Abstract

A simple theory of enhanced backscattering from weakly rough dielectric and metallic surfaces is presented in which constructive interference of time-reversed scattering paths is explicitly displayed. We show that, when starting from a perturbative second-order electromagnetic theory, one may calculate the backscattering intensity distribution from a single analytic formula. Excellent agreement is obtained with published results that were achieved with extensive numerical simulations. In the case of scattering from dielectric surfaces or loss-less metallic surfaces, the angular width of the backscattering peak is determined solely by diffraction.

© 1993 Optical Society of America

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References

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  1. A. A. Maradudin, E. R. Mendez, T. Michel, in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), p. 157.
  2. S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
    [CrossRef]
  3. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  4. G. Bergman, “Weak localization in thin films,” Phys. Lett. 107, 2–55 (1984).
  5. A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [CrossRef]
  6. K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  7. R. T. Deck, R. K. Grygier, “Surface-plasmon enhanced harmonic generation at a rough metal surface,” Appl. Opt. 23, 3202–3213 (1984).
    [CrossRef] [PubMed]
  8. M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
    [CrossRef] [PubMed]
  9. S. Etemad, R. Thompson, M. J. Andresco, “Weak localization of photons, universal fluctuations, and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
    [CrossRef] [PubMed]
  10. P. Tran, A. A. Maradudin, V. Celli, “Backscattering enhancement from a dielectric surface,” J. Opt. Soc. Am. B 8, 1526–1530 (1991).
    [CrossRef]
  11. A. R. McGurn, A. A. Maradudin, “Localization effects in the scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
    [CrossRef]
  12. A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
    [CrossRef]

1991

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

P. Tran, A. A. Maradudin, V. Celli, “Backscattering enhancement from a dielectric surface,” J. Opt. Soc. Am. B 8, 1526–1530 (1991).
[CrossRef]

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

1989

1987

1986

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

S. Etemad, R. Thompson, M. J. Andresco, “Weak localization of photons, universal fluctuations, and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

1985

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

1984

Andresco, M. J.

S. Etemad, R. Thompson, M. J. Andresco, “Weak localization of photons, universal fluctuations, and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Bergman, G.

G. Bergman, “Weak localization in thin films,” Phys. Lett. 107, 2–55 (1984).

Celli, V.

P. Tran, A. A. Maradudin, V. Celli, “Backscattering enhancement from a dielectric surface,” J. Opt. Soc. Am. B 8, 1526–1530 (1991).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Dainty, J. C.

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

Deck, R. T.

Edrei, I.

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Etemad, S.

S. Etemad, R. Thompson, M. J. Andresco, “Weak localization of photons, universal fluctuations, and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Freund, I.

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Grygier, R. K.

Gu, Z.-H.

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

John, S.

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

Kaveh, M.

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Lu, J. Q.

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

P. Tran, A. A. Maradudin, V. Celli, “Backscattering enhancement from a dielectric surface,” J. Opt. Soc. Am. B 8, 1526–1530 (1991).
[CrossRef]

A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef]

A. R. McGurn, A. A. Maradudin, “Localization effects in the scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

A. A. Maradudin, E. R. Mendez, T. Michel, in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), p. 157.

McGurn, A. R.

A. R. McGurn, A. A. Maradudin, “Localization effects in the scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Mendez, E. R.

Méndez, E. R.

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

Michel, T.

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef]

A. A. Maradudin, E. R. Mendez, T. Michel, in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), p. 157.

Nieto-Vesperinas, M.

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

O’Donnell, K. A.

Rosenbluh, M.

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Sant, A. J.

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

Thompson, R.

S. Etemad, R. Thompson, M. J. Andresco, “Weak localization of photons, universal fluctuations, and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Tran, P.

Appl. Opt.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Lett.

G. Bergman, “Weak localization in thin films,” Phys. Lett. 107, 2–55 (1984).

Phys. Rev. B

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Phys. Rev. Lett.

M. Kaveh, M. Rosenbluh, I. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

S. Etemad, R. Thompson, M. J. Andresco, “Weak localization of photons, universal fluctuations, and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Phys. Today

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

Waves Random Media

A. A. Maradudin, J. Q. Lu, T. Michel, Z.-H. Gu, J. C. Dainty, A. J. Sant, E. R. Méndez, M. Nieto-Vesperinas, “Enhanced backscattering and transmission of light from random surfaces on semi-infinite substrates and thin films,” Waves Random Media 3, S129–S141 (1991).
[CrossRef]

Other

A. A. Maradudin, E. R. Mendez, T. Michel, in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), p. 157.

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Figures (2)

Fig. 1
Fig. 1

Diagram of second-order scattering from one-dimensional rough surface illustrating the change in phase between incident and scattered wave fronts.

Fig. 2
Fig. 2

Second-order scattering intensity versus scattering angle for the scattering of p-polarized light from a one-dimensional dielectric surface at normal incidence. The filled squares represent data taken from Fig. 1 of Ref. 10; the dashed curve is calculated from Eq. (19).

Equations (31)

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ϕ a a = k x 1 + ϕ 12 + K x ( L - x 2 ) ,
ϕ b b = k x 2 + ϕ 21 + K x ( L - x 1 ) .
Δ ϕ = ( k + K x ) ( x 1 - x 2 ) .
E ( 2 ) ( r ) = d K x ( k ^ s + y ^ ) E ( 2 ) ( K x ) exp ( i k s · r ) ,
k s = K x x ^ + ( 0 ω 2 c 2 - K x 2 ) 1 / 2 z ^ .
ζ ( K x ) = 1 / 2 π d x Z ( x ) exp ( - i K x x )
E ( 2 ) ( K x ) = E 0 d K x ζ ( K x - K x ) ζ ( K x - k ) f ( K x , K x ; k ) .
S ( 2 ) ( r ) = c 2 8 π ω d K x d K x E ( 2 ) ( K x ) E ( 2 ) * ( K x ) k s × exp [ i ( k s - k s ) · r ] .
K x = ( 2 π / L ) n ,             n = 0 , ± 1 , ± 2 ,
- d K x « 2 π L K x .
S ( 2 ) ( r ) = ( 2 π / L ) K x S ( 2 ) ( K x ) ,
S ( 2 ) ( K x ) = ( c 2 / 8 π ω ) ( 2 π / L ) E ( 2 ) ( K x ) 2 k s
ζ ( K x - K x ) = 1 / 2 π d x Z ( x ) exp [ - i ( K x - K x ) x ] .
E ( 2 ) ( K x ) = 1 2 π L E 0 K x f ( K x , K x ; k ) × - L / 2 L / 2 d x 1 - L / 2 L / 2 d x 2 Z ( x 1 ) Z ( x 2 ) × exp [ i ( k - K x ) x 1 ] exp [ i ( K x - K x ) x 2 ] .
E ( 2 ) ( K x ) = E 0 2 π L K x f ( K x , K x ; k ) × x 1 x 2 ( Δ x ) 2 Z ( x 1 ) Z ( x 2 ) exp [ i ( k - K x ) x 1 ] × exp [ i ( K x - K x ) x 2 ] .
E ( 2 ) ( K x ) 2 = E 0 2 ( 2 π L ) 2 K x x 1 x 2 > x 1 ( Δ x ) 4 Z 2 ( x 1 ) Z 2 ( x 2 ) × F ( k , K x , K x ; x 1 , x 2 ) ,
F ( k , K x , K x ; x 1 , x 2 ) = f ( K x , K x ; k ) exp [ i ( k - K x ) x 1 ] exp [ i ( K x - K x ) x 2 ] + f ( K x , - K x ; k ) exp [ i ( k + K x ) x 2 ] exp [ - i ( K x + K x ) x 1 ] 2 .
F ( k , K x , K x ; x 1 , x 2 ) = f ( K x ) 2 + f ( - K x 2 + 2 Re { f ( K x ) f * ( - K x ) exp [ i ( k + K x ) ( x 1 - x 2 ) ] } ,
Z ( x 1 ) Z ( x 2 ) = δ 2 exp [ - ( x 1 - x 2 ) 2 σ 2 ] , Z ( x 1 ) Z ( x 2 ) Z ( x 3 ) = Z ( x 1 ) = 0.
Z 2 ( x ) = δ 2 , Z 2 ( x 1 ) Z 2 ( x 2 ) = Z 2 ( x 1 ) Z 2 ( x 2 ) = δ 4 .
S ( 2 ) ( K x ) = c 2 8 π ω E 0 2 k s δ 4 ( Δ x ) 4 2 π L 3 × k x x 1 x 2 > x 1 F ( k , K x , K x ; x 1 , x 2 ) .
( Δ x ) 2 x 1 x 2 > x 1 - L / 2 L / 2 d x 1 x 1 L / 2 d x 2 ,
S ( 2 ) ( K x ) = c 2 k s 8 π ω E 0 2 δ 4 ( Δ x ) 2 2 ( 2 π ) 2 × d K x { [ f ( K x ) 2 + f ( - K x ) 2 ] + 2 Re [ f ( K x ) f * ( - K x ) ] sinc 2 ( Q L 2 ) + 4 Im [ f ( K x ) f * ( - K x ) ] [ 1 - sinc ( Q L ) ] Q L } ,
Δ θ λ / D ,
peak / background = 1 + 2 Re [ f ( K x ) f * ( - K x ) ] / [ f ( K x ) 2 + f ( - K x ) 2 ] ,
f ( K x , K x ; k ) = f ˜ ( K x ) ( K x - K p ) ( K x + K p ) ,
E ( 2 ) ( K x ) = ( Δ x ) 2 ( 2 π ) 2 E 0 x 1 x 2 Z ( x 1 ) Z ( x 2 ) I ( x 1 , x 2 ; k , K x ) ,
1 2 π i ( x 1 , x 2 ; k , K x ) = { exp [ i ( k x 1 - K x x 2 ) ] exp [ i K ( x 2 - x 2 ) ] f ˜ ( K p ) 2 K p x 2 > x 1 exp [ i ( k x 1 - K x x 2 ) ] exp [ - i K p ( x 2 - x 1 ) ] f ˜ ( - K p ) 2 K p x 2 < x 1 .
E ( 2 ) ( K x ) 2 = ( Δ x ) 4 ( 2 π ) 4 E 0 2 x 1 x 2 > x 1 Z 2 ( x 1 ) Z 2 ( x 2 ) × I ( x 1 , x 2 ; k , K x ) + I ( x 2 , x 1 ; k , K x ) 2 = ( Δ x ) 4 ( 2 π ) 2 E 0 2 4 K p 2 x 1 x 2 > x 1 Z 2 ( x 1 ) Z 2 ( x 2 ) × exp [ - 2 K p ( x 2 - x 1 ) ] { [ f ˜ ( K p ) 2 + f ˜ ( - K p ) 2 ] + 2 Re [ f ˜ ( K p ) f ˜ * ( - K p ) ] × exp [ i ( k + K x ) ( x 1 - x 2 ) ] } .
S ( 2 ) ( K x ) = c 2 k s 8 π ω ( Δ x ) 2 δ 4 8 π K p 2 E 0 2 { } ,
{ } [ f ˜ ( K p ) 2 + f ˜ ( - K p ) 2 ] 2 K p { 1 - [ 1 - exp ( - 2 i K p L ) ] 2 K p L } + 2 Re ( f ˜ ( K p ) f ˜ * ( - K p ) ( 2 K p + i Q ) { 1 - 1 - exp [ - 2 K p + i Q ) L ( 2 K p + i Q ) L } ) .

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