Abstract

The concept of band-limited fractals is introduced and used to describe the diffraction of electromagnetic and optical waves by irregular structures. This concept is demonstrated through the example of plane-wave diffraction by a fractal phase screen of finite extent. The effect of the fractal phase screen is noted on the evolution of an incident wave with the fractal dimension and other descriptors used as parameters. Of particular interest is the result that, for random fractal phase screens, the diffraction pattern from a single realization of the model phase screen can be identical to the pattern averaged over an ensemble of screens. In these cases an orderly pattern emerges from a chaotic one. This problem of fractal diffraction is of intrinsic interest because of the variety of problems found to be described by fractal, as opposed to Euclidean, geometry. The results have potential applications to the propagation of waves through random media, the reflection of waves from rough surfaces, and the characterization of these processes through remote means.

© 1987 Optical Society of America

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References

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1983).
  2. M. V. Berry, Z. V. Lewis, “On the Weierstrass–Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
    [Crossref]
  3. H. Bohr, Almost Periodic Functions, translated by H. Cohn, F. Steinhardt (Chelsea, New York, 1947).
  4. A. Besicovitch, Almost Periodic Functions (Cambridge U. Press, London, 1932).
  5. See, e.g., N. Weiner, Generalized Harmonic Analysis-Tauberian Theorems (MIT Press, Cambridge, Mass., 1964); A. Fink, Almost Periodic Differential Equations (Springer-Verlag, Berlin, 1974).
  6. A. Mickelson, D. L. Jaggard, “Wave propagation in almost-periodic media,”IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
    [Crossref]
  7. D. L. Jaggard, A. Mickelson, “The reflection of electromagnetic waves from almost-periodic structures,” Appl. Phys. 19, 405–412 (1980).
    [Crossref]
  8. Y. Kim, D. L. Jaggard, “A band-limited fractal model of atmospheric refractivity fluctuations,” J. Opt. Soc. Am. A (submitted).
  9. M. V. Berry, “Diffractals,”J. Phys. A 12, 781–797 (1979).
    [Crossref]
  10. M. V. Berry, T. M. Blackwell, “Diffractal echoes,”J. Phys. A 14, 3101–3110 (1981).
    [Crossref]
  11. E. Jakeman, “Scattering by a corrugated random surface with fractal slope,”J. Phys. A 15, L55–L59 (1982); “Fresnel scattering by a corrugated random surface with fractal slope,”J. Opt. Soc. Am. 72, 1034–1041 (1982).
    [Crossref]
  12. E. Jakeman, “Fraunhofer scattering by a sub-fractal diffuser,” Opt. Acta 30, 1207–1212 (1983).
    [Crossref]
  13. The limitation 1 < D< 2 is needed only for the case of an infinite number of tones. However, we will restrict ourselves to this limitation here even for a countable number of tones.
  14. Y. Kim, D. L. Jaggard, “Fractal random arrays,” Proc. IEEE 74, 1278–1280 (1986); Y. Kim, D. L. Jaggard, “Fractal viewpoint of linear array theory,” presented at the 1986 Institute of Electrical and Electronics Engineers AP-S/International Union of Radio Science Meeting, Philadelphia, Pa., June 9–13, 1986.
    [Crossref]

1986 (1)

Y. Kim, D. L. Jaggard, “Fractal random arrays,” Proc. IEEE 74, 1278–1280 (1986); Y. Kim, D. L. Jaggard, “Fractal viewpoint of linear array theory,” presented at the 1986 Institute of Electrical and Electronics Engineers AP-S/International Union of Radio Science Meeting, Philadelphia, Pa., June 9–13, 1986.
[Crossref]

1983 (1)

E. Jakeman, “Fraunhofer scattering by a sub-fractal diffuser,” Opt. Acta 30, 1207–1212 (1983).
[Crossref]

1982 (1)

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,”J. Phys. A 15, L55–L59 (1982); “Fresnel scattering by a corrugated random surface with fractal slope,”J. Opt. Soc. Am. 72, 1034–1041 (1982).
[Crossref]

1981 (1)

M. V. Berry, T. M. Blackwell, “Diffractal echoes,”J. Phys. A 14, 3101–3110 (1981).
[Crossref]

1980 (2)

M. V. Berry, Z. V. Lewis, “On the Weierstrass–Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[Crossref]

D. L. Jaggard, A. Mickelson, “The reflection of electromagnetic waves from almost-periodic structures,” Appl. Phys. 19, 405–412 (1980).
[Crossref]

1979 (2)

M. V. Berry, “Diffractals,”J. Phys. A 12, 781–797 (1979).
[Crossref]

A. Mickelson, D. L. Jaggard, “Wave propagation in almost-periodic media,”IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
[Crossref]

Berry, M. V.

M. V. Berry, T. M. Blackwell, “Diffractal echoes,”J. Phys. A 14, 3101–3110 (1981).
[Crossref]

M. V. Berry, Z. V. Lewis, “On the Weierstrass–Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[Crossref]

M. V. Berry, “Diffractals,”J. Phys. A 12, 781–797 (1979).
[Crossref]

Besicovitch, A.

A. Besicovitch, Almost Periodic Functions (Cambridge U. Press, London, 1932).

Blackwell, T. M.

M. V. Berry, T. M. Blackwell, “Diffractal echoes,”J. Phys. A 14, 3101–3110 (1981).
[Crossref]

Bohr, H.

H. Bohr, Almost Periodic Functions, translated by H. Cohn, F. Steinhardt (Chelsea, New York, 1947).

Jaggard, D. L.

Y. Kim, D. L. Jaggard, “Fractal random arrays,” Proc. IEEE 74, 1278–1280 (1986); Y. Kim, D. L. Jaggard, “Fractal viewpoint of linear array theory,” presented at the 1986 Institute of Electrical and Electronics Engineers AP-S/International Union of Radio Science Meeting, Philadelphia, Pa., June 9–13, 1986.
[Crossref]

D. L. Jaggard, A. Mickelson, “The reflection of electromagnetic waves from almost-periodic structures,” Appl. Phys. 19, 405–412 (1980).
[Crossref]

A. Mickelson, D. L. Jaggard, “Wave propagation in almost-periodic media,”IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
[Crossref]

Y. Kim, D. L. Jaggard, “A band-limited fractal model of atmospheric refractivity fluctuations,” J. Opt. Soc. Am. A (submitted).

Jakeman, E.

E. Jakeman, “Fraunhofer scattering by a sub-fractal diffuser,” Opt. Acta 30, 1207–1212 (1983).
[Crossref]

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,”J. Phys. A 15, L55–L59 (1982); “Fresnel scattering by a corrugated random surface with fractal slope,”J. Opt. Soc. Am. 72, 1034–1041 (1982).
[Crossref]

Kim, Y.

Y. Kim, D. L. Jaggard, “Fractal random arrays,” Proc. IEEE 74, 1278–1280 (1986); Y. Kim, D. L. Jaggard, “Fractal viewpoint of linear array theory,” presented at the 1986 Institute of Electrical and Electronics Engineers AP-S/International Union of Radio Science Meeting, Philadelphia, Pa., June 9–13, 1986.
[Crossref]

Y. Kim, D. L. Jaggard, “A band-limited fractal model of atmospheric refractivity fluctuations,” J. Opt. Soc. Am. A (submitted).

Lewis, Z. V.

M. V. Berry, Z. V. Lewis, “On the Weierstrass–Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[Crossref]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1983).

Mickelson, A.

D. L. Jaggard, A. Mickelson, “The reflection of electromagnetic waves from almost-periodic structures,” Appl. Phys. 19, 405–412 (1980).
[Crossref]

A. Mickelson, D. L. Jaggard, “Wave propagation in almost-periodic media,”IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
[Crossref]

Weiner, N.

See, e.g., N. Weiner, Generalized Harmonic Analysis-Tauberian Theorems (MIT Press, Cambridge, Mass., 1964); A. Fink, Almost Periodic Differential Equations (Springer-Verlag, Berlin, 1974).

Appl. Phys. (1)

D. L. Jaggard, A. Mickelson, “The reflection of electromagnetic waves from almost-periodic structures,” Appl. Phys. 19, 405–412 (1980).
[Crossref]

IEEE Trans. Antennas Propag. (1)

A. Mickelson, D. L. Jaggard, “Wave propagation in almost-periodic media,”IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
[Crossref]

J. Phys. A (3)

M. V. Berry, “Diffractals,”J. Phys. A 12, 781–797 (1979).
[Crossref]

M. V. Berry, T. M. Blackwell, “Diffractal echoes,”J. Phys. A 14, 3101–3110 (1981).
[Crossref]

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,”J. Phys. A 15, L55–L59 (1982); “Fresnel scattering by a corrugated random surface with fractal slope,”J. Opt. Soc. Am. 72, 1034–1041 (1982).
[Crossref]

Opt. Acta (1)

E. Jakeman, “Fraunhofer scattering by a sub-fractal diffuser,” Opt. Acta 30, 1207–1212 (1983).
[Crossref]

Proc. IEEE (1)

Y. Kim, D. L. Jaggard, “Fractal random arrays,” Proc. IEEE 74, 1278–1280 (1986); Y. Kim, D. L. Jaggard, “Fractal viewpoint of linear array theory,” presented at the 1986 Institute of Electrical and Electronics Engineers AP-S/International Union of Radio Science Meeting, Philadelphia, Pa., June 9–13, 1986.
[Crossref]

Proc. R. Soc. London Ser. A (1)

M. V. Berry, Z. V. Lewis, “On the Weierstrass–Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[Crossref]

Other (6)

H. Bohr, Almost Periodic Functions, translated by H. Cohn, F. Steinhardt (Chelsea, New York, 1947).

A. Besicovitch, Almost Periodic Functions (Cambridge U. Press, London, 1932).

See, e.g., N. Weiner, Generalized Harmonic Analysis-Tauberian Theorems (MIT Press, Cambridge, Mass., 1964); A. Fink, Almost Periodic Differential Equations (Springer-Verlag, Berlin, 1974).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1983).

Y. Kim, D. L. Jaggard, “A band-limited fractal model of atmospheric refractivity fluctuations,” J. Opt. Soc. Am. A (submitted).

The limitation 1 < D< 2 is needed only for the case of an infinite number of tones. However, we will restrict ourselves to this limitation here even for a countable number of tones.

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Figures (5)

Fig. 1
Fig. 1

Configuration under consideration. A square fractal phase screen of side L in the x′–y′ plane at z = 0 is illuminated by an incident plane wave ψ(x′, y′). The transfer function of the phase screen is described by t(x′, y′). The diffracted field ψ(x, y) or intensity I(x, y) is observed a distance z away in the xy plane.

Fig. 2
Fig. 2

Power spectral density S(ν) as a function of frequency ν (top line) for fractal dimension D = 1.01, 5/3, and 1.99 (left to right). Accompanying Weierstrass functions Φ(x) as a function of coordinate x (lower four lines) with increasing tone numbers N = 1, 5, 10, and 20 (top to bottom) for the indicated fractal dimensions.

Fig. 3
Fig. 3

The evolution of the average-intensity distribution (left) for a fractal phase screen of finite extent (λ/L = 10−5) as a function of transverse normalized coordinate (x/L) for various distances from the screen (z/L = 10, 100, 200, 400, 800, and 1600) going from top to bottom. The analogous intensity distribution for a single realization is given on the right-hand side. Here D = 1.01 and all intensities are normalized to their maximum height.

Fig. 4
Fig. 4

The evolution of the average-intensity distribution (left) for a fractal phase screen of finite extent (λ/L = 105) as a function of transverse normalized coordinate (x/L) for various distances from the screen (z/L = 10, 100, 200, 400, 800, and 1600) going from top to bottom. The analogous intensity distribution for a single realization is given on the right-hand side. Here D = 5/3 and all intensities are normalized to their maximum height.

Fig. 5
Fig. 5

The evolution of the average-intensity distribution (left) for a fractal phase screen of finite extent (λ/L = 10−5) as a function of transverse normalized coordinate (x/L) for various distances from the screen (z/L = 10, 100, 200, 400, 800, and 1600) going from top to bottom. The analogous intensity distribution for a single realization is given on the right-hand side. Here D = 1.99 and all intensities are normalized to their maximum height.

Equations (42)

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Φ ( x ) = η [ 1 - b ( 2 D - 4 ) ] 1 / 2 n = N 1 N 2 b ( D - 2 ) n cos ( 2 π s b n x + ϕ n ) ,
R Φ ( x + τ ; x ) = Φ ( x + τ ) Φ ( x ) ,
R Φ ( x + τ ; x ) = η 2 2 [ 1 - b ( 2 D - 4 ) ] n = N 1 N 2 b 2 ( D - 2 ) n cos ( 2 π s b n τ ) .
S Φ ( ν ) = - R Φ ( τ ) exp ( - i 2 π ν τ ) d τ
S Φ ( ν ) = η 2 4 [ 1 - b ( 2 D - 4 ) ] n = N 1 N 2 b 2 ( D - 2 ) n [ δ ( ν - s b n ) + δ ( ν + s b n ) ] .
σ Φ 2 = R Φ ( 0 ) .
Φ ( x ) = 2 σ Φ [ 1 - b ( 2 D - 4 ) ] 1 / 2 [ b ( 2 D - 4 ) N 1 - b ( 2 D - 4 ) ( N 2 + 1 ) ] 1 / 2 × n = N 1 N 2 b ( D - 2 ) n cos ( 2 π s b n x + ϕ n ) .
S ¯ Φ ( ν ) = 1 Δ ν ν - Δ ν / 2 ν + Δ ν / 2 S Φ ( ν ) d ν .
S ¯ Φ ( ν ) = 1 Δ ν η 2 ( ν / s ) ( 2 D - 4 ) 4 [ 1 - b ( 2 D - 4 ) ] ( d n / d ν ) Δ ν .
S ¯ Φ ( ν ) = η 2 s - ( 2 D - 4 ) 4 [ 1 - b ( 2 D - 4 ) ] ln ( b ) ν ( 2 D - 5 ) .
t ( x ) = t ( x , y ) = exp [ i Φ ( x ) ] ,
P ( x ) = P ( x , y ) = rect ( x / L ) rect ( y / L ) .
ψ ( x ) = t ( x ) .
ψ ( x ) = q 1 = - q 2 = - q N = - { J q 1 ( C N 1 ) J q 2 ( C N 1 + 1 J q N ( C N 2 ) } × exp { i 2 π s [ q 1 b N 1 + q 2 b ( N 1 + 1 ) + + q N b N 2 ] x } × exp [ i ( q 1 ϕ ¯ N 1 + q 2 ϕ ¯ N 1 + 1 + + q N ϕ ¯ N ) ] ,
C n = 2 σ Φ [ 1 - b ( 2 D - 4 ) b ( D - 2 ) n ] 1 / 2 [ b ( 2 D - 4 ) N 1 - b ( 2 D - 4 ) ( N 2 + 1 ] 1 / 2 ,             ϕ ¯ n = ϕ n + π / 2.
ψ ( x ) = ( - 2 i k ) - - ψ ( x ) P ( x ) G ( x ; x ) d x
G ( x ; x ) = exp ( i k x - x ) 4 π x - x ,
G ( x ; x ) ( { exp ( i k z ) exp [ i k ( x 2 + y 2 ) / 2 z ] × exp [ - i k ( x x + y y ) / z ] } / ( 4 π z ) ) .
ψ ( x ) = { ( - i L 2 ) exp ( i k z ) exp [ i k ( x 2 + y 2 ) / 2 z ] λ z } × q 1 = - q 2 = - q N = - { J q 1 ( C N 1 ) J q 2 ( C N 1 + 1 ) × J q N ( C N 2 ) } { exp [ i ( q 1 ϕ ¯ N 1 + q 2 ϕ ¯ N 1 + 1 + + q N ϕ ¯ N 2 ) ] } × ( sinc { L [ x / λ z - s q 1 b N 1 - s q 2 b ( N 1 + 1 ) - - s q N b N 2 } ) { sinc [ L ( y / λ z ) ] } ,
I ( x ) = ψ ( x ) ψ * ( x ) ,
I ( x ) = L 4 λ 2 z 2 q 1 = - q 2 = - q N = - × [ J q 1 2 ( C N 1 ) J q 2 2 ( C N 1 + 1 ) J q N 2 ( C N 2 ) ] × ( sinc 2 { L [ x / λ z - s q 1 b - s q 2 b ( N 1 + 1 ) - - s q N b N 2 ] } ) [ sinc 2 ( L y / λ z ) ] .
I ( x ) = L 4 λ 2 z 2 { ( 1 - σ Φ 2 ) sinc 2 ( L x / λ z ) + n = - ( C n 2 / 4 ) sinc 2 [ L ( x / λ z - s b n ) ] } sinc 2 ( L y / λ z ) .
G ( x ; x ) ( exp ( i k z ) exp { i k [ ( x - x ) 2 + ( y - y ) 2 ] / 2 z } ) / ( 4 π z ) .
ψ ( x ) = { exp ( i k z ) exp [ i k ( x 2 + y 2 ) / 2 z ] 2 i } q 1 = - × q 2 = - q N = - [ J q 1 ( C N 1 ) J q 2 ( C N 1 + 1 ) J q N ( C N 2 ) ] × { exp [ i ( q 1 ϕ ¯ N 1 + q 2 ϕ ¯ N 1 + 1 + + q N ϕ ¯ N 2 ) ] } × [ exp ( ( - i k / 2 z ) { x - z λ s [ q 1 b N 1 + q 2 b ( N 1 + 1 ) + + q N b N 2 ] } 2 ) ] { [ C ( ξ 2 ) - C ( ξ 1 ) ] + i [ S ( ξ 2 ) - S ( ξ 1 ) ] } × { [ C ( η 2 ) - C ( η 1 ) ] + i [ S ( η 2 ) - S ( η 1 ) ] } ,
ξ 1 = - k / π z { L / 2 + x - z λ s [ q 1 b N 1 + q 2 b ( N 1 + 1 ) + + q N b N 2 ] } ,
ξ 2 = - k / π z { L / 2 - x + z λ s [ q 1 b N 1 + q 2 b ( N 1 + 1 ) + + q N b N 2 ] } ,
η 1 = - k / π z ( L / 2 + y ) ,
η 2 = - k / π z ( L / 2 - y ) ,
I ( x ) = 1 4 q 1 = - q 2 = - q N = - × [ J q 1 2 ( C N 1 ) J q 2 2 ( C N 1 + 1 ) J q N 2 ( C N 2 ) ] × { [ C ( ξ 2 ) - C ( ξ 1 ) ] 2 + [ S ( ξ 2 ) - S ( ξ 1 ) ] 2 } × { [ C ( η 2 ) - C ( η 1 ) ] 2 + [ S ( η 2 ) - S ( η 1 ) ] 2 } .
I ( x ) = 1 4 ( ( 1 - σ Φ 2 ) { [ C ( ξ 4 ) - C ( ξ 3 ) ] 2 + [ S ( ξ 4 ) - S ( ξ 3 ) ] 2 } + n = N 1 N 2 ( C n 2 / 4 ) { [ C ( ξ 6 ) - C ( ξ 5 ) ] 2 + [ S ( ξ 6 ) - S ( ξ 5 ) ] 2 } ) × { [ C ( η 2 ) - C ( η 1 ) ] 2 + [ S ( η 2 ) - S ( η 1 ) ] 2 } ,
ξ 3 = - k / π z ( L / 2 + x ) ,
ξ 4 = k / π z ( L / 2 - x ) ,
ξ 5 = - k / π z ( L / 2 + x - z λ s b n ) ,
ξ 6 = k / π z ( L / 2 - x + z λ s b n ) ,
I ( x , y , z ) coh d x d y I ( x , y , z ) tot d x d y 1 - σ Φ 2 exp ( - σ Φ 2 ) .
z > L / [ λ s b N 1 ( b - 1 ) ] .
s b N 1 ( b - 1 ) > 2 / L .
ψ ( x ) = exp [ i n = N 1 N 2 C n cos ( 2 π s b n x + ϕ n ) ]
ψ ( x ) = n = N 1 N 2 exp [ i C n cos ( 2 π s b n x + ϕ n ) ] ,
exp ( i w cos ρ ) = q = - i q J q ( w ) exp ( i q ρ ) ,
ψ ( x ) = n = N 1 N 2 { q = - exp ( i q π / 2 ) J q ( C n ) exp [ i q ( 2 π s b n x + ϕ n ) ] }
ψ ( x ) = n = N 1 N 2 { q = - J q ( C n ) exp ( i 2 π s q b n x ) × exp [ i q ( ϕ n + π / 2 ) ] } .

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