Abstract

Using the full wave approach, the scattering cross sections for finitely conducting particles with very rough surfaces are expressed as weighted sums of specular point (physical optics) and diffuse scattering cross sections. Through judicious use of the forward-scattering theorem and the observation that for large particles the forward-scattered “shadow forming wave is the same for all surfaces which have the same shadow line,” the albedos and the extinction cross sections for particles with rough surfaces are evaluated. These computations are essential to solve the equation of radiative transfer for the specific intensities (Stokes parameters) in media consisting of random distributions of particles with rough surfaces. The particle surface roughness has a significant effect on the diffuse specific intensities.

© 1986 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  3. G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
  4. J. M. Greenberg, “Scattering by Nonspherical Particles,” J. Appl. Phys. 31, 82 (1960).
    [CrossRef]
  5. P. Chylek, “Depolarization of Electromagnetic Radiation Scattered by Nonspherical Particles,” J. Opt. Soc. Am. 67, 175 (1977).
    [CrossRef]
  6. D. W. Scheurman, Ed., Light Scattering by Irregular Shaped Particles (Plenum, New York, 1980).
    [CrossRef]
  7. E. Bahar, M. A. Fitzwater, “Backscatter Cross Sections for Randomly Oriented Metallic Flakes at Optical Frequencies: Full Wave Approach,” Appl. Opt. 22, 3813 (1983).
    [CrossRef] [PubMed]
  8. E. Bahar, S. Chakrabarti, “Scattering and Depolarization by Large Conducting Spheres with Very Rough Surfaces,” Appl. Opt. 24, 1820 (1985).
    [CrossRef] [PubMed]
  9. C. Yeh, K. K. Mei, “On the Scattering from Arbitrarily Shaped Inhomogeneous Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
    [CrossRef]
  10. S. O. Rice, “Reflection of Electromagnetic Waves from a Slightly Rough Surface,” Commun. Pure Appl. Math. 4, 351 (1951).
    [CrossRef]
  11. J. T. Kiehl, M. W. Ko, A. Mugnai, P. Chylek, “Perturbation Approach to Light Scattering by Nonspherical Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
    [CrossRef]
  12. E. Bahar, M. Fitzwater, “Multiple Scattering by Finitely Conducting Particles with Rough Surfaces at Infrared and Optical Frequencies,” Radio Sci. in press (1986).
    [CrossRef]
  13. A. Ishimaru, R. L.-T. Cheung, “Multiple Scattering Effects in Wave Propagation Due to Rain,” Ann. Telecommun. 35, 373 (1980).
  14. P. Chylek “Extinction Cross Sections of Arbitrarily Shaped Randomly Oriented Nonspherical Particles,” Opt. Soc. Am., 67, 1348 (1977).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  16. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  17. E. Bahar, “Full Wave Solutions for the Depolarization of the Scattered Radiation Fields by Rough Surfaces of Arbitrary Slope,” IEEE Trans. Antennas Propag. AP-24, 443 (1981).
    [CrossRef]
  18. H. Ehrenreich, H. R. Philipp, B. Segall, “Optical Properties of Aluminum,” Phys. Rev. 132, 1918 (1963).
    [CrossRef]
  19. M. K. Abdelazeez, “Wave Scattering from a Large Sphere with Rough Surface,” IEEE Trans. Antennas Propag. AP-31, 375 (1983).
    [CrossRef]
  20. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, DC, 1964).

1985

1983

1981

E. Bahar, “Full Wave Solutions for the Depolarization of the Scattered Radiation Fields by Rough Surfaces of Arbitrary Slope,” IEEE Trans. Antennas Propag. AP-24, 443 (1981).
[CrossRef]

1980

A. Ishimaru, R. L.-T. Cheung, “Multiple Scattering Effects in Wave Propagation Due to Rain,” Ann. Telecommun. 35, 373 (1980).

1977

P. Chylek “Extinction Cross Sections of Arbitrarily Shaped Randomly Oriented Nonspherical Particles,” Opt. Soc. Am., 67, 1348 (1977).
[CrossRef]

P. Chylek, “Depolarization of Electromagnetic Radiation Scattered by Nonspherical Particles,” J. Opt. Soc. Am. 67, 175 (1977).
[CrossRef]

1963

H. Ehrenreich, H. R. Philipp, B. Segall, “Optical Properties of Aluminum,” Phys. Rev. 132, 1918 (1963).
[CrossRef]

1960

J. M. Greenberg, “Scattering by Nonspherical Particles,” J. Appl. Phys. 31, 82 (1960).
[CrossRef]

1951

S. O. Rice, “Reflection of Electromagnetic Waves from a Slightly Rough Surface,” Commun. Pure Appl. Math. 4, 351 (1951).
[CrossRef]

Abdelazeez, M. K.

M. K. Abdelazeez, “Wave Scattering from a Large Sphere with Rough Surface,” IEEE Trans. Antennas Propag. AP-31, 375 (1983).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, DC, 1964).

Bahar, E.

E. Bahar, S. Chakrabarti, “Scattering and Depolarization by Large Conducting Spheres with Very Rough Surfaces,” Appl. Opt. 24, 1820 (1985).
[CrossRef] [PubMed]

E. Bahar, M. A. Fitzwater, “Backscatter Cross Sections for Randomly Oriented Metallic Flakes at Optical Frequencies: Full Wave Approach,” Appl. Opt. 22, 3813 (1983).
[CrossRef] [PubMed]

E. Bahar, “Full Wave Solutions for the Depolarization of the Scattered Radiation Fields by Rough Surfaces of Arbitrary Slope,” IEEE Trans. Antennas Propag. AP-24, 443 (1981).
[CrossRef]

E. Bahar, M. Fitzwater, “Multiple Scattering by Finitely Conducting Particles with Rough Surfaces at Infrared and Optical Frequencies,” Radio Sci. in press (1986).
[CrossRef]

Barrick, D. E.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Chakrabarti, S.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950).

Cheung, R. L.-T.

A. Ishimaru, R. L.-T. Cheung, “Multiple Scattering Effects in Wave Propagation Due to Rain,” Ann. Telecommun. 35, 373 (1980).

Chylek, P.

P. Chylek “Extinction Cross Sections of Arbitrarily Shaped Randomly Oriented Nonspherical Particles,” Opt. Soc. Am., 67, 1348 (1977).
[CrossRef]

P. Chylek, “Depolarization of Electromagnetic Radiation Scattered by Nonspherical Particles,” J. Opt. Soc. Am. 67, 175 (1977).
[CrossRef]

J. T. Kiehl, M. W. Ko, A. Mugnai, P. Chylek, “Perturbation Approach to Light Scattering by Nonspherical Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
[CrossRef]

Ehrenreich, H.

H. Ehrenreich, H. R. Philipp, B. Segall, “Optical Properties of Aluminum,” Phys. Rev. 132, 1918 (1963).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fitzwater, M.

E. Bahar, M. Fitzwater, “Multiple Scattering by Finitely Conducting Particles with Rough Surfaces at Infrared and Optical Frequencies,” Radio Sci. in press (1986).
[CrossRef]

Fitzwater, M. A.

Greenberg, J. M.

J. M. Greenberg, “Scattering by Nonspherical Particles,” J. Appl. Phys. 31, 82 (1960).
[CrossRef]

Ishimaru, A.

A. Ishimaru, R. L.-T. Cheung, “Multiple Scattering Effects in Wave Propagation Due to Rain,” Ann. Telecommun. 35, 373 (1980).

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

Kiehl, J. T.

J. T. Kiehl, M. W. Ko, A. Mugnai, P. Chylek, “Perturbation Approach to Light Scattering by Nonspherical Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
[CrossRef]

Ko, M. W.

J. T. Kiehl, M. W. Ko, A. Mugnai, P. Chylek, “Perturbation Approach to Light Scattering by Nonspherical Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
[CrossRef]

Krichbaum, C. K.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Mei, K. K.

C. Yeh, K. K. Mei, “On the Scattering from Arbitrarily Shaped Inhomogeneous Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Mugnai, A.

J. T. Kiehl, M. W. Ko, A. Mugnai, P. Chylek, “Perturbation Approach to Light Scattering by Nonspherical Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
[CrossRef]

Philipp, H. R.

H. Ehrenreich, H. R. Philipp, B. Segall, “Optical Properties of Aluminum,” Phys. Rev. 132, 1918 (1963).
[CrossRef]

Rice, S. O.

S. O. Rice, “Reflection of Electromagnetic Waves from a Slightly Rough Surface,” Commun. Pure Appl. Math. 4, 351 (1951).
[CrossRef]

Ruck, G. T.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Segall, B.

H. Ehrenreich, H. R. Philipp, B. Segall, “Optical Properties of Aluminum,” Phys. Rev. 132, 1918 (1963).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, DC, 1964).

Stuart, W. D.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Yeh, C.

C. Yeh, K. K. Mei, “On the Scattering from Arbitrarily Shaped Inhomogeneous Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
[CrossRef]

Ann. Telecommun.

A. Ishimaru, R. L.-T. Cheung, “Multiple Scattering Effects in Wave Propagation Due to Rain,” Ann. Telecommun. 35, 373 (1980).

Appl. Opt.

Commun. Pure Appl. Math.

S. O. Rice, “Reflection of Electromagnetic Waves from a Slightly Rough Surface,” Commun. Pure Appl. Math. 4, 351 (1951).
[CrossRef]

IEEE Trans. Antennas Propag.

E. Bahar, “Full Wave Solutions for the Depolarization of the Scattered Radiation Fields by Rough Surfaces of Arbitrary Slope,” IEEE Trans. Antennas Propag. AP-24, 443 (1981).
[CrossRef]

M. K. Abdelazeez, “Wave Scattering from a Large Sphere with Rough Surface,” IEEE Trans. Antennas Propag. AP-31, 375 (1983).
[CrossRef]

J. Appl. Phys.

J. M. Greenberg, “Scattering by Nonspherical Particles,” J. Appl. Phys. 31, 82 (1960).
[CrossRef]

J. Opt. Soc. Am.

Opt. Soc. Am.

P. Chylek “Extinction Cross Sections of Arbitrarily Shaped Randomly Oriented Nonspherical Particles,” Opt. Soc. Am., 67, 1348 (1977).
[CrossRef]

Phys. Rev.

H. Ehrenreich, H. R. Philipp, B. Segall, “Optical Properties of Aluminum,” Phys. Rev. 132, 1918 (1963).
[CrossRef]

Other

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, DC, 1964).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

D. W. Scheurman, Ed., Light Scattering by Irregular Shaped Particles (Plenum, New York, 1980).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950).

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

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

J. T. Kiehl, M. W. Ko, A. Mugnai, P. Chylek, “Perturbation Approach to Light Scattering by Nonspherical Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
[CrossRef]

E. Bahar, M. Fitzwater, “Multiple Scattering by Finitely Conducting Particles with Rough Surfaces at Infrared and Optical Frequencies,” Radio Sci. in press (1986).
[CrossRef]

C. Yeh, K. K. Mei, “On the Scattering from Arbitrarily Shaped Inhomogeneous Particles,” in Light Scattering by Irregular Shaped Particles, D. W. Scheurman, Ed. (Plenum, New York, 1980).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Scattering cross sections σS1, σS2, and σS vs β (roughness parameter) (Table I).

Fig. 2
Fig. 2

Extinction cross section σt, σt0, albedos A and A1 vs β (roughness parameter) (Table I).

Fig. 3
Fig. 3

Scattering cross sections σS1, σS2 and σS vs β (roughness parameter) (Table III).

Fig. 4
Fig. 4

Extinction cross sections σt and albedo A vs β (roughness parameter) (Table III).

Fig. 5
Fig. 5

Scattering cross sections σS1, σS2, and σS vs β (roughness parameter) (Table III).

Fig. 6
Fig. 6

Extinction cross section σt and albedo A vs β (roughness parameter) (Table III).

Fig. 7
Fig. 7

Scattering cross section σS1, σS2, σS, and σS0 vs D/λ (Table IV).

Fig. 8
Fig. 8

Extinction cross section σt, σt0, albedos A and A0 vs D/λ (Table IV).

Tables (4)

Tables Icon

Table I Extinction Cross Sections and Albedos for Spheres with Rough Surfaces (0 ≤ β ≤ 1)

Tables Icon

Table II Extinction Cross Sections and Albedos for Spheres with Rough Surfaces (1 ≤ β ≤ 10)

Tables Icon

Table III Extinction Cross Sections and Albedos for Spheres with Rough Surfaces (0 ≤ β ≤ 10)

Tables Icon

Table IV Extinction Cross Sections and Albedos for Spheres with Rough Surfaces (5λ ≤ D ≤ 8λ)

Equations (21)

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d [ I ] d s = - [ T ] [ I ] + [ S ] [ I ] d μ d ϕ + [ I i ] ,
[ E l E r ] = [ f 11 f 12 f 21 f 22 ] [ E l E r ] exp ( - i k 0 r ) r .
[ I i ] = [ S ] [ I r i ] d μ d ϕ ,
σ t = 4 k 0 a 2 Im [ f i i ( n ¯ f , n ¯ i ) ] , i = 1 or 2.
[ S ] = χ ( v ¯ · a ¯ r ) 2 [ S Mie ] + [ S R ] ,
χ ( v r ) 2 = exp [ v r 2 h 2 ) ] ,
v ¯ = k 0 ( n ¯ f - n ¯ i ) ,             v r = v ¯ · a ¯ r .
A = σ S / σ t = σ S / ( σ S + σ a ) ,
σ S = 1 π a 2 χ 2 σ M d Ω + 1 4 π σ R d Ω σ S 1 + σ S 2 ;
σ S 1 = ( k 0 a ) - 2 χ 2 [ S 1 ( θ ) 2 + S 2 ( θ ) 2 sin θ d θ .
σ S 2 = 0.25 ( σ R V V + σ R V H + σ R H V + σ R H H ) sin θ d θ ,
A = σ S σ t = σ S σ t 0 σ t 0 σ t σ S σ t 0 ( σ t 0 σ t ) P . C . .
( σ t ) P . C . = ( σ S ) P . C . ,
A = ( σ S t 0 ) / ( σ S σ t 0 ) P . C . = A 1 / A 2 .
W ( v T ) = 2 C / π v T 4             v d < v T < v c ,
h 2 = π 2 v d v c W ( v T ) v T d v T = C 2 ( 1 v d 2 - 1 v c 2 ) .
σ s 2 = π 2 v d v c W ( v T ) v T 3 d v T = C ln ( v c / v d ) ,
W ( v T ) = 2 C π [ v T v T 2 + v m 2 ] 8 .
R ( ξ ) = [ 1 - 3 ξ 2 8 + 3 ξ 4 32 + ξ 6 3072 ] ξ K 1 ( ξ ) + [ 1 2 - ξ 2 4 - ξ 4 96 ] ξ 2 K 0 ( ξ ) .
ξ = v m r d .
h 2 = C / 210 v m 6 ,             σ s 2 = C / 84 v m 4 .

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