Abstract

The fourth-order moment of the scattered light, namely, the correlation function of the scattered intensity fluctuation from two-dimensional optically weak homogeneous and isotropic rough surfaces obeying Gaussian distribution are investigated based on Beckmann theory and Gaussian moment theorem. Analytical and numerical results are given for the correlation functions of the scattered intensity fluctuation. Also two important special cases, two-frequency correlation and angular correlation, are discussed, as well as the influence of the incident and observation conditions and the characteristic parameters of the rough surfaces on the correlation function which could lead to a more clear understanding of the scattering property of the rough surface and provide a theoretical basis for the 3D target recognition.

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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. 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]
  4. 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]
  5. E. Bahar and S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt. 24(12), 1820–1825 (1985).
    [CrossRef] [PubMed]
  6. E. Bahar and M. A. Fitzwater, “Scattering and depolarization by conducting cylinders with rough surfaces,” Appl. Opt. 25(11), 1826–1832 (1986).
    [CrossRef] [PubMed]
  7. 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]
  8. P. Phu, A. Ishimaru, and Y. Kuga, “Controlied milimeter-wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. 28(4), 533–548 (1993).
    [CrossRef]
  9. 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]
  10. 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]
  11. D. J. Schertler and N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A 11(8), 2286–2297 (1994).
    [CrossRef]
  12. T. Michel and K. A. O'Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9(8), 1374–1384 (1992).
    [CrossRef]
  13. G. Zhang, L. Tsang, and Y. Kuga, “Studies of the angular correlation function of scattering by random rough surfaces with and without a buried object,” IEEE Trans. Geosci. Rem. Sens. 35(2), 444–453 (1997).
    [CrossRef]
  14. G. Zhang, L. Tsang, and K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional rough surface,” J. Opt. Soc. Am. A 15(12), 2995–3002 (1998).
    [CrossRef]
  15. C. T. C. Le, Y. Kuga, and A. Ishimaru, “Angular correlation function based on the second-order Kirchhoff approximation and comparison with experiments,” J. Opt. Soc. Am. A 13(5), 1057–1067 (1996).
    [CrossRef]
  16. Y. Kuga, C. T. C. Le, A. Ishimaru, and L. Ailes-Sengers, “Analytical, experimental, and numerical studies of angular memory signatures of waves scattered from one-dimensional rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 34(6), 1300–1307 (1996).
    [CrossRef]
  17. C. T. C. Le, A. Ishimaru, Y. Kuga, and J.-H. Yea, “Angular memory and frequency interferometry for mean height profiling of a rough surface,” IEEE Trans. Geosci. Rem. Sens. 36(1), 61–71 (1998).
    [CrossRef]
  18. Z. S. 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]
  19. M. E. Knotts, T. R. Michel, and K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 9(10), 1822–1831 (1992).
    [CrossRef]
  20. M. Nieto-Vesperinas and J. A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10(1), 150–157 (1993).
    [CrossRef]
  21. 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. Technol. 76(11), 697–701 (2009).
    [CrossRef]
  22. 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. Physica. Sinica. 58(4), 2390–2396 (2009).
  23. 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]
  24. H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28(6), 1139–1144 (2011).
    [CrossRef] [PubMed]
  25. 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]
  26. L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6(6), 765–781 (1989).
    [CrossRef]
  27. J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

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. Technol. 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. Physica. Sinica. 58(4), 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. S. 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 (1)

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]

1998 (2)

C. T. C. Le, A. Ishimaru, Y. Kuga, and J.-H. Yea, “Angular memory and frequency interferometry for mean height profiling of a rough surface,” IEEE Trans. Geosci. Rem. Sens. 36(1), 61–71 (1998).
[CrossRef]

G. Zhang, L. Tsang, and K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional rough surface,” J. Opt. Soc. Am. A 15(12), 2995–3002 (1998).
[CrossRef]

1997 (1)

G. Zhang, L. Tsang, and Y. Kuga, “Studies of the angular correlation function of scattering by random rough surfaces with and without a buried object,” IEEE Trans. Geosci. Rem. Sens. 35(2), 444–453 (1997).
[CrossRef]

1996 (2)

Y. Kuga, C. T. C. Le, A. Ishimaru, and L. Ailes-Sengers, “Analytical, experimental, and numerical studies of angular memory signatures of waves scattered from one-dimensional rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 34(6), 1300–1307 (1996).
[CrossRef]

C. T. C. Le, Y. Kuga, and A. Ishimaru, “Angular correlation function based on the second-order Kirchhoff approximation and comparison with experiments,” J. Opt. Soc. Am. A 13(5), 1057–1067 (1996).
[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]

1993 (2)

P. Phu, A. Ishimaru, and Y. Kuga, “Controlied milimeter-wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. 28(4), 533–548 (1993).
[CrossRef]

M. Nieto-Vesperinas and J. A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10(1), 150–157 (1993).
[CrossRef]

1992 (4)

1989 (1)

1986 (1)

1985 (1)

Ailes-Sengers, L.

Y. Kuga, C. T. C. Le, A. Ishimaru, and L. Ailes-Sengers, “Analytical, experimental, and numerical studies of angular memory signatures of waves scattered from one-dimensional rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 34(6), 1300–1307 (1996).
[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 Media 4(2), 139–148 (1994).
[CrossRef]

Bahar, E.

Bronnikov, V. N.

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.

C. T. C. Le, A. Ishimaru, Y. Kuga, and J.-H. Yea, “Angular memory and frequency interferometry for mean height profiling of a rough surface,” IEEE Trans. Geosci. Rem. Sens. 36(1), 61–71 (1998).
[CrossRef]

C. T. C. Le, Y. Kuga, and A. Ishimaru, “Angular correlation function based on the second-order Kirchhoff approximation and comparison with experiments,” J. Opt. Soc. Am. A 13(5), 1057–1067 (1996).
[CrossRef]

Y. Kuga, C. T. C. Le, A. Ishimaru, and L. Ailes-Sengers, “Analytical, experimental, and numerical studies of angular memory signatures of waves scattered from one-dimensional rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 34(6), 1300–1307 (1996).
[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 Media 4(2), 139–148 (1994).
[CrossRef]

P. Phu, A. Ishimaru, and Y. Kuga, “Controlied milimeter-wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. 28(4), 533–548 (1993).
[CrossRef]

Jacks, H. C.

Kalugin, M. M.

Knotts, M. E.

Korotkova, O.

Kuga, Y.

C. T. C. Le, A. Ishimaru, Y. Kuga, and J.-H. Yea, “Angular memory and frequency interferometry for mean height profiling of a rough surface,” IEEE Trans. Geosci. Rem. Sens. 36(1), 61–71 (1998).
[CrossRef]

G. Zhang, L. Tsang, and Y. Kuga, “Studies of the angular correlation function of scattering by random rough surfaces with and without a buried object,” IEEE Trans. Geosci. Rem. Sens. 35(2), 444–453 (1997).
[CrossRef]

C. T. C. Le, Y. Kuga, and A. Ishimaru, “Angular correlation function based on the second-order Kirchhoff approximation and comparison with experiments,” J. Opt. Soc. Am. A 13(5), 1057–1067 (1996).
[CrossRef]

Y. Kuga, C. T. C. Le, A. Ishimaru, and L. Ailes-Sengers, “Analytical, experimental, and numerical studies of angular memory signatures of waves scattered from one-dimensional rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 34(6), 1300–1307 (1996).
[CrossRef]

P. Phu, A. Ishimaru, and Y. Kuga, “Controlied milimeter-wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. 28(4), 533–548 (1993).
[CrossRef]

Le, C. T. C.

C. T. C. Le, A. Ishimaru, Y. Kuga, and J.-H. Yea, “Angular memory and frequency interferometry for mean height profiling of a rough surface,” IEEE Trans. Geosci. Rem. Sens. 36(1), 61–71 (1998).
[CrossRef]

C. T. C. Le, Y. Kuga, and A. Ishimaru, “Angular correlation function based on the second-order Kirchhoff approximation and comparison with experiments,” J. Opt. Soc. Am. A 13(5), 1057–1067 (1996).
[CrossRef]

Y. Kuga, C. T. C. Le, A. Ishimaru, and L. Ailes-Sengers, “Analytical, experimental, and numerical studies of angular memory signatures of waves scattered from one-dimensional rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 34(6), 1300–1307 (1996).
[CrossRef]

Li, J.

Li, Q.

Z. S. 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. Physica. Sinica. 58(4), 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]

Michel, T.

Michel, T. R.

Nieto-Vesperinas, M.

O’Donnell, K. A.

O'Donnell, K. A.

Pak, K.

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]

P. Phu, A. Ishimaru, and Y. Kuga, “Controlied milimeter-wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. 28(4), 533–548 (1993).
[CrossRef]

Sanchez-Gil, J. A.

Schertler, D. J.

Shirley, L. G.

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]

Tsang, L.

G. Zhang, L. Tsang, and K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional rough surface,” J. Opt. Soc. Am. A 15(12), 2995–3002 (1998).
[CrossRef]

G. Zhang, L. Tsang, and Y. Kuga, “Studies of the angular correlation function of scattering by random rough surfaces with and without a buried object,” IEEE Trans. Geosci. Rem. Sens. 35(2), 444–453 (1997).
[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. Physica. Sinica. 58(4), 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. S. 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. 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. Physica. Sinica. 58(4), 2390–2396 (2009).

Z. S. 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. S.

Z. S. 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]

Yea, J.-H.

C. T. C. Le, A. Ishimaru, Y. Kuga, and J.-H. Yea, “Angular memory and frequency interferometry for mean height profiling of a rough surface,” IEEE Trans. Geosci. Rem. Sens. 36(1), 61–71 (1998).
[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. Physica. Sinica. 58(4), 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. Physica. Sinica. 58(4), 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. Physica. Sinica. (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. Physica. Sinica. 58(4), 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. S. 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. (3)

G. Zhang, L. Tsang, and Y. Kuga, “Studies of the angular correlation function of scattering by random rough surfaces with and without a buried object,” IEEE Trans. Geosci. Rem. Sens. 35(2), 444–453 (1997).
[CrossRef]

Y. Kuga, C. T. C. Le, A. Ishimaru, and L. Ailes-Sengers, “Analytical, experimental, and numerical studies of angular memory signatures of waves scattered from one-dimensional rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 34(6), 1300–1307 (1996).
[CrossRef]

C. T. C. Le, A. Ishimaru, Y. Kuga, and J.-H. Yea, “Angular memory and frequency interferometry for mean height profiling of a rough surface,” IEEE Trans. Geosci. Rem. Sens. 36(1), 61–71 (1998).
[CrossRef]

Int. J. Infrared Millim. Waves (2)

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]

J. Opt. Soc. Am. A (9)

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

L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6(6), 765–781 (1989).
[CrossRef]

T. Michel and K. A. O'Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9(8), 1374–1384 (1992).
[CrossRef]

C. T. C. Le, Y. Kuga, and A. Ishimaru, “Angular correlation function based on the second-order Kirchhoff approximation and comparison with experiments,” J. Opt. Soc. Am. A 13(5), 1057–1067 (1996).
[CrossRef]

M. Nieto-Vesperinas and J. A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10(1), 150–157 (1993).
[CrossRef]

M. E. Knotts, T. R. Michel, and K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 9(10), 1822–1831 (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]

G. Zhang, L. Tsang, and K. Pak, “Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional rough surface,” J. Opt. Soc. Am. A 15(12), 2995–3002 (1998).
[CrossRef]

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28(6), 1139–1144 (2011).
[CrossRef] [PubMed]

J. Opt. Technol. (1)

Opt. Express (1)

Opt. Lett. (1)

Radio Sci. (1)

P. Phu, A. Ishimaru, and Y. Kuga, “Controlied milimeter-wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. 28(4), 533–548 (1993).
[CrossRef]

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 (3)

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).

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

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Figures (11)

Fig. 1
Fig. 1

Geometry of rough surface scattering.

Fig. 2
Fig. 2

Correlation function γ 12 versus second angle θ 2 .

Fig. 3
Fig. 3

Correlation function γ 12 versus wavelength difference Δλ .

Fig. 4
Fig. 4

TFCF γ 12 versus RMSδwith different wavelength-difference Δλ .

Fig. 7
Fig. 7

TFCF γ 12 versus wavelength difference Δλ with differentκ.

Fig. 8
Fig. 8

ACF γ 12 versus RMSδwith different angles.

Fig. 11
Fig. 11

ACF γ 12 versus angle θ 2 with differentκand RMSδ.

Fig. 5
Fig. 5

TFCF γ 12 versus RMSδwith differentκand angleθ.

Fig. 6
Fig. 6

TFCF γ 12 versus wavelength difference Δλ with RMSδand angleθ.

Fig. 9
Fig. 9

ACF γ 12 versus RMSδwith different Δθ andκ.

Fig. 10
Fig. 10

ACF γ 12 versus angle θ 2 with different angle θ 1 and RMSδ.

Equations (31)

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

E s ( k i , k s )=K(ω)F( θ i ; θ s , φ s ) p(x,y)exp(i υ r )dxdy
υ = k i k s = υ x x ^ + υ y y ^ + υ z z ^
υ r = υ x x+ υ y y+ υ z ζ(x,y)
E s ( k i , k s )=Δ E s ( k i , k s )+ E s ( k i , k s )
C 12 = Δ E s1 Δ E s1 Δ E s2 Δ E s2 Δ E s1 Δ E s1 Δ E s2 Δ E s2 + E s1 E s2 ×( Δ E s1 Δ E s2 + Δ E s1 Δ E s2 + Δ E s1 Δ E s2 + Δ E s1 Δ E s2 )
Δ E s1 Δ E s1 Δ E s2 Δ E s2 = Δ E s1 Δ E s1 Δ E s2 Δ E s2 + Δ E s1 Δ E s2 Δ E s1 Δ E s2 + Δ E s1 Δ E s2 Δ E s1 Δ E s2
C 12 = | E s1 E s2 | 2 + | E s1 E s2 | 2 2 E s1 2 E s2 2
γ 12 = I 1 I 2 I 1 I 2 ( I 1 2 I 1 2 )( I 2 2 I 2 2 ) = C 12 C 11 C 22
E s1 E s2 * = C 1 p( x 1 , y 1 )p( x 2 , y 2 ) exp(i υ 1 r 1 i υ 2 r 2 ) d x 1 d y 1 d x 2 d y 2
exp(i υ 1 r 1 i υ 2 r 2 ) =exp[ i( υ 1 r 1 υ 2 r 2 ) ] exp[ i( υ z1 ζ 1 υ z2 ζ 2 ) ]
exp[ i( υ z1 ζ 1 υ z2 ζ 2 ) ] =exp{ 1 2 δ 2 [ υ z1 2 2 υ z1 υ z2 ρ( r 1 r 2 )+ υ z2 2 ] }
ρ( r 1 r 2 )=exp( | r 1 r 2 | 2 / l c 2 )
r d = r 1 r 2 r c =( r 1 + r 2 )/2 υ d = υ 1 υ 2 υ c =( υ 1 + υ 2 )/2
E s1 E s2 * = C 1 exp[ 1 2 δ 2 ( υ z1 2 + υ z2 2 ) ] d r c d r d exp( 2 | r c | 2 D 2 ) ×exp( | r d | 2 2 D 2 )exp[ i( υ d r c + υ c r d ) ]exp[ υ z1 υ z2 δ 2 ρ( r d ) ]
exp[ υ z1 υ z2 δ 2 ρ( r d ) ]= n=0 ( υ z1 υ z2 δ 2 ) n n! ρ n ( r d )
E s1 E s2 * = C 1 π 2 D 4 exp[ 1 2 δ 2 ( υ z1 2 + υ z2 2 ) ]exp[ D 2 | υ d | 2 8 ] × n=0 l c 2 ( υ z1 υ z2 δ 2 ) n n!( l c 2 +2n D 2 ) exp( D 2 l c 2 | υ c | 2 2 l c 2 +4n D 2 )
E s1 E s2 = C 2 d r 1 d r 2 p( r 1 )p( r 2 ) ×exp[ i( υ 1 r 1 + υ 2 r 2 ) ] exp[ i( υ z1 ζ 1 + υ z2 ζ 2 ) ]
exp[ i( υ z1 ζ 1 + υ z2 ζ 2 ) ] =exp{ 1 2 δ 2 [ υ z1 2 +2 υ z1 υ z2 ρ( r d )+ υ z2 2 ] }
E s1 E s2 = C 2 π D 2 2 exp[ 1 2 δ 2 ( υ z1 2 + υ z2 2 ) ]exp( D 2 | υ c | 2 2 ) × d r d exp( | r d | 2 2 D 2 )exp( i υ cd r d /2 )exp[ υ z1 υ z2 δ 2 ρ( r d ) ]
E s1 E s2 = C 2 π D 2 2 exp[ 1 2 δ 2 ( υ z1 2 + υ z2 2 ) ]exp( D 2 | υ c | 2 2 ) 0 rdrexp( r 2 2 D 2 ) ×exp[ υ z1 υ z2 δ 2 ρ( r ) ]{ 0 2π dθexp[ ir 2 ( υ xd cosθ+ υ yd sinθ ) ] }
0 2π exp( iAcosθ )dθ =2π J 0 ( | A | )
E s1 E s2 = C 2 π 2 D 2 exp[ 1 2 δ 2 ( υ z1 2 + υ z2 2 ) ]exp( D 2 | υ c | 2 2 ) × 0 rdrexp( r 2 2 D 2 )exp[ υ z1 υ z2 δ 2 ρ( r ) ] J 0 ( r| υ d | 2 )
E s1 E s2 = C 2 π 2 D 2 exp[ 1 2 δ 2 ( υ z1 2 + υ z2 2 ) ]exp( D 2 | υ c | 2 2 ){ exp( υ z1 υ z2 δ 2 ) × 0 r d rdrexp( r 2 2 D 2 ) J 0 ( r| υ d | 2 )+ r d rdrexp( r 2 2 D 2 ) J 0 ( r| υ d | 2 ) }
r d = l c { ln( υ z1 υ z2 δ 2 ln2ln[ 1+exp( υ z1 υ z2 δ 2 ) ] ) } 1/2
J 0 ( z )= i=0 ( 1 ) k z 2k 2 2k ( k! ) 2
γ( a,x )= 0 x e t t a1 dt Γ( a,x )= x e t t a1 dt [Re a>0]
E s1 E s2 = C 2 π 2 D 4 exp[ 1 2 δ 2 ( υ z1 2 + υ z2 2 ) ]exp( D 2 | υ c | 2 2 ) × k=0 ( 1 ) k | υ d | 2k D 2k 2 3k ( k! ) 2 [ exp( υ z1 υ z2 δ 2 )γ( k+1, r d 2 2 D 2 )+Γ( k+1, r d 2 2 D 2 ) ]
E s =KF p( r )exp( i υ r ) exp[ i υ z ζ( r ) ] d r
exp[ i υ z ζ( r ) ] =exp( 1 2 υ z 2 δ 2 )
exp( A 2 x 2 )exp( iBx )dx = π A exp( B 2 4 A 2 )
E s =KFπ D 2 exp( 1 2 υ z 2 δ 2 )exp( D 2 | υ | 2 4 )

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