lett. 15, 58–60 (1979).

A. W. Snyder, "Radiation losses due to variations of radius on dielectric or optical fibres," IEEE Trans. Microwave Theory Tech. MTT-18, 608–615(1970).

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides," IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).

G. L. Yip, "Launching efficiency of the HE_{11} surface wave mode on a dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-18, 1033–1041 (1970).

To do this we assume ψ(β) =ψ^{FS}(β + Δβ), where β = *k*_{c1}cosθ, i.e., we assume the radiation at observation angle θ equals that of free-space radiation at angle θ + Δθ. Now β enters Eq. (14) in the combination *k*^{2}(*r*) - β^{2}, whereas in free space it appears as *k*^{2}_{c1} - β^{2}. Thus for the step fiber, we rewrite *k*^{2}(*r*) - β^{2} as *k*^{2}_{c1} - {β^{2} +*k*^{2}_{c1} - *k*^{2}_{c0}} calling the bracketed quantity (β + Δβ)^{2} or β for short. By replacing β in the free-space radiation expressions with (β^{2} + *k*^{2}_{c1} - *k*^{2}_{c0})^{½}, or replacing *Q* by *U* we account for the shift. Equivalently, we can replace θ in the free-space radiation expressions by θ˜= θ + Δθ, where cos^{2}θ ≅ cos^{2}θ - θ^{2}_{c}(1 + θ^{2}_{c}) provided cos^{2}θ >θ^{2}_{c}(1 + θ^{2}_{c}) and 0 < *r*≤ ≤ ρ.

I. A. White, Radiation Losses in Dielectric Optical Waveguides, Ph.D. thesis, Australian National University, 1977 (unpublished).

P. J. B. Clarricoats, "Optical fibre waveguides—a review" in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1976).

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965).

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

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

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

H. G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).