Abstract

The angular correlation function (ACF) of scattering amplitudes is presented using the second-order Kirchhoff approximation (KA) with angular and propagation shadowing functions. The theory is applicable to surfaces with large radii of curvature and high slopes of the order of unity. The correlation consists of contributions from single and second-order scattering. The single scattering provides the necessary condition for substantial correlation to occur. The second-order scattering yields high peaks in the correlation function. The ladder term gives a peak when two waves that have the same incident and scattering angles are traveling in the same direction. The cyclic term gives another peak in the time-reversed direction. These two peaks are related by the reciprocity condition. Although the second-order KA contains several approximations and the solution is simplified to yield a numerically tractable form, its agreement with experimental results is excellent. The theory correctly shows the peaks in the ACF observed in both co-polarization and cross-polarization responses. The width of the memory line is also very close to the value predicted by the theory.

© 1996 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  2. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).
  3. A. Ishimaru, “From radar cross section and light localizations to rough surface scattering,” IEEE Trans. Antennas Propag. Magazine 33(5), 7–11 (1991).
    [Crossref]
  4. S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1985).
    [Crossref]
  5. I. Freud, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
    [Crossref]
  6. R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
    [Crossref]
  7. R. Berkovits, M. Kaveh, “Angular correlations of waves in disordered systems: new numerical results,” Phys. Rev. B 41, 7308–7310 (1990).
    [Crossref]
  8. I. Freud, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1990).
    [Crossref]
  9. M. Kaveh, “New phenomena in the propagation of optical waves through random media,” Waves Random Media 3, S121–S128 (1991).
    [Crossref]
  10. I. Endrei, M. Kaveh, “Wavelength intensity correlation functions for transmitted waves through a slab: numerical results,” Phys. Rev. B 40, 9419–9422 (1989).
    [Crossref]
  11. A. Ishimaru, L. Ailes-Sengers, P. Phu, D. Winebrenner, “Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces,” Waves Random Media 4, 139–148 (1994).
    [Crossref]
  12. M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
    [Crossref]
  13. A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1888 (1990).
    [Crossref]
  14. P. Phu, A. Ishimaru, Y. Kuga, “Co-polarized and cross-polarized enhanced backscattering from two-dimensional very rough surfaces at millimeter wave frequencies,” Radio Sci. 29, 1275–1291 (1994).
    [Crossref]
  15. L. Tsang, C. H. Chan, K. Pak, “Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulation,” J. Opt. Soc. Am. A 11, 711–715 (1994).
    [Crossref]
  16. M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
    [Crossref]
  17. T. R. Michel, K. A. O’Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
    [Crossref]
  18. M. E. Knots, T. R. Michel, K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 10, 1822–1831 (1992).
    [Crossref]
  19. A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” in Progress in Electromagnetics Research (Elsevier, Cambridge, Mass., to be published).
  20. Y. Kuga, “Millimeter-wave scattering from rough surfaces,” presented at PIERS’95, Seattle, Wash., July 1995.

1994 (3)

P. Phu, A. Ishimaru, Y. Kuga, “Co-polarized and cross-polarized enhanced backscattering from two-dimensional very rough surfaces at millimeter wave frequencies,” Radio Sci. 29, 1275–1291 (1994).
[Crossref]

L. Tsang, C. H. Chan, K. Pak, “Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulation,” J. Opt. Soc. Am. A 11, 711–715 (1994).
[Crossref]

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

1993 (1)

1992 (3)

T. R. Michel, K. A. O’Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
[Crossref]

M. E. Knots, T. R. Michel, K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 10, 1822–1831 (1992).
[Crossref]

M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
[Crossref]

1991 (2)

A. Ishimaru, “From radar cross section and light localizations to rough surface scattering,” IEEE Trans. Antennas Propag. Magazine 33(5), 7–11 (1991).
[Crossref]

M. Kaveh, “New phenomena in the propagation of optical waves through random media,” Waves Random Media 3, S121–S128 (1991).
[Crossref]

1990 (3)

R. Berkovits, M. Kaveh, “Angular correlations of waves in disordered systems: new numerical results,” Phys. Rev. B 41, 7308–7310 (1990).
[Crossref]

I. Freud, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1990).
[Crossref]

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1888 (1990).
[Crossref]

1989 (2)

I. Endrei, M. Kaveh, “Wavelength intensity correlation functions for transmitted waves through a slab: numerical results,” Phys. Rev. B 40, 9419–9422 (1989).
[Crossref]

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[Crossref]

1988 (1)

I. Freud, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[Crossref]

1985 (1)

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1985).
[Crossref]

Ailes-Sengers, L.

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

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” in Progress in Electromagnetics Research (Elsevier, Cambridge, Mass., to be published).

Berkovits, R.

I. Freud, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1990).
[Crossref]

R. Berkovits, M. Kaveh, “Angular correlations of waves in disordered systems: new numerical results,” Phys. Rev. B 41, 7308–7310 (1990).
[Crossref]

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[Crossref]

Chan, C. H.

Chan, T. K.

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” in Progress in Electromagnetics Research (Elsevier, Cambridge, Mass., to be published).

Chen, J. S.

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1888 (1990).
[Crossref]

Endrei, I.

I. Endrei, M. Kaveh, “Wavelength intensity correlation functions for transmitted waves through a slab: numerical results,” Phys. Rev. B 40, 9419–9422 (1989).
[Crossref]

Feng, S.

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[Crossref]

I. Freud, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[Crossref]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1985).
[Crossref]

Freud, I.

I. Freud, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1990).
[Crossref]

I. Freud, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[Crossref]

Ishimaru, A.

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

P. Phu, A. Ishimaru, Y. Kuga, “Co-polarized and cross-polarized enhanced backscattering from two-dimensional very rough surfaces at millimeter wave frequencies,” Radio Sci. 29, 1275–1291 (1994).
[Crossref]

A. Ishimaru, “From radar cross section and light localizations to rough surface scattering,” IEEE Trans. Antennas Propag. Magazine 33(5), 7–11 (1991).
[Crossref]

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1888 (1990).
[Crossref]

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

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” in Progress in Electromagnetics Research (Elsevier, Cambridge, Mass., to be published).

Kane, C.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1985).
[Crossref]

Kaveh, M.

M. Kaveh, “New phenomena in the propagation of optical waves through random media,” Waves Random Media 3, S121–S128 (1991).
[Crossref]

I. Freud, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1990).
[Crossref]

R. Berkovits, M. Kaveh, “Angular correlations of waves in disordered systems: new numerical results,” Phys. Rev. B 41, 7308–7310 (1990).
[Crossref]

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[Crossref]

I. Endrei, M. Kaveh, “Wavelength intensity correlation functions for transmitted waves through a slab: numerical results,” Phys. Rev. B 40, 9419–9422 (1989).
[Crossref]

Knots, M. E.

M. E. Knots, T. R. Michel, K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 10, 1822–1831 (1992).
[Crossref]

Kong, J. A.

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

Kuga, Y.

P. Phu, A. Ishimaru, Y. Kuga, “Co-polarized and cross-polarized enhanced backscattering from two-dimensional very rough surfaces at millimeter wave frequencies,” Radio Sci. 29, 1275–1291 (1994).
[Crossref]

Y. Kuga, “Millimeter-wave scattering from rough surfaces,” presented at PIERS’95, Seattle, Wash., July 1995.

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” in Progress in Electromagnetics Research (Elsevier, Cambridge, Mass., to be published).

Le, C.

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” in Progress in Electromagnetics Research (Elsevier, Cambridge, Mass., to be published).

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1985).
[Crossref]

Michel, T. R.

T. R. Michel, K. A. O’Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
[Crossref]

M. E. Knots, T. R. Michel, K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 10, 1822–1831 (1992).
[Crossref]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
[Crossref]

M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
[Crossref]

O’Donnell, K. A.

T. R. Michel, K. A. O’Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
[Crossref]

M. E. Knots, T. R. Michel, K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 10, 1822–1831 (1992).
[Crossref]

Pak, K.

Phu, P.

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

P. Phu, A. Ishimaru, Y. Kuga, “Co-polarized and cross-polarized enhanced backscattering from two-dimensional very rough surfaces at millimeter wave frequencies,” Radio Sci. 29, 1275–1291 (1994).
[Crossref]

Rosenbluh, M.

I. Freud, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1990).
[Crossref]

I. Freud, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[Crossref]

Sanchez-Gil, J. A.

M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
[Crossref]

M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
[Crossref]

Shin, R. T.

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

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1985).
[Crossref]

Tsang, L.

Winebrenner, D.

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

IEEE Trans. Antennas Propag. Magazine (1)

A. Ishimaru, “From radar cross section and light localizations to rough surface scattering,” IEEE Trans. Antennas Propag. Magazine 33(5), 7–11 (1991).
[Crossref]

J. Acoust. Soc. Am. (1)

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1888 (1990).
[Crossref]

J. Opt. Soc. Am. A (4)

Phys. Rev. B (5)

I. Endrei, M. Kaveh, “Wavelength intensity correlation functions for transmitted waves through a slab: numerical results,” Phys. Rev. B 40, 9419–9422 (1989).
[Crossref]

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[Crossref]

R. Berkovits, M. Kaveh, “Angular correlations of waves in disordered systems: new numerical results,” Phys. Rev. B 41, 7308–7310 (1990).
[Crossref]

I. Freud, M. Kaveh, R. Berkovits, M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1990).
[Crossref]

M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
[Crossref]

Phys. Rev. Lett. (2)

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1985).
[Crossref]

I. Freud, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[Crossref]

Radio Sci. (1)

P. Phu, A. Ishimaru, Y. Kuga, “Co-polarized and cross-polarized enhanced backscattering from two-dimensional very rough surfaces at millimeter wave frequencies,” Radio Sci. 29, 1275–1291 (1994).
[Crossref]

Waves Random Media (2)

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

M. Kaveh, “New phenomena in the propagation of optical waves through random media,” Waves Random Media 3, S121–S128 (1991).
[Crossref]

Other (4)

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

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

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” in Progress in Electromagnetics Research (Elsevier, Cambridge, Mass., to be published).

Y. Kuga, “Millimeter-wave scattering from rough surfaces,” presented at PIERS’95, Seattle, Wash., July 1995.

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Figures (11)

Fig. 1
Fig. 1

Condition for angular correlation. A strong correlation between E s ( θ i , θ s ) and Es(θi, θs) exists if k i x - k i x = k s x - k s x k 0 sin θ i - k 0 sin θ i = k 0 sin θ s - k 0 sin θ s sin θ i - sin θ i = sin θ s - sin θ s.

Fig. 2
Fig. 2

Three terms of the second-order KA.

Fig. 3
Fig. 3

Angular memory line. Reference angles: (θi, θ) = (20°, −40°).

Fig. 4
Fig. 4

Dependence of ACF on propagation distance Dp for (σ, l) = (1λ, 2λ).

Fig. 5
Fig. 5

(a) 3-D plot of ACF, (b) contour plot of ACF. (σ, l) = (1λ, 4λ); reference angles (θi, θ) = (20°, −40°).

Fig. 6
Fig. 6

ACF ’s of co-polarization and cross-polarization fields; (θi, θ) = (20°, −40°). The following contributions are shown: total (solid curves), first-order (dotted curves), ladder (dotted–dashed curves), and cyclic (dashed curves).

Fig. 7
Fig. 7

Comparisons between theory (solid curves) and experiments (dotted curves) for co-polarization ACF’s; (θi, θ) = (20°, −40°).

Fig. 8
Fig. 8

Comparisons between theory (solid curves) and experiments (dotted curves) for cross-polarization ACF’s; (θi, θ) = (20°, −40°).

Fig. 9
Fig. 9

Same as Fig. 7, but for (θi, θ) = (20°, −20°)

Fig. 10
Fig. 10

Same as Fig. 8, but for (θi, θ) = (20°, −20°).

Fig. 11
Fig. 11

ACF’s along the line perpendicular to the memory line: analytical (solid curves) and experimental (dotted curves); (θi, θ) = (20°, −40°).

Equations (79)

Equations on this page are rendered with MathJax. Learn more.

Γ ( θ i , θ ; θ i , θ ) = E s ( θ i , θ ) E s * ( θ i , θ ) ,
E s = exp ( i k R ) R F , F = [ f ] E i ,
[ f ] = [ f 11 f 12 f 21 f 22 ] ,
F = i 4 π [ - ( K × K × n ^ 1 × e 1 s ) 1 k + ( K × n ^ 1 × e 1 s ) ] × exp [ - i ( K - K i ) · r 1 ] d S 1 ,
n ^ 1 = K - K i K - K i .
B × C = [ B ] [ C ] = [ 0 - B z B y B z 0 - B x - B y B x 0 ] [ C x C y C z ] ,
F = ( [ H ] E i ) J 1 ,
[ H ] = i 4 π ( - [ K ] [ K ] [ n 1 ] [ r ] 1 k + [ K ] [ n 1 ] ) [ e 1 s ] ,
[ e 1 s ] = R [ p r ] [ p ] + R [ q ] [ q ] ,
E i = E θ θ ^ i + E ϕ ϕ ^ i ,
J 1 = exp [ - i ( K - K i ) · r 1 ] I inc S ( θ i , θ ) d S 1 ,
I inc = exp ( - x 2 L x 2 - y 2 L y 2 ) ,
L y = L 0 ,
L x = L 0 / cos θ i .
I 0 = I inc d S = exp ( - x 2 L x 2 - y 2 L y 2 ) d S = π L x L y = A ,
f 11 = θ ^ · [ H ] · θ ^ i J 1 = H 11 J 1 , f 12 = θ ^ · [ H ] · ϕ ^ i J 1 = H 12 J 1 , f 21 = ϕ ^ · [ H ] · θ ^ i J 1 = H 21 J 1 , f 22 = ϕ ^ · [ H ] · ϕ ^ i J 1 = H 22 J 1 .
f i j f k l * I 0 I 0 = H i j H k l * I ( 1 ) I 0 I 0 ,
I ( 1 ) = J 1 ( k , θ i , θ ) J 1 * ( k , θ i , θ ) - J 1 ( k , θ i , θ ) J 1 * ( k , θ i , θ ) .
I ( 1 ) = J 1 J 1 * = exp [ - i ( K - K i ) · r 1 ] exp [ - i ( K - K i ) · r 2 ] × I inc I inc * [ S ( θ i , θ ) S ( θ i , θ ) ] 1 / 2 d S 1 d S 2 .
J 1 J 1 * = Φ 1 Φ s 1 S ( θ i , θ ) S ( θ i , θ ) ,
Φ 1 = exp [ - σ 2 2 ( ν z - ν z ) 2 ] N z N z ( π l 2 ν z ν z σ 2 ) × exp ( - ν c 2 l 2 4 ν z ν z σ 2 ) ,
Φ s 1 = ( π L x eq L y eq ) × exp ( - ν d x 2 L x eq 2 4 - ν d y 2 L y eq 2 4 ) ,
K - K i = ν + ν z z ^ , K - K i = ν + ν z z ^ , ν c = 1 2 ( ν + ν ) , ν d = ν - ν , N z = ν z K - K i , N z = ν z K - K i ,
exp ( - i v 1 f 1 - i v 2 f 2 ) = exp [ - 1 2 ( v 1 2 σ 1 2 - 2 v 1 v 2 σ 1 σ 2 C + v 2 2 σ 2 2 ) ] ,
C = exp ( - x d 2 l 2 ) ( 1 - x d 2 l 2 ) ,
I 0 I 0 = exp ( - 1 2 x 2 L x 2 - 1 2 y 2 L y 2 - 1 2 x 2 L x 2 - 1 2 y 2 L x 2 ) ,
1 L x eq 2 = 1 2 ( 1 L x 2 + 1 L x 2 ) , 1 L y eq 2 = 1 2 ( 1 L y 2 + 1 L y 2 ) , L x = L 0 / cos θ i , L y = L 0 , L x = L 0 / cos θ i , L y = L 0 .
K 1 = k ( cos α 0 cos ψ 0 ) x ^ + k ( cos α 0 sin ψ 0 ) y ^ + k ( sin α 0 ) z ^ .
F KA 2 = { L [ H ] ( 2 ) E i } J 2 ,
L = i k 2 π 0 2 π d ψ 0 - π / 2 π / 2 cos α 0 d α 0 .
f i j f k l * I 0 I 0 = L L * [ H ] i j ( 2 ) [ H ] k l ( 2 ) I ( 2 ) I 0 I 0 ,
I ( 2 ) = J 2 J 2 * - J 2 J 2 * .
J 2 J 2 * = J 2 + J 2 + * + J 2 - J 2 - *
+ J 2 + J 2 - * + J 2 - J 2 + * .
J 2 + J 2 * = 1 N z 1 N z 2 N z 1 N z 2 Φ 1 l Φ 2 l Φ s 2 l F d S 2 ,
Φ 1 l = exp [ - σ 2 2 ( ν 1 - ν 1 ) 2 ] π l 2 σ 2 ν 1 ν 1 exp ( - c 1 2 l 2 4 σ 2 ν 1 ν 1 ) ,
Φ 2 l = exp [ - σ 2 2 ( ν 2 - ν 2 ) 2 ] π l 2 σ 2 ν 2 ν 2 exp ( - c 1 2 l 2 4 σ 2 ν 2 ν 2 ) ,
Φ s 2 l = ( π L x eq L y eq ) exp ( - ν d x 2 L x eq 2 4 - ν d y 2 L y eq 2 4 ) ,
F l = π D p 2 2 exp ( - A 2 + B 2 4 D p 2 ) ,
S 2 = [ S ( π 2 - θ i ) S ( π 2 - θ ) S ( π 2 - θ i ) × S ( π 2 - θ ) ] 1 / 2 [ 1 - S ( α 0 ) ] [ 1 - S ( α 0 ) ] ,
K - K 1 = u 1 + ν 1 z ^ , K 1 - K i = u 2 + ν 2 z ^ , K - K 1 = u 1 + ν 1 z ^ , K 1 - K i = u 2 + ν 2 z ^ , c 1 = ½ ( u 1 + u 1 ) , c 2 = ½ ( u 2 + u 2 ) , d 1 = u 1 - u 1 , d 2 = u 2 - u 2 , d c = ½ ( d 1 + d 2 ) , d d = d 1 - d 2 , ν d x = ( 2 d c ) x , ν d y = ( 2 d c ) y , A = ( d d 2 ) x , B = ( d d 2 ) y .
J 2 + J 2 - * = 1 N z 1 N z 2 N z 1 N z 2 Φ 1 c Φ 2 c Φ s 2 c F c S 2 ,
Φ 1 c = exp [ - σ 2 2 ( ν 1 - ν 2 ) 2 ] π l 2 σ 2 ν 1 ν 2 × exp ( - c 1 2 l 2 4 σ 2 ν 1 ν 2 ) ,
Φ 2 c = exp [ - σ 2 2 ( ν 2 - ν 1 ) 2 ] π l 2 σ 2 ν 2 ν 1 × exp ( - c 1 2 l 2 4 σ 2 ν 2 ν 1 ) ,
Φ s 2 c = ( π L x eq L y eq ) exp ( - ν d x 2 L x eq 2 4 - ν d y 2 L y eq 2 4 ) ,
F c = π D p 2 2 exp ( - A 2 + B 2 4 D p 2 ) ,
K - K 1 + = u 1 + ν 1 z ^ , K 1 + - K i = u 2 + ν 2 z ^ , K - K 1 - = u 1 + ν 1 z ^ , K 1 - - K i = u 2 + ν 2 z ^ , c 1 = 1 2 ( u 1 + u 2 ) , c 2 = 1 2 ( u 2 + u 1 ) , d 1 = u 1 - u 2 , d 2 = u 2 - u 1 .
F d = L * F l , c = ( π D p 2 / 2 ) 4 π ( k D p ) 3 / 2 Γ ( 1 / 4 ) exp ( Λ ) ,
Λ = - D p 2 4 ( P x 2 + P y 2 ) + D p 2 32 P x y k 2 - 1 16 D p 2 P x y α 0 2 - 1 32 k 2 D p 2 α 0 4 ,
P = { ( K + K i ) - ( K + K i ) 2 ladder + + , - - ( K + K i ) + ( K + K i ) 2 cyclic + - , - + ,
P x y = { 2 k ( P x cos ψ 0 + P y sin ψ 0 ) ladder + + , cyclic + - 2 k ( - P x cos ψ 0 - P y sin ψ 0 ) ladder - - , cyclic - + .
ACF = H i j H k l * I ( 1 ) I 0 I 0 + L L * [ H ] i j ( 2 ) [ H ] k l ( 2 ) I ( 2 ) I 0 I 0 .
ν d = ν - ν = ( K - K i ) x y - ( K - K i ) x y = 0.
sin θ i - sin θ i = sin θ - sin θ .
F d exp [ - D p 2 4 ( P x 2 + P y 2 ) + D p 2 32 P x y k 2 - 1 16 D p 2 P x y α 0 2 - 1 32 k 2 D p 2 α 0 4 ] ,
( K + K i ) x , y = ( K + K i ) x , y .
( K - K i ) x , y = ( K - K i ) x , y .
sin θ + sin θ i = sin θ + sin θ i ,
sin θ - sin θ i = sin θ - sin θ i ,
cos θ = cos θ ,
cos θ i = cos θ i
θ i = θ i ,             θ = θ ,
θ i = - θ ,             θ = - θ i ,
L x = L 0 / cos θ i .
Φ s 1 I 0 I 0 = π L x eq L y eq I 0 I 0 exp ( - ν d x 2 L x eq 2 4 - ν d y 2 L y eq 2 4 ) .
N = π L x eq L y eq I 0 I 0 = ( 2 L x L x L x 2 + L x 2 ) 1 / 2 = ( 2 cos θ i cos θ i cos 2 θ i + cos 2 θ i ) 1 / 2
R = 0 cos θ 1 - 1 cos θ 0 0 cos θ 1 + 1 cos θ 0 ,
R = 0 cos θ 0 - 1 cos θ 1 0 cos θ 0 + 1 cos θ 1 ,
[ e 1 s ] = R [ p r ] [ p ] + R [ q ] [ q ] ,
[ q ] = [ i ] [ n 1 ] { 1 - ( [ i ] [ n 1 ] ) 2 } 1 / 2 ,
[ p ] = [ q ] [ i ] ,
[ p r ] = [ r ] [ q ] ,
[ r ] = [ i ] - 2 [ n 1 ] ( [ n 1 ] [ i ] ) ,
[ n 1 ] = [ o ] - [ i ] { ( [ o ] - [ i ] ) ( [ o ] - [ i ] ) } 1 / 2 ,
S = S ( θ 1 )             0 θ θ i ,
S = S ( θ 2 )             θ i θ 0 , θ 1 = π / 2 - θ i ,             θ 2 = π / 2 - θ .
S = S ( θ 1 , θ 2 ) .
S ( θ k ) = [ 1 + erf ( ν k ) ] [ 1 - exp ( - F k ) ] 1 2 F k ,
S ( θ 1 , θ 2 ) = { 1 - exp [ - ( F 1 + F 2 ) ] } erf ( ν 1 ) + erf ( ν 2 ) 2 ( F 1 + F 2 ) , ν k = tan θ k 2 σ / l , F k = 1 2 { exp ( - 9 ν k 2 / 8 ) 3 π ν k + exp ( - ν k 2 ) π ν k - [ 1 - erf ( ν k ) ] } ,

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