Abstract

Interesting applications arising in optical and chemical engineering, environmental science, and biology motivate the investigation of electromagnetic wave scattering problems by radially inhomogeneous obstacles. Our main purpose is the investigation of plane-wave scattering by quasi-homogeneous obstacles, that is, obstacles with wavenumbers not exhibiting large variations from a specific average value k¯. The analysis is presented separately for a slab (1D), a cylindrical (2D), and a spherical (3D) scatterer. First, we consider a step approximation of the wavenumber and express the field coefficients by applying a T-matrix method for the corresponding piecewise homogeneous scatterer. Then, by performing an appropriate Taylor expansion, we express the field coefficients as linear combinations of the distances of the wavenumber samples from k¯. The combinations’ weights are called layer-factors, because each one describes the contribution of a specific layer in the scattered field. Furthermore, it is shown that the far-field pattern of the quasi-homogeneous scatterer is decomposed into that of the respective homogeneous scatterer plus the perturbation far-field pattern, depending on the wavenumber’s deviations from k¯. Several numerical results are presented concerning the comparison of the far-field patterns computed by the proposed technique and the T-matrix method, as well as investigations of the perturbation far-field pattern and the layer-factors. Linear, sinusoidal, Lunenburg type, and triangular wavenumber profiles are analyzed.

© 2009 Optical Society of America

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References

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  1. M. Kerker, L. H. Kauffman, and W. A. Farone, “Scattering of electromagnetic waves from two concentric spheres when the outer shell has a variable refractive index. numerical results,” J. Opt. Soc. Am. 56, 1053-1056 (1966).
    [CrossRef]
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  3. L. Shafai, “Scattering by spherically symmetrical objects,” Can. J. Phys. 50, 749-753 (1972).
    [CrossRef]
  4. Y. Nomura and K. Takaku, “On the propagation of the electromagnetic waves in an inhomogeneous atmosphere,” Res. Inst. Electron. Commun. Tohoku Univ. 7B, 107-114 (1955).
  5. K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, 1988).
  6. Z.-F. Sang and Z.-Y. Li, “Partial resonant response of composites containing coated particles with graded shells,” Phys. Lett. A 332, 376-381 (2004).
    [CrossRef]
  7. S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
    [CrossRef]
  8. F. Onofri, D. Blondel, G. Gréhan, and G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayered particles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104-111 (1996).
    [CrossRef]
  9. V. N. Lopatin, N. V. Shepelevich, and I. V. Prostakova, “Modelling optical properties of organicmineral complexes in water ecosystems,” J. Phys. D 38, 2556-2563 (2005).
    [CrossRef]
  10. L. Kai and P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501-511 (1994).
    [CrossRef] [PubMed]
  11. I. Gurwich, N. Shiloah, and M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder,” J. Quant. Spectrosc. Radiat. Transf. 63, 217-229 (1999).
    [CrossRef]
  12. A. Y. Perelman, “Scattering by particles with radially variable refractive indices,” Appl. Opt. 35, 5452-5460 (1996).
    [CrossRef] [PubMed]
  13. W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere,” Appl. Opt. 42, 1710-1720 (2003).
    [CrossRef] [PubMed]
  14. B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286-3296 (1996).
    [CrossRef] [PubMed]
  15. N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
    [CrossRef]
  16. T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631-1638 (2006).
    [CrossRef]
  17. E. Baleine and A. Dogariu, “Variable-coherence tomography for inverse scattering problems,” J. Opt. Soc. Am. A 21, 1917-1923 (2004).
    [CrossRef]
  18. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).
  19. P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, 1953).
  20. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

2007 (1)

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

2006 (2)

N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
[CrossRef]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631-1638 (2006).
[CrossRef]

2005 (1)

V. N. Lopatin, N. V. Shepelevich, and I. V. Prostakova, “Modelling optical properties of organicmineral complexes in water ecosystems,” J. Phys. D 38, 2556-2563 (2005).
[CrossRef]

2004 (2)

Z.-F. Sang and Z.-Y. Li, “Partial resonant response of composites containing coated particles with graded shells,” Phys. Lett. A 332, 376-381 (2004).
[CrossRef]

E. Baleine and A. Dogariu, “Variable-coherence tomography for inverse scattering problems,” J. Opt. Soc. Am. A 21, 1917-1923 (2004).
[CrossRef]

2003 (1)

1999 (1)

I. Gurwich, N. Shiloah, and M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder,” J. Quant. Spectrosc. Radiat. Transf. 63, 217-229 (1999).
[CrossRef]

1996 (3)

A. Y. Perelman, “Scattering by particles with radially variable refractive indices,” Appl. Opt. 35, 5452-5460 (1996).
[CrossRef] [PubMed]

B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286-3296 (1996).
[CrossRef] [PubMed]

F. Onofri, D. Blondel, G. Gréhan, and G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayered particles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104-111 (1996).
[CrossRef]

1994 (1)

1972 (1)

L. Shafai, “Scattering by spherically symmetrical objects,” Can. J. Phys. 50, 749-753 (1972).
[CrossRef]

1966 (1)

1955 (1)

Y. Nomura and K. Takaku, “On the propagation of the electromagnetic waves in an inhomogeneous atmosphere,” Res. Inst. Electron. Commun. Tohoku Univ. 7B, 107-114 (1955).

Athanasiadis, C.

N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

Baleine, E.

Biscos, Y.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Blondel, D.

F. Onofri, D. Blondel, G. Gréhan, and G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayered particles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104-111 (1996).
[CrossRef]

Charinpanitkul, T.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Dogariu, A.

Farone, W. A.

Feshbach, H.

P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, 1953).

Fischer, D. G.

Garcia, N.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Gouesbet, G.

F. Onofri, D. Blondel, G. Gréhan, and G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayered particles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104-111 (1996).
[CrossRef]

Gousebet, G.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Grehan, G.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Gréhan, G.

F. Onofri, D. Blondel, G. Gréhan, and G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayered particles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104-111 (1996).
[CrossRef]

Gurwich, I.

I. Gurwich, N. Shiloah, and M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder,” J. Quant. Spectrosc. Radiat. Transf. 63, 217-229 (1999).
[CrossRef]

Johnson, B. R.

Kai, L.

Kauffman, L. H.

Kerker, M.

Kleiman, M.

I. Gurwich, N. Shiloah, and M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder,” J. Quant. Spectrosc. Radiat. Transf. 63, 217-229 (1999).
[CrossRef]

Lavergne, G.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Li, Z.-Y.

Z.-F. Sang and Z.-Y. Li, “Partial resonant response of composites containing coated particles with graded shells,” Phys. Lett. A 332, 376-381 (2004).
[CrossRef]

Lopatin, V. N.

V. N. Lopatin, N. V. Shepelevich, and I. V. Prostakova, “Modelling optical properties of organicmineral complexes in water ecosystems,” J. Phys. D 38, 2556-2563 (2005).
[CrossRef]

Massoli, P.

Mees, L.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Morse, P.

P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, 1953).

Nomura, Y.

Y. Nomura and K. Takaku, “On the propagation of the electromagnetic waves in an inhomogeneous atmosphere,” Res. Inst. Electron. Commun. Tohoku Univ. 7B, 107-114 (1955).

Onofri, F.

F. Onofri, D. Blondel, G. Gréhan, and G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayered particles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104-111 (1996).
[CrossRef]

Perelman, A. Y.

Prostakova, I. V.

V. N. Lopatin, N. V. Shepelevich, and I. V. Prostakova, “Modelling optical properties of organicmineral complexes in water ecosystems,” J. Phys. D 38, 2556-2563 (2005).
[CrossRef]

Saengkaew, S.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Sang, Z.-F.

Z.-F. Sang and Z.-Y. Li, “Partial resonant response of composites containing coated particles with graded shells,” Phys. Lett. A 332, 376-381 (2004).
[CrossRef]

Shafai, L.

L. Shafai, “Scattering by spherically symmetrical objects,” Can. J. Phys. 50, 749-753 (1972).
[CrossRef]

Shepelevich, N. V.

V. N. Lopatin, N. V. Shepelevich, and I. V. Prostakova, “Modelling optical properties of organicmineral complexes in water ecosystems,” J. Phys. D 38, 2556-2563 (2005).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, 1988).

Shiloah, N.

I. Gurwich, N. Shiloah, and M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder,” J. Quant. Spectrosc. Radiat. Transf. 63, 217-229 (1999).
[CrossRef]

Tai, C. T.

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

Takaku, K.

Y. Nomura and K. Takaku, “On the propagation of the electromagnetic waves in an inhomogeneous atmosphere,” Res. Inst. Electron. Commun. Tohoku Univ. 7B, 107-114 (1955).

Tanthapanichakoon, W.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Tsitsas, N. L.

N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
[CrossRef]

Vanisri, H.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

Visser, T. D.

Wolf, E.

Yang, W.

Appl. Opt. (4)

Can. J. Phys. (1)

L. Shafai, “Scattering by spherically symmetrical objects,” Can. J. Phys. 50, 749-753 (1972).
[CrossRef]

Exp. Fluids (1)

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, Y. Biscos, N. Garcia, G. Lavergne,L. Mees, G. Gousebet, and G. Grehan, “Rainbow refractrometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595-601 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

V. N. Lopatin, N. V. Shepelevich, and I. V. Prostakova, “Modelling optical properties of organicmineral complexes in water ecosystems,” J. Phys. D 38, 2556-2563 (2005).
[CrossRef]

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

I. Gurwich, N. Shiloah, and M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder,” J. Quant. Spectrosc. Radiat. Transf. 63, 217-229 (1999).
[CrossRef]

Part. Part. Syst. Charact. (1)

F. Onofri, D. Blondel, G. Gréhan, and G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayered particles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104-111 (1996).
[CrossRef]

Phys. Lett. A (1)

Z.-F. Sang and Z.-Y. Li, “Partial resonant response of composites containing coated particles with graded shells,” Phys. Lett. A 332, 376-381 (2004).
[CrossRef]

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

N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
[CrossRef]

Res. Inst. Electron. Commun. Tohoku Univ. (1)

Y. Nomura and K. Takaku, “On the propagation of the electromagnetic waves in an inhomogeneous atmosphere,” Res. Inst. Electron. Commun. Tohoku Univ. 7B, 107-114 (1955).

Other (5)

K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, 1988).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, 1953).

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

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Figures (14)

Fig. 1
Fig. 1

Geometrical configuration of the slab studied in Subsection 2A.

Fig. 2
Fig. 2

Geometrical configuration of the cylinder studied in Subsection 2B.

Fig. 3
Fig. 3

Reflection coefficient error for the 1D slab geometry: relative difference between g 1 D ex and g 1 D ap as a function of k ¯ Δ k max for U = 20 and (a) k 0 a 1 = π (circles), 2 π (squares), 4 π (stars) with k 0 = 2 π and k ¯ = 3 k 0 , and (b) k ¯ k 0 = 2 (circles), 4 (squares), 6 (stars) with k 0 a 1 = 2 π .

Fig. 4
Fig. 4

Far-field pattern relative error for the 2D cylindrical geometry: relative difference between g 2 D ex and g 2 D ap as a function of k ¯ Δ k max for k 0 = 2 π , k ¯ = 3 k 0 , U = 20 and (a) k 0 a 1 = π (circles), 2 π (squares), 4 π (stars) with ϕ = 60 ° , and (b) ϕ = 0 ° (circles), 90° (squares), 270° (stars) with k 0 a 1 = 2 π .

Fig. 5
Fig. 5

Far-field pattern relative error for the 3D spherical geometry: relative difference between g 3 D ex and g 3 D ap as a function of k ¯ Δ k max for ϕ = 60 ° , k 0 = 2 π , k ¯ = 3 k 0 , U = 20 and (a) k 0 a 1 = π (circles), 2 π (squares), 4 π (stars) with θ = 45 ° , and (b) θ = 0 ° (circles), 90° (squares), 180° (stars) with k 0 a 1 = 2 π .

Fig. 6
Fig. 6

Wavenumber functions k 1 ( z ) and k 2 ( z ) for linear and sinusoidal profiles, respectively, with k ¯ = 3 π , a 1 = 1 , U = 25 , and Δ k max = 0.02 k ¯ .

Fig. 7
Fig. 7

Reflection coefficients g 1 D per due to the wavenumber profile perturbations as functions of k 0 a 1 for (a) linear k 1 ( z ) and (b) sinusoidal k 2 ( z ) profile, with Δ k max k ¯ = 0.02 (solid), 0.06 (dashed–dotted), 0.1 (dashed) and k ¯ = 3 π , k 0 = 2 π , U = 25 .

Fig. 8
Fig. 8

Wavenumber functions k 2 ( ρ ) and k 3 ( ρ ) for sinusoidal and Lunenburg-type profiles with k ¯ = 3 π , a 1 = 1 , U = 25 , and Δ k max = 0.02 k ¯ .

Fig. 9
Fig. 9

Far-field patterns g 2 D per due to the wavenumber profile perturbations as functions of ϕ for (a) sinusoidal k 2 ( ρ ) and (b) Lunenburg k 3 ( ρ ) profile with Δ k max k ¯ = 0.02 (solid), 0.06 (dashed–dotted), 0.1 (dashed) and k ¯ = 3 π , k 0 a 1 = 2 π , U = 25 .

Fig. 10
Fig. 10

Wavenumber functions k 3 ( r ) and k 4 ( r ) for Lunenburg-type and triangular profiles with k ¯ = 3 π , a 1 = 1 , U = 25 , and Δ k max = 0.02 k ¯ .

Fig. 11
Fig. 11

Far-field patterns g 3 D per due to the wavenumber profile perturbations as functions of θ for (a) Lunenburg k 3 ( r ) and (b) triangular k 4 ( r ) profiles with ϕ = 60 ° , Δ k max k ¯ = 0.02 (solid), 0.06 (dashed–dotted), 0.1 (dashed), and k ¯ = 3 π , k 0 a 1 = 2 π , U = 25 .

Fig. 12
Fig. 12

Layer factor L F j as a function of the layer index j with ( j = 2 , , U 1 ) for a quasi-homogeneous slab with k 0 = 2 π , U = 120 and (a) k 0 a 1 = 0.4 π (solid), π (dashed–dotted), 1.6 π (dashed) with k ¯ = 6 π , (b) k ¯ k 0 = 1.1 (solid), 2.5 (dashed–dotted), 5 (dashed) with k 0 a 1 = 2 π .

Fig. 13
Fig. 13

Layer-factor L F j as a function of the layer index j with ( j = 2 , , U 1 ) for a quasi-homogeneous cylinder with k 0 = 2 π , U = 120 , ϕ = 90 ° and (a) k 0 a 1 = 0.4 π (solid), π (dashed–dotted), 1.6 π (dashed) with k ¯ = 3 π , (b) k ¯ k 0 = 1.1 (solid), 2.5 (dashed–dotted), 5 (dashed) with k 0 a 1 = 2 π .

Fig. 14
Fig. 14

Layer factor L F j as a function of the layer index j with ( j = 2 , , U 1 ) for a quasi-homogeneous sphere with k 0 = 2 π , U = 120 , θ = 0 ° , ϕ = 90 ° and (a) k 0 a 1 = 0.8 π (solid), 1.2 π (dashed–dotted), 1.6 π (dashed) with k ¯ = 3 π , (b) k ¯ k 0 = 1.1 (solid), 2.5 (dashed–dotted), 5 (dashed) with k 0 a 1 = 2 π .

Equations (61)

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E inc = E inc ( z ) x ̂ = exp ( i k 0 z ) x ̂ , z > a .
E j = E j ( z ) x ̂ = [ α j exp ( i k j z ) + β j exp ( i k j z ) ] x ̂ , a j + 1 < z < a j .
[ α j β j ] = A j [ α j 1 β j 1 ]
A j = A j ( k j 1 , k j ) = 1 2 k j [ exp [ i ( k j 1 k j ) a j ] ( k j + k j 1 ) exp [ i ( k j 1 + k j ) a j ] ( k j k j 1 ) exp [ i ( k j 1 + k j ) a j ] ( k j k j 1 ) exp [ i ( k j 1 k j ) a j ] ( k j + k j 1 ) ] .
[ 0 β U + 1 ] = A U + 1 A U A 2 A 1 [ α 0 1 ] .
a 0 = ( A U + 1 A U A 2 A 1 ) 12 ( A U + 1 A U A 2 A 1 ) 11 ,
( A U A 2 ) k j = A U A j + 2 ( A j + 1 A j ) k j A j 1 A 2 .
( A U A 2 ) k j k ¯ = A j + 1 k j ( k ¯ , k ¯ ) + A j k j ( k ¯ , k ¯ ) .
A U A 2 I + j = 2 U 1 ( k j k ¯ ) C j = I + j = 2 U 1 ( k j k ¯ ) [ A j + 1 k j ( k ¯ , k ¯ ) + A j k j ( k ¯ , k ¯ ) ] .
α 0 ( A U + 1 A 1 ) 12 + j = 2 U 1 ( k j k ¯ ) ( A U + 1 C j A 1 ) 12 ( A U + 1 A 1 ) 11 + j = 2 U 1 ( k j k ¯ ) ( A U + 1 C j A 1 ) 11 ,
α 0 ( A U + 1 A 1 ) 12 ( A U + 1 A 1 ) 11 + A 1 [ ( A U + 1 A 1 ) 11 ] 2 j = 2 U 1 ( k j k ¯ ) C j ,
C j = ( A U + 1 ) 11 ( A U + 1 ) 12 [ ( C j ) 11 ( C j ) 22 ] [ ( A U + 1 ) 11 ] 2 ( C j ) 12 + [ ( A U + 1 ) 12 ] 2 ( C j ) 21 .
g 1 D = α 0 ,
g 1 D = g 1 D hom + g 1 D per ,
g 1 D hom = ( A U + 1 A 1 ) 12 ( A U + 1 A 1 ) 11 , g 1 D per = A 1 [ ( A U + 1 A 1 ) 11 ] 2 j = 2 U 1 ( k j k ¯ ) C j .
g 1 D per = j = 2 U 1 L F j ( k j k ¯ ) ,
L F j = A 1 [ ( A U + 1 A 1 ) 11 ] 2 C j , ( j = 2 , , U 1 ) .
E inc = E inc ( ρ , ϕ ) z ̂ = exp ( i k 0 ρ sin ϕ ) z ̂ = n = + ( 1 ) n exp ( i n ϕ ) J n ( k 0 ρ ) z ̂ , ( ρ > a ) .
E j = E j ( ρ , ϕ ) z ̂ = n = + ( 1 ) n exp ( i n ϕ ) [ α n j H n ( k j ρ ) + β n j J n ( k j ρ ) ] z ̂ , a j + 1 < ρ < a j ,
[ α n j β n j ] = A n j [ α n j 1 β n j 1 ]
A n j = A n j ( k j 1 , k j ) = π 2 i [ y j J n ( x j ) H n ( y j ) + x j J n ( x j ) H n ( y j ) y j J n ( x j ) J n ( y j ) + x j J n ( x j ) J n ( y j ) y j H n ( x j ) H n ( y j ) x j H n ( x j ) H n ( y j ) y j H n ( x j ) J n ( y j ) x j J n ( x j ) H n ( y j ) ] ,
[ 0 β n U ] = A n U A n U 1 A n 2 A n 1 [ α n 0 1 ] ,
α n 0 = ( A n U A n U 1 A n 2 A n 1 ) 12 ( A n U A n U 1 A n 2 A n 1 ) 11
α n 0 ( A n 1 ) 12 + j = 2 U 1 ( k j k ¯ ) ( C n j A n 1 ) 12 ( A n 1 ) 11 + j = 2 U 1 ( k j k ¯ ) ( C n j A n 1 ) 11 ,
C n j = A n j + 1 k j ( k ¯ , k ¯ ) + A n j k j ( k ¯ , k ¯ ) ,
α n 0 ( A n 1 ) 12 ( A n 1 ) 11 1 [ ( A n 1 ) 11 ] 2 j = 2 U 1 ( k j k ¯ ) ( C n j ) 12 .
lim ρ E sc ( ρ , ϕ ) = 2 π k 0 ρ exp [ i ( k 0 ρ π 4 ) ] g 2 D ( ϕ ) ,
g 2 D ( ϕ ) = n = + i n exp ( i n ϕ ) α n 0 ,
g 2 D = g 2 D hom + g 2 D per ,
g 2 D hom = n = + i n exp ( i n ϕ ) ( A n 1 ) 12 ( A n 1 ) 11 ,
g 2 D per = n = + i n exp ( i n ϕ ) [ ( A n 1 ) 11 ] 2 j = 2 U 1 ( k j k ¯ ) ( C n j ) 12 .
g 2 D per = j = 2 U 1 L F j ( k j k ¯ ) ,
L F j = n = + i n exp ( i n ϕ ) [ ( A n 1 ) 11 ] 2 ( C n j ) 12 , j = 2 , , U 1 .
E inc ( r ) = n = 1 + ( i ) n 2 n + 1 n ( n + 1 ) [ M o 1 n 1 ( r , k 0 ) + i N e 1 n 1 ( r , k 0 ) ] ,
E j ( r ) = n = 1 + ( i ) n 2 n + 1 n ( n + 1 ) [ β ̃ n j M o 1 n 1 ( r , k j ) + β n j M o 1 n 3 ( r , k j ) + i α ¯ n j N e 1 n 1 ( r , k j ) + i α n j N e 1 n 3 ( r , k j ) ] ,
[ α n j α ̃ n j ] = A n j [ α n j 1 α ̃ n j 1 ] , [ β n j β ̃ n j ] = B n j [ β n j 1 β ̃ n j 1 ] .
A n j = i x j [ x j y j j n ( x j ) h ̂ n ( y j ) y j x j j ̂ n ( x j ) h n ( y j ) x j y j j n ( x j ) j ̂ n ( y j ) y j x j j ̂ n ( x j ) j n ( y j ) y j x j h ̂ n ( x j ) h n ( y j ) x j y j h n ( x j ) h ̂ n ( y j ) y j x j h ̂ n ( x j ) j n ( y j ) x j y j h n ( x j ) j ̂ n ( y j ) ] ,
B n j = i x j [ j n ( x j ) h ̂ n ( y j ) j ̂ n ( x j ) h n ( y j ) j n ( x j ) j ̂ n ( y j ) j ̂ n ( x j ) j n ( y j ) h ̂ n ( x j ) h n ( y j ) h n ( x j ) h ̂ n ( y j ) h ̂ n ( x j ) j n ( y j ) h n ( x j ) j ̂ n ( y j ) ] ,
[ 0 α ̃ n N ] = A n U A n U 1 A n 2 A n 1 [ α n 0 1 ] ,
[ 0 β ̃ n N ] = B n U B n U 1 B n 2 B n 1 [ β n 0 1 ] ,
α n 0 = ( A n U A n U 1 A n 2 A n 1 ) 12 ( A n U A n U 1 A n 2 A n 1 ) 11 , β n 0 = ( B n U B n U 1 B n 2 B n 1 ) 12 ( B n U B n U 1 B n 2 B n 1 ) 11 .
α n 0 ( A n 1 ) 12 + j = 2 U 1 ( k j k ¯ ) ( C n j A n 1 ) 12 ( A n 1 ) 11 + j = 2 U 1 ( k j k ¯ ) ( C n j A n 1 ) 11 ,
β n 0 ( B n 1 ) 12 + j = 2 U 1 ( k j k ¯ ) ( D n j B n 1 ) 12 ( B n 1 ) 11 + j = 2 U 1 ( k j k ¯ ) ( D n j B n 1 ) 11 ,
C n j = A n j + 1 k j ( k ¯ , k ¯ ) + A n j k j ( k ¯ , k ¯ ) ,
D n j = B n j + 1 k j ( k ¯ , k ¯ ) + B n j k j ( k ¯ , k ¯ ) .
α n 0 ( A n 1 ) 12 ( A n 1 ) 11 k ¯ k 0 [ ( A n 1 ) 11 ] 2 j = 2 U 1 ( k j k ¯ ) ( C n j ) 12 ,
β n 0 ( B n 1 ) 12 ( B n 1 ) 11 k ¯ k 0 [ ( B n 1 ) 11 ] 2 j = 2 U 1 ( k j k ¯ ) ( D n j ) 12 .
lim r E sc ( r , θ , ϕ ) = exp ( i k 0 r ) k 0 r g 3 D ( θ , ϕ ) ,
g 3 D ( θ , ϕ ) = n = 1 + ( 1 ) n 2 n + 1 n ( n + 1 ) i [ α n 0 B e 1 n ( θ , ϕ ) β n 0 C o 1 n ( θ , ϕ ) ] ,
g 3 D = g 3 D hom + g 3 D per ,
g 3 D hom = n = 1 + ( 1 ) n 2 n + 1 n ( n + 1 ) i [ ( A n 1 ) 12 ( A n 1 ) 11 B e 1 n ( θ , ϕ ) ( B n 1 ) 12 ( B n 1 ) 11 C o 1 n ( θ , ϕ ) ] ,
g 3 D per = n = 1 + ( 1 ) n 2 n + 1 n ( n + 1 ) i k ¯ k 0 [ 1 [ ( A n 1 ) 11 ] 2 j = 2 U 1 ( k j k ¯ ) ( C n j ) 12 B e 1 n ( θ , ϕ ) 1 [ ( B n 1 ) 11 ] 2 j = 2 U 1 ( k j k ¯ ) ( D n j ) 12 C o 1 n ( θ , ϕ ) ] .
g 3 D per = j = 2 U 1 L F j ( k j k ¯ ) ,
L F j = n = 1 + ( 1 ) n 2 n + 1 n ( n + 1 ) i k ¯ k 0 [ 1 [ ( A n 1 ) 11 ] 2 ( C n j ) 12 B e 1 n ( θ , ϕ ) 1 [ ( B n 1 ) 11 ] 2 ( D n j ) 12 C o 1 n ( θ , ϕ ) ] .
k 1 ( z ) = k ¯ Δ k max + 2 Δ k max ( z a 1 ) ,
k 2 ( z ) = k ¯ Δ k max + 2 Δ k max sin ( π z a 1 ) .
k ( z ) = 1 a 0 a k ( z ) d z .
k 2 ( ρ ) = k ¯ Δ k max + 2 Δ k max sin ( π ρ a 1 ) ,
k 3 ( ρ ) = k ¯ + Δ k max 2 Δ k max ( ρ a 1 ) 2 .
k 3 ( r ) = k ¯ + Δ k max 2 Δ k max ( r a 1 ) 2 , 0 r < a 1 ,
k 4 ( r ) = { k ¯ Δ k max + 2 Δ k max ( r a 1 2 ) , 0 r a 1 2 k ¯ + Δ k max 2 Δ k max ( r a 1 2 a 1 2 ) , a 1 2 r a 1 } .

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