Abstract

We derive the Debye-series expansion of the partial-wave scattering and interior amplitudes for the interaction of a diagonally incident beam of arbitrary profile with an infinitely long homogeneous dielectric circular cylinder. We then discuss the physical interpretation of the various terms of the series. We also consider the one-internal-reflection Debye-series terms for diagonal plane-wave incidence and examine the first-order rainbow extinction transition as a function of the tilt angle of the incident plane wave. We experimentally observe the first-order rainbow extinction transition and compare our observations with the Debye-series predictions.

© 1997 Optical Society of America

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References

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  1. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
    [CrossRef]
  2. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5, pp. 372–378.
  3. M. Kerker, D. Cooke, W. A. Farone, R. A. Jacobsen, “Electromagnetic scattering from an infinite circular cylinder at oblique incidence. I. Radiance functions for m=1.46,” J. Opt. Soc. Am. 56, 487–491 (1966).
    [CrossRef]
  4. A. Cohen, C. Acquista, “Light scattering by tilted cylinders: properties of the partial wave coefficients,” J. Opt. Soc. Am. 72, 531–534 (1982).
    [CrossRef]
  5. Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders: a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
    [CrossRef] [PubMed]
  6. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 12.33, pp. 210–214.
  7. K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
    [CrossRef]
  8. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. (N.Y.) 10, 82–124 (1969).
    [CrossRef]
  9. M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
    [CrossRef]
  10. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
    [CrossRef]
  11. J. A. Lock, “Morphology-dependent resonances of an infinitely long circular cylinder illuminated by a diagonally incident plane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A 14, 653–661 (1997).
    [CrossRef]
  12. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
    [CrossRef]
  13. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.3, pp. 92–113.
  14. R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 20–21, 122–123.
  15. Ref. 6, Sec. 12-32, pp. 209–210.
  16. P. Debye, “Das Elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering, Milestone Series Vol. MS89 (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 198–204.
  17. B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finite conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).
  18. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  19. K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
    [CrossRef] [PubMed]
  20. J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air Technical Digest Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.
  21. J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690, 4960 (1994).
    [CrossRef] [PubMed]
  22. J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 4, pp. 179–296, especially p. 241.
  23. C. L. Adler, J. A. Lock, B. R. Stone, C. J. Garcia, “High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1305–1315 (1997).
    [CrossRef]
  24. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), App. C.
  25. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
    [CrossRef]
  26. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), p. 92, Fig. 40.
  27. R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), p. 45, plate III.
  28. M. V. Berry, C. Upstill, “Catastrophe optics:morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
    [CrossRef]

1997

1994

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690, 4960 (1994).
[CrossRef] [PubMed]

1993

1992

1988

1982

1980

1979

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

1972

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
[CrossRef]

1969

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. (N.Y.) 10, 82–124 (1969).
[CrossRef]

1966

1959

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[CrossRef]

1955

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

1937

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finite conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

1908

P. Debye, “Das Elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering, Milestone Series Vol. MS89 (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 198–204.

Acquista, C.

Adler, C. L.

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics:morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
[CrossRef]

Bohren, C. F.

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

Bremmer, H.

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finite conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5, pp. 372–378.

Cohen, A.

Cooke, D.

Debye, P.

P. Debye, “Das Elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering, Milestone Series Vol. MS89 (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 198–204.

Farone, W. A.

Feynman, R. P.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 20–21, 122–123.

Ford, K. W.

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[CrossRef]

Fuller, K. A.

Garcia, C. J.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.3, pp. 92–113.

Hibbs, A. R.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 20–21, 122–123.

Hovenac, E. A.

Huffman, D. R.

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

Jacobsen, R. A.

Jamison, J. M.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690, 4960 (1994).
[CrossRef] [PubMed]

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air Technical Digest Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.

Kerker, M.

Lin, C.-Y.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690, 4960 (1994).
[CrossRef] [PubMed]

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air Technical Digest Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.

Lock, J. A.

Marston, P. L.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), p. 92, Fig. 40.

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5, pp. 372–378.

Mount, K. E.

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. (N.Y.) 10, 82–124 (1969).
[CrossRef]

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5, pp. 372–378.

Sakurai, J. J.

J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 4, pp. 179–296, especially p. 241.

Stone, B. R.

Takano, Y.

Tanaka, M.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), p. 45, plate III.

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics:morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 12.33, pp. 210–214.

van der Pol, B.

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finite conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

Wheeler, J. A.

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[CrossRef]

Ann. Phys. (N.Y.)

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[CrossRef]

Appl. Opt.

Can. J. Phys.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

J. Math. Phys. (N.Y.)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. (N.Y.) 10, 82–124 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Philos. Mag.

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finite conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Phys. Acoust.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), p. 92, Fig. 40.

Phys. Z.

P. Debye, “Das Elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering, Milestone Series Vol. MS89 (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 198–204.

Prog. Opt.

M. V. Berry, C. Upstill, “Catastrophe optics:morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

Rep. Prog. Phys.

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
[CrossRef]

Other

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.3, pp. 92–113.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 20–21, 122–123.

Ref. 6, Sec. 12-32, pp. 209–210.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” in Light and Color in the Open Air Technical Digest Vol. 13 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 8–11.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5, pp. 372–378.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 12.33, pp. 210–214.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), p. 45, plate III.

J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 4, pp. 179–296, especially p. 241.

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

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Figures (9)

Fig. 1
Fig. 1

Pictorial representation of the partial-wave summed interior reflection amplitude Rij11 for initial polarization state i and final polarization state j. Each dot is an R11 factor.

Fig. 2
Fig. 2

Pictorial representation of the partial-wave interior amplitudes (a) cl(h) for the polarization channel, (b) dl(h) for the μμ channel, and (c) pl(h) for the μ and μ channels. Each dot is a T21 factor, and each star is an R11 factor.

Fig. 3
Fig. 3

Pictorial representation of the partial-wave scattering amplitudes (a) al(h) for scattering, (b) bl(h) for μμ scattering, and (c) ql(h) for the μ and μ scatterings. Each dot is a T21 or T12 factor, each star is an R11 factor, each triangle is an R22 factor, and D represents diffraction.  

Fig. 4
Fig. 4

One-internal-reflection portion of the scattered intensity as a function of the scattering angle θ for n=1.484, 2πa/λ =1000.0, and (a) ξ=39.0°, (b) ξ=45.0°, (c) ξe=50.72°, and (d) ξ=56.0°. The points labeled R in (a) and (b) are the two branches of the first-order rainbow.

Fig. 5
Fig. 5

Relation between the scattering angle θ and the impact parameter of a diagonally incident ray parameterized by the angle ϕi0 for n=1.484 and (a) ξ=45.0°, (b) ξe=50.72°, and (c) ξ=56.0°. The points labeled R in (a) are the two branches of the first-order rainbow, the points labeled D are the two dominant rays in the supernumerary region, and the points labeled T are the third ray.

Fig. 6
Fig. 6

Scattered intensity as a function of the scattering angle θ for n=1.484, 2πa/λ=1000.0, and (a) ξ=39.0°, (b) ξ=45.0°, (c) ξe=50.72°, and (d) ξ=56.0°. These graphs are to be compared with the one-internal-reflection portion of the intensity of Figs. 4(a), 4(b), 4(c), and 4(d), respectively.

Fig. 7
Fig. 7

A laser beam, expanded by a series of lenses, is incident on a 7.6-mm-radius glass rod. Light scattered near 180° is recorded on unexposed camera film 72 cm above the rod. A black card with a narrow slit cut into it is placed at AA so as to illuminate a narrow band BB on the rod.

Fig. 8
Fig. 8

Photographs of the first-order rainbow region for (a) ξ =33°, (b) ξ=40°, (c) ξe=45°, and (d) ξ=50°.

Fig. 9
Fig. 9

Photographs of the first-order rainbow region for narrow-band illumination of the glass rod for (a) ξ=40°, (b) ξe =45°, and (c) ξ=50°. These photographs are to be compared with the ray theory predictions of Figs. 5(a), 5(b), and 5(c), respectively.  

Equations (82)

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k^inc=(cos ξ)u^x-(sin ξ)u^z,
al(h)=U2W1-nU3W3D,
cl(h)=-2inxW1πy2D,
xka(1-h2)1/2,ynka(1-h2/n2)1/2,
U1=n2xy Jl(x)Jl(y)-Jl(x)Jl(y),
U2=nxy Jl(x)Jl(y)-nJl(x)Jl(y),
U3=hl(y2-x2)xy2 Jl(x)Jl(y),
W1=n2xy Hl(1)(x)Jl(y)-Hl(1)(x)Jl(y),
W2=nxy Hl(1)(x)Jl(y)-nHl(1)(x)Jl(y),
W3=hl(y2-x2)xy2 Hl(1)(x)Jl(y),
D=W1W2-nW32,
bl(h)=U1W2-nU3W3D,
dl(h)=-2inxW2πy2D,
ql(h)=2nlh(y2-x2)Jl2(y)πx2y2D,
pl(h)=-2nxW3πy2D,
limr Escatt(r, θ, z)=E0cos ξ 2πkr1/2[exp(-iπ/4)]×(T5u^r-T2u^θ-T3u^z),
limr Bscatt(r, θ, z)=E0/ccos ξ 2πkr1/2[exp(-iπ/4)]×(T6u^r+T1u^θ-T4u^z),
T1=- dh l=-(1-h2)1/4βl(h)Φ(h, l),
T2=- dh l=-(1-h2)1/4αl(h)Φ(h, l),
T3=- dh l=-(1-h2)3/4βl(h)Φ(h, l),
T4=- dh l=-(1-h2)3/4αl(h)Φ(h, l),
T5=- dh l=- h(1-h2)1/4βl(h)Φ(h, l),
T6=- dh l=- h(1-h2)1/4αl(h)Φ(h, l).
Φ(h, l)=exp[ikr(1-h2)1/2]exp(ikhz)exp(ilθ),
αl(h)=al(h)Al(h)+ql(h)Bl(h),
βl(h)=-ql(h)Al(h)+bl(h)Bl(h).
Einterior(r, θ, z)=E0cos ξ - dh l=- il+1×exp(ikhz)exp(ilθ)ilkr γl(h)Jl(y¯)+h(1-h2/n2)1/2δl(h)Jl(y¯)u^r-n(1-h2/n2)1/2γl(h)Jl(y¯)-ihlnkr δl(h)Jl(y¯)u^θ-[in(1-h2/n2)δl(h)Jl(y¯)]u^z,
y¯=nkr(1-h2/n2)1/2
γl(h)=cl(h)Al(h)+pl(h)Bl(h),
δl(h)=-npl(h)Al(h)+dl(h)Bl(h).
ψinc(r, θ, z)=Hl(2)(kr(1-h2)1/2)exp(ikhz)exp(ilθ).
ψref(r, θ, z)=R22Hl(1)(kr(1-h2)1/2)exp(ikhz)exp(ilθ),
ψrefμ(r, θ, z)=Rμ22Hl(1)(kr(1-h2)1/2)exp(ikhz)exp(ilθ),
ψtrans(r, θ, z)=T21Hl(2)(nkr(1-h2/n2)1/2)×exp(ikhz)exp(ilθ),
ψtransμ(r, θ, z)=Tμ21Hl(2)(nkr(1-h2/n2)1/2)×exp(ikhz)exp(ilθ).
R22=nG12G22-E22F12D12,
Rμμ22=nG12G22-F22E12D12,
Rμ22=iD12 (E12G22-E22G12),
Rμ22=-inD12 (F12G22-F22G12),
R11=nG12G11-E11F12D12,
Rμμ11=nG12G11-F11E12D12,
Rμ11=inD12 (E12G11-E11G12),
Rμ11=-iD12 (F12G11-F11G12),
T21=-4in2xF12πy2D12,Tμμ21=-4in2xE12πy2D12,
Tμ21=4n3xG12πy2D12,Tμ21=-n2xG12πy2D12,
T12=-4iF12πxD12,Tμμ12=-4iE12πxD12,
Tμ12=4G12πxD12,Tμ12=-4nG12πxD12,
Eij=nxy Hl(i)(x)Hl(j)(y)-nHl(i)(x)Hl(j)(y),
Fij=n2xy Hl(i)(x)Hl(j)(y)-Hl(i)(x)Hl(j)(y),
Gij=hlxy2 (y2-x2)Hl(i)(x)Hl(j)(y),
D12=E12F12-nG122.
Rij11=Rij11+k Rik11Rkj11+kl Rik11Rkl11Rlj11+klm Rik11Rkl11Rlm11Rmj11+ .
R11=r=1s=0rt=0 r!s!(r-s)!×(R11)r-s(Rμ11)s(Rμμ11)st(Rμ11)s=R11(1-Rμμ11)+Rμ11Rμ11(1-R11)(1-Rμμ11)-Rμ11Rμ11.
Rμ11=Rμ11(1-R11)(1-Rμμ11)-Rμ11Rμ11.
cl(h)=1n T21+k Tk21Rk11,
dl(h)=1n Tμμ21+kTμk21Rkμ11,
pl(h)=1n Tμ21+kTμk21Rk11=-1n2 Tμ21+kTk21Rkμ11,
al(h)=12-12 R22+kTk21Tk12+klTk21Rkl11Tl12,
bl(h)=12-12 Rμμ22+kTμk21Tkμ12+klTμk21Rkl11Tlμ12,
ql(h)=-12 Rμ22+kTμk21Tk12+klTμk21Rkl11Tl12=12 Rμ22+kTk21Tkμ12+klTk21Rkl11Tlμ12,
R11=R111-R11,Rμμ11=Rμμ111-Rμμ11,
Rμ11=Rμ11=0.
cl(0)=1n p=1T21(R11)p-1,
dl(0)=1n p=1Tμμ21(Rμμ11)p-1,
pl(0)=0,
al(0)=12-12 p=1T21(R11)p-1T12,
bl(0)=12-12 p=1Tμμ21(Rμμ11)p-1Tμμ12,
ql(0)=0,
al(h)=Jl(x)Hl(1)(x)=12 (1-R22),
bl(h)=Jl(x)Hl(1)(x)=12 (1-Rμμ22),
ql(h)=0,
R22=-Hl(2)(x)/Hl(1)(x),Rμμ22=-Hl(2)(x)/Hl(1)(x).
sin ξe=4-n231/2.
Al(h)=δ(h+sin ξ),Bl(h)=0;
Al(h)=0,Bl(h)=δ(h+sin ξ).
limr Iscatt(r,θ,z)=E022μ0c1πkr cos ξ[|S(θ)|2+|Sμ(θ)|2+2|Sq(θ)|2],
S(θ)=a0+2 l=1 al cos(lθ),
Sμ(θ)=b0+2 l=1 bl cos(lθ),
Sq(θ)=2i l=1 ql sin(lθ),
al-12(T21R11T12+Tμ21Rμ11T12+T21Rμ11Tμ12+Tμ21Rμμ11Tμ12),
bl-12(Tμμ21Rμμ11Tμμ12+Tμ21Rμ11Tμμ12+Tμμ21Rμ11Tμ12+Tμ21R11Tμ12),
ql-12(Tμμ21Rμμ11Tμ12+Tμ21Rμ11Tμ12+Tμμ21Rμ11T12+Tμ21R11T12).

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