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).
  3. E. Bahar and S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt.24(12), 1820–1825 (1985).
    [CrossRef] [PubMed]
  4. E. Bahar and M. A. Fitzwater, “Scattering and depolarization by conducting cylinders with rough surfaces,” Appl. Opt.25(11), 1826–1832 (1986).
    [CrossRef] [PubMed]
  5. W. Zhensen and C. Suomin, “Bistatic scattering by arbitrarily shaped objects with rough surface at optical and infrared frequencies,” Int. J. Infrared Millim. Waves13(4), 537–549 (1992).
    [CrossRef]
  6. 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 Media4(2), 139–148 (1994).
    [CrossRef]
  7. D. J. Schertler and N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A9(11), 2056–2066 (1992).
    [CrossRef]
  8. D. J. Schertler and N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A11(8), 2286–2297 (1994).
    [CrossRef]
  9. C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves25(8), 1211–1219 (2004).
    [CrossRef]
  10. G. Zhang and Z. S. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express19(8), 7007–7019 (2011).
    [CrossRef] [PubMed]
  11. 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]
  12. 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]
  13. 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 ED58, 2390–2396 (2009).
  14. Y. Xin, Y. J. He, Y. R. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett.35(23), 4000–4002 (2010).
    [CrossRef] [PubMed]
  15. H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A28(6), 1139–1144 (2011).
    [CrossRef] [PubMed]
  16. 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]
  17. G. Zhang and Z. Wu, “Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces,” Opt. Express20(2), 1491–1502 (2012).
    [CrossRef] [PubMed]
  18. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing, Vol. 2 (Addison-Wesley Publishing,1982).
  19. 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]
  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

G. Zhang and Z. Wu, “Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces,” Opt. Express20(2), 1491–1502 (2012).
[CrossRef] [PubMed]

2011

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A28(6), 1139–1144 (2011).
[CrossRef] [PubMed]

G. Zhang and Z. S. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express19(8), 7007–7019 (2011).
[CrossRef] [PubMed]

2010

Y. Xin, Y. J. He, Y. R. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett.35(23), 4000–4002 (2010).
[CrossRef] [PubMed]

2009

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 ED58, 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

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

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. Waves25(8), 1211–1219 (2004).
[CrossRef]

1994

D. J. Schertler and N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A11(8), 2286–2297 (1994).
[CrossRef]

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 Media4(2), 139–148 (1994).
[CrossRef]

1992

D. J. Schertler and N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A9(11), 2056–2066 (1992).
[CrossRef]

W. Zhensen and C. Suomin, “Bistatic scattering by arbitrarily shaped objects with rough surface at optical and infrared frequencies,” Int. J. Infrared Millim. Waves13(4), 537–549 (1992).
[CrossRef]

1986

E. Bahar and M. A. Fitzwater, “Scattering and depolarization by conducting cylinders with rough surfaces,” Appl. Opt.25(11), 1826–1832 (1986).
[CrossRef] [PubMed]

1985

E. Bahar and S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt.24(12), 1820–1825 (1985).
[CrossRef] [PubMed]

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 Media4(2), 139–148 (1994).
[CrossRef]

Bahar, E.

E. Bahar and M. A. Fitzwater, “Scattering and depolarization by conducting cylinders with rough surfaces,” Appl. Opt.25(11), 1826–1832 (1986).
[CrossRef] [PubMed]

E. Bahar and S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt.24(12), 1820–1825 (1985).
[CrossRef] [PubMed]

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.

E. Bahar and S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt.24(12), 1820–1825 (1985).
[CrossRef] [PubMed]

Chen, Y. R.

Y. Xin, Y. J. He, Y. R. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett.35(23), 4000–4002 (2010).
[CrossRef] [PubMed]

Fitzwater, M. A.

E. Bahar and M. A. Fitzwater, “Scattering and depolarization by conducting cylinders with rough surfaces,” Appl. Opt.25(11), 1826–1832 (1986).
[CrossRef] [PubMed]

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.

D. J. Schertler and N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A11(8), 2286–2297 (1994).
[CrossRef]

D. J. Schertler and N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A9(11), 2056–2066 (1992).
[CrossRef]

He, Y. J.

Y. Xin, Y. J. He, Y. R. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett.35(23), 4000–4002 (2010).
[CrossRef] [PubMed]

Hui, C.

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves25(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 Media4(2), 139–148 (1994).
[CrossRef]

Jacks, H. C.

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A28(6), 1139–1144 (2011).
[CrossRef] [PubMed]

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.

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A28(6), 1139–1144 (2011).
[CrossRef] [PubMed]

Li, J.

Y. Xin, Y. J. He, Y. R. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett.35(23), 4000–4002 (2010).
[CrossRef] [PubMed]

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 ED58, 2390–2396 (2009).

Lu, B.

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves25(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 Media4(2), 139–148 (1994).
[CrossRef]

Schertler, D. J.

D. J. Schertler and N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A11(8), 2286–2297 (1994).
[CrossRef]

D. J. Schertler and N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A9(11), 2056–2066 (1992).
[CrossRef]

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. Waves13(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 ED58, 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 Media4(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.

G. Zhang and Z. Wu, “Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces,” Opt. Express20(2), 1491–1502 (2012).
[CrossRef] [PubMed]

Wu, Z. S.

G. Zhang and Z. S. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express19(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 ED58, 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.

Y. Xin, Y. J. He, Y. R. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett.35(23), 4000–4002 (2010).
[CrossRef] [PubMed]

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.

G. Zhang and Z. Wu, “Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces,” Opt. Express20(2), 1491–1502 (2012).
[CrossRef] [PubMed]

G. Zhang and Z. S. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express19(8), 7007–7019 (2011).
[CrossRef] [PubMed]

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 ED58, 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 ED58, 2390–2396 (2009).

Zhensen, W.

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves25(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. Waves13(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

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 ED58, 2390–2396 (2009).

Appl. Opt.

E. Bahar and S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt.24(12), 1820–1825 (1985).
[CrossRef] [PubMed]

E. Bahar and M. A. Fitzwater, “Scattering and depolarization by conducting cylinders with rough surfaces,” Appl. Opt.25(11), 1826–1832 (1986).
[CrossRef] [PubMed]

Chin. Phys. Lett.

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.

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.

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

C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves25(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. Waves13(4), 537–549 (1992).
[CrossRef]

J. Opt. Soc. Am. A

D. J. Schertler and N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A9(11), 2056–2066 (1992).
[CrossRef]

D. J. Schertler and N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A11(8), 2286–2297 (1994).
[CrossRef]

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A28(6), 1139–1144 (2011).
[CrossRef] [PubMed]

J. Opt. Tech.

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

G. Zhang and Z. S. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express19(8), 7007–7019 (2011).
[CrossRef] [PubMed]

G. Zhang and Z. Wu, “Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces,” Opt. Express20(2), 1491–1502 (2012).
[CrossRef] [PubMed]

Opt. Lett.

Y. Xin, Y. J. He, Y. R. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett.35(23), 4000–4002 (2010).
[CrossRef] [PubMed]

Waves Random Media

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 Media4(2), 139–148 (1994).
[CrossRef]

Other

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 (15)

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exp[ β 12 ρ( r d ) ]= n=0 ( β 12 ) n n! ρ n ( r d )
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 ) ]
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 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 )
Λ( ξ )= 0 2π dαcosαexp[ iξ| V d | 2 cos( αε ) ]
Φ( ξ )= 0 2π dαsinαexp[ iξ| V d | 2 cos( αε ) ]
Λ( ξ )=i2π J 1 ( ξ 2 | V d | )cosε
Φ( ξ )=i2π J 1 ( ξ 2 | V d | )sinε
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 | ] }
exp[ i V zj h( r ) ] =exp( V zj 2 δ 2 /2 )=exp( β jj /2 )

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