Abstract

The intensity scattered by particles randomly placed beneath a rough interface is studied with rigorous simulations. It is shown that the angular intensity pattern is close to that obtained by adding the intensity scattered by particles under a flat surface to that scattered by a rough homogeneous surface whose permittivity is evaluated with an effective-medium theory. This heuristic splitting rule is accurate for a large range of parameters that are well beyond any perturbative treatment.

© 2002 Optical Society of America

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References

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  1. L. Tsang, G. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).
  2. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).
  3. C. Lam, A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate surface roughness,” Math. Gen. 260, 111–125 (1993).
  4. J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
    [CrossRef]
  5. K. Sarabandi, Y. Oh, F. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
    [CrossRef]
  6. S. Mudaliar, “Electromagnetic wave scattering from a random medium layer with random interface,” Waves Random Media 4, 167–176 (1994).
    [CrossRef]
  7. S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
    [CrossRef]
  8. S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
    [CrossRef]
  9. A. Sentenac, J.-J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1998).
    [CrossRef]
  10. O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
    [CrossRef]
  11. A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
    [CrossRef]
  12. G. Zhang, L. Tsang, K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional surface,” J. Opt. Soc. Am. A 15, 2995–3002 (1998).
    [CrossRef]
  13. K. Pak, L. Tsang, L. Li, C. Chan, “Combined random rough surface and volume scattering based on Monte-Carlo solutions of Maxwell’s equation,” Radio Sci. 28, 331–338 (1993).
    [CrossRef]
  14. G. Pelosi, R. Coccioli, “A finite element approach for scattering from inhomogeneous media with a rough interface,” Waves Random Media 7, 119–127 (1997).
    [CrossRef]
  15. G. Zhang, L. Tsang, Y. Kuga, “Angular correlation function of wave scattering by a buried object embedded in random discrete scatterers under a rough surface,” Microwave Opt. Technol. Lett. 14, 144–151 (1997).
    [CrossRef]
  16. H. Giovannini, M. Saillard, A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A 15, 1182–1191 (1998).
    [CrossRef]
  17. O. Calvo, “Diffusion des ondes électromagnétiques par un film rugueux hétérogène,” Ph.D. thesis (Ecole Centrale Paris, 1999).
  18. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  19. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  20. C. Bohren, D. Huffman, Absorption and Scattering by Small Particles (Wiley, New York, 1983).
  21. U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), pp. 75–197.
  22. J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
    [CrossRef]
  23. A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  24. M. Saillard, “A characterization tool for dielectric random rough surface: Brewster phenomenon,” Waves Random Media 2, 67–79 (1992).
    [CrossRef]
  25. C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 25, 2695–2702 (1986).
    [CrossRef] [PubMed]
  26. H. Giovannini, C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36, 5574–5579 (1997).
    [CrossRef] [PubMed]
  27. H. Giovannini, C. Amra, “Dielectric thin films for maximized absorption with standard quality black surfaces,” Appl. Opt. 37, 103–105 (1998).
    [CrossRef]

1999 (1)

O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
[CrossRef]

1998 (4)

1997 (4)

H. Giovannini, C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36, 5574–5579 (1997).
[CrossRef] [PubMed]

A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
[CrossRef]

G. Pelosi, R. Coccioli, “A finite element approach for scattering from inhomogeneous media with a rough interface,” Waves Random Media 7, 119–127 (1997).
[CrossRef]

G. Zhang, L. Tsang, Y. Kuga, “Angular correlation function of wave scattering by a buried object embedded in random discrete scatterers under a rough surface,” Microwave Opt. Technol. Lett. 14, 144–151 (1997).
[CrossRef]

1996 (3)

1995 (1)

S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
[CrossRef]

1994 (1)

S. Mudaliar, “Electromagnetic wave scattering from a random medium layer with random interface,” Waves Random Media 4, 167–176 (1994).
[CrossRef]

1993 (2)

C. Lam, A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate surface roughness,” Math. Gen. 260, 111–125 (1993).

K. Pak, L. Tsang, L. Li, C. Chan, “Combined random rough surface and volume scattering based on Monte-Carlo solutions of Maxwell’s equation,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

1992 (1)

M. Saillard, “A characterization tool for dielectric random rough surface: Brewster phenomenon,” Waves Random Media 2, 67–79 (1992).
[CrossRef]

1988 (1)

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

1986 (1)

1984 (1)

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

1982 (1)

1964 (1)

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

Albrand, G.

Amra, C.

Bohren, C.

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

Calvo, O.

O. Calvo, “Diffusion des ondes électromagnétiques par un film rugueux hétérogène,” Ph.D. thesis (Ecole Centrale Paris, 1999).

Calvo-Perez, O.

O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
[CrossRef]

Chan, C.

K. Pak, L. Tsang, L. Li, C. Chan, “Combined random rough surface and volume scattering based on Monte-Carlo solutions of Maxwell’s equation,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

Coccioli, R.

G. Pelosi, R. Coccioli, “A finite element approach for scattering from inhomogeneous media with a rough interface,” Waves Random Media 7, 119–127 (1997).
[CrossRef]

Dietrich, S.

S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
[CrossRef]

Elson, J. M.

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

Frish, U.

U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), pp. 75–197.

Fung, K.

K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).

Garoff, S.

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Giovannini, H.

Greffet, J.-J.

O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
[CrossRef]

A. Sentenac, J.-J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1998).
[CrossRef]

Haase, A.

S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
[CrossRef]

Huffman, D.

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

Ishimaru, A.

C. Lam, A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate surface roughness,” Math. Gen. 260, 111–125 (1993).

A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

Keller, J. B.

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

Kong, G.

L. Tsang, G. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

Kuga, Y.

G. Zhang, L. Tsang, Y. Kuga, “Angular correlation function of wave scattering by a buried object embedded in random discrete scatterers under a rough surface,” Microwave Opt. Technol. Lett. 14, 144–151 (1997).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

Lam, C.

C. Lam, A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate surface roughness,” Math. Gen. 260, 111–125 (1993).

Li, L.

Madrazo, A.

Mudaliar, S.

S. Mudaliar, “Electromagnetic wave scattering from a random medium layer with random interface,” Waves Random Media 4, 167–176 (1994).
[CrossRef]

Nieto-Vesperinas, M.

Oh, Y.

K. Sarabandi, Y. Oh, F. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

Pak, K.

G. Zhang, L. Tsang, K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional surface,” J. Opt. Soc. Am. A 15, 2995–3002 (1998).
[CrossRef]

K. Pak, L. Tsang, L. Li, C. Chan, “Combined random rough surface and volume scattering based on Monte-Carlo solutions of Maxwell’s equation,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

Pelosi, G.

G. Pelosi, R. Coccioli, “A finite element approach for scattering from inhomogeneous media with a rough interface,” Waves Random Media 7, 119–127 (1997).
[CrossRef]

Roche, P.

Saillard, M.

H. Giovannini, M. Saillard, A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A 15, 1182–1191 (1998).
[CrossRef]

M. Saillard, “A characterization tool for dielectric random rough surface: Brewster phenomenon,” Waves Random Media 2, 67–79 (1992).
[CrossRef]

Sarabandi, K.

K. Sarabandi, Y. Oh, F. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

Sentenac, A.

Shin, R.

L. Tsang, G. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

Sinha, S. K.

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Sirota, E. B.

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Tsang, L.

G. Zhang, L. Tsang, K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional surface,” J. Opt. Soc. Am. A 15, 2995–3002 (1998).
[CrossRef]

G. Zhang, L. Tsang, Y. Kuga, “Angular correlation function of wave scattering by a buried object embedded in random discrete scatterers under a rough surface,” Microwave Opt. Technol. Lett. 14, 144–151 (1997).
[CrossRef]

K. Pak, L. Tsang, L. Li, C. Chan, “Combined random rough surface and volume scattering based on Monte-Carlo solutions of Maxwell’s equation,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

L. Tsang, G. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

Ulaby, F.

K. Sarabandi, Y. Oh, F. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

Zhang, G.

G. Zhang, L. Tsang, K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional surface,” J. Opt. Soc. Am. A 15, 2995–3002 (1998).
[CrossRef]

G. Zhang, L. Tsang, Y. Kuga, “Angular correlation function of wave scattering by a buried object embedded in random discrete scatterers under a rough surface,” Microwave Opt. Technol. Lett. 14, 144–151 (1997).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Geosci. Remote Sens. (1)

K. Sarabandi, Y. Oh, F. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Math. Gen. (1)

C. Lam, A. Ishimaru, “Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate surface roughness,” Math. Gen. 260, 111–125 (1993).

Microwave Opt. Technol. Lett. (1)

G. Zhang, L. Tsang, Y. Kuga, “Angular correlation function of wave scattering by a buried object embedded in random discrete scatterers under a rough surface,” Microwave Opt. Technol. Lett. 14, 144–151 (1997).
[CrossRef]

Phys. Rep. (1)

S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
[CrossRef]

Phys. Rev. B (2)

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

Proc. Symp. Appl. Math. (1)

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

Radio Sci. (2)

O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
[CrossRef]

K. Pak, L. Tsang, L. Li, C. Chan, “Combined random rough surface and volume scattering based on Monte-Carlo solutions of Maxwell’s equation,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

Waves Random Media (3)

G. Pelosi, R. Coccioli, “A finite element approach for scattering from inhomogeneous media with a rough interface,” Waves Random Media 7, 119–127 (1997).
[CrossRef]

M. Saillard, “A characterization tool for dielectric random rough surface: Brewster phenomenon,” Waves Random Media 2, 67–79 (1992).
[CrossRef]

S. Mudaliar, “Electromagnetic wave scattering from a random medium layer with random interface,” Waves Random Media 4, 167–176 (1994).
[CrossRef]

Other (5)

O. Calvo, “Diffusion des ondes électromagnétiques par un film rugueux hétérogène,” Ph.D. thesis (Ecole Centrale Paris, 1999).

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

U. Frish, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), pp. 75–197.

L. Tsang, G. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, Mass., 1994).

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Figures (7)

Fig. 1
Fig. 1

Geometry and notation.

Fig. 2
Fig. 2

Scattered intensity of various geometries illuminated under normal incidence; s polarization, w=5λ. Binder, ϵb=2, σ=0.1λ, l=λ; scatterers, d=0.2, a=0.035λ, ϵsc=9+i3.5. Solid curve, rough inhomogeneous medium; dash-dot-dot curve, rough effective surface. Same roughness parameters as the inhomogeneous medium. Dotted curve, particles under a flat surface; same parameters as that of the inhomogeneous medium, but the binder is flat. The effective permittivity is numerically evaluated, ϵeff=2.35+i0.65.

Fig. 3
Fig. 3

Rough inhomogeneous medium illuminated under 20° of incidence; s polarization, w=3λ. Binder, ϵb=3+i0.4 σ=0.3λ, l=λ. The cylindrical scatterers are perfectly conducting: a=0.035λ, d=0.03λ; the effective permittivity is calculated with Eq. (17) ϵeff=3+i0.6.

Fig. 4
Fig. 4

Various geometries illuminated under 20° incidence; s polarization, w=4λ. Binder, ϵb=2. The cylindrical scatterers are perfectly conducting a=0.035λ. (a) d=0.05, l=λ, various rms heights. The volume contribution dominates in such a way that changing the roughness does not affect the scattering pattern. (b) d=0.1, σ=0.2λ, l=λ. The effective surface contribution is increased and the penetration depth of the wave inside the volume is limited by the diffusion losses. The effective permittivity calculated with Eq. (17) is ϵeff=0.67+i1.95. (c) d=0.2, σ=0.2λ, l=0.5λ. The diffusion losses are important, the penetration depth is small, and the effective surface contribution dominates. The effective permittivity is obtained numerically by studying the coherent reflected energy of the inhomogeneous medium with a flat interface ϵeff=-7+i3.

Fig. 5
Fig. 5

Various geometries illuminated under s polarization; w=4λ. Binder, ϵb=5+i0.1; perfectly conducting scatterers: d=0.05, a=0.035λ. Effective permittivity calculated with Eq. (17) is ϵeff=4.4+i1.3. (a) σ=0.3λ, l=λ, normal incidence; the splitting rule works. (b) σ=0.7λ, l=λ, 30° of incidence; the splitting rule is not accurate.

Fig. 6
Fig. 6

Scattered patterns of various geometries illuminated under 55° of incidence; p polarization, w=3λ. A zoom about the Brewster angle of the binder (54°76) is given in (b) and (d). Binder, ϵb=2+i0.1; σ=0.05λ, l=0.5λ. Scatterers, d=0.03, a=0.05. (a) and (b), ϵsc=1; (c) and (d), ϵsc=5. Solid curves, reflected intensity of the rough inhomogeneous medium; dotted curves, particles under a flat surface (volume); dashed–dotted curves, effective surface; dashed curves, scattered intensity of the volume plus that of the effective surface, minus the reflection from a plane with the effective permittivity. (b) and (d), solely the dash and solid curves are plotted.

Fig. 7
Fig. 7

Scattering-reduction effect obtained with two different coatings deposited on a rough inhomogeneous medium. Parameters of the medium are the same as in Fig. 2. The value of the effective permittivity ϵeff was determined numerically by studying the specularly reflected and transmitted field of slabs of the random medium with different thicknesses ϵeff=2.35+i0.65. When the antireflection (AR) layer is optimized for the binder, its permittivity is ϵ=ϵb0.5=1.41 and thickness h=λ/4ϵb0.5=0.21λ. When the layer is optimized for the effective medium Eqs. (18), ϵ=1.627 and h=0.159λ.

Equations (21)

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Finc(x, z)=-+p(α-α0)exp[iαx-iγ(α)z]dα,
p(α)=w exp(-w2α2/2),
α0=k0 sin θinc
γ=(k02-α2)1/2,Im(γ)0.
Fd(x, z)= B(α)exp[iαx+iγ(α)z]dα.
Rθs=|γ(αs)B(αs)|2Pinc,
Pinc= γ(α)|p(α-α0)|2dα.
2e(α, z)z2+ Δκ(α-α, z)e(α, z)dα=0,
ddz eez(z)=T(z)eez(z),
F(P)=Finc+C1G(P, M)ϕ1(M)ds,
FjR(P)=n=-+bn(j)Hn(1)(kbrj)exp(inθj),
F(P)=C1K(P, M)ϕ1(M)ds+j>1FjR(P),
ϵeff=ϵb(1-d)+dϵsc.
ϵeff2+(1-2d)(ϵsc-ϵb)ϵeff-ϵscϵb=0.
|ϵsc-ϵb|1,
2π2a|ϵsc-ϵb|/λ1.
Ed(r, θ)2πkbr exp(ikbr-iπ/4)f(θ).
K2=kb2+4iρf-(4ρf)2kb-1 0 exp(ikbr)g(r)sin g(kbr)dr,
h=λ2πϵ arctanϵ(ns-1)ks,
ϵ=ks2ns-1+ns,
ns+iks=ϵs.

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