Abstract

We consider Fraunhofer diffraction by an ensemble of large arbitrary-shaped screens that are randomly oriented in the plane of a wavefront and have edges of arbitrary shape. It is shown that far outside the main diffraction peak the differential scattering cross section behaves asymptotically as θ3, where θ is the diffraction angle. Moreover, the differential scattering cross section depends only on the length of the contours bordering the screens and does not depend on the shape of the obstacles. As both strictly forward and total diffraction cross sections are specified by obstacle area only, the differential cross section of size-distributed obstacles is expected to be nearly independent of obstacle shape over the entire region of the diffraction angles.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  2. B. A. Baum, A. J. Heymsfield, P. Yang, and S. T. Bedka, “Bulk scattering models for the remote sensing of ice clouds. 1: Microphysical data and models,” J. Appl. Meteorol. 44, 1885-1895 (2005).
    [Crossref]
  3. G. L. Abramyan, “Theory of the diffraction by an opaque disk with a randomly rough edge,” Radiophys. Quantum Electron. 24, 132-138 (1981).
    [Crossref]
  4. M. K. Abdelazecz, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antennas Propag. AP-31, 375-377 (1983).
    [Crossref]
  5. K. S. Shifrin, Y. S. Shifrin, and I. A. Mikulinsky, “Diffraction of electromagnetic wave on a screen of a random shape,” Tech. Phys. Lett. 10, 68-72 (1984) (in Russian).
  6. A. L. Jones, “Fraunhofer diffraction by random irregular particles,” Part. Charact. 4, 123-127 (1987).
    [Crossref]
  7. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. 2 of Course of Theoretical Physics, 4th ed. (Butterworth-Heinemann, 1987).
  8. N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).
  9. C. F. Bohren and G. Koh, “Forward-scattering corrected extinction by nonspherical particles,” Appl. Opt. 24, 1023-1029 (1985).
    [Crossref] [PubMed]
  10. E. P. Zege, I. L. Katsev, A. S. Prikhach, G. Gilbert, and N. Witherspoon, “Simple model of the optical characteristics of bubbles and sediments in seawater of the surf zone,” Appl. Opt. 45, 6577-6585 (2006).
    [Crossref] [PubMed]

2006 (1)

2005 (1)

B. A. Baum, A. J. Heymsfield, P. Yang, and S. T. Bedka, “Bulk scattering models for the remote sensing of ice clouds. 1: Microphysical data and models,” J. Appl. Meteorol. 44, 1885-1895 (2005).
[Crossref]

1987 (1)

A. L. Jones, “Fraunhofer diffraction by random irregular particles,” Part. Charact. 4, 123-127 (1987).
[Crossref]

1985 (1)

1984 (1)

K. S. Shifrin, Y. S. Shifrin, and I. A. Mikulinsky, “Diffraction of electromagnetic wave on a screen of a random shape,” Tech. Phys. Lett. 10, 68-72 (1984) (in Russian).

1983 (1)

M. K. Abdelazecz, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antennas Propag. AP-31, 375-377 (1983).
[Crossref]

1981 (1)

G. L. Abramyan, “Theory of the diffraction by an opaque disk with a randomly rough edge,” Radiophys. Quantum Electron. 24, 132-138 (1981).
[Crossref]

Abdelazecz, M. K.

M. K. Abdelazecz, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antennas Propag. AP-31, 375-377 (1983).
[Crossref]

Abramyan, G. L.

G. L. Abramyan, “Theory of the diffraction by an opaque disk with a randomly rough edge,” Radiophys. Quantum Electron. 24, 132-138 (1981).
[Crossref]

Baum, B. A.

B. A. Baum, A. J. Heymsfield, P. Yang, and S. T. Bedka, “Bulk scattering models for the remote sensing of ice clouds. 1: Microphysical data and models,” J. Appl. Meteorol. 44, 1885-1895 (2005).
[Crossref]

Bedka, S. T.

B. A. Baum, A. J. Heymsfield, P. Yang, and S. T. Bedka, “Bulk scattering models for the remote sensing of ice clouds. 1: Microphysical data and models,” J. Appl. Meteorol. 44, 1885-1895 (2005).
[Crossref]

Bleistein, N.

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).

Bohren, C. F.

Gilbert, G.

Handelsman, R. A.

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).

Heymsfield, A. J.

B. A. Baum, A. J. Heymsfield, P. Yang, and S. T. Bedka, “Bulk scattering models for the remote sensing of ice clouds. 1: Microphysical data and models,” J. Appl. Meteorol. 44, 1885-1895 (2005).
[Crossref]

Jones, A. L.

A. L. Jones, “Fraunhofer diffraction by random irregular particles,” Part. Charact. 4, 123-127 (1987).
[Crossref]

Katsev, I. L.

Koh, G.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. 2 of Course of Theoretical Physics, 4th ed. (Butterworth-Heinemann, 1987).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. 2 of Course of Theoretical Physics, 4th ed. (Butterworth-Heinemann, 1987).

Mikulinsky, I. A.

K. S. Shifrin, Y. S. Shifrin, and I. A. Mikulinsky, “Diffraction of electromagnetic wave on a screen of a random shape,” Tech. Phys. Lett. 10, 68-72 (1984) (in Russian).

Prikhach, A. S.

Shifrin, K. S.

K. S. Shifrin, Y. S. Shifrin, and I. A. Mikulinsky, “Diffraction of electromagnetic wave on a screen of a random shape,” Tech. Phys. Lett. 10, 68-72 (1984) (in Russian).

Shifrin, Y. S.

K. S. Shifrin, Y. S. Shifrin, and I. A. Mikulinsky, “Diffraction of electromagnetic wave on a screen of a random shape,” Tech. Phys. Lett. 10, 68-72 (1984) (in Russian).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

Witherspoon, N.

Yang, P.

B. A. Baum, A. J. Heymsfield, P. Yang, and S. T. Bedka, “Bulk scattering models for the remote sensing of ice clouds. 1: Microphysical data and models,” J. Appl. Meteorol. 44, 1885-1895 (2005).
[Crossref]

Zege, E. P.

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

M. K. Abdelazecz, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antennas Propag. AP-31, 375-377 (1983).
[Crossref]

J. Appl. Meteorol. (1)

B. A. Baum, A. J. Heymsfield, P. Yang, and S. T. Bedka, “Bulk scattering models for the remote sensing of ice clouds. 1: Microphysical data and models,” J. Appl. Meteorol. 44, 1885-1895 (2005).
[Crossref]

Part. Charact. (1)

A. L. Jones, “Fraunhofer diffraction by random irregular particles,” Part. Charact. 4, 123-127 (1987).
[Crossref]

Radiophys. Quantum Electron. (1)

G. L. Abramyan, “Theory of the diffraction by an opaque disk with a randomly rough edge,” Radiophys. Quantum Electron. 24, 132-138 (1981).
[Crossref]

Tech. Phys. Lett. (1)

K. S. Shifrin, Y. S. Shifrin, and I. A. Mikulinsky, “Diffraction of electromagnetic wave on a screen of a random shape,” Tech. Phys. Lett. 10, 68-72 (1984) (in Russian).

Other (3)

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. 2 of Course of Theoretical Physics, 4th ed. (Butterworth-Heinemann, 1987).

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Example of an arbitrary contour and the points where the tangent line is perpendicular to Δ k .

Fig. 2
Fig. 2

Differential cross section of Fraunhofer diffraction by a randomly oriented rectangular screen with sides of 10 and 15 μ m (thin curve) and the first term of Eq. (29) (thick curve). Wavelength is equal to 532 nm .

Fig. 3
Fig. 3

(a) Phase functions of size-distributed rectangles and circles and (b) their relative differences.

Equations (46)

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

θ 1 .
Δ k = k k 0 = ( 2 π λ θ cos φ , 2 π λ θ sin φ ) ,
Δ k = Δ k = 2 π λ θ .
r = r ( l ) = ( x ( l ) , y ( l ) ) .
d r d l = n ( l ) = ( cos ψ ( l ) , sin ψ ( l ) ) ,
f = 1 λ S e i k r d 2 r ,
d C d Ω = f 2
d C d Ω d Ω = 0 2 π 0 f 2 θ d θ d φ = S ,
f = i λ Δ k 2 e i Δ k r Δ k × d r ,
a × b = a x b y b x a y .
f = i λ Δ k 2 e i Δ k r Δ k × n d l .
r λ .
Δ k r = 2 π λ θ r = x θ 1 .
d Δ k r d l = Δ k n = 0 ,
Δ k × d r d l = Δ k × n = Δ k .
I n = exp ( i Δ k r n + i Δ k d 2 r n d l 2 l 2 2 ) d l ,
I n = exp ( i Δ k r n + π i 4 sgn n ) 2 π Δ k d 2 r n d l 2 .
d 2 r n d l 2 = d n d l .
Δ k d 2 r n d l 2 = Δ k R n ,
I n = exp ( i Δ k r n + π i 4 sgn n ) 2 π R n Δ k .
f = i λ Δ k 2 n I n Δ k = i λ Δ k 2 n exp ( i Δ k r n + π i 4 sgn n ) 2 π Δ k R n ,
f 2 = 2 π λ 2 Δ k 3 n m exp ( i Δ k ( r n r m ) + π i 4 ( sgn n sgn m ) ) R n R m .
f 2 = 2 π λ 2 Δ k 3 n R n + Int .
R = d l d ψ .
d C d Ω = f 2 ¯ = 1 2 π 0 2 π f 2 d φ = 2 π λ 2 Δ k 3 n 0 π R n d ψ π + Int ¯ .
R n d ψ = d l d ψ d ψ = l n ,
l n = L ,
0 π n R n d ψ π = L π .
d C d Ω = 2 L λ 2 Δ k 3 + Int ¯ = 2 λ L ( 2 π ) 3 θ 3 + Int ¯ .
d C d Ω circ = r 2 J 1 2 ( x θ ) θ 2 ,
x = 2 π r λ .
x θ ,
J 1 ( z 1 ) = 2 π z cos ( z 3 π 4 ) ,
d C d Ω circ = 2 r 2 π x θ 3 cos 2 ( x θ 3 π 4 ) .
d C d Ω circ = 2 λ ( 2 π r ) ( 2 π ) 3 θ 3 ( 1 + cos ( 2 x θ 3 π 2 ) ) .
d C d Ω = 2 λ L ( 2 π ) 3 θ 3 + Int ¯ ,
Int ¯ = Int ¯ = 2 π λ 2 Δ k 3 1 2 π 0 2 π n m exp ( i Δ k ( r n r m ) + π i 4 ( sgn n sgn m ) ) R n R m = 0 .
d C d Ω = 2 λ L ( 2 π ) 3 θ 3 .
θ < λ L .
d C d Ω θ = 0 = S 2 λ 2 ,
d C d Ω θ λ L = 2 λ L ( 2 π ) 3 θ 3 ,
d C d Ω d Ω = S .
p ( θ ) = 2 π C d C d Ω
d w rect d a a m exp ( a a 0 ) ,
d w circ d r r μ exp ( r r 0 ) ,
δ = p circ ( θ ) p rec ( θ ) 1 ,

Metrics