Abstract

The analytical expression for the Bi-frequency correlation function of the intensity scattered from two-dimensional dielectric randomly rough surfaces obeying Gaussian distribution are presented based on the scalar Kirchhoff approximation theory with the root-mean-square (rms) slope of the surface less than 0.25 and the Gaussian moment theorem. The results show that the bi-frequency correlation properties of the scattered intensity closely depend on the incident and scattered conditions as well as on the statistical parameters and complex refractive index of the surface. Especially, the correlation function mainly comes from the specular direction, and the coherence bandwidth and the function decrease with the increase of the roughness of the rough surface. In addition, comparing with the real part, the imagery of the complex refractive index has a greater impact on the bi-frequency correlation function.

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).
  2. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).
  21. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1958).

2012 (1)

2011 (2)

2010 (1)

2009 (3)

V. N. Bronnikov and M. M. Kalugin, “Measuring the parameters of vibrations and surface roughness, using the frequency spectrum of the intensity fluctuations of scattered radiation,” J. Opt. Tech. 76(11), 697–701 (2009).
[CrossRef]

M. J. Wang, Z. S. Wu, Y. L. Li, X. A. Zhang, and H. Zhang, “The fourth order moment statistical characteristic of the laser pulse scattering on random rough surface,” Acta Phys. Sin-CH ED 58, 2390–2396 (2009).

W. Zhen-Sen and Z. Geng “Intensity Correlation Function of Light Scattering from a Weakly One-Dimensional Random Rough Surface,” Chin. Phys. Lett. 26(11), 114208 (2009).
[CrossRef]

2007 (1)

Z. W. Xu, J. Wu, Z. S. Wu, and Q. Li, “Solution for the Fourth Moment Equation of Waves in Random Continuum Under Strong Fluctuations: General Theory and Plane Wave Solution,” IEEE Trans. Antenn. Propag. 55(6), 1613–1621 (2007).
[CrossRef]

2004 (2)

C. Bourlier, “Azimuthal Harmonic Coefficients of the Microwave Backscattering from a Non-Gaussian Ocean Surface With the First-Order SSA Model,” IEEE Trans. Geosci. Rem. Sens. 42(11), 2600–2611 (2004).
[CrossRef]

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves 25(8), 1211–1219 (2004).
[CrossRef]

1994 (2)

A. Ishimaru, L. Ailes-Sengers, P. Phu, and D. Winebrenner, “Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces,” Waves Random Media 4(2), 139–148 (1994).
[CrossRef]

D. J. Schertler and N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A 11(8), 2286–2297 (1994).
[CrossRef]

1992 (2)

W. Zhensen and C. Suomin, “Bistatic scattering by arbitrarily shaped objects with rough surface at optical and infrared frequencies,” Int. J. Infrared Millim. Waves 13(4), 537–549 (1992).
[CrossRef]

D. J. Schertler and N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A 9(11), 2056–2066 (1992).
[CrossRef]

1986 (1)

1985 (1)

Ailes-Sengers, L.

A. Ishimaru, L. Ailes-Sengers, P. Phu, and D. Winebrenner, “Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces,” Waves Random Media 4(2), 139–148 (1994).
[CrossRef]

Bahar, E.

Bourlier, C.

C. Bourlier, “Azimuthal Harmonic Coefficients of the Microwave Backscattering from a Non-Gaussian Ocean Surface With the First-Order SSA Model,” IEEE Trans. Geosci. Rem. Sens. 42(11), 2600–2611 (2004).
[CrossRef]

Bronnikov, V. N.

V. N. Bronnikov and M. M. Kalugin, “Measuring the parameters of vibrations and surface roughness, using the frequency spectrum of the intensity fluctuations of scattered radiation,” J. Opt. Tech. 76(11), 697–701 (2009).
[CrossRef]

Chakrabarti, S.

Chen, Y. R.

Fitzwater, M. A.

Geng, Z.

W. Zhen-Sen and Z. Geng “Intensity Correlation Function of Light Scattering from a Weakly One-Dimensional Random Rough Surface,” Chin. Phys. Lett. 26(11), 114208 (2009).
[CrossRef]

George, N.

He, Y. J.

Hui, C.

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves 25(8), 1211–1219 (2004).
[CrossRef]

Ishimaru, A.

A. Ishimaru, L. Ailes-Sengers, P. Phu, and D. Winebrenner, “Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces,” Waves Random Media 4(2), 139–148 (1994).
[CrossRef]

Jacks, H. C.

Kalugin, M. M.

V. N. Bronnikov and M. M. Kalugin, “Measuring the parameters of vibrations and surface roughness, using the frequency spectrum of the intensity fluctuations of scattered radiation,” J. Opt. Tech. 76(11), 697–701 (2009).
[CrossRef]

Korotkova, O.

Li, J.

Li, Q.

Z. W. Xu, J. Wu, Z. S. Wu, and Q. Li, “Solution for the Fourth Moment Equation of Waves in Random Continuum Under Strong Fluctuations: General Theory and Plane Wave Solution,” IEEE Trans. Antenn. Propag. 55(6), 1613–1621 (2007).
[CrossRef]

Li, Y. L.

M. J. Wang, Z. S. Wu, Y. L. Li, X. A. Zhang, and H. Zhang, “The fourth order moment statistical characteristic of the laser pulse scattering on random rough surface,” Acta Phys. Sin-CH ED 58, 2390–2396 (2009).

Lu, B.

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves 25(8), 1211–1219 (2004).
[CrossRef]

Phu, P.

A. Ishimaru, L. Ailes-Sengers, P. Phu, and D. Winebrenner, “Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces,” Waves Random Media 4(2), 139–148 (1994).
[CrossRef]

Schertler, D. J.

Suomin, C.

W. Zhensen and C. Suomin, “Bistatic scattering by arbitrarily shaped objects with rough surface at optical and infrared frequencies,” Int. J. Infrared Millim. Waves 13(4), 537–549 (1992).
[CrossRef]

Wang, M. J.

M. J. Wang, Z. S. Wu, Y. L. Li, X. A. Zhang, and H. Zhang, “The fourth order moment statistical characteristic of the laser pulse scattering on random rough surface,” Acta Phys. Sin-CH ED 58, 2390–2396 (2009).

Winebrenner, D.

A. Ishimaru, L. Ailes-Sengers, P. Phu, and D. Winebrenner, “Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces,” Waves Random Media 4(2), 139–148 (1994).
[CrossRef]

Wu, J.

Z. W. Xu, J. Wu, Z. S. Wu, and Q. Li, “Solution for the Fourth Moment Equation of Waves in Random Continuum Under Strong Fluctuations: General Theory and Plane Wave Solution,” IEEE Trans. Antenn. Propag. 55(6), 1613–1621 (2007).
[CrossRef]

Wu, Z.

Wu, Z. S.

G. Zhang and Z. S. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express 19(8), 7007–7019 (2011).
[CrossRef] [PubMed]

M. J. Wang, Z. S. Wu, Y. L. Li, X. A. Zhang, and H. Zhang, “The fourth order moment statistical characteristic of the laser pulse scattering on random rough surface,” Acta Phys. Sin-CH ED 58, 2390–2396 (2009).

Z. W. Xu, J. Wu, Z. S. Wu, and Q. Li, “Solution for the Fourth Moment Equation of Waves in Random Continuum Under Strong Fluctuations: General Theory and Plane Wave Solution,” IEEE Trans. Antenn. Propag. 55(6), 1613–1621 (2007).
[CrossRef]

Xin, Y.

Xu, Z. W.

Z. W. Xu, J. Wu, Z. S. Wu, and Q. Li, “Solution for the Fourth Moment Equation of Waves in Random Continuum Under Strong Fluctuations: General Theory and Plane Wave Solution,” IEEE Trans. Antenn. Propag. 55(6), 1613–1621 (2007).
[CrossRef]

Zhang, G.

Zhang, H.

M. J. Wang, Z. S. Wu, Y. L. Li, X. A. Zhang, and H. Zhang, “The fourth order moment statistical characteristic of the laser pulse scattering on random rough surface,” Acta Phys. Sin-CH ED 58, 2390–2396 (2009).

Zhang, X. A.

M. J. Wang, Z. S. Wu, Y. L. Li, X. A. Zhang, and H. Zhang, “The fourth order moment statistical characteristic of the laser pulse scattering on random rough surface,” Acta Phys. Sin-CH ED 58, 2390–2396 (2009).

Zhensen, W.

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves 25(8), 1211–1219 (2004).
[CrossRef]

W. Zhensen and C. Suomin, “Bistatic scattering by arbitrarily shaped objects with rough surface at optical and infrared frequencies,” Int. J. Infrared Millim. Waves 13(4), 537–549 (1992).
[CrossRef]

Zhen-Sen, W.

W. Zhen-Sen and Z. Geng “Intensity Correlation Function of Light Scattering from a Weakly One-Dimensional Random Rough Surface,” Chin. Phys. Lett. 26(11), 114208 (2009).
[CrossRef]

Acta Phys. Sin-CH ED (1)

M. J. Wang, Z. S. Wu, Y. L. Li, X. A. Zhang, and H. Zhang, “The fourth order moment statistical characteristic of the laser pulse scattering on random rough surface,” Acta Phys. Sin-CH ED 58, 2390–2396 (2009).

Appl. Opt. (2)

Chin. Phys. Lett. (1)

W. Zhen-Sen and Z. Geng “Intensity Correlation Function of Light Scattering from a Weakly One-Dimensional Random Rough Surface,” Chin. Phys. Lett. 26(11), 114208 (2009).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

Z. W. Xu, J. Wu, Z. S. Wu, and Q. Li, “Solution for the Fourth Moment Equation of Waves in Random Continuum Under Strong Fluctuations: General Theory and Plane Wave Solution,” IEEE Trans. Antenn. Propag. 55(6), 1613–1621 (2007).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (1)

C. Bourlier, “Azimuthal Harmonic Coefficients of the Microwave Backscattering from a Non-Gaussian Ocean Surface With the First-Order SSA Model,” IEEE Trans. Geosci. Rem. Sens. 42(11), 2600–2611 (2004).
[CrossRef]

Int. J. Infrared Millim. Waves (2)

W. Zhensen and C. Suomin, “Bistatic scattering by arbitrarily shaped objects with rough surface at optical and infrared frequencies,” Int. J. Infrared Millim. Waves 13(4), 537–549 (1992).
[CrossRef]

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves 25(8), 1211–1219 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Tech. (1)

V. N. Bronnikov and M. M. Kalugin, “Measuring the parameters of vibrations and surface roughness, using the frequency spectrum of the intensity fluctuations of scattered radiation,” J. Opt. Tech. 76(11), 697–701 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Waves Random Media (1)

A. Ishimaru, L. Ailes-Sengers, P. Phu, and D. Winebrenner, “Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces,” Waves Random Media 4(2), 139–148 (1994).
[CrossRef]

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).

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

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing, Vol. 2 (Addison-Wesley Publishing,1982).

J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1958).

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Figures (9)

Fig. 1
Fig. 1

Bi-frequency correlation function C 12 versus scattering angle and frequency difference with the parameters δ=0.6μm and l c =5.89μm

Fig. 4
Fig. 4

Bi-frequency correlation function C 12 versus scattering angle and frequency difference with the parameters θ i = 10 ° , δ=0.9μm and l c =7.5μm

Fig. 3
Fig. 3

Bi-frequency correlation function C 12 versus scattering angle and frequency difference with the parameters δ=0.9μm and l c =5.89μm

Fig. 2
Fig. 2

Bi-frequency correlation function C 12 versus scattering angle and frequency difference with the parameters δ=0.8μm and l c =5.89μm

Fig. 5
Fig. 5

Bi-frequency correlation function C 12 versus scattering azimuth angle and frequency difference with the parameters δ=0.6μm and l c =5.89μm , HH-polarization

Fig. 6
Fig. 6

Bi-frequency correlation function C 12 versus scattering azimuth angle and frequency difference with the parameters δ=0.8μm and l c =5.89μm , HH-polarization

Fig. 7
Fig. 7

Bi-frequency correlation function C 12 versus scattering azimuth angle and frequency difference with the parameters δ=0.6μm and l c =5.89μm , VH-polarization

Fig. 8
Fig. 8

Bi-frequency correlation function C 12 versus scattering azimuth angle and frequency difference with δ=0.8μm and l c =5.89μm , VH-polarization

Fig. 9
Fig. 9

Bi-frequency correlation function C 12 versus frequency difference with different refractive indexes

Equations (63)

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E pq s ( f )= E 0 K( f ) p( x,y ) U pq exp[ i( k s k i )r ]dS
E pq s ( f )=Δ E pq s ( f )+ E pq s ( f )
C 12 = | I pq1 ( f 1 , f 2 ) | 2 + | I pq2 ( f 1 , f 2 ) | 2 2 E pq1 s 2 E pq2 s 2
I pq1 = U pq ( f 1 ) U pq * ( f 2 )exp[ i( V 1 r 1 V 2 r 2 ) ] d S 1 d S 2
U pq ( f 1 ) U pq * ( f 2 )= a 01 a 02 * + a 11 a 02 * Z x1 + a 01 a 12 * Z x2 + a 21 a 02 * Z y1 + a 01 a 22 * Z y2
I pq1 ( f 1 , f 2 )= I pq10 ( f 1 , f 2 )+ I pq1s ( f 1 , f 2 )
I pq10 = a 01 a 02 * exp[ i( V z1 h 1 V z2 h 2 ) ] exp[ i( V 1 r 1 V 2 r 2 ) ]d S 1 d S 2
I pq1s ( f 1 , f 2 )= I pq1x ( f 1 , f 2 )+ I pq1y ( f 1 , f 2 )
I pq1x = a 11 a 02 * I pq1x1 + a 01 a 12 * I pq1x2
I pq1y = a 21 a 02 * I pq1y1 + a 01 a 22 * I pq1y2
I pq1xj = Z xj exp[ i( V z1 h 1 V z2 h 2 ) ] exp[ i( V 1 r 1 V 2 r 2 ) ]d S 1 d S 2
I pq1yj = Z yj exp[ i( V z1 h 1 V z2 h 2 ) ] exp[ i( V 1 r 1 V 2 r 2 ) ]d S 1 d S 2
F( V z1 , V z2 ,ρ )= exp[ i( V z1 h 1 V z2 h 2 ) ] =exp[ ( β 11 + β 22 ) /2 ]exp[ β 12 ρ( r 1 r 2 ) ]
Z xj exp[ i( V z1 h 1 V z2 h 2 ) ] =i V z(3j) δ 2 ρ x d F( V z1 , V z2 ,ρ )
Z yj exp[ i( V z1 h 1 V z2 h 2 ) ] =i V z(3j) δ 2 ρ y d F( V z1 , V z2 ,ρ )
r c = ( r 1 + r 2 ) /2 r d = r 1 r 2 V c = ( V 1 + V 2 ) /2 V d = V 1 V 2
I pq10 = a 01 a 02 * β d r c d r d exp( 2 | r c | 2 D 2 )exp( | r d | 2 2 D 2 )exp( β 12 ρ )exp[ i( V d r c + V c r d ) ]
exp[ β 12 ρ( r d ) ]= n=0 ( β 12 ) n n! ρ n ( r d )
I pq10 = a 01 a 02 * π 2 D 4 l c 2 βexp[ D 2 | V d | 2 8 ] n=0 ( β 12 ) n n!( l c 2 +2n D 2 ) exp( D 2 l c 2 | V c | 2 2 l c 2 +4n D 2 )
x d =ξcosα y d =ξsinα
ρ x d = ρ ξ cosα ρ y d = ρ ξ sinα
I pq1xj =i V z(3j) π D 2 δ 2 β 2 exp[ D 2 | V d | 2 8 ] 0 ξdξexp( β 12 ρ )exp( ξ 2 2 D 2 ) ρ ξ Γ( ξ )
Γ( ξ )= 0 2π dαcosα exp[ iξ| V c |cos( αχ ) ]
exp[ ibcos( αχ ) ]= n= i n J n ( b )exp[ in( αχ ) ]
0 2π dαcosαexp( inα ) ={ π n=±1 0 n±1
0 2π dαsinαexp( inα ) ={ ±iπ n=±1 0 n±1
Γ( ξ )=i2π J 1 ( ξ| V c | )cosχ
I pq1xj =2 π 2 D 6 V z(3j) δ 2 β V cx exp( D 2 | V d | 2 /8 ) × n=0 l c 2 β 12 n n! [ l c 2 +2( 1+n ) D 2 ] 2 exp{ D 2 l c 2 | V c | 2 2[ l c 2 +2( 1+n ) D 2 ] }
I pq1yj =2 π 2 D 6 V z(3j) δ 2 β V cy exp( D 2 | V d | 2 /8 ) × n=0 l c 2 β 12 n n! [ l c 2 +2( 1+n ) D 2 ] 2 exp{ D 2 l c 2 | V c | 2 2[ l c 2 +2( 1+n ) D 2 ] }
I pq1s =[ ( a 11 a 02 * V z2 + a 01 a 12 * V z1 ) V cx +( a 21 a 02 * V z2 + a 01 a 22 * V z1 ) V cy ]2 π 2 D 6 δ 2 β ×exp[ D 2 | V d | 2 8 ] n=0 l c 2 β 12 n n! [ l c 2 +2( 1+n ) D 2 ] 2 exp{ D 2 l c 2 | V c | 2 2[ l c 2 +2( 1+n ) D 2 ] }
I pq2 = U pq1 U pq2 exp[ i( V 1 r 1 + V 2 r 2 ) ] d S 1 d S 2
U pq1 U pq2 = a 01 a 02 + a 11 a 02 Z x1 + a 01 a 12 Z x2 + a 21 a 02 Z y1 + a 01 a 22 Z y2
I pq2 ( f 1 , f 2 )= I pq20 ( f 1 , f 2 )+ I pq2s ( f 1 , f 2 )
I pq20 = a 01 a 02 exp[ i( V 1 r 1 + V 2 r 2 ) ] exp[ i( V z1 h 1 + V z2 h 2 ) ] d r 1 d r 2
I pq2s ( f 1 , f 2 )= I pq2x ( f 1 , f 2 )+ I pq2y ( f 1 , f 2 )
I pq2x = a 11 a 02 I pq2x1 + a 01 a 12 I pq2x2
I pq2y = a 21 a 02 I pq2y1 + a 01 a 22 I pq2y2
I pq2xj = Z xj exp[ i( V z1 h 1 + V z2 h 2 ) ] exp[ i( V 1 r 1 + V 2 r 2 ) ]d r 1 d r 2
I pq2yj = Z yj exp[ i( V z1 h 1 + V z2 h 2 ) ] exp[ i( V 1 r 1 + V 2 r 2 ) ]d r 1 d r 2
F( V z1 , V z2 ,ρ )= exp[ i( V z1 h 1 + V z2 h 2 ) ] =βexp( β 12 ρ )
Z xj exp[ i( V z1 h 1 + V z2 h 2 ) ] = ( 1 ) 3j i V z(3j) δ 2 ρ ξ cosαF( V z1 , V z2 ,ρ )
Z yj exp[ i( V z1 h 1 + V z2 h 2 ) ] = ( 1 ) 3j i V z(3j) δ 2 ρ ξ sinαF( V z1 , V z2 ,ρ )
I pq20 = a 01 a 02 πβ D 2 2 exp( D 2 | V c | 2 2 ) 0 ξdξexp( ξ 2 2 D 2 ) exp( β 12 ρ )Η( ξ )
I pq2xj = ( 1 ) 3j i δ 2 V z( 3j ) πβ D 2 2 exp( D 2 | V c | 2 2 ) 0 ξdξexp( ξ 2 2 D 2 ) ρ ξ exp( β 12 ρ )Λ( ξ )
I pq2yj = ( 1 ) 3j i δ 2 V z( 3j ) πβ D 2 2 exp( D 2 | V c | 2 2 ) 0 ξdξexp( ξ 2 2 D 2 ) ρ ξ exp( β 12 ρ )Φ( ξ )
Η( ξ )= 0 2π dαexp[ iξ| V d | 2 cos( αε ) ]
Λ( ξ )= 0 2π dαcosαexp[ iξ| V d | 2 cos( αε ) ]
Φ( ξ )= 0 2π dαsinαexp[ iξ| V d | 2 cos( αε ) ]
Η( ξ )=2π J 0 [ ξ 2 | V d | ]
Λ( ξ )=i2π J 1 ( ξ 2 | V d | )cosε
Φ( ξ )=i2π J 1 ( ξ 2 | V d | )sinε
I pq20 = π 2 D 2 a 01 a 02 βexp( D 2 | V c | 2 2 ){ exp( β 12 ) 0 ξ 0 ξdξexp( ξ 2 2 D 2 ) × J 0 [ ξ 2 | V d | ]+ ξ 0 ξdξexp( ξ 2 2 D 2 ) J 0 [ ξ 2 | V d | ] }
I pq2xj = ( 1 ) 4j V z( 3j ) π 2 D 2 δ 2 βcosεexp( D 2 | V c | 2 2 ){ exp( β 12 ) 0 ξ 0 ξdξ ×exp( ξ 2 2 D 2 ) ρ ξ J 1 [ ξ 2 | V d | ]+ ξ 0 ξdξexp( ξ 2 2 D 2 ) ρ ξ J 1 [ ξ 2 | V d | ] }
I pq2yj = ( 1 ) 4j V z( 3j ) π 2 D 2 δ 2 βsinεexp( D 2 | V c | 2 2 ){ exp( β 12 ) 0 ξ 0 ξdξ × exp( ξ 2 2 D 2 ) ρ ξ J 1 [ ξ 2 | V d | ]+ ξ 0 ξdξexp( ξ 2 2 D 2 ) ρ ξ J 1 [ ξ 2 | V d | ] }
ξ 0 = l c { ln( β 12 ln2ln[ 1+exp( β 12 ) ] ) } 1/2
J n ( z )= k=0 ( 1 ) k z 2k+n 2 2k+n k!( k+n )!
γ( a,x )= 0 x e t t a1 dt Γ( a,x )= x e t t a1 dt [Re a>0]
I pq20 = π 2 D 4 a 01 a 02 βexp( D 2 | V c | 2 2 ) k=0 ( 1 ) k ( | V d |D ) 2k 2 3k ( k! ) 2 ×[ exp( β 12 )γ( k+1, ξ 0 2 2 D 2 )+Γ( k+1, ξ 0 2 2 D 2 ) ]
I pq2s =[ ( a 11 a 02 V z2 a 01 a 12 V z1 ) V dx +( a 21 a 02 V z2 a 01 a 22 V z1 ) V dy ] π 2 D 6 δ 2 β ×exp( D 2 | V c | 2 2 ) k=0 ( 1 ) k | V d | 2k D 2k l c 2k+2 2 3k k!( k+1 )! ( l c 2 +2 D 2 ) k+2 [ exp( β 12 )γ( k+2, x 0 )+Γ( k+2, x 0 ) ]
E pqj s = ( a 0j + a 1j Z x + a 2j Z y )exp( i V zj h ) exp( i V j r )dS
exp[ i V zj h( r ) ] =exp( V zj 2 δ 2 /2 )=exp( β jj /2 )
Z x exp[ i V zj h( r ) ] = Z y exp[ i V zj h( r ) ] =0
E pqj s =π D 2 a 0j exp( β jj /2 )exp( D 2 | V j | 2 /4 )

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