Abstract

We present a caustic model of morphology-dependent resonances based on geometrical optics, which describes the electromagnetic field in cylinders or spheres by families of ray congruences. A ray congruence in this model is basically a family of rays that osculate in phase on a common circle, the caustic. The circumference of this circle is in the plane case an integer multiple of the wavelength. This integer number corresponds to the mode number in the multipole expansion. In the spherical case two families of caustics exist. The mode numbers l, m of the multipole expansion for spherical particles define the corresponding radii of the caustics of the two ray families. The localization principle follows in this model simply by conservation of angular momentum. The condition for narrow modes caused by total internal reflection and the leaking of these modes is explained. The excitation of narrow resonances can also be explained in a straightforward manner by requiring that rays propagating from the caustic to the surface couple in phase to the caustic after reflection. Comparison with the exact solution shows excellent agreement.

© 1998 Optical Society of America

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    [CrossRef]
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1997 (1)

1996 (4)

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection,” Appl. Opt. 35, 500–513 (1996).
[CrossRef] [PubMed]

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35, 515–531 (1996).
[CrossRef] [PubMed]

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by gaussian particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 55, 577–601 (1996).
[CrossRef]

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7282 (1996).
[CrossRef]

1995 (1)

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, R. K. Chang, “Ray chaos and Q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

1994 (3)

1993 (1)

1992 (2)

1991 (4)

1989 (3)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonance,” Comments At. Mol. Phys. 23, 175–187 (1989).

M. A. Jarzembski, V. Srivastava, “Electromagnetic field enhancement in small liquid droplets using geometric optics,” Appl. Opt. 28, 4962–4965 (1989).
[CrossRef] [PubMed]

1984 (1)

M. Robnik, “Quantising a generic family of billiards with analytic boundaries,” J. Phys. A 17, 1049–1074 (1984).
[CrossRef]

1981 (1)

1980 (1)

J. D. Murphy, P. J. Moser, A. Nagl, H. Überall, “A surface wave interpretation for the resonances of a dielectric sphere,” IEEE Trans. Antennas Propag. 28, 924–927 (1980).
[CrossRef]

1979 (1)

1977 (2)

D. S. Jones, “Electromagnetic tunnelling,” Q. J. Mech. Appl. Math. 31, 409–434 (1977).

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

1975 (1)

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface: electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. 23, 134–141 (1975).
[CrossRef]

1974 (1)

W. D. Wang, G. A. Deschamps, “Application of complex ray tracing to scattering problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

1970 (1)

H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,” IEEE Trans. Antennas Propag. 18, 89–99 (1970).
[CrossRef]

1967 (1)

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

1960 (1)

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

1951 (1)

R. Landauer, “Associated Legendre polynomial approximations,” J. Appl. Phys. 22, 87–89 (1951).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Aker, P. M.

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7282 (1996).
[CrossRef]

Aker, W. P.

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Barber, P. W.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Benner, R. E.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, UK, 1993).

Chang, R. K.

Chen, G.

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, R. K. Chang, “Ray chaos and Q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

Chen, G. C.

Chiao, R. Y.

R. Y. Chiao, P. G. Kwiat, A. M. Steinberg, “Analogies between electron and photon tunneling,” Phys. Rev. B 175, 257–262 (1991).

Ching, E. S. C.

E. S. C. Ching, P. T. Leung, K. Young, “Optical processes in microcavities—the role of quasinormal modes,” in Optical Processes in Microcavities, R. K. Chang, A. J. Campillo, eds. (World Scientific, Singapore, 1996), p. 13.

Chodhury, D. Q.

Chowdhury, D. Q.

Deschamps, G. A.

W. D. Wang, G. A. Deschamps, “Application of complex ray tracing to scattering problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

Esam, E.

Fast, P.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by gaussian particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 55, 577–601 (1996).
[CrossRef]

Felsen, L. B.

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

Gouesbet, G.

Gréhan, G.

Hill, S. C.

Hovenac, E. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Inada, H.

H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,” IEEE Trans. Antennas Propag. 18, 89–99 (1970).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jarzembski, M. A.

Johnson, B. R.

Jones, D. S.

D. S. Jones, “Electromagnetic tunnelling,” Q. J. Mech. Appl. Math. 31, 409–434 (1977).

Kaiser, T.

Keller, J. B.

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Mathematics, L. M. Graves, ed. (McGraw-Hill, New York, 1958), Vol. 8.

Khaled, M.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Kwiat, P. G.

R. Y. Chiao, P. G. Kwiat, A. M. Steinberg, “Analogies between electron and photon tunneling,” Phys. Rev. B 175, 257–262 (1991).

Lai, H. M.

Landauer, R.

R. Landauer, “Associated Legendre polynomial approximations,” J. Appl. Phys. 22, 87–89 (1951).
[CrossRef]

Leach, D. H.

Leung, P. T.

H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Characterization of the internal energy density in Mie scattering,” J. Opt. Soc. Am. A 8, 1553–1558 (1991).
[CrossRef]

E. S. C. Ching, P. T. Leung, K. Young, “Optical processes in microcavities—the role of quasinormal modes,” in Optical Processes in Microcavities, R. K. Chang, A. J. Campillo, eds. (World Scientific, Singapore, 1996), p. 13.

Lock, J. A.

Love, J. D.

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface: electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. 23, 134–141 (1975).
[CrossRef]

Lumme, K.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by gaussian particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 55, 577–601 (1996).
[CrossRef]

Maurer, S. J.

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

Mekis, A.

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, R. K. Chang, “Ray chaos and Q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

Moortgat, P. A.

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7282 (1996).
[CrossRef]

Moser, P. J.

J. D. Murphy, P. J. Moser, A. Nagl, H. Überall, “A surface wave interpretation for the resonances of a dielectric sphere,” IEEE Trans. Antennas Propag. 28, 924–927 (1980).
[CrossRef]

Muinonen, K.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by gaussian particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 55, 577–601 (1996).
[CrossRef]

Murphy, J. D.

J. D. Murphy, P. J. Moser, A. Nagl, H. Überall, “A surface wave interpretation for the resonances of a dielectric sphere,” IEEE Trans. Antennas Propag. 28, 924–927 (1980).
[CrossRef]

Nagl, A.

J. D. Murphy, P. J. Moser, A. Nagl, H. Überall, “A surface wave interpretation for the resonances of a dielectric sphere,” IEEE Trans. Antennas Propag. 28, 924–927 (1980).
[CrossRef]

Nöckel, J. U.

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, R. K. Chang, “Ray chaos and Q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

J. U. Nöckel, A. D. Stone, R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19, 1693–1695 (1994).
[CrossRef]

Nousiainen, T.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by gaussian particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 55, 577–601 (1996).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonance,” Comments At. Mol. Phys. 23, 175–187 (1989).

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Peltoniemi, J. I.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by gaussian particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 55, 577–601 (1996).
[CrossRef]

Plonus, M. A.

H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,” IEEE Trans. Antennas Propag. 18, 89–99 (1970).
[CrossRef]

Poon, K. L.

Robnik, M.

M. Robnik, “Quantising a generic family of billiards with analytic boundaries,” J. Phys. A 17, 1049–1074 (1984).
[CrossRef]

Rubinow, S. I.

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Schweiger, G.

Snyder, A. W.

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface: electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. 23, 134–141 (1975).
[CrossRef]

Srivastava, V.

Stegun, I. E.

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Steinberg, A. M.

R. Y. Chiao, P. G. Kwiat, A. M. Steinberg, “Analogies between electron and photon tunneling,” Phys. Rev. B 175, 257–262 (1991).

Stone, A. D.

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, R. K. Chang, “Ray chaos and Q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

J. U. Nöckel, A. D. Stone, R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19, 1693–1695 (1994).
[CrossRef]

Überall, H.

J. D. Murphy, P. J. Moser, A. Nagl, H. Überall, “A surface wave interpretation for the resonances of a dielectric sphere,” IEEE Trans. Antennas Propag. 28, 924–927 (1980).
[CrossRef]

Ungut, A.

Vainshtein, L. A.

L. A. Vainshtein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

Velesco, N.

Wang, W. D.

W. D. Wang, G. A. Deschamps, “Application of complex ray tracing to scattering problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, UK, 1993).

Young, K.

H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Characterization of the internal energy density in Mie scattering,” J. Opt. Soc. Am. A 8, 1553–1558 (1991).
[CrossRef]

E. S. C. Ching, P. T. Leung, K. Young, “Optical processes in microcavities—the role of quasinormal modes,” in Optical Processes in Microcavities, R. K. Chang, A. J. Campillo, eds. (World Scientific, Singapore, 1996), p. 13.

Zhang, J. X.

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7282 (1996).
[CrossRef]

Ann. Phys. (N.Y.) (1)

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

Appl. Opt. (7)

Comments At. Mol. Phys. (1)

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonance,” Comments At. Mol. Phys. 23, 175–187 (1989).

IEEE Trans. Antennas Propag. (2)

H. Inada, M. A. Plonus, “The geometric optics contribution to the scattering from a large dense dielectric sphere,” IEEE Trans. Antennas Propag. 18, 89–99 (1970).
[CrossRef]

J. D. Murphy, P. J. Moser, A. Nagl, H. Überall, “A surface wave interpretation for the resonances of a dielectric sphere,” IEEE Trans. Antennas Propag. 28, 924–927 (1980).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface: electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. 23, 134–141 (1975).
[CrossRef]

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

R. Landauer, “Associated Legendre polynomial approximations,” J. Appl. Phys. 22, 87–89 (1951).
[CrossRef]

J. Chem. Phys. (1)

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7282 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Phys. A (1)

M. Robnik, “Quantising a generic family of billiards with analytic boundaries,” J. Phys. A 17, 1049–1074 (1984).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by gaussian particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 55, 577–601 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (1)

R. Y. Chiao, P. G. Kwiat, A. M. Steinberg, “Analogies between electron and photon tunneling,” Phys. Rev. B 175, 257–262 (1991).

Phys. Rev. Lett. (2)

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, R. K. Chang, “Ray chaos and Q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

Proc. IEEE (2)

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

W. D. Wang, G. A. Deschamps, “Application of complex ray tracing to scattering problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

Q. J. Mech. Appl. Math. (1)

D. S. Jones, “Electromagnetic tunnelling,” Q. J. Mech. Appl. Math. 31, 409–434 (1977).

Other (10)

M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, UK, 1993).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Mathematics, L. M. Graves, ed. (McGraw-Hill, New York, 1958), Vol. 8.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

E. S. C. Ching, P. T. Leung, K. Young, “Optical processes in microcavities—the role of quasinormal modes,” in Optical Processes in Microcavities, R. K. Chang, A. J. Campillo, eds. (World Scientific, Singapore, 1996), p. 13.

L. A. Vainshtein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988).

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (11)

Fig. 1
Fig. 1

(a) Refractive and (b) diffractive coupling of internal and external rays. Upper parts, ray-path geometry; lower parts, plots of kr2 as a function of radial position. All drawn rays have the same angular momentum, and thus there are two circular caustics to which the external and internal rays are tangential. If the impact distance b is smaller than the particle radius, there is an evanescent gap (kr imaginary) that the rays have to bridge by tunneling.

Fig. 2
Fig. 2

Wave vector k and the local coordinate system at an arbitrary point (r, η) in polar coordinates for a ray that is tangential to a circle of radius rc.

Fig. 3
Fig. 3

Family of phase-coupled rays and associated wave fronts in reversed shading. Every ray is in phase with its neighbors, because they are normal to the same wave fronts, leading to a closed annular phase pattern.

Fig. 4
Fig. 4

Geometrical determination of the phase at an arbitrary point (r, η). Each point is intersected by two rays. The phase can be determined with respect to a reference point P0 by proceeding to the point of interest along caustic and straight ray segments.

Fig. 5
Fig. 5

Three-dimensional diagram to illustrate the orientation of the plane of propagation (indicated by the two-dimensional phase pattern) within a sphere. The direction of the normal vector n is described by angles ϑn and φn.

Fig. 6
Fig. 6

Illustration of the arrangement of a manifold of plane-ray congruences (lines of intersection with the spherical surface are drawn in reversed shading) to a self-consistent three-dimensional phase distribution. Shown is the phase distribution in an arbitrary radial distance r. The hole on top of the sphere, where the field is evanescent (kϑ is imaginary), allows a look into the sphere. The inner radius is the caustic radius rc=Λ/k. The circumference of the hole has to be an integer number of wavelength m to result in a continuous phase distribution. The depicted situation was calculated with Λ=50 and m=24.

Fig. 7
Fig. 7

Diagram to illustrate the strategy for determining the ϑ dependence of the phase distribution geometrically. The phase progress from point A to point C is determined by proceeding first from A to B and then from B to C along caustics and great-circle segments, where the phase progress per unit length is constant.

Fig. 8
Fig. 8

Plot of the spherical Bessel function j100(kr) for 80<kr<160. Solid curve, exact solution; dashed curve, Debye expansion. The Debye expansion is infinite if the argument and the order increased by 1/2 are identical to kr=l+1/2=100.5. The graph clearly shows the turn from exponential (drawn on a logarithmic scale) to oscillatory behavior (drawn on a linear scale) at x=l+1/2. The inset is a linear plot of j100(kr) for 0<kr<160; dotted–dashed line, kr=100.5.

Fig. 9
Fig. 9

Plot of the Legendre function P6015(cos ϑ) for ϑ in the interval 0.03<ϑ<π/2. Solid curve, exact solution; dashed curve, asymptotic solution. At ϑ=arcsin(15/60)=0.253 the behavior of the function changes from evanescent to oscillatory. Analogously to Figs. 7 and 8, the evanescent interval is plotted on a logarithmic scale. The inset shows the function in the full interval 0<ϑ<π.

Fig. 10
Fig. 10

Illustration of the ray path and polarization geometry of two rays propagating in the (r, η) plane. The field direction for the TE case is given by e and that of TM polarization is given by e for each ray. (a) The two rays have identically signed kη (both rays propagate counterclockwise) but differ in the sign of kr (one ray propagates inward and one outward). The radial dependence of the field produced by these rays is identical to that of the total field (radially standing wave). (b) The rays differ in the sign of kη, whereas the signs of kr are identical. The azimuthal dependence of the field produced by these rays is identical to that of the total field (standing wave in the η direction).

Fig. 11
Fig. 11

Gray-scale plot of |E|2 of the multipole field 60,3 in meridional planes of a dielectric sphere computed by Mie theory and the geometrical optics approximation. TE (φ=π/2) and TM (φ=0) cases are shown.

Equations (90)

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(2+k2)E(r)=0.
(2+k2)f(r)=0.
f(r)=jIj(r)exp[iΦj(r)],
(Φj)2=k2
·(Ij2k)=0,
Φ(P)=Φ(P0)+Ck(r)·dr,
ku=1guΦu,
f=fr(kr)fϑ(ϑ)fφ(φ),
I=IrIϑIφ,Φ=Φr+Φϑ+Φφ.
Φu=Φu(u0)+u0ukugudu;
u(g/gu kuIu2)=0,
Φ=φ+iψ,
(ϕ)2-(ψ)2=k2,
ϕ·ψ=0,
·(I2ϕ)=0
·(I2ψ)=0,
Ir1/kr.
L˜=rcek0=rciki=rck
kη=±L˜r=±rcrk,
kr=k2-kη2=±k2r2-L˜2r=±nk01-rcr2,
cos γ=kηk=L˜kr=rcr,
cos γicos γe=rcirce=neni.
L˜<x<niL˜
L˜=l=0, 1, 2, 3,.
Φ(r, η)=Φ(rc, 0)+krcηc+ks,
ηc=ηarccos(rc/r)=ηγ,
s=±r2-rc2,(rrc).
ηc=ηi arccosh(rc/r),
s=±irc2-r2,(r<rc).
ϕη(η)=lη.
ϕr(kr)=k2r2-l2-l arccoslkr0,
ψr(kr)=0,rrc,
ψr(kr)=l2-k2r2-l arccoshlkr<0,
ϕr(kr)=0,r<rc.
ϕη(η)=η0ηkηrdη=0ηldη,
ϕr(r)=r0rkrdr=rcr k2r2-l2rdr,
krc cos ϑn=m,
cos ϑn=m/Λ=sin ϑc,
kη=Λ/r,kr=k2r2-Λ2r.
kφ=mr sin ϑ.
kϑ=Λ2r2-m2r2 sin2 ϑ1/2=Λr1-sin2 ϑcsin2 ϑ1/2.
ϕr(kr)=k2r2-Λ2-Λ arccos(Λ/kr),
ϕϑ(ϑ)=Λ arccosΛ cos ϑΛ2-m2-m arccosm cot ϑΛ2-m2,
ϕφ(φ)=mφ,
ϕϑ=±mΔφΛΔη,
Δη=arccoscos ϑcos ϑc,Δφ=arccoscot ϑcot ϑc.
Ir(r)(krk2r2-Λ2)-1/2,
Iϑ(ϑ)(Λ2 sin2 ϑ-m2)-1/4,
Iφ=const.
f=frfϑfφexp[i(ϕr-π/4)](krk2r2-Λ2)1/2exp[i(ϕϑ-π/4)]Λ2 sin2 ϑ-m24exp(iϕφ),
fu+Dufu+ku2gu2fu=0,
f=zl(kr)Plm(cos ϑ)exp(imφ),
hl(1)(kr)exp[+i(ϕr-π/4)](krk2r2-Λ2)1/2krΛ.
jl(kr)cos(ϕr-π/4)(krk2r2-Λ2)1/2krΛ12exp(ψr)(krΛ2-k2r2)1/2kr<Λ.
Plm(cos ϑ)
AΛ2-m2Λ2 sin2 ϑ-m24 cosϕϑ-π4sin ϑ>mΛBΛ2-m2m2-Λ2 sin2 ϑ4 exp(ψϑ)sin ϑ<mΛ,
ϕϑ=Λ arccosΛ cos ϑΛ2-m2-m arccosm cot ϑΛ2-m2,
ψϑ=Λ arccoshΛ|cos ϑ|Λ2-m2-m arccoshm|cot ϑ|Λ2-m2,
A=2mπΓ(l/2+m/2+1/2)Γ(l/2-m/2+1),
B=12cos ϑ|cos ϑ|l-mA,
e=(er×ek)kr/Λ,
e=ek×e.
E,rcos γ{exp[i(+ϕr+ϕη)]+exp[i(-ϕr+ϕη)]+exp[i(-ϕr-ϕη)]+exp[i(+ϕr-ϕη)]},
E,ηsin γ{-exp[i(+ϕr+ϕη)]+exp[i(-ϕr+ϕη)]-exp[i(-ϕr-ϕη)]+exp[i(+ϕr-ϕη)]}.
E,rcos γ cos ϕr cos ϕη,
E,ηsin γ sin ϕr sin ϕη.
e=(-kφreϑ+kϑreφ) 1Λ,
e=Λkrer-krkrΛ(kϑeϑ+kφeφ),
kϑ2+kφ2=k2-kr2=kη2=(Λ/r)2.
k-krer=kηeη=kϑeϑ+kφeφ,
Er=0,
Eϑi IrIϑIφΛcos ϕˆr msin ϑcos ϕˆϑ cos ϕφ,
Eφi IrIϑIφΛcos ϕˆrkϑr sin ϕˆϑ sin ϕφ.
ErIrIϑIφΛΛ2krcos ϕˆr cos ϕˆϑ cos ϕφ,
EϑIrIϑIφΛkrksin ϕˆrkϑr sin ϕˆϑ cos ϕφ,
EφIrIϑIφΛkrksin ϕˆr msin ϑcos ϕˆϑ sin ϕφ.
Er=0,
Eϑi 2l+1l(l+1)jl(kr) Pl1(cos ϑ)sin ϑcos φ,
Eφ-i 2l+1l(l+1)jl(kr) dPl1(cos ϑ)dϑsin φ
Er2l+1l(l+1)l(l+1)krj(kr)Pl1(cos ϑ)cos φ,
Eϑ2l+1l(l+1)[jl(kr)kr]krdPl1(cos ϑ)dϑcos φ,
Eφ-2l+1l(l+1)[jl(kr)kr]krPl1(cos ϑ)sin ϑsin φ
Ir cos ϕˆr(kr)jl(kr),
Iϑ cos ϕˆϑPl1(cos ϑ).
[jl(kr)kr]kr=jl(kr)+jl(kr)krjl(kr),
ddu(Iu cos ϕu)-Iu sin ϕukugu,
Ir krksin ϕˆr(kr)-jl(kr),
Iϑkϑr sin ϕˆϑ(ϑ)-dPl1(cos ϑ)dϑ,
Iϑ 1sin ϑcos ϕˆϑ(ϑ)Pl1(cos ϑ)sin ϑ,
2l+1l(l+1)=2ΛΛ2-1/42Λ.

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