Abstract

Differential light scattering techniques appear to represent an attractive physical method for the rapid identification of various microorganisms. Certain general results of inverse scattering theory suggest strongly that characteristic of each distinct microorganism that scatters light is an essentially unique scattering pattern, i.e., unique differential scattered intensity and polarization. Although a mathematically rigorous inversion procedure seems impractical at this time, the use of detailed differential scattered intensity data as an identification fingerprint shows considerable. promise. Published measurements on nonbiological scatterers confirm this possibility. A variety of calculations are presented that contrast the expected scattering characteristics of certain microorganisms such as Bacillus subtilis, B. anthracis, Staphylococcus epidermidis, S. aureus, Escherichia coli, and the spores of B. megaterium and B. cereus. Experimental and instrumentation difficulties and possible procedures are discussed. A review and laboration of some applicable features of Rayleigh-Gans scattering are included as an appendix.

© 1968 Optical Society of America

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1966

L. D. Maxim, A. Klein, M. E. Meyer, C. H. Kuist, Amer. Chem. Soc., Div. Polymer Chem., Preprints 7, 783 (1966).

S. H. Maron, P. E. Pierce, Amer. Chem. Soc., Div. Polymer. Chem., Preprints 7, 773 (1966).

H. G. Jerrard, B. R. Jennings, Amer. Chem. Soc., Div. Polymer Chem., Preprints 7, 1184 (1966).

D. K. Carpenter, J. Polym. Sci., Pt. A-2, 4, 923 (1966).
[CrossRef]

J. P. Kratohvil, Anal. Chem. 38, 517 (1966).
[CrossRef]

W. A. Farone, W. Kerker, J. Opt. Soc. Amer. 56, 481 (1966).
[CrossRef]

R. Mireles, J. Math. Phys. 45, 179 (1966).

1965

N. A. Logan, Proc. IEEE 53, 773 (1965).
[CrossRef]

V. I. Klenin, Biofizika 10, 387 (1965).
[PubMed]

V. K. Petukhov, Biofizika 10, 993 (1965).
[PubMed]

G. Hind, A. T. Jagendorf, J. Biol. Chem. 240, 3195 (1965).

G. Hind, A. T. Jagendorf, J. Biol. Chem. 240, 3202 (1965).

I. R. Gibbons, J. Cell Biol. 26, 707 (1965).
[CrossRef] [PubMed]

1964

J. P. Kratohvil, Anal. Chem. 36, 458 (1964).
[CrossRef]

R. A. Dilley, L. P. Vernon, Biochem. 3, 817 (1964).
[CrossRef]

P. Gerhardt, E. Ribi, J. Bacteriol. 88, 1774 (1964).

R. W. Hart, E. P. Gray, J. Appl. Phys. 35, 1408 (1964).
[CrossRef]

D. Arnush, IEEE Trans. Antennas and Prop. AP12, 86 (1964).
[CrossRef]

1963

P. J. Wyatt, J. Appl. Phys. 34, 2078 (1963).
[CrossRef]

M. Kerker, W. A. Farone, E. Matijevic, J. Opt. Soc. Amer. 53, 758 (1963).
[CrossRef]

L. D. Faddeyev, J. Math. Phys. 4, 72 (1963).
[CrossRef]

L. Packer, Biochem. Biophys. Acta 75, 12 (1963).
[CrossRef]

L. Packer, R. H. Marchant, Y. Mukohata, Biochem. Biophys. Acta 75, 23 (1963).
[CrossRef]

B. A. Fikhman, Biophys. 8, 441 (1963).

H. Wesslau, Makromol. Chem. 69, 213 (1963).
[CrossRef]

1962

P. J. Wyatt, Phys. Rev. 127, 1837 (1962); Phys. Rev. 134, AB1 (1964).
[CrossRef]

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

J. G. Negi, Geophys. 27, 480 (1962).
[CrossRef]

1961

J. Shmoys, J. Appl. Phys. 32, 689 (1961).
[CrossRef]

F. T. Gucker, J. J. Egan, J. Coll. Sci. 16, 68 (1961).
[CrossRef]

A. L. Koch, Biochem. Biophys. Acta 51, 429 (1961).
[CrossRef]

L. Packer, M. Perry, Arch. Biochein. Biophys. 95, 379 (1961).
[CrossRef]

G. S. Gotterer, T. E. Thompson, A. L. Lehninger, J. Biophys. Biochem. Cytol. 10, 15 (1961).
[CrossRef] [PubMed]

1958

C. T. Tai, Appl. Sci. Res. Sect. B7, 113 (1958).
[CrossRef]

1957

B. H. Mayall, C. F. Robinow, J. Appl. Bacteriol. 20, 333 (1957).

K. F. A. Ross. Quart J. Microscop. Sci. 98, 435 (1957).

K. F. A. Ross, E. Billing, J. Gen. Microbiol. 16, 418. (1957).
[PubMed]

J. G. Franklin, D. E. Bradley, J. Appl. Bacteriol. 20, 467 (1957).

1955

Y. Nomura, K. Takaku, Sci. Rep. Res. Inst. of Tohoku Univ. 7, 107 (1955).

1954

H. G. Davies, J. H. F. Wilkins, J. Cahyen, L. F. LaCour, Quart. J. Microscop. Sci. 95, 271 (1954).

R. Barer, S. Joseph, Quart. J. Microscop. Sci. 95, 399 (1954).

1953

R. Barer, Nature 172, 1097 (1953).
[CrossRef] [PubMed]

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature 171, 720 (1953).
[CrossRef] [PubMed]

1952

It. Barer, K. F. A. Ross, Proc. Physiol. Soc. 118, 38 (1952).

R. Barer, Nature 169, 366 (1952).
[CrossRef] [PubMed]

1950

P. Doty, R. F. Steiner, J. Chem. Phys. 18, 1211 (1950)
[CrossRef]

1928

L. N. G. Filon, Proc. Roy. Soc. Edinburgh 49, 38 (1928).

1908

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

1890

L. V. Lorenz, Videnski. Selsk. Skrifter 6, 1 (1890). [In Danish. Translated into French in 1896. Oeuvres sciéntifiques de L. Lorenz. Libraire Lehmann. Reprinted 1964 Johnson, New York. pp 405–502. Notes by H. Valentines, pp 503–29.]

Arnush, D.

D. Arnush, IEEE Trans. Antennas and Prop. AP12, 86 (1964).
[CrossRef]

Barer, It.

It. Barer, K. F. A. Ross, Proc. Physiol. Soc. 118, 38 (1952).

Barer, R.

R. Barer, S. Joseph, Quart. J. Microscop. Sci. 95, 399 (1954).

R. Barer, Nature 172, 1097 (1953).
[CrossRef] [PubMed]

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature 171, 720 (1953).
[CrossRef] [PubMed]

R. Barer, Nature 169, 366 (1952).
[CrossRef] [PubMed]

Bell, R. J. T.

R. J. T. Bell, Coordinate Solid Geometry (Macmillan and Company, Ltd., London, 1948).

Billing, E.

K. F. A. Ross, E. Billing, J. Gen. Microbiol. 16, 418. (1957).
[PubMed]

Bradley, D. E.

J. G. Franklin, D. E. Bradley, J. Appl. Bacteriol. 20, 467 (1957).

Cahyen, J.

H. G. Davies, J. H. F. Wilkins, J. Cahyen, L. F. LaCour, Quart. J. Microscop. Sci. 95, 271 (1954).

Carpenter, D. K.

D. K. Carpenter, J. Polym. Sci., Pt. A-2, 4, 923 (1966).
[CrossRef]

Chamot, E. M.

E. M. Chamot, C. W. Mason, Handbook of Chemical Microscopy (Chapman and Hall, Ltd., London, 1938). Vol. 1.

Crozier, D.

D. Crozier, U. S. Army Medical Unit, Walter Reed Army Medical Center; private communication (1967).

Davies, H. G.

H. G. Davies, J. H. F. Wilkins, J. Cahyen, L. F. LaCour, Quart. J. Microscop. Sci. 95, 271 (1954).

Dilley, R. A.

R. A. Dilley, L. P. Vernon, Biochem. 3, 817 (1964).
[CrossRef]

Ditchburn, R. W.

R. W. Ditchburn, Light (Blackie & Sons, Ltd., Glasgow, 1953).

Doty, P.

P. Doty, R. F. Steiner, J. Chem. Phys. 18, 1211 (1950)
[CrossRef]

Egan, J. J.

F. T. Gucker, J. J. Egan, J. Coll. Sci. 16, 68 (1961).
[CrossRef]

Faddeyev, L. D.

L. D. Faddeyev, J. Math. Phys. 4, 72 (1963).
[CrossRef]

Farone, W. A.

W. A. Farone, W. Kerker, J. Opt. Soc. Amer. 56, 481 (1966).
[CrossRef]

M. Kerker, W. A. Farone, E. Matijevic, J. Opt. Soc. Amer. 53, 758 (1963).
[CrossRef]

Fikhman, B. A.

B. A. Fikhman, Biophys. 8, 441 (1963).

Filon, L. N. G.

L. N. G. Filon, Proc. Roy. Soc. Edinburgh 49, 38 (1928).

Franklin, J. G.

J. G. Franklin, D. E. Bradley, J. Appl. Bacteriol. 20, 467 (1957).

Gerhardt, P.

P. Gerhardt, E. Ribi, J. Bacteriol. 88, 1774 (1964).

Gibbons, I. R.

I. R. Gibbons, J. Cell Biol. 26, 707 (1965).
[CrossRef] [PubMed]

Gotterer, G. S.

G. S. Gotterer, T. E. Thompson, A. L. Lehninger, J. Biophys. Biochem. Cytol. 10, 15 (1961).
[CrossRef] [PubMed]

Gray, E. P.

R. W. Hart, E. P. Gray, J. Appl. Phys. 35, 1408 (1964).
[CrossRef]

Green, S. L.

S. L. Green, Algebraic Solid Geometry: An Introduction (Cambridge University Press, Cambridge, 1941).

Gucker, F. T.

F. T. Gucker, J. J. Egan, J. Coll. Sci. 16, 68 (1961).
[CrossRef]

Halvorson, H. O.

A. Sussman, H. O. Halvorson, Spores: Their Dormancy and Germination (Harper & Rowe, Inc., New York, 1966).

Hart, R. W.

R. W. Hart, E. P. Gray, J. Appl. Phys. 35, 1408 (1964).
[CrossRef]

Heller, W.

W. Heller, “Apparatus for determining the shape of colloidal particles using light scattering,” U.S. Pat. No. 3,296,446 (1967).

Hind, G.

G. Hind, A. T. Jagendorf, J. Biol. Chem. 240, 3195 (1965).

G. Hind, A. T. Jagendorf, J. Biol. Chem. 240, 3202 (1965).

Jagendorf, A. T.

G. Hind, A. T. Jagendorf, J. Biol. Chem. 240, 3202 (1965).

G. Hind, A. T. Jagendorf, J. Biol. Chem. 240, 3195 (1965).

Jennings, B. R.

H. G. Jerrard, B. R. Jennings, Amer. Chem. Soc., Div. Polymer Chem., Preprints 7, 1184 (1966).

Jerrard, H. G.

H. G. Jerrard, B. R. Jennings, Amer. Chem. Soc., Div. Polymer Chem., Preprints 7, 1184 (1966).

Joseph, S.

R. Barer, S. Joseph, Quart. J. Microscop. Sci. 95, 399 (1954).

Kerker, M.

M. Kerker, W. A. Farone, E. Matijevic, J. Opt. Soc. Amer. 53, 758 (1963).
[CrossRef]

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

Kerker, W.

W. A. Farone, W. Kerker, J. Opt. Soc. Amer. 56, 481 (1966).
[CrossRef]

Klein, A.

L. D. Maxim, A. Klein, M. E. Meyer, C. H. Kuist, Amer. Chem. Soc., Div. Polymer Chem., Preprints 7, 783 (1966).

Klenin, V. I.

V. I. Klenin, Biofizika 10, 387 (1965).
[PubMed]

Koch, A. L.

A. L. Koch, Biochem. Biophys. Acta 51, 429 (1961).
[CrossRef]

Kratohvil, J. P.

J. P. Kratohvil, Anal. Chem. 38, 517 (1966).
[CrossRef]

J. P. Kratohvil, Anal. Chem. 36, 458 (1964).
[CrossRef]

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

Kuist, C. H.

L. D. Maxim, A. Klein, M. E. Meyer, C. H. Kuist, Amer. Chem. Soc., Div. Polymer Chem., Preprints 7, 783 (1966).

LaCour, L. F.

H. G. Davies, J. H. F. Wilkins, J. Cahyen, L. F. LaCour, Quart. J. Microscop. Sci. 95, 271 (1954).

Lehninger, A. L.

G. S. Gotterer, T. E. Thompson, A. L. Lehninger, J. Biophys. Biochem. Cytol. 10, 15 (1961).
[CrossRef] [PubMed]

Logan, N. A.

N. A. Logan, Proc. IEEE 53, 773 (1965).
[CrossRef]

Lorenz, L. V.

L. V. Lorenz, Videnski. Selsk. Skrifter 6, 1 (1890). [In Danish. Translated into French in 1896. Oeuvres sciéntifiques de L. Lorenz. Libraire Lehmann. Reprinted 1964 Johnson, New York. pp 405–502. Notes by H. Valentines, pp 503–29.]

Luria, S. E.

S. E. Luria, in Ref. 42, Chap. 1.

Marchant, R. H.

L. Packer, R. H. Marchant, Y. Mukohata, Biochem. Biophys. Acta 75, 23 (1963).
[CrossRef]

Maron, S. H.

S. H. Maron, P. E. Pierce, Amer. Chem. Soc., Div. Polymer. Chem., Preprints 7, 773 (1966).

Mason, C. W.

E. M. Chamot, C. W. Mason, Handbook of Chemical Microscopy (Chapman and Hall, Ltd., London, 1938). Vol. 1.

Matijevic, E.

M. Kerker, W. A. Farone, E. Matijevic, J. Opt. Soc. Amer. 53, 758 (1963).
[CrossRef]

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

Maxim, L. D.

L. D. Maxim, A. Klein, M. E. Meyer, C. H. Kuist, Amer. Chem. Soc., Div. Polymer Chem., Preprints 7, 783 (1966).

Mayall, B. H.

B. H. Mayall, C. F. Robinow, J. Appl. Bacteriol. 20, 333 (1957).

Meyer, M. E.

L. D. Maxim, A. Klein, M. E. Meyer, C. H. Kuist, Amer. Chem. Soc., Div. Polymer Chem., Preprints 7, 783 (1966).

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Mireles, R.

R. Mireles, J. Math. Phys. 45, 179 (1966).

Mukohata, Y.

L. Packer, R. H. Marchant, Y. Mukohata, Biochem. Biophys. Acta 75, 23 (1963).
[CrossRef]

Murray, R. G. E.

R. G. E. Murray, Ref. 42, Chap. 2.

Negi, J. G.

J. G. Negi, Geophys. 27, 480 (1962).
[CrossRef]

Nomura, Y.

Y. Nomura, K. Takaku, Sci. Rep. Res. Inst. of Tohoku Univ. 7, 107 (1955).

Packer, L.

L. Packer, Biochem. Biophys. Acta 75, 12 (1963).
[CrossRef]

L. Packer, R. H. Marchant, Y. Mukohata, Biochem. Biophys. Acta 75, 23 (1963).
[CrossRef]

L. Packer, M. Perry, Arch. Biochein. Biophys. 95, 379 (1961).
[CrossRef]

Perry, M.

L. Packer, M. Perry, Arch. Biochein. Biophys. 95, 379 (1961).
[CrossRef]

Petukhov, V. K.

V. K. Petukhov, Biofizika 10, 993 (1965).
[PubMed]

Pierce, P. E.

S. H. Maron, P. E. Pierce, Amer. Chem. Soc., Div. Polymer. Chem., Preprints 7, 773 (1966).

Rayleigh,

J. W. Strutt, Rayleigh, Scientific Papers (Dover, New York, 1964).

Ribi, E.

P. Gerhardt, E. Ribi, J. Bacteriol. 88, 1774 (1964).

Robinow, C. F.

B. H. Mayall, C. F. Robinow, J. Appl. Bacteriol. 20, 333 (1957).

C. F. Robinow, in The Bacteria: a Treatise on Structure and Function, Vol. I, Structure, I. C. Gunsalus, R. Y. Stanier, eds., (Academic Press Inc., New York, 1960), Chap. 5.

Ross, K. F. A.

K. F. A. Ross, E. Billing, J. Gen. Microbiol. 16, 418. (1957).
[PubMed]

K. F. A. Ross. Quart J. Microscop. Sci. 98, 435 (1957).

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature 171, 720 (1953).
[CrossRef] [PubMed]

It. Barer, K. F. A. Ross, Proc. Physiol. Soc. 118, 38 (1952).

Salton, W. R. J.

W. R. J. Salton, Ref. 42, Chap. 3.

Shmoys, J.

J. Shmoys, J. Appl. Phys. 32, 689 (1961).
[CrossRef]

Steiner, R. F.

P. Doty, R. F. Steiner, J. Chem. Phys. 18, 1211 (1950)
[CrossRef]

Strutt, J. W.

J. W. Strutt, Rayleigh, Scientific Papers (Dover, New York, 1964).

Sussman, A.

A. Sussman, H. O. Halvorson, Spores: Their Dormancy and Germination (Harper & Rowe, Inc., New York, 1966).

Taherzadeh, M.

M. Taherzadeh, to be published, probably in Physics of Fluids.

Tai, C. T.

C. T. Tai, Appl. Sci. Res. Sect. B7, 113 (1958).
[CrossRef]

Takaku, K.

Y. Nomura, K. Takaku, Sci. Rep. Res. Inst. of Tohoku Univ. 7, 107 (1955).

Thompson, T. E.

G. S. Gotterer, T. E. Thompson, A. L. Lehninger, J. Biophys. Biochem. Cytol. 10, 15 (1961).
[CrossRef] [PubMed]

Tkaczyk, S.

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature 171, 720 (1953).
[CrossRef] [PubMed]

van de Hulst, H. C.

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Figures (19)

Fig. 1
Fig. 1

Schematic setup of a scattering measurement. Radiation source (S) illuminates scatterer (O), and detector (D) records scattered intensity at an angle θ.

Fig. 2
Fig. 2

Differential scattering characteristics of homogeneous diotyl phthalate sphere of 1.45-μ radius, relative refractive index 1.5, and illuminated at a wavelength λ = 436 mμ. Solid curve corresponds to measured data of Gucker and Egan. Dashed curve is best-fit predicted from Lorenz scattering theory. The parameter α = 2πr/λ.

Fig. 3
Fig. 3

Polarization ratio ρ as a function of scattering angle θ. Experimental measurements (circles) are contrasted to theoretical predictions (solid curves) for a silica fiber at an incident wavelength λ = 546 mμ and various (curves A–F) choices of structural parameters m (refractive index) and α (=2πλ/r).

Fig. 4
Fig. 4

Drawings of spore of B. cereus and spore of B. megaterium, respectively, based on electron micrographs. (After Mayall and Robinow.)

Fig. 5
Fig. 5

Hypothetical radial variation of refractive index of spores of (a)B. cereus and (b)B. megaterium.

Fig. 6
Fig. 6

Polarization ratio ρ as a function of scattering angle θ based on hypothetical refractive index distributions of Fig. 5.

Fig. 7
Fig. 7

Typical members of several bacterial species whose probable scattering characteristics are discussed in the text. A special case of B. anthracis is also shown (see text).

Fig. 8
Fig. 8

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from E. coli (see Fig. 7) illuminated perpendicular to its major axis. Bacterium is assumed to be a prolate spheroid of semiaxes 1.5 μ and 0.27 μ, respectively. The scattering pattern is symmetric about 180°.

Fig. 9
Fig. 9

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from E. coli illuminated at 45° with respect to its major axis. Characteristics of bacterium are identical to those indicated in Fig. 8.

Fig. 10
Fig. 10

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from S. aureus. Bacterium is assumed spherical of radius 0.5 μ and thus all aspects are equivalent. Scattering from such an object is always symmetric about 180°.

Fig. 11
Fig. 11

Relative intensity of scattered 0.63 μ unpolarized light as a function of scattering angle from B. subtilis (see Fig. 7) illuminated perpendicular to its major axis. Bacterium is assumed to be a prolate super spheroid of semiaxes 2.5 μ and 0.38 μ, respectively. Exponent factor [(Eq. (1)] was chosen as 3.5. The included spore is of prolate spheroid form with semiaxes 0.6 μ and 0.33 μ, respectively, and located 1.5 μ from the center of the bacterium on its major axis. Scattering is symmetric about 180°.

Fig. 12
Fig. 12

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from B. sublilis illuminated at 45° with respect to its major axis. Characteristics of bacterium are identical to those indicated in Fig. 11.

Fig. 13
Fig. 13

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from S. epidermidis. Bacterium is assumed spherical of radius 0.3 μ.

Fig. 14
Fig. 14

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from B. anthracis (see Fig. 7) illuminated perpendicular to its major axis. Bacterium is assumed to be a prolate super spheroid of semiaxes 4 μ and 0.6 μ, respectively. Exponent factor was chosen as 8. The included spore is of prolate spheroid form with semiaxes 0.7 μ and 0.45 μ, respectively, and located 1.5 μ from the center of the bacterium on its major axis. Scattering is symmetric about 180°.

Fig. 15
Fig. 15

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from B. anthracis illuminated at 45° with respect to its major axis. Characteristics of the bacterium are identical to those indicated in Fig. 14.

Fig. 16
Fig. 16

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from a special model of B. anthracis (see Fig 7) illuminated perpendicular to its major axis. This special example was chosen as a super prolate spheroid of semiaxes 2.5 μ and 0.5 μ, respectively. Exponent factor was chosen as 6. All other characteristics are the same as Fig. 14.

Fig. 17
Fig. 17

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from a special model of B. anthracis illuminated at 45° with respect to its major axis. Characteristics of this special model are identical to those indicated in Fig. 16.

Fig. 18
Fig. 18

Relative intensity of scattered 0.63-μ unpolarized light as a function of scattering angle from E. coli averaged over all planar aspects. Characteristics of the bacterium are identical to those indicated in Fig. 8.

Fig. 19
Fig. 19

Scattering geometry for the Rayleigh-Gans approximation. The directions n and m refer to the incident and scattered directions, respectively.

Equations (48)

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( x / a ) n + ( y / b ) n = 1
1 - n / n 0 1 ,
2 k a 1 - n / n 0 1.
d S ( θ ) d S ( θ ) } = i k 3 ( n / n 0 - 1 ) 2 π e i δ d V { 1 cos θ .
S ( θ ) S ( θ ) } = i k 3 ( n / n 0 - 1 ) 2 π R ( θ , ϕ ) { 1 cos θ .
R ( θ , ϕ ) = e i δ d V .
S ( θ ) S ( θ ) } = i k 3 2 π R ( θ , ϕ ) { 1 cos θ ,
R ( θ , ϕ ) = [ n ( r , θ , ϕ ) - n 0 ] n 0 e i δ d V .
s = r · m - r · n .
δ = k s = k r · ( m - n ) .
δ = k b m - n = 2 k b sin ( θ / 2 ) ,
m - n 2 = 2 - 2 m · n = 2 ( 1 - cos θ ) = 4 sin 2 ( θ / 2 ) .
R ( θ , ϕ ) = - A exp [ 2 i k b sin ( θ / 2 ) ] d b .
I ( θ , ϕ ) = 1 2 I 0 [ S ( θ , ϕ ) 2 + S ( θ , ϕ ) 2 ] / ( k r ) 2 = k 4 I 0 8 π 2 r 2 ( n / n 0 - 1 ) 2 R ( θ , ϕ ) 2 ( 1 + cos 2 θ ) .
r = [ a 2 - b 2 ] 1 2 ,
R ( θ , ϕ ) = - b b π ( a 2 - b 2 ) exp ( 2 i k b sin θ / 2 ) d b .
R ( θ , ϕ ) = π a 2 - 1 1 e i u z ( 1 - z 2 ) d z = 2 π a 3 0 1 cos ( u z ) ( 1 - z 2 ) d z = [ ( 4 / 3 ) π a 3 ] [ ( 3 / u 3 ) ( sin u - u cos u ) ] = V G ( u ) ,
G ( u ) = ( 3 / u 3 ) ( sin u - u cos u ) = ( 9 π / 2 u 3 ) 1 2 J 3 / 2 ( u ) .
S ( θ ) S ( θ ) } = i k 3 2 π [ ( m 1 - 1 ) R 1 ( θ , ϕ ) + ( m 2 - m 1 ) R 2 ( θ , ϕ ) ] { 1 cos θ ,
R 1 ( θ , ϕ ) = ( 4 / 3 ) π a 3 G [ 2 k a sin ( θ / 2 ) ] , R 2 ( θ , ϕ ) = ( 4 / 3 ) π ( a - t ) 3 G [ 2 k ( a - t ) sin ( θ / 2 ) ] .
( x / a ) 2 + ( y / b ) 2 + ( z / c ) 2 = 1.
x x 1 / a 2 + y y 1 / b 2 + z z 1 / c 2 = 1.
l x + m y + n z = ξ ,
x 1 = l a 2 / ξ , y 1 = m b 2 / ξ , z 1 = n c 2 / ξ .
ξ = [ l 2 a 2 + m 2 b 2 + n 2 c 2 ] 1 2 .
l x + m y + n z = p ( p ξ )
A = π a b c ( 1 - p 2 / ξ 2 ) / ξ .
R ( θ , ϕ ) = π a b c ξ - ξ ξ ( 1 - p 2 / ξ 2 ) exp [ 2 i k p sin ( θ / 2 ) ] d p = π a b c - 1 1 ( 1 - z 2 ) e i u z d z = V G ( u ) ,
cos β = - cos α sin θ / 2 + sin α cos θ / 2 cos ϕ .
x = constant .
y 2 + z 2 = [ 1 - ( x / a ) 2 ] b 2 .
z = ± b [ 1 - ( x / a ) 2 - ( y / b ) 2 ] 1 2
δ 1 = 2 k sin ( θ / 2 ) ( y sin β ) ,
δ 2 = 2 k sin ( θ / 2 ) ( x cos β ) .
R ( θ , ϕ ) = 2 - b [ 1 - ( x / a ) 2 ] 1 2 b [ 1 - ( x / a ) 2 ] 1 2 e i δ 1 c [ 1 - ( x / a ) 2 - ( y / b ) 2 ] 1 2 d y - a a e i δ 2 d x .
2 - b [ 1 - ( x / a ) 2 ] 1 2 b [ 1 - ( x / a ) 2 ] 1 2 e i δ 1 b [ 1 - ( x / a ) 2 - ( y / b ) 2 ] 1 2 d y = 2 b 2 - 1 1 e i u 1 ξ [ η 2 - ξ 2 ] 1 2 d ξ = 2 b 2 η 2 - 1 1 e i u 1 η p [ 1 - p 2 ] 1 2 d p = 4 b 2 η 2 0 1 cos ( u 1 p η ) [ 1 - p 2 ] 1 2 d p = 4 b 2 η 2 2 u 1 η J 1 ( u 1 η ) π = 2 π b 2 η J 1 ( u 1 η ) u 1 .
R ( θ , ϕ ) = - a a 2 π b 2 [ 1 - ( x / a ) 2 ] 1 2 u 1 J 1 [ u 1 ( 1 - ( x / a ) 2 ) 1 2 ] e i δ 2 d x .
R ( θ , ϕ ) = 2 π b 2 a - 1 1 [ 1 - q 2 ] 1 2 J 1 [ u 1 ( 1 - q 2 ) 1 2 ] cos u 2 q d q .
I = - 1 1 [ 1 - q 2 ] 1 2 J 1 [ u 1 ( 1 - q 2 ) 1 2 ] cos u 2 q d q .
I = 0 π sin 2 θ cos ( u 2 cos θ ) J 1 ( u 1 sin θ ) d θ
= 0 π sin 2 θ cos ( z cos ϕ cos θ ) J 1 ( z sin ϕ sin θ ) d θ ,
I = ( 2 π / z ) 1 2 sin ϕ J 3 / 2 ( z ) = ( 2 π / u ) 1 2 ( u 1 / u ) J 3 / 2 ( u ) ,
z = [ u 1 2 + u 2 2 ] 1 2 = 2 k sin θ / 2 [ a 2 cos 2 β + b 2 sin 2 β ] 1 2 = 2 k ξ sin ( θ / 2 ) = u .
R ( θ , ϕ ) = 4 π a b c u ( π 2 u ) 1 2 J 3 / 2 ( u ) = V ( 9 π / 2 u 3 ) 1 2 J 3 / 2 ( u ) = V G ( u ) ,
( x / a ) n + ( y / b ) n = 1.
R ( θ , ϕ ) = 4 π a b 2 u 1 0 1 J 1 [ u 1 ( 1 - ξ n ) 1 2 ] ( 1 - ξ n ) 1 2 cos u 2 ξ d ξ ,
( i k 3 / 2 π ) ( m 3 - m 2 ) R ( θ , ϕ ) e i C ,
C = 2 k c sin ( θ / 2 ) cos β .

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