Abstract

We present in this paper, approximate analytical expressions for the intensity of light scattered by a rough surface, whose elevation ξ(x,y) in the z-direction is a zero mean stationary Gaussian random variable. With (x,y) and (x,y) being two points on the surface, we have ξ(x,y)=0 with a correlation, ξ(x,y)ξ(x,y)=σ2g(r), where r=[(xx)2+(yy)2]1/2 is the distance between these two points. We consider g(r)=exp[(r/l)β] with 1β2, showing that g(0)=1 and g(r)0 for rl. The intensity expression is sought to be expressed as f(vxy)={1+(c/2y)[vx2+vy2]}y, where vx and vy are the wave vectors of scattering, as defined by the Beckmann notation. In the paper, we present expressions for c and y, in terms of σ, l, and β. The closed form expressions are verified to be true, for the cases β=1 and β=2, for which exact expressions are known. For other cases, i.e., β1, 2 we present approximate expressions for the scattered intensity, in the range, vxy=(vx2+vy2)1/26.0 and show that the relation for f(vxy), given above, expresses the scattered intensity quite accurately, thus providing a simple computational methods in situations of practical importance.

© 2013 Optical Society of America

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References

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  1. M. Zamani, M. Salami, S. M. Fazelli, and G. R. Zafari, “Analytical expression for wave scattering from exponential height correlated rough surfaces,” J. Mod. Opt. 59, 1448–1452 (2012).
    [CrossRef]
  2. L. M. Sanchez-Brea and F. J. Torcal-Milla, “Self imaging of the grating with two roughness levels,” Opt. Commun. 285, 13–17 (2012).
    [CrossRef]
  3. N. C. Bruce, “Control of the backscatter intensity in random-groove surfaces with variation in groove depth,” Appl. Opt. 44, 784–791 (2005).
    [CrossRef]
  4. J. C. Jin, C. Jin, C. Li, and Y. Chang, “Fabrication anti reflection (AR) coatings for polarized 193 nm laser light at an incidence angle of 74°,” Opt. Commun. 298, 171–175 (2013).
    [CrossRef]
  5. B. S. H. Burlison, P. D. Ruiz, and J. M. Huntley, “Evaluation of the performance of tilt scanning interferometry for tomographic scanning,” Opt. Commun. 285, 1654–1661 (2012).
    [CrossRef]
  6. J. Alvarez-Borrego, “1D rough surfaces: glitter function for remote sensing,” Opt. Commun. 113, 353–356 (1995).
    [CrossRef]
  7. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
  8. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North Holland, 1967), Vol. 6, pp. 55–69.
  9. V. C. Vani and S. Chatterjee, “Detection of a periodic structure hidden in random background: the role of signal amplitude in the matched filter detection method,” Phys. Scr. 81, 055402 (2010).
    [CrossRef]
  10. V. C. Vani and S. Chatterjee, “Detection of a periodic structure embedded in surface roughness, for various correlation functions,” Pramana 77, 611–626 (2011).
    [CrossRef]
  11. R. N. Bracewell, The Fourier Transformation and Its Applications (McGraw-Hill, 1985).
  12. A. Papoulis, The Fourier Integral and Its Application (McGraw-Hill, 1962).
  13. M. Abramowicz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).
  14. I. S. Geadsteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).
  15. R. Elliot, “On the theory of corrugated plane surfaces,” Trans. I. R. E. Antennas Propag. 2, 71–81 (1954).
    [CrossRef]
  16. See, for example, Eq. 6.623.2, listed in reference [10].
  17. See, for example, Eq. 6.631.4, listed in reference [10].
  18. R. B. Dingle, Asymptotic Expansion: Their Derivation and Interpretation (Academic, 1973).
  19. F. W. J. Olver, Introduction to Asymptotics and Special Functions (Academic, 1974).
  20. T. J. Rivlin, An Introduction to the Theory of Functions (Blaisdell,1969).
  21. A. Yu. Luchka, Method of Averaging Functional Corrections (Academic, 1965).
  22. See, for example, Eq. 6.621.1 of reference [10] and Chapter 13 of reference [9].
  23. See, for example, Eq. 6.631.1 of reference [10] and Chapter 8 of reference [9].
  24. S. Chatterjee and R. K. Banyal, “Scattering of light by rough surfaces: high and low roughness approximations,” Proceedings of the International Conference on Trends in Optics and Photonics, Kolkata, India, 2009, pp. 462–470.
  25. M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
    [CrossRef]
  26. V. V. Vosiekovich and S. Cuevas, “Adaptive optics and the outer scale of turbulence,” J. Opt. Soc. Am. A 12, 2523–2531 (2005).
  27. S. Wang, Y. Tian, C. J. Tay, and C. Quan, “Development of a laser-scattering-based probe for on-line measurement of surface roughness,” Appl. Opt. 42, 1318–1324 (2003).
    [CrossRef]
  28. C. J. Tay, S. H. Wang, C. Quan, and H. M. Shang, “In situ surface roughness measurement using a laser scattering method,” Opt. Commun. 218, 1–10 (2003).
    [CrossRef]
  29. Y. Quinsa and C. Tournier, “In situ non-contact measurements of surface roughness,” Precis. Eng. 36, 97–103 (2012).
    [CrossRef]
  30. Y. K. Fuh, K. C. Hsu, and J. R. Fan, “Rapid in-process measurement of surface roughness using adaptive optics,” Opt. Lett. 37, 848–850 (2012).
    [CrossRef]
  31. J. M. Tamkin and T. D. Milster, “Effects of structured mid-spatial frequency surface errors on image performance,” Appl. Opt. 49, 6522–6536 (2010).
    [CrossRef]

2013 (1)

J. C. Jin, C. Jin, C. Li, and Y. Chang, “Fabrication anti reflection (AR) coatings for polarized 193 nm laser light at an incidence angle of 74°,” Opt. Commun. 298, 171–175 (2013).
[CrossRef]

2012 (5)

B. S. H. Burlison, P. D. Ruiz, and J. M. Huntley, “Evaluation of the performance of tilt scanning interferometry for tomographic scanning,” Opt. Commun. 285, 1654–1661 (2012).
[CrossRef]

M. Zamani, M. Salami, S. M. Fazelli, and G. R. Zafari, “Analytical expression for wave scattering from exponential height correlated rough surfaces,” J. Mod. Opt. 59, 1448–1452 (2012).
[CrossRef]

L. M. Sanchez-Brea and F. J. Torcal-Milla, “Self imaging of the grating with two roughness levels,” Opt. Commun. 285, 13–17 (2012).
[CrossRef]

Y. Quinsa and C. Tournier, “In situ non-contact measurements of surface roughness,” Precis. Eng. 36, 97–103 (2012).
[CrossRef]

Y. K. Fuh, K. C. Hsu, and J. R. Fan, “Rapid in-process measurement of surface roughness using adaptive optics,” Opt. Lett. 37, 848–850 (2012).
[CrossRef]

2011 (1)

V. C. Vani and S. Chatterjee, “Detection of a periodic structure embedded in surface roughness, for various correlation functions,” Pramana 77, 611–626 (2011).
[CrossRef]

2010 (2)

V. C. Vani and S. Chatterjee, “Detection of a periodic structure hidden in random background: the role of signal amplitude in the matched filter detection method,” Phys. Scr. 81, 055402 (2010).
[CrossRef]

J. M. Tamkin and T. D. Milster, “Effects of structured mid-spatial frequency surface errors on image performance,” Appl. Opt. 49, 6522–6536 (2010).
[CrossRef]

2005 (2)

2003 (2)

S. Wang, Y. Tian, C. J. Tay, and C. Quan, “Development of a laser-scattering-based probe for on-line measurement of surface roughness,” Appl. Opt. 42, 1318–1324 (2003).
[CrossRef]

C. J. Tay, S. H. Wang, C. Quan, and H. M. Shang, “In situ surface roughness measurement using a laser scattering method,” Opt. Commun. 218, 1–10 (2003).
[CrossRef]

1997 (1)

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

1995 (1)

J. Alvarez-Borrego, “1D rough surfaces: glitter function for remote sensing,” Opt. Commun. 113, 353–356 (1995).
[CrossRef]

1954 (1)

R. Elliot, “On the theory of corrugated plane surfaces,” Trans. I. R. E. Antennas Propag. 2, 71–81 (1954).
[CrossRef]

Abramowicz, M.

M. Abramowicz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Alvarez-Borrego, J.

J. Alvarez-Borrego, “1D rough surfaces: glitter function for remote sensing,” Opt. Commun. 113, 353–356 (1995).
[CrossRef]

Banyal, R. K.

S. Chatterjee and R. K. Banyal, “Scattering of light by rough surfaces: high and low roughness approximations,” Proceedings of the International Conference on Trends in Optics and Photonics, Kolkata, India, 2009, pp. 462–470.

Beckmann, P.

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

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North Holland, 1967), Vol. 6, pp. 55–69.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transformation and Its Applications (McGraw-Hill, 1985).

Bruce, N. C.

Burlison, B. S. H.

B. S. H. Burlison, P. D. Ruiz, and J. M. Huntley, “Evaluation of the performance of tilt scanning interferometry for tomographic scanning,” Opt. Commun. 285, 1654–1661 (2012).
[CrossRef]

Chang, Y.

J. C. Jin, C. Jin, C. Li, and Y. Chang, “Fabrication anti reflection (AR) coatings for polarized 193 nm laser light at an incidence angle of 74°,” Opt. Commun. 298, 171–175 (2013).
[CrossRef]

Chatterjee, S.

V. C. Vani and S. Chatterjee, “Detection of a periodic structure embedded in surface roughness, for various correlation functions,” Pramana 77, 611–626 (2011).
[CrossRef]

V. C. Vani and S. Chatterjee, “Detection of a periodic structure hidden in random background: the role of signal amplitude in the matched filter detection method,” Phys. Scr. 81, 055402 (2010).
[CrossRef]

S. Chatterjee and R. K. Banyal, “Scattering of light by rough surfaces: high and low roughness approximations,” Proceedings of the International Conference on Trends in Optics and Photonics, Kolkata, India, 2009, pp. 462–470.

Cuevas, S.

Dingle, R. B.

R. B. Dingle, Asymptotic Expansion: Their Derivation and Interpretation (Academic, 1973).

Elliot, R.

R. Elliot, “On the theory of corrugated plane surfaces,” Trans. I. R. E. Antennas Propag. 2, 71–81 (1954).
[CrossRef]

Fan, J. R.

Fazelli, S. M.

M. Zamani, M. Salami, S. M. Fazelli, and G. R. Zafari, “Analytical expression for wave scattering from exponential height correlated rough surfaces,” J. Mod. Opt. 59, 1448–1452 (2012).
[CrossRef]

Fugate, R. Q.

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

Fuh, Y. K.

Geadsteyn, I. S.

I. S. Geadsteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Hsu, K. C.

Huntley, J. M.

B. S. H. Burlison, P. D. Ruiz, and J. M. Huntley, “Evaluation of the performance of tilt scanning interferometry for tomographic scanning,” Opt. Commun. 285, 1654–1661 (2012).
[CrossRef]

Jin, C.

J. C. Jin, C. Jin, C. Li, and Y. Chang, “Fabrication anti reflection (AR) coatings for polarized 193 nm laser light at an incidence angle of 74°,” Opt. Commun. 298, 171–175 (2013).
[CrossRef]

Jin, J. C.

J. C. Jin, C. Jin, C. Li, and Y. Chang, “Fabrication anti reflection (AR) coatings for polarized 193 nm laser light at an incidence angle of 74°,” Opt. Commun. 298, 171–175 (2013).
[CrossRef]

Li, C.

J. C. Jin, C. Jin, C. Li, and Y. Chang, “Fabrication anti reflection (AR) coatings for polarized 193 nm laser light at an incidence angle of 74°,” Opt. Commun. 298, 171–175 (2013).
[CrossRef]

Luchka, A. Yu.

A. Yu. Luchka, Method of Averaging Functional Corrections (Academic, 1965).

Milster, T. D.

Olver, F. W. J.

F. W. J. Olver, Introduction to Asymptotics and Special Functions (Academic, 1974).

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Application (McGraw-Hill, 1962).

Quan, C.

S. Wang, Y. Tian, C. J. Tay, and C. Quan, “Development of a laser-scattering-based probe for on-line measurement of surface roughness,” Appl. Opt. 42, 1318–1324 (2003).
[CrossRef]

C. J. Tay, S. H. Wang, C. Quan, and H. M. Shang, “In situ surface roughness measurement using a laser scattering method,” Opt. Commun. 218, 1–10 (2003).
[CrossRef]

Quinsa, Y.

Y. Quinsa and C. Tournier, “In situ non-contact measurements of surface roughness,” Precis. Eng. 36, 97–103 (2012).
[CrossRef]

Rivlin, T. J.

T. J. Rivlin, An Introduction to the Theory of Functions (Blaisdell,1969).

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

Ruiz, P. D.

B. S. H. Burlison, P. D. Ruiz, and J. M. Huntley, “Evaluation of the performance of tilt scanning interferometry for tomographic scanning,” Opt. Commun. 285, 1654–1661 (2012).
[CrossRef]

Ryzhik, I. M.

I. S. Geadsteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Salami, M.

M. Zamani, M. Salami, S. M. Fazelli, and G. R. Zafari, “Analytical expression for wave scattering from exponential height correlated rough surfaces,” J. Mod. Opt. 59, 1448–1452 (2012).
[CrossRef]

Sanchez-Brea, L. M.

L. M. Sanchez-Brea and F. J. Torcal-Milla, “Self imaging of the grating with two roughness levels,” Opt. Commun. 285, 13–17 (2012).
[CrossRef]

Shang, H. M.

C. J. Tay, S. H. Wang, C. Quan, and H. M. Shang, “In situ surface roughness measurement using a laser scattering method,” Opt. Commun. 218, 1–10 (2003).
[CrossRef]

Spizzichino, A.

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

Stegun, I.

M. Abramowicz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Tamkin, J. M.

Tay, C. J.

S. Wang, Y. Tian, C. J. Tay, and C. Quan, “Development of a laser-scattering-based probe for on-line measurement of surface roughness,” Appl. Opt. 42, 1318–1324 (2003).
[CrossRef]

C. J. Tay, S. H. Wang, C. Quan, and H. M. Shang, “In situ surface roughness measurement using a laser scattering method,” Opt. Commun. 218, 1–10 (2003).
[CrossRef]

Tian, Y.

Torcal-Milla, F. J.

L. M. Sanchez-Brea and F. J. Torcal-Milla, “Self imaging of the grating with two roughness levels,” Opt. Commun. 285, 13–17 (2012).
[CrossRef]

Tournier, C.

Y. Quinsa and C. Tournier, “In situ non-contact measurements of surface roughness,” Precis. Eng. 36, 97–103 (2012).
[CrossRef]

Vani, V. C.

V. C. Vani and S. Chatterjee, “Detection of a periodic structure embedded in surface roughness, for various correlation functions,” Pramana 77, 611–626 (2011).
[CrossRef]

V. C. Vani and S. Chatterjee, “Detection of a periodic structure hidden in random background: the role of signal amplitude in the matched filter detection method,” Phys. Scr. 81, 055402 (2010).
[CrossRef]

Vosiekovich, V. V.

Wang, S.

Wang, S. H.

C. J. Tay, S. H. Wang, C. Quan, and H. M. Shang, “In situ surface roughness measurement using a laser scattering method,” Opt. Commun. 218, 1–10 (2003).
[CrossRef]

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

Zafari, G. R.

M. Zamani, M. Salami, S. M. Fazelli, and G. R. Zafari, “Analytical expression for wave scattering from exponential height correlated rough surfaces,” J. Mod. Opt. 59, 1448–1452 (2012).
[CrossRef]

Zamani, M.

M. Zamani, M. Salami, S. M. Fazelli, and G. R. Zafari, “Analytical expression for wave scattering from exponential height correlated rough surfaces,” J. Mod. Opt. 59, 1448–1452 (2012).
[CrossRef]

Appl. Opt. (3)

J. Mod. Opt. (1)

M. Zamani, M. Salami, S. M. Fazelli, and G. R. Zafari, “Analytical expression for wave scattering from exponential height correlated rough surfaces,” J. Mod. Opt. 59, 1448–1452 (2012).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (5)

L. M. Sanchez-Brea and F. J. Torcal-Milla, “Self imaging of the grating with two roughness levels,” Opt. Commun. 285, 13–17 (2012).
[CrossRef]

C. J. Tay, S. H. Wang, C. Quan, and H. M. Shang, “In situ surface roughness measurement using a laser scattering method,” Opt. Commun. 218, 1–10 (2003).
[CrossRef]

J. C. Jin, C. Jin, C. Li, and Y. Chang, “Fabrication anti reflection (AR) coatings for polarized 193 nm laser light at an incidence angle of 74°,” Opt. Commun. 298, 171–175 (2013).
[CrossRef]

B. S. H. Burlison, P. D. Ruiz, and J. M. Huntley, “Evaluation of the performance of tilt scanning interferometry for tomographic scanning,” Opt. Commun. 285, 1654–1661 (2012).
[CrossRef]

J. Alvarez-Borrego, “1D rough surfaces: glitter function for remote sensing,” Opt. Commun. 113, 353–356 (1995).
[CrossRef]

Opt. Lett. (1)

Phys. Scr. (1)

V. C. Vani and S. Chatterjee, “Detection of a periodic structure hidden in random background: the role of signal amplitude in the matched filter detection method,” Phys. Scr. 81, 055402 (2010).
[CrossRef]

Pramana (1)

V. C. Vani and S. Chatterjee, “Detection of a periodic structure embedded in surface roughness, for various correlation functions,” Pramana 77, 611–626 (2011).
[CrossRef]

Precis. Eng. (1)

Y. Quinsa and C. Tournier, “In situ non-contact measurements of surface roughness,” Precis. Eng. 36, 97–103 (2012).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground based telescopes,” Rev. Mod. Phys. 69, 437–506 (1997).
[CrossRef]

Trans. I. R. E. Antennas Propag. (1)

R. Elliot, “On the theory of corrugated plane surfaces,” Trans. I. R. E. Antennas Propag. 2, 71–81 (1954).
[CrossRef]

Other (15)

See, for example, Eq. 6.623.2, listed in reference [10].

See, for example, Eq. 6.631.4, listed in reference [10].

R. B. Dingle, Asymptotic Expansion: Their Derivation and Interpretation (Academic, 1973).

F. W. J. Olver, Introduction to Asymptotics and Special Functions (Academic, 1974).

T. J. Rivlin, An Introduction to the Theory of Functions (Blaisdell,1969).

A. Yu. Luchka, Method of Averaging Functional Corrections (Academic, 1965).

See, for example, Eq. 6.621.1 of reference [10] and Chapter 13 of reference [9].

See, for example, Eq. 6.631.1 of reference [10] and Chapter 8 of reference [9].

S. Chatterjee and R. K. Banyal, “Scattering of light by rough surfaces: high and low roughness approximations,” Proceedings of the International Conference on Trends in Optics and Photonics, Kolkata, India, 2009, pp. 462–470.

R. N. Bracewell, The Fourier Transformation and Its Applications (McGraw-Hill, 1985).

A. Papoulis, The Fourier Integral and Its Application (McGraw-Hill, 1962).

M. Abramowicz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

I. S. Geadsteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

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

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North Holland, 1967), Vol. 6, pp. 55–69.

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Figures (9)

Fig. 1.
Fig. 1.

(a) Typical scattering geometry where S represents the rough surface, K1 is the wave vector of incidence, and K2 is the wave vector of scattering, A is the plane of incidence, and B is the plane of scattering. We have designated θ1 as the angle of incidence, i.e., the angle between Z and K1; θ2 is the angle of scattering, i.e., the angle between Z and K2; while θ3 is the azimuthal angle of scattering. (b) A microscopic view of the rough surface showing the ξ(x,y) profile variation with x, for a fixed y. The drawing on the right shows a comparison between the local radius of curvature rc of the rough surface sss, with s being the point of incidence. The psp is the local tangent plane, with the normal being in the direction n^. Under Kirchhoff’s approximation, rcλ, the wavelength of light, i.e., the direction n^ does not vary significantly, within a distance λ.

Fig. 2.
Fig. 2.

Variation of g(ρ)=exp(ρβ) for various values of β, where we have chosen β=1.0, 1.2, 1.4, 1.6, 1.8 and 2.0. Note that for ρ<1, variation is faster for smaller β, while for ρ>1, the variation of g(ρ) is faster for higher β.

Fig. 3.
Fig. 3.

Comparison between g(ρ) represented by the dotted curve and g1(ρ) represented by the continuous curve for β=1.0, 1.1 and 1.2. The two curves overlap for all these cases.

Fig. 4.
Fig. 4.

Comparison between ρ on the x axis and g(ρ) represented by the dotted curve g1(ρ) represented by the continuous curve on the y axis, for β=1.3, 1.4 and 1.5. The differences are striking in certain regions.

Fig. 5.
Fig. 5.

Comparison between ρ on the x axis and g(ρ) represented by the dotted curve and g2(ρ) represented by the continuous curve on the y axis for β=1.3, 1.4, 1.5, and 1.6. The two curves nearly overlap.

Fig. 6.
Fig. 6.

Comparison between ρ on the x axis and g(ρ) represented by the dotted curve and g2(ρ) represented by the continuous curve on the y axis, for β=1.7, 1.8, 1.9, and 2.0. The two curves nearly overlap.

Fig. 7.
Fig. 7.

(a) Variation of qn versus f(qn) for β=1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0. For low qn the fall is faster for lower β. (b) The variation of log(qn) versus log(f(qn)) for the above β values. For high qn values the fall in log[f(qn)] is faster.

Fig. 8.
Fig. 8.

Comparison between f(qn) and f0(qn) for different values of β. There is a marginal difference between the two curves at the tails for some cases. f0(qn) is the dotted curve and f(qn) is the continuous curve.

Fig. 9.
Fig. 9.

Comparison between f(qn) and f0(qn) for different values of β. There is a marginal difference between the two curves at the tails for some cases. f0(qn) is the dotted curve and f(qn) is the continuous curve.

Equations (77)

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

ξ(x,y)=0andξ(x,y)ξ(x,y)=σg(r).
g(r)=exp[(r/l)β],
r=[(xx)2+(yy)2+(zz)2]1/2.
vx=k(sinθ1sinθ2sinθ3)vy=ksinθ2sinθ3,vz=k(cosθ1cosθ2)vxy2=vx2+vy2,
ρρ*0=Intensity of light scattered in the direction(θ2,θ3)Intensity of light scattered in the specular direction(θ1=θ2,θ3=0).
ρρ*0=2πAf(vx,vy;g),
f(vx,vy;g)=exp[g(1g(r))]J0(vxyr)rdr,
g=(σvz)22.
f(vx,vy;g)=exp(g)n=0(gnn!)0[g(r)]nJ0(vxyr)rdr=exp(g)n=00(gnn!)exp[n(rl)β]J0(vxyr)rdr.
f(vx,vy;g)=exp(g)n=0(gnn!)fn(vxy),
fn(vxy)=0exp[(rl)β]J0(vxyr)rdr=ln20exp(ρβ)J0(qnρ)ρdρ,
ln=1n1/β.
qn=vxyln.
fn(vxy)=ln20exp(ρβ)J0(vxylnρ)ρdρ=ln2m=0(1)m(vxy2ln2/4)m0exp(ρβ)ρ2mρdρfn(vxy)=(ln2β)0[qn24]mΓ(2m+2)[Γ(m+1)]2.
fn(vxy)=ln20exp(ρ)J0(qnρ)ρdρ=ln2[1+qn2]3/2,
fn(vxy)=ln2m=0(qn24)mΓ(2m+2)[Γ(m+1)]2=ln2m=0(qn24)m(2m+1)!(m!)2=ln2[111!32(vxy2ln2)+12!3252(vxy2ln2)2].
fn(vxy)=ln20exp(ρ2)J0(qnρ)ρdρ=ln22[exp(qn24)].
fn(vxy)=ln22[vxy2ln24]mΓ(m+1)[Γ(m+1)]2
=ln22exp(vxy2ln24),
F(qn)=fn(vxy)fn(0).
fn(vxy)=(1β)a(1+cyqn2)y.
fn(vxy)=(ln2aβ){111!ycqn2y+12!y(y+1)y2c2qn4},
fn(vxy)=(ln2β)Γ(2β){111!r(4β)Γ(2β)Γ(2)(qn24)+12!Γ(6β)Γ(2β)Γ(3)(qn24)},
a=Γ(2β),
c=(14)Γ(4β)Γ(2β),
y=2[Γ(4β)]2Γ(6β)Γ(2β)2[Γ(4β)]2,
a=Γ(2)=1,c=(14)Γ(4)Γ(2)=34,y=2.6.61202.6.6=7248=32,
a=Γ(22)=Γ(1)=1,c=(14)Γ(4)Γ(2)=14
y=2[Γ(2)]2{Γ(3)Γ(2)2[Γ(2)]2}=2[Γ(2)][Γ(2)]22.1.1.
fn(vxy)=ln22limy1[1+1vxy2ln2y4]y=ln22evxy2ln24,
f(qn)=f0(qn)1[1+c2yqn2]y.
1<β1.2,
1.3<β<2.0
g(ρ)=eρβ=eρ+(ρρβ)eρ[1+11!(ρρβ)+12!(ρρβ)2+13!(ρρβ)3]
g1(ρ)=eρ+Δg1(ρ),
Δg1(ρ)exp(ρ)[(ρρβ)+12(ρρβ)2+16(ρρβ)3].
g(ρ)=exp[ρ2]{1+(ρ2ρβ)+(1/2!)(ρ2ρβ)2+(1/3!)(ρ2ρβ)3}g2(ρ)=exp[ρ2]+Δg2(ρ),
Δg2(ρ)exp(ρ2)[(ρ2ρβ)+12(ρ2ρβ)2+16(ρ2ρβ)3].
g(ρ)g1(ρ)=exp(ρ)+Δg1(ρ).
g(ρ)g2(ρ)=exp(ρ2)+Δg2(ρ).
fn(vxy)=0exp(ρ)J0(qnρ)ρdρ+0Δg1(ρ)J0(qnρ)ρdρ.
0eαxJv(βx)xμ1dx=(α2+β2)μ2Γ(v+μ)Pμ1v[α(α2+β2)]12,
fn(vxy)=T00+m=13r=0m(1)rΓ(m+2+(β1)rΓ(r+1)Γ(mr+1)(1+q2)(m+2+(β1)r)2×Γ(m+2+(β1)r)Pm+2+(β1)r10[(1+qn2)12].
fn(vxy)=T00+ΔT,
WithT00=(1+qn2)3/2,
WithΔT=T1+T2+T3,
T1=T10+T11,
T2=T20+T21+T22,
T3=T30+T31+T32+T33,
T10=Γ(3)Γ(1)Γ(2)(1+qn2)32P20(1+qn2)12,
T11=Γ(2+β)Γ(2)Γ(1)(1+qn2)(2+β2)P(1+β)0[(1+qn2)12],
T20=Γ(4)Γ(1)Γ(3)(1+qn2)2P30[(1+qn2)12],
T21=Γ(3+β)Γ(2)Γ(2)(1+qn2)(3+β2)P(2+β)0[(1+qn2)12],
T22=Γ(2+2β)Γ(3)Γ(1)(1+qn2)(1+β)P(1+2β)0[(1+qn2)12],
T30=Γ(5)Γ(1)Γ(4)(1+qn2)52P40[(1+qn2)12],
T31=Γ(4+β)Γ(2)Γ(3)(1+qn2)(4+β2)P(3+β)0[(1+qn2)12],
T32=Γ(3+2β)Γ(3)Γ(2)(1+qn2)(3+2β2)P(2+2β)0[(1+qn2)12],
T33=Γ(2+3β)Γ(4)Γ(1)(1+qn2)(2+3β2)P(1+3β)0[(1+qn2)12].
fn(vxy)=0exp(ρ2)J0(qnρ)ρdρ+0Δg2(ρ)J0(qnρ)ρdρ
0xμexp(αx2)Jv(βx)dx=Γ(v2+μ2+12)βαμ2Γ(v+1)eβ28αMμ2,v2(β24α)
Mλ,μ(z)=zμ+12ez2Φ(μλ+12,2μ+1;z),
fn(vxy)=t00+12Δt,
t00=12eqn24,
Δt=t1+t2+t3,
t1=t10+t11,
t2=t20+t21+t22,
t3=t30+t31+t32+t33,
t10=A(1,1;qn24),
t11=Γ(2+β2)A(β2,1;qn24),
t20=A(2,1;qn24),
t21=Γ(4+β2)A(2β22,1;qn24),
t22=Γ(β+1)Γ(3)Γ(1)A(β,1;qn24),
t30=A(3,1;qn24),
t31=Γ(6+β2)Γ(2)Γ(3)A(4β2,1;qn24),
t32=Γ(2+β)Γ(3)Γ(2)A(1β,1;qn24),
t33=Γ(2+3β2)Γ(4)Γ(1)A(3β2,1;qn24),
A(a,b;z)=ezΦ(a,b;z),

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