Abstract

Mie scattering theory is used to calculate the efficiency factors for absorption by microscopic dielectric spheres. Resonances in the efficiency factors for absorption and resonances in the amplitudes of the electric and magnetic multipoles which occur in an expansion of the fields inside the dielectric sphere are discussed. Several trends in the strengths and width of the various resonances as functions of the absorption coefficient of the sphere, size parameter, multipole order, and multipole resonance number are given. A formal solution for elastic (Mie) and inelastic (Raman) scattering by microscopic particles is derived from the extinction theorem. With this background, some implications of the resonances in the interpretation of absorption, fluorescence, and Raman scattering by microscopic particles are discussed.

© 1978 Optical Society of America

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  1. A. Rosencwaig and A. Gersho, J. Appl. Phys. 47, 64 (1976); H. S. Bennett and R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
    [Crossref]
  2. “Aerosol Measurements: Proceedings of a Conference,” edited by W. A. Cassatt and R. S. Maddock, NBS Special Pub. 412, U.S. Govt. Printing Office (1974); W. A. Bonner, H. R. Hulett, R. G. Sweet, and L. A. Herzenberg, Rev. Sci. Instr. 43, 404 (1972).
    [Crossref]
  3. R. G. Stafford, R. K. Chang, and P. J. Kindlmann, in “Proc. of the 8th Materials Research Symposium on Methods and Standards for Environmental Measurements,” edited by W. H. Kirchhoff, NBS Special Pub. 464, U.S. Govt. Printing Office (1977), p. 659.
  4. H. Rosen and T. Novakov, Nature 266, 708 (1977); R. dos Santos and W. H. Stevenson, Appl. Phys. Lett. 30, 236 (1977).
    [Crossref]
  5. G. J. Rosasco and E. S. Etz, Ind. Res. Dev. 28, 20 (1977).
  6. H. S. Bennett and G. J. Rosasco, Appl. Opt. 17, 491 (1978).
    [Crossref] [PubMed]
  7. R. Fuchs and K. L. Kliewer, J. Opt. Soc. Am. 58, 319 (1968).
    [Crossref]
  8. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  9. M. Kerker, The Scattering of Light (Academic, New York, 1969).
  10. H. C. Van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).
  11. J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” IBM Report No. 320-3237, May1968 (unpublished).
  12. H. Oser (private communication).
  13. Milton Kerker, Clarkson College of Technology, has graciously verified our numerical results for x≤ 10 utilizing his routines, which differ from ours, for Mie scattering calculations (private communication).
  14. H. J. Metz and H. K. Dettmar, Kolloid Z. Z. Polym. 192, 107 (1963).
    [Crossref]
  15. G. W. Kattawar and G. N. Plass, Appl. Opt. 9, 1549 (1967); G. N. Plass, Appl. Opt. 5, 279 (1966).
    [Crossref] [PubMed]
  16. P. Chylek, J. Opt. Soc. Am. 66, 285 (1976).
    [Crossref]
  17. H. M. Nussenzveig, J. Math. Phys. 10, 125 (1969).
    [Crossref]
  18. H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396 (1976).
    [Crossref]
  19. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, N.J., 1960), p. 81.
  20. F. W. J. Olver, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U.S. Government Printing Office, Washington, D.C., 1964), p. 364.
  21. A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 38, 351, 1977.
    [Crossref]
  22. H. S. Bennett and G. J. Rosasco, “Heating microscopic particles with laser beams,” J. Appl. Phys. 49, 640 (1978).
    [Crossref]
  23. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  24. A. B. Bhatia and W. J. Noble, Proc. R. Soc. London A220, 356 (1953).
  25. G. J. Rosasco, Thesis, Fordham Univ., 1970 (Univ. Microfilms, Ann Arbor, Michigan) (unpublished).
  26. P. W. Barber, IEEE Trans. Microwave Theory Tech. MTT-25, 373 (1977).
    [Crossref]
  27. R. Ruppin and R. Engleman, Rep. Prog. Phys. 33, 149 (1970).
    [Crossref]
  28. R. Ruppin, J. Phys. C 8, 1969 (1975).
    [Crossref]

1978 (2)

H. S. Bennett and G. J. Rosasco, Appl. Opt. 17, 491 (1978).
[Crossref] [PubMed]

H. S. Bennett and G. J. Rosasco, “Heating microscopic particles with laser beams,” J. Appl. Phys. 49, 640 (1978).
[Crossref]

1977 (4)

A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 38, 351, 1977.
[Crossref]

P. W. Barber, IEEE Trans. Microwave Theory Tech. MTT-25, 373 (1977).
[Crossref]

H. Rosen and T. Novakov, Nature 266, 708 (1977); R. dos Santos and W. H. Stevenson, Appl. Phys. Lett. 30, 236 (1977).
[Crossref]

G. J. Rosasco and E. S. Etz, Ind. Res. Dev. 28, 20 (1977).

1976 (3)

A. Rosencwaig and A. Gersho, J. Appl. Phys. 47, 64 (1976); H. S. Bennett and R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[Crossref]

P. Chylek, J. Opt. Soc. Am. 66, 285 (1976).
[Crossref]

H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396 (1976).
[Crossref]

1975 (1)

R. Ruppin, J. Phys. C 8, 1969 (1975).
[Crossref]

1970 (1)

R. Ruppin and R. Engleman, Rep. Prog. Phys. 33, 149 (1970).
[Crossref]

1969 (1)

H. M. Nussenzveig, J. Math. Phys. 10, 125 (1969).
[Crossref]

1968 (1)

1967 (1)

G. W. Kattawar and G. N. Plass, Appl. Opt. 9, 1549 (1967); G. N. Plass, Appl. Opt. 5, 279 (1966).
[Crossref] [PubMed]

1963 (1)

H. J. Metz and H. K. Dettmar, Kolloid Z. Z. Polym. 192, 107 (1963).
[Crossref]

1953 (1)

A. B. Bhatia and W. J. Noble, Proc. R. Soc. London A220, 356 (1953).

Ashkin, A.

A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 38, 351, 1977.
[Crossref]

Barber, P. W.

P. W. Barber, IEEE Trans. Microwave Theory Tech. MTT-25, 373 (1977).
[Crossref]

Bennett, H. S.

H. S. Bennett and G. J. Rosasco, “Heating microscopic particles with laser beams,” J. Appl. Phys. 49, 640 (1978).
[Crossref]

H. S. Bennett and G. J. Rosasco, Appl. Opt. 17, 491 (1978).
[Crossref] [PubMed]

Bhatia, A. B.

A. B. Bhatia and W. J. Noble, Proc. R. Soc. London A220, 356 (1953).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Chang, R. K.

R. G. Stafford, R. K. Chang, and P. J. Kindlmann, in “Proc. of the 8th Materials Research Symposium on Methods and Standards for Environmental Measurements,” edited by W. H. Kirchhoff, NBS Special Pub. 464, U.S. Govt. Printing Office (1977), p. 659.

Chew, H.

H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396 (1976).
[Crossref]

Chylek, P.

Dave, J. V.

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” IBM Report No. 320-3237, May1968 (unpublished).

Dettmar, H. K.

H. J. Metz and H. K. Dettmar, Kolloid Z. Z. Polym. 192, 107 (1963).
[Crossref]

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 38, 351, 1977.
[Crossref]

Edmonds, R.

R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, N.J., 1960), p. 81.

Engleman, R.

R. Ruppin and R. Engleman, Rep. Prog. Phys. 33, 149 (1970).
[Crossref]

Etz, E. S.

G. J. Rosasco and E. S. Etz, Ind. Res. Dev. 28, 20 (1977).

Fuchs, R.

Gersho, A.

A. Rosencwaig and A. Gersho, J. Appl. Phys. 47, 64 (1976); H. S. Bennett and R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[Crossref]

Kattawar, G. W.

G. W. Kattawar and G. N. Plass, Appl. Opt. 9, 1549 (1967); G. N. Plass, Appl. Opt. 5, 279 (1966).
[Crossref] [PubMed]

Kerker, M.

H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396 (1976).
[Crossref]

M. Kerker, The Scattering of Light (Academic, New York, 1969).

Kerker, Milton

Milton Kerker, Clarkson College of Technology, has graciously verified our numerical results for x≤ 10 utilizing his routines, which differ from ours, for Mie scattering calculations (private communication).

Kindlmann, P. J.

R. G. Stafford, R. K. Chang, and P. J. Kindlmann, in “Proc. of the 8th Materials Research Symposium on Methods and Standards for Environmental Measurements,” edited by W. H. Kirchhoff, NBS Special Pub. 464, U.S. Govt. Printing Office (1977), p. 659.

Kliewer, K. L.

McNulty, P. J.

H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396 (1976).
[Crossref]

Metz, H. J.

H. J. Metz and H. K. Dettmar, Kolloid Z. Z. Polym. 192, 107 (1963).
[Crossref]

Noble, W. J.

A. B. Bhatia and W. J. Noble, Proc. R. Soc. London A220, 356 (1953).

Novakov, T.

H. Rosen and T. Novakov, Nature 266, 708 (1977); R. dos Santos and W. H. Stevenson, Appl. Phys. Lett. 30, 236 (1977).
[Crossref]

Nussenzveig, H. M.

H. M. Nussenzveig, J. Math. Phys. 10, 125 (1969).
[Crossref]

Olver, F. W. J.

F. W. J. Olver, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U.S. Government Printing Office, Washington, D.C., 1964), p. 364.

Oser, H.

H. Oser (private communication).

Plass, G. N.

G. W. Kattawar and G. N. Plass, Appl. Opt. 9, 1549 (1967); G. N. Plass, Appl. Opt. 5, 279 (1966).
[Crossref] [PubMed]

Rosasco, G. J.

H. S. Bennett and G. J. Rosasco, “Heating microscopic particles with laser beams,” J. Appl. Phys. 49, 640 (1978).
[Crossref]

H. S. Bennett and G. J. Rosasco, Appl. Opt. 17, 491 (1978).
[Crossref] [PubMed]

G. J. Rosasco and E. S. Etz, Ind. Res. Dev. 28, 20 (1977).

G. J. Rosasco, Thesis, Fordham Univ., 1970 (Univ. Microfilms, Ann Arbor, Michigan) (unpublished).

Rosen, H.

H. Rosen and T. Novakov, Nature 266, 708 (1977); R. dos Santos and W. H. Stevenson, Appl. Phys. Lett. 30, 236 (1977).
[Crossref]

Rosencwaig, A.

A. Rosencwaig and A. Gersho, J. Appl. Phys. 47, 64 (1976); H. S. Bennett and R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[Crossref]

Ruppin, R.

R. Ruppin, J. Phys. C 8, 1969 (1975).
[Crossref]

R. Ruppin and R. Engleman, Rep. Prog. Phys. 33, 149 (1970).
[Crossref]

Stafford, R. G.

R. G. Stafford, R. K. Chang, and P. J. Kindlmann, in “Proc. of the 8th Materials Research Symposium on Methods and Standards for Environmental Measurements,” edited by W. H. Kirchhoff, NBS Special Pub. 464, U.S. Govt. Printing Office (1977), p. 659.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Van de Hulst, H. C.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Appl. Opt. (2)

H. S. Bennett and G. J. Rosasco, Appl. Opt. 17, 491 (1978).
[Crossref] [PubMed]

G. W. Kattawar and G. N. Plass, Appl. Opt. 9, 1549 (1967); G. N. Plass, Appl. Opt. 5, 279 (1966).
[Crossref] [PubMed]

IEEE Trans. Microwave Theory Tech. (1)

P. W. Barber, IEEE Trans. Microwave Theory Tech. MTT-25, 373 (1977).
[Crossref]

Ind. Res. Dev. (1)

G. J. Rosasco and E. S. Etz, Ind. Res. Dev. 28, 20 (1977).

J. Appl. Phys. (2)

A. Rosencwaig and A. Gersho, J. Appl. Phys. 47, 64 (1976); H. S. Bennett and R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[Crossref]

H. S. Bennett and G. J. Rosasco, “Heating microscopic particles with laser beams,” J. Appl. Phys. 49, 640 (1978).
[Crossref]

J. Math. Phys. (1)

H. M. Nussenzveig, J. Math. Phys. 10, 125 (1969).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. C (1)

R. Ruppin, J. Phys. C 8, 1969 (1975).
[Crossref]

Kolloid Z. Z. Polym. (1)

H. J. Metz and H. K. Dettmar, Kolloid Z. Z. Polym. 192, 107 (1963).
[Crossref]

Nature (1)

H. Rosen and T. Novakov, Nature 266, 708 (1977); R. dos Santos and W. H. Stevenson, Appl. Phys. Lett. 30, 236 (1977).
[Crossref]

Phys. Rev. A (1)

H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396 (1976).
[Crossref]

Phys. Rev. Lett. (1)

A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 38, 351, 1977.
[Crossref]

Proc. R. Soc. London (1)

A. B. Bhatia and W. J. Noble, Proc. R. Soc. London A220, 356 (1953).

Rep. Prog. Phys. (1)

R. Ruppin and R. Engleman, Rep. Prog. Phys. 33, 149 (1970).
[Crossref]

Other (12)

G. J. Rosasco, Thesis, Fordham Univ., 1970 (Univ. Microfilms, Ann Arbor, Michigan) (unpublished).

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, N.J., 1960), p. 81.

F. W. J. Olver, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U.S. Government Printing Office, Washington, D.C., 1964), p. 364.

“Aerosol Measurements: Proceedings of a Conference,” edited by W. A. Cassatt and R. S. Maddock, NBS Special Pub. 412, U.S. Govt. Printing Office (1974); W. A. Bonner, H. R. Hulett, R. G. Sweet, and L. A. Herzenberg, Rev. Sci. Instr. 43, 404 (1972).
[Crossref]

R. G. Stafford, R. K. Chang, and P. J. Kindlmann, in “Proc. of the 8th Materials Research Symposium on Methods and Standards for Environmental Measurements,” edited by W. H. Kirchhoff, NBS Special Pub. 464, U.S. Govt. Printing Office (1977), p. 659.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

M. Kerker, The Scattering of Light (Academic, New York, 1969).

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

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” IBM Report No. 320-3237, May1968 (unpublished).

H. Oser (private communication).

Milton Kerker, Clarkson College of Technology, has graciously verified our numerical results for x≤ 10 utilizing his routines, which differ from ours, for Mie scattering calculations (private communication).

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Figures (7)

FIG. 1
FIG. 1

Efficiency factors for scattering and absorption as a function of the absorption coefficient for a microparticle. The complex refractive index is m and the grid for the size parameter, x, is x = 0.2 + (100.0 − 0.2) (l − 1)/499, where 1 ≤ l ≤ 500 and the grid increment Δx = 0.2. The solid curve is the efficiency factor for scattering, Qs, and the dashed curve is the scaled efficiency factor for absorption (1000/αp)Qa, where αp is the absorption coefficient of the microparticle in cm−1. All other quantities are dimensionless. These plots accurately represent the base-line trends, however, only a few resonances appear for this size grid increment (see text).

FIG. 2
FIG. 2

Electric and magnetic multipole contributions to the efficiency factor for absorption. The complex refractive index is m and the grid for the size parameter, x, is x = 2.5 + (4.91 − 2.5)(l − 1)/241, where 1 ≤ l ≤ 242 and the grid increment is Δx = 0.01. The solid curve is the efficiency factor for absorption, Qa; the broken solid curve is the contribution from electric multipoles, Qel; the dashed curve is the contribution from magnetic multipoles, Qmag; and Qa = Qel + Qmag. The absorption coefficient of the microparticle, αp, is 1000 cm−1. All other quantities are dimensionless. The numbers adjacent to the arrows indicate the multipole order, n, whose first resonance contributes most to the peak. For example, the first peak shown arises essentially from the n = 4 magnetic multipole term.

FIG. 3
FIG. 3

Normalized efficiency factors of absorption as a function of the absorption coefficient for the microparticle. The complex refractive index is m and the grid for the size parameter is x = 7.141 + (7.179 − 7.141) (l − 1)/76, where 1 ≤ l ≤ 77, Δx = 0.0005. The normalized efficiency factor for absorption when xminxxmax is Qa(x, norm) = Qa(x)/Qa(max; xmin,xmax), where Qa(x) is the efficiency factor for absorption and Qa(max;xmin,xmax) is the maximum value of Qa(x) in the interval xminxxmax.

FIG. 4
FIG. 4

Width at one-half the maximum value as a function of the absorption coefficients. The width (in size parameter from Fig. 3), Δx1/2, at one-half the maximum value of the efficiency factor for absorpion when the size parameter x = 7.16043 and when the real part of the refractive index m = 2.2 is plotted as a function of the absorption coefficient, αp, in the range 1 cm−1αp ≤ 500 cm−1. This particular peak in the efficiency factor for absorption corresponds to the first resonance at x = 7.16043 of the n = 11 electric multipole term.

FIG. 5
FIG. 5

Location of the maxima for the electric multipole coefficient, {1/En(ρ1,ρ2,m)}. The ordinate, mx, gives the value of mx for which En(ρ1,ρ2,m) has a minimum value and the abscissa gives the order of the electric multipole demoninator. The dimensionless quantities ρ1 = mx, ρ2 = x where x is the size parameter and m is the refractive index pure real in this example. The solid curve corresponds to the case for which m = 1.5 and the dashed curve corresponds to the case for which m = 2.2. The lines with slopes in the mx-n plane of 1.74 and 2.53 give the locations where the periodicity in mx for the resonances changes significantly. Refer to the text for a discussion of the increments in size parameter used in this calculation.

FIG. 6
FIG. 6

Peak heights and widths of the first two magnetic resonances in the multipole coefficient, {1/Mn}, as a function of multipole order, n. All quantities are dimensionless. The heights of the first and second resonance peaks of order n are denoted, respectively, by lm(n, 1) and lm(n,2) and are given, respectively, by dots (·) and pluses (+). The widths in terms of size parameter of the first and second resonance peaks at (½) maximum value relative to the base-line trend are denoted, respectively, by Γm(n, 1) and Γm(n, 2) and are given, respectively, by · and +. Refer to the text for a discussion of the increments in size parameter used in this calculation.

FIG. 7
FIG. 7

The electric multipole coefficients, {1/En(ρ1,ρ2,m)}, and the magnetic multipole coefficients, {1/Mn(ρ1,ρ2,m)}, as functions of size parameter, x, for the value m = 2.2. The grid for the size parameter is x = 0.1 + (10.1 − 0.1) [(l − 1)/100] where 1 ≤ l ≤ 101, Δx = 0.1. The top curves correspond to n = 3 and the bottom curves correspond to n = 8. The electric multipole coefficients are given by the solid curves and the magnetic multipole coefficients are given by the dashed curves. All quantities are dimensionless. The first peaks on the left for the n = 8 case are truncated. The height of the first peak for {1/M8} (dashed) is 7.8 and the height of the first peak for {1/E8} (solid) is 5.5.

Equations (46)

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Q e = 2 x 2 n = 0 ( 2 n + 1 ) ( Re { a n } + Re { b n } ) ,
Q s = 2 x 2 n = 0 ( 2 n + 1 ) ( a n 2 + b n 2 ) ,
Q a = Q e - Q s ,
Q a ( x , norm ) = Q a ( x ) / Q a ( max ; x min , x max ) .
E in = n , μ { ( i c / m 1 2 ω 0 ) γ E ( n , μ ) × [ j n ( ρ , η ) Y n n μ ( θ , ϕ ) ] + γ M ( n , μ ) j n ( ρ , η ) Y n n μ ( θ , ϕ ) }
B in = n , μ { γ E ( n , μ ) j n ( ρ , η ) Y n n μ ( θ , ϕ ) - ( i c / ω 0 ) γ M ( n , μ ) × [ j n ( ρ , η ) Y n n μ ( θ , ϕ ) ] } ,
γ E ( n , μ ) = ( - i m 1 2 / ρ 2 ) α E ( n , μ ) / E n ( ρ 1 , ρ 2 , m )
γ M ( n , μ ) = ( - i / ρ 2 ) α M ( n , μ ) / M n ( ρ 1 , ρ 2 , m ) ,
α E ( n , μ ) = ± i α M ( n , μ )
α M ( n , μ ) = i n [ 4 π ( 2 n + 1 ) ] 1 / 2 δ μ , ± 1 ,
E n ( ρ 1 , ρ 2 , m ) = h n ( 1 ) ( ρ 2 ) ( d / d ρ 1 ) [ ρ 1 j n ( ρ 1 ) ] - m 2 j n ( ρ 1 ) ( d / d ρ 2 ) [ ρ 2 h n ( 1 ) ( ρ 2 ) ]
M n ( ρ 1 , ρ 2 , m ) = h n ( 1 ) ( ρ 2 ) ( d / d ρ 1 ) [ ρ 1 j n ( ρ 1 ) ] - j n ( ρ 1 ) ( d / d ρ 2 ) [ ρ 2 h n ( 1 ) ( ρ 2 ) ] ,
lim x 0 E n ( m x , x , m ) = - i m n [ ( n + 1 ) + n m 2 ] 2 n + 1
lim x 0 M n ( m x , x , m ) = i m n + 1 ,
E n ( m x , x , m ) ~ x e i σ ( - i cos ν - m sin ν ) = E n ( m x )
M n ( m x , x , m ) ~ x e i σ ( - i m cos ν - sin ν ) = M n ( m x )
E 0 ( m x , x , m ) = E 0 ( m x )
M 0 ( m x , x , m ) = M 0 ( m x ) .
Im { Z k } = x i 0 - ( m 2 / m 1 ) x r 0
Re { Z k } = x r 0 - ( m 2 / m 1 ) · { ( m 1 2 - 1 ) - 1 - ( x i 0 / m 1 ) }
x i 0 = lim m 2 0 Im { Z k } = ( - 1 2 ) m 1 ln { ( m 1 + 1 ) / ( m 1 - 1 ) }
x r 0 = lim m 2 0 Re { Z k } = k π / 2 m 1 ,
F ( ρ ) = 2 - 4 sin ρ ρ + 4 ( 1 - cos ρ ) ρ 2
ρ = 2 x ( m - 1 )
Q a = ( 4 / 3 ) ( λ 0 / 2 π ) α p x
ν ˜ 0 = 1 / λ 0 19 436 cm - 1
E ( i ) = A ( i ) ( r ) e - i ω 0 t .
A ( i ) ( r ) + curl curl Σ ( Q ( r ) G ν - G Q ( r ) ν ) d S = 0 ,
G = [ exp ( i k 0 R / R ) ]
k 0 = ω 0 / c
R = r - r ;
E = P / N α = ( 4 π / 3 ) ( m 2 + 2 ) k 0 2 Q e - i ω t ,
( 4 π / 3 ) N α = ( m 2 - 1 ) / ( m 2 + 2 ) .
E in = 4 π k 0 2 Q e - i ω 0 t
E out = E ( i ) + E ( s )
E ( s ) = A ( s ) e - i ω 0 t
A ( s ) = curl curl Σ ( Q G ν - G Q ν ) d S .
P = N ( α + Δ α e - i ω p t ) [ E ( i ) + E ( d ) ]
E d = σ Σ curl curl P ( r , t - R / c ) R d V ,
Q = Q 0 ( r ) e - i ω 0 t + Δ Q ( r ) e - i ω s t ,
Δ α N α 2 ( m 2 - 1 ) k 0 2 Q 0 + curl curl × Σ ( Δ Q G s ν - G s Δ Q ν ) d S = 0
G s = e i k s R / R , k s = ω s / c ,
ω s = ω 0 + ω p .
( Δ α / N α 2 ) ( m 2 - 1 ) k 0 2 Q = ( Δ α / α ) E 0 ,
E out = E ( i ) + E 0 ( s ) + Δ E ( s ) ,
Δ A ( s ) = curl curl Σ ( Δ Q G s ν - G s Δ Q ν ) d S .