Abstract

A model is developed that calculates the mean field and the specular scattering from rough dielectric surfaces. The roughness is regarded as random fluctuations of the permittivity whose averaged value depends on z. A Dyson equation (in its volume formulation) is solved within the Bourret approximation to yield the mean field and the coherent reflection and transmission. Comparisons of the latter three items with the results of numerical simulations are presented in the case of one-dimensional rough surfaces illuminated by an s-polarized plane wave. Good agreement is obtained for rms heights as great as λ/2 and a wide range of correlation lengths (from λ/5 to 2λ) as long as the dielectric contrast remains small (<4).

© 1998 Optical Society of America

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References

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  1. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  2. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
    [CrossRef]
  3. K. P. Pak, L. Tsang, C. H. Chan, J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte-Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
    [CrossRef]
  4. P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from randomly rough 2D metallic surface,” Opt. Commun. 110, 269–273 (1994).
    [CrossRef]
  5. K. Pak, L. Tsang, J. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two-dimensional dielectric random rough surfaces with the sparce-matrix canonical method,” J. Opt. Soc. Am. A 14, 1515–1529 (1997).
    [CrossRef]
  6. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  7. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).
  8. E. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
    [CrossRef]
  9. L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of the extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell equations,” Opt. Lett. 17, 314–316 (1992).
    [CrossRef] [PubMed]
  10. U. Frisch, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968).
  11. L. Tsang, J. A. Kong, T. Shin, eds., Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  12. B. Michel, “Statistical method to calculate extinction by small irregularly shaped particles,” J. Opt. Soc. Am. A 12, 2471–2581 (1995).
    [CrossRef]
  13. S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
    [CrossRef]
  14. V. Holy, J. Kubena, I. Ohlidal, “X-ray reflection from rough layered systems,” Phys. Rev. B 47, 15896–15903 (1993).
    [CrossRef]
  15. S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep.260, (1995).
    [CrossRef]
  16. A. Sentenac, J.-J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1997).
    [CrossRef]
  17. J. A. Sanchez, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces,” J. Opt. Soc. Am. A 12, 1547–1557 (1995).
    [CrossRef]
  18. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
    [CrossRef]
  19. T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial frequency rectangular groove dielectric surface relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
    [CrossRef]
  20. P. Beckmann, A. Spizzichino, eds., The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  21. C. Baylard, J.-J. Greffet, A. A. Maradudin, “Coherent reflection factor of a random rough surface: applications,” J. Opt. Soc. Am. A 10, 2637–2647 (1993).
    [CrossRef]
  22. D. A. Wolf, “Effective-medium permittivity in particulate media at low densities and frequencies: a unified approach,” J. Opt. Soc. Am. A 10, 1544–1548 (1993).
    [CrossRef]

1997 (2)

1995 (3)

1994 (1)

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from randomly rough 2D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[CrossRef]

1993 (3)

1992 (1)

1990 (2)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

1988 (2)

E. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

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

1987 (1)

1986 (1)

Baird, W. E.

Baylard, C.

Chan, C. H.

Dietrich, S.

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

Ding, K. H.

Frisch, U.

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

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]

Gaylord, T. K.

Greffet, J.-J.

Haase, A.

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

Holy, V.

V. Holy, J. Kubena, I. Ohlidal, “X-ray reflection from rough layered systems,” Phys. Rev. B 47, 15896–15903 (1993).
[CrossRef]

Johnson, J.

Kubena, J.

V. Holy, J. Kubena, I. Ohlidal, “X-ray reflection from rough layered systems,” Phys. Rev. B 47, 15896–15903 (1993).
[CrossRef]

Mandt, C. E.

Maradudin, A. A.

J. A. Sanchez, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces,” J. Opt. Soc. Am. A 12, 1547–1557 (1995).
[CrossRef]

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from randomly rough 2D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[CrossRef]

C. Baylard, J.-J. Greffet, A. A. Maradudin, “Coherent reflection factor of a random rough surface: applications,” J. Opt. Soc. Am. A 10, 2637–2647 (1993).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Maystre, D.

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Mendez, E. R.

Michel, B.

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Moharam, M. G.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

Ohlidal, I.

V. Holy, J. Kubena, I. Ohlidal, “X-ray reflection from rough layered systems,” Phys. Rev. B 47, 15896–15903 (1993).
[CrossRef]

Pak, K.

Pak, K. P.

Saillard, M.

Sanchez, J. A.

Sentenac, A.

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]

Sipe, J. E.

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]

Thorsos, E.

E. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Tran, P.

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from randomly rough 2D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[CrossRef]

Tsang, L.

Voronovich, G.

G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).

Wolf, D. A.

Ann. Phys. (N.Y.) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (1)

J. Acoust. Soc. Am. (1)

E. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

J. Opt. Soc. Am. A (8)

K. P. Pak, L. Tsang, C. H. Chan, J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte-Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

K. Pak, L. Tsang, J. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two-dimensional dielectric random rough surfaces with the sparce-matrix canonical method,” J. Opt. Soc. Am. A 14, 1515–1529 (1997).
[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 (1997).
[CrossRef]

J. A. Sanchez, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces,” J. Opt. Soc. Am. A 12, 1547–1557 (1995).
[CrossRef]

B. Michel, “Statistical method to calculate extinction by small irregularly shaped particles,” J. Opt. Soc. Am. A 12, 2471–2581 (1995).
[CrossRef]

C. Baylard, J.-J. Greffet, A. A. Maradudin, “Coherent reflection factor of a random rough surface: applications,” J. Opt. Soc. Am. A 10, 2637–2647 (1993).
[CrossRef]

D. A. Wolf, “Effective-medium permittivity in particulate media at low densities and frequencies: a unified approach,” J. Opt. Soc. Am. A 10, 1544–1548 (1993).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from randomly rough 2D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[CrossRef]

Opt. Lett. (1)

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]

V. Holy, J. Kubena, I. Ohlidal, “X-ray reflection from rough layered systems,” Phys. Rev. B 47, 15896–15903 (1993).
[CrossRef]

Other (6)

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

P. Beckmann, A. Spizzichino, eds., The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

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

L. Tsang, J. A. Kong, T. Shin, eds., Theory of Microwave Remote Sensing (Wiley, New York, 1985).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the problem. (a) Randomly rough monodimensional surface.(b) Averaged geometry, the permittivity of the system is 〈〉. In the inset the shape of |〈〉| is plotted as a function of z. This system is equivalent, for the coherent field, to the rough surface when the correlation length tends toward 0. (c) Randomly elevated planes placed at z=h; h is a random variable whose density probability denoted p1 is represented in the inset. This system is equivalent, for the coherent field, to the rough surface when the correlation length tends toward infinity.

Fig. 2
Fig. 2

Modulus of the coherent transmission factor as a function of the correlation length δ=0.2λ, θinc=0, and b=2.25.

Fig. 3
Fig. 3

Modulus of the mean field inside the roughness for δ=0.2λ, σ=0.2λ, θinc=0, and b=2.25. Solid curve, rigorous calculations; short-dashed curve, MFT; long-dashed curve, homogenization; dotted curve, Kirchhoff model.

Fig. 4
Fig. 4

Same as Fig. 3 with δ=0.2λ, σ=λ, θinc=0, b=2.25.

Fig. 5
Fig. 5

Imaginary part of the effective complex permittivity calculated from the values of the mean field given in Figs. 2 and 3 with relation (24). In the positive half-space is plotted +Im(eff) for δ=0.2λ, σ=λ, θinc=0, and b=2.25. In the negative half-space is plotted -Im(eff) for δ=0.2λ, σ=0.2λ, θinc=0, and b=2.25. Solid curve, rigorous calculations; short-dashed curve, MFT; dotted curve, Kirchhoff model.

Fig. 6
Fig. 6

Modulus of the mean field inside the roughness for δ=0.2λ, σ=λ, θinc=0, b=2.25+i0.3. Solid curve, rigorous calculations; short-dashed curve, MFT.

Fig. 7
Fig. 7

Imaginary part of the effective complex permittivity calculated from the values of the mean field given in Figs. 3, 4, and 6, minus that of 〈〉. The positive half-space is devoted to +Im(eff-) for δ=0.2λ, σ=λ, θinc=0, and b=2.25. In the negative half-space are plotted -Im(eff-) for δ=0.2λ, σ=λ, θinc=0, and b=2.25+i0.3. Solid curve, rigorous calculations; short-dashed curve, MFT.

Equations (35)

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(x, z)=aifz>h(x),
(x, z)=bifz<h(x).
ΔE+(x, z)k02E=S(x, z),
ΔE+ref(z)k02E=S(x, z)-L1(x, z)E(x, z),
ΔG+ref(z)k02G=δ(x-x)δ(z-z)
E=Eref-GL1E,
L1 : f(x, z)k02[(x, z)-ref(z)]f(x, z)
G : f(x, z)G(x-x, z, z)f(x, z)dxdz.
δE=n=1[-G(1-P)L1]nE
E=Eref+GME,
M=-n=1PL1[-G(1-P)L1]nP.
δE-GL1E,
EEref+GL1GL1E.
Eref(x, z)=eref(z)exp(iκinc x),
E(x, z)=e(z)exp(iκincx).
G(x-x, z, z)=12πG˜(k, z, z)exp[ik(x-x)]dk
L(x-x, z, z)=L1L1(x-x, z, z)/k04=[(x, z)-ref(z)]×[(x, z)-ref(z)]=12πL˜(k, z, z)exp[ik(x-x)]dk,
e(z)=eref(z)+k044π2G˜(κinc, z, z1)e(z2)dz1dz2×G˜(κ, z1, z2)L˜(κinc-κ, z1, z2)dk.
[e]=[eref]+Mat[e],
Mat(i, j)=Δzjk=1NG˜(κinc, zi, zk)Δzk×G˜(κ, zk, zj)L˜(κinc-κ, zk, zj)dκ.
e(za)=einc(za)+rexp[i2γa(κinc)za]einc(za)
e(zb)=texp{i[γa(κinc)-γb(κinc)]zb}einc(zb)
γa(b)(κ)=(a(b)k02-κ2)1/2,Im(γa(b))>0,
p1(h)=1δ2πexp-h22δ2,
p2(h, h, r)=12πδ21-C2(r)×exp-h2+h2-2hhC(r)2δ2[1-C2(r)]
d2edz2+eff(z)k02e=0,
(z)=12a+b+(a-b)erfz2δ.
L(r, z, z)=A22πδ-z exp(-h2/2δ2)×erfz-C(r)hδ[2-2C2(r)]1/2-erfz2δdh.
e(z)=einc(z)+r(h)exp[iγa(κinc)z]=Ea(z, h)for z>h,
E(z)=t(h)exp[-iγb(κinc)z]=Eb(z, h)for z<h,
r(h)=γa(κinc)-γb(κinc)γa(κinc)+γb(κinc)exp[-2iγa(κinc)h],
t(h)=2γa(κinc)γa(κinc)+γb(κinc)×exp{i[γb(κinc)-γa(κinc)]h}.
e(z)=-zEa(z, h)p1(h)dh+z+Eb(z, h)p1(h)dh,
r=-+r(h)p1(h)dh=γa(κinc)-γb(κinc)γa(κinc)+γb(κinc)exp[-2γa2(κinc)δ2],
t=-+t(h)p1(h)dh=2γa(κinc)γa(κinc)+γb(κinc)×exp{-[γb(κinc)-γa(κinc)]2δ2/2}.

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