Abstract

Scattering by a small object located close to an interface is analyzed according to the exact-image theory formulation. The scatterer is assumed to be small compared with wavelength, permitting the electric-dipole approximation, and to have a scalar polarizability. After the derivation of the dipole moment, investigations concentrate on far-field scattering. Backscattering enhancement and reversal of linear polarization are confirmed through statistical averaging over scatterer height and system orientation.

© 1991 Optical Society of America

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References

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  1. I. V. Lindell, A. H. Sihvola, K. O. Muinonen, P. W. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991).
    [CrossRef]
  2. K. O. Muinonen, “Electromagnetic scattering by two interacting dipoles,” in Proceedings of the 1989 URSI EM Theory Symposium (International Union of Radio Science, Brussels, 1989), pp. 428–430.
  3. H. von Seeliger, “Zur Theorie der Beleuchtung der grossen Planeten, Insbesondere des Saturn,” Abh. Bayer. Akad. Wissen. Math. Naturwiss. Kl. 16, 405–516 (1887).
  4. A. Rougier, “Photométrie photoélectrique globale de la Lune,” Ann. Obs. Strasbourg 2, 205–339 (1933).
  5. B. Lyot, “Recherches sur la polarisation de la lumière des planètes et de quelques substances terrestres,” Ann. Obs. Paris 8, 1–161 (1929).
  6. Y. G. Shkuratov, “Diffractional model of the brightness surge of complex structure surfaces,” Kinemat. Fis. Nebesnyh Tel 4, 33–39 (1988).
  7. Y. G. Shkuratov, N. V. Opanasenko, L. Y. Melkumova, “Interference surge of backscattering and negative polarization of light reflected by complex structure,” Preprint 361 (Institute of Radiophysics and Electronics, Academy of Sciences, Kharkov, USSR, 1989), pp. 1–26.
  8. S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 1-4 (Springer-Verlag, Berlin, 1989).
  9. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  10. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  11. I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. 32, 1027–1032 (1989).
    [CrossRef]
  12. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

1991

1989

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. 32, 1027–1032 (1989).
[CrossRef]

1988

Y. G. Shkuratov, “Diffractional model of the brightness surge of complex structure surfaces,” Kinemat. Fis. Nebesnyh Tel 4, 33–39 (1988).

1933

A. Rougier, “Photométrie photoélectrique globale de la Lune,” Ann. Obs. Strasbourg 2, 205–339 (1933).

1929

B. Lyot, “Recherches sur la polarisation de la lumière des planètes et de quelques substances terrestres,” Ann. Obs. Paris 8, 1–161 (1929).

1887

H. von Seeliger, “Zur Theorie der Beleuchtung der grossen Planeten, Insbesondere des Saturn,” Abh. Bayer. Akad. Wissen. Math. Naturwiss. Kl. 16, 405–516 (1887).

Alanen, E.

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. 32, 1027–1032 (1989).
[CrossRef]

Barber, P. W.

Bohren, C. F.

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

Huffman, D. R.

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

Kong, J. A.

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

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 1-4 (Springer-Verlag, Berlin, 1989).

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, K. O. Muinonen, P. W. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. 32, 1027–1032 (1989).
[CrossRef]

Lyot, B.

B. Lyot, “Recherches sur la polarisation de la lumière des planètes et de quelques substances terrestres,” Ann. Obs. Paris 8, 1–161 (1929).

Melkumova, L. Y.

Y. G. Shkuratov, N. V. Opanasenko, L. Y. Melkumova, “Interference surge of backscattering and negative polarization of light reflected by complex structure,” Preprint 361 (Institute of Radiophysics and Electronics, Academy of Sciences, Kharkov, USSR, 1989), pp. 1–26.

Muinonen, K. O.

I. V. Lindell, A. H. Sihvola, K. O. Muinonen, P. W. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991).
[CrossRef]

K. O. Muinonen, “Electromagnetic scattering by two interacting dipoles,” in Proceedings of the 1989 URSI EM Theory Symposium (International Union of Radio Science, Brussels, 1989), pp. 428–430.

Opanasenko, N. V.

Y. G. Shkuratov, N. V. Opanasenko, L. Y. Melkumova, “Interference surge of backscattering and negative polarization of light reflected by complex structure,” Preprint 361 (Institute of Radiophysics and Electronics, Academy of Sciences, Kharkov, USSR, 1989), pp. 1–26.

Rougier, A.

A. Rougier, “Photométrie photoélectrique globale de la Lune,” Ann. Obs. Strasbourg 2, 205–339 (1933).

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 1-4 (Springer-Verlag, Berlin, 1989).

Shin, R. T.

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

Shkuratov, Y. G.

Y. G. Shkuratov, “Diffractional model of the brightness surge of complex structure surfaces,” Kinemat. Fis. Nebesnyh Tel 4, 33–39 (1988).

Y. G. Shkuratov, N. V. Opanasenko, L. Y. Melkumova, “Interference surge of backscattering and negative polarization of light reflected by complex structure,” Preprint 361 (Institute of Radiophysics and Electronics, Academy of Sciences, Kharkov, USSR, 1989), pp. 1–26.

Sihvola, A. H.

Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 1-4 (Springer-Verlag, Berlin, 1989).

Tsang, L.

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

von Seeliger, H.

H. von Seeliger, “Zur Theorie der Beleuchtung der grossen Planeten, Insbesondere des Saturn,” Abh. Bayer. Akad. Wissen. Math. Naturwiss. Kl. 16, 405–516 (1887).

Abh. Bayer. Akad. Wissen. Math. Naturwiss. Kl.

H. von Seeliger, “Zur Theorie der Beleuchtung der grossen Planeten, Insbesondere des Saturn,” Abh. Bayer. Akad. Wissen. Math. Naturwiss. Kl. 16, 405–516 (1887).

Ann. Obs. Paris

B. Lyot, “Recherches sur la polarisation de la lumière des planètes et de quelques substances terrestres,” Ann. Obs. Paris 8, 1–161 (1929).

Ann. Obs. Strasbourg

A. Rougier, “Photométrie photoélectrique globale de la Lune,” Ann. Obs. Strasbourg 2, 205–339 (1933).

IEEE Trans. Antennas Propag.

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. 32, 1027–1032 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Kinemat. Fis. Nebesnyh Tel

Y. G. Shkuratov, “Diffractional model of the brightness surge of complex structure surfaces,” Kinemat. Fis. Nebesnyh Tel 4, 33–39 (1988).

Other

Y. G. Shkuratov, N. V. Opanasenko, L. Y. Melkumova, “Interference surge of backscattering and negative polarization of light reflected by complex structure,” Preprint 361 (Institute of Radiophysics and Electronics, Academy of Sciences, Kharkov, USSR, 1989), pp. 1–26.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 1-4 (Springer-Verlag, Berlin, 1989).

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

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

K. O. Muinonen, “Electromagnetic scattering by two interacting dipoles,” in Proceedings of the 1989 URSI EM Theory Symposium (International Union of Radio Science, Brussels, 1989), pp. 428–430.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (8)

Fig. 1
Fig. 1

Scattering geometry. The incident and the reflected fields propagate in the yz plane and scatter from a small object. Notice the fixed and incidence-dependent (primed) coordinate systems.

Fig. 2
Fig. 2

Total and second-order scattering by a spherical object at kh = π/2 according to Eq. (18) for an unpolarized incident field with incidence angle θi = 45°. See text for scattering into grazing angles.

Fig. 3
Fig. 3

Same as Fig. 2 for kh = π. Note the asymmetry of total scattering in the yz plane of incidence.

Fig. 4
Fig. 4

Same as Fig. 2 for kh = 2π. Notice the formation of backward enhancement in second-order scattering.

Fig. 5
Fig. 5

Slight backward enhancement in averaged total scattering. The enhancement is due mainly to second-order scattering.

Fig. 6
Fig. 6

Backward enhancement in averaged second-order scattering. Note the sharpening of the peak with increasing mean height.

Fig. 7
Fig. 7

Reversal of linear polarization in averaged total scattering. The horizontal lines indicate zero-polarization levels.

Fig. 8
Fig. 8

Reversal of linear polarization in averaged second-order scattering. Note the widening of the negative branch with decreasing mean height.

Equations (28)

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E i ( u z h ) = E 0 exp ( j β i h ) , E r ( u z h ) = R i ( β i ) · E 0 exp ( - j β i h ) , β i = k cos θ i ,             k 2 = ω 2 μ 0 0 ,
R i ( β i ) = u x u x R TE ( β i ) + u y u y R TM ( β i ) - u z u z R TM ( β i ) ,
R TE ( β i ) = β i - β 1 β i + β 1 , R TM ( β i ) = - β i - β 1 β 1 + β 1 ,             β 1 = k ( - 1 + β i 2 k 2 ) 1 / 2 .
α = α J = 4 π 0 s - 1 s + 2 a 3 J ,
p = [ J - ω 2 μ 0 α · K ( u z 2 h ) · C ] - 1 · α · [ E i ( u z h ) + E r ( u z h ) ] ,
C = J t - u z u z , J t = u x u x + u y u y .
K ( u z 2 h ) = J t K t ( 2 h ) + u z u z K z ( 2 h ) ,
p = Q ( u z 2 h ) · [ E i ( u z h ) + E r ( u z h ) ] ,
Q ( u z 2 h ) = J t Q t ( 2 h ) + u z u z Q z ( 2 h ) , Q t ( 2 h ) = α 1 - ω 2 μ 0 α K t ( 2 h ) , Q z ( 2 h ) = α 1 + ω 2 μ 0 α K z ( 2 h ) .
p = [ exp ( j β i h ) Q + exp ( - j β i h ) Q · R i ] · E 0 .
E 1 s ( r ) = - j ω μ 0 V d V G ( r - r ) · J ( r ) = - j ω μ 0 V d V G ( r - r ) · j ω p δ ( r - u z h ) ω 2 μ 0 G ( r ) exp ( j β s h ) J s · p ,             k r 1 ,
G ( r ) = ( J + 1 k 2 ) · G ( r ) , G ( r ) = exp [ - j k D ( r ) ] 4 π D ( r ) ,             D ( r ) = ( r · r ) 1 / 2 .
J s = u ϑ u ϑ + u φ u φ , β s = k cos ϑ .
E 2 s ( r ) = - j ω μ 0 V d V K ( r - r ) · J c ( r ) = - j ω μ 0 V d V K ( r - r ) · C · j ω p δ ( r + u z h ) ω 2 μ 0 G ( r ) exp ( - j β s h ) R s ( β s ) · p ,             k r 1 ,
R s ( β s ) = [ u ϑ u ϑ R TM ( β s ) + u φ u φ R TE ( β s ) ] · C .
E s ( r ) = E 1 s ( r ) + E 2 s ( r ) = ω 2 μ 0 G ( r ) [ exp ( j β s h ) J s + exp ( - j β s h ) R s ( β s ) ] · p ω 2 μ 0 G ( r ) A ,
A = { exp [ j ( β i + β s ) h ] J s · Q + exp [ - j ( β i - β s ) h ] J s · Q · R i + exp [ j ( β i - β s ) h ] R s · Q + exp [ - j ( β i + β s ) h ] R s · Q · R i } · E 0 .
S ( ϑ , φ ) = 1 2 A TE 2 + A TM 2 E 0 2 .
p h ( h ) d h = 1 ( γ - 1 ) ! ( γ h 0 ) γ h γ - 1 exp ( - γ h h 0 ) d h ,
p t ( t ) d t d ϕ i = 1 2 π ρ 2 exp ( - t 2 2 ρ 2 ) t d t d ϕ i = 1 2 π ρ 2 exp ( - tan 2 θ i 2 ρ 2 ) sin θ i cos 3 θ i d θ i d ϕ i ,
A θ Θ 1 exp [ j ( β i + β s ) h ] + Θ 2 exp [ - j ( β i - β s ) h ] + Θ 3 exp [ j ( β i - β s ) h ] + Θ 4 exp [ - j ( β i + β s ) h ] , A ϕ Φ 1 exp [ j ( β i + β s ) h ] + Φ 2 exp [ - j ( β i - β s ) h ] + Φ 3 exp [ j ( β i - β s ) h ] + Φ 4 exp [ - j ( β i + β s ) h ] ,
Θ 1 = u θ · Q · E 0 , Θ 2 = u 0 · Q · R i · E 0 , Θ 3 = u θ · R s · Q · E 0 , Θ 4 = u θ · R s · Q · R i · E 0 .
sin ϑ cos φ = sin θ cos ϕ , sin ϑ sin φ = sin θ sin ϕ cos θ i - cos θ sin θ i , cos ϑ = sin θ sin ϕ sin θ i + cos θ cos θ i
S ( θ ) = 1 2 E 0 2 A θ TE 2 + A θ TM 2 + A ϕ TE 2 + A ϕ TM 2 , P ( θ ) = - A θ TE 2 + A θ TM 2 - A ϕ TE 2 - A ϕ TM 2 A θ TE 2 + A θ TM 2 + A ϕ TE 2 + A ϕ TM 2 ,
A θ TE 2 = 0 d h p h ( h ) 0 d t 0 2 π d ϕ i p t ( t , ϕ i ) A θ TE 2 .
ω 2 μ 0 α K t = 4 π k 3 a 3 s - 1 s + 2 K t k < 10 - 4 1 ,
A θ 2 h = Θ 1 2 + Θ 2 2 + Θ 3 2 + Θ 4 2 + 2 Re { ( Θ 1 * Θ 2 + Θ 3 * Θ 4 ) × exp ( - j 2 β i h ) h + ( Θ 1 * Θ 3 + Θ 2 * Θ 4 ) exp ( - j 2 β s h ) h + Θ 1 * Θ 4 exp [ - j 2 ( β i + β s ) h ] h + Θ 2 * Θ 3 exp [ j 2 ( β i - β s ) h ] h }
exp ( j 2 β h ) h = 0 d h p h ( h ) exp ( j 2 β h ) = 1 ( 1 - j 2 β h 0 γ ) γ ,

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