Abstract

It is shown that optical diffraction can be used to reveal the scaling features not only of deterministic but also of random fractal fields and to determine their main characteristics. The analysis of fractals can be experimentally realized through first-order optical systems.

© 1998 Optical Society of America

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References

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature, 2nd ed. (Freeman, New York, 1982), Chap. 1, pp. 1–5.
  2. A. Lakhtakia, N. S. Holter, V. K. Varadan, V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Trans. Antennas Propag. AP-35, 236–239 (1987).
    [CrossRef]
  3. C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
    [CrossRef]
  4. Y. Kim, H. Grebel, D. L. Jaggard, “Diffraction by fractally serrated apertures,” J. Opt. Soc. Am. A 8, 20–26 (1991).
    [CrossRef]
  5. Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
    [CrossRef]
  6. J. Uozumi, Y. Sakurada, T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995), and references therein.
    [CrossRef]
  7. T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
    [CrossRef]
  8. M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
    [CrossRef]
  9. T. Alieva, “Fractional Fourier transform as a tool for investigation of fractal objects,” J. Opt. Soc. Am. A 13, 1189–1192 (1996).
    [CrossRef]
  10. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, Calif., 1966), Chap. 4, pp. 217–268.
  11. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), pp. 381–393.
  12. M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  13. C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).
  14. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  15. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  16. T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
    [CrossRef]
  17. T. Alieva, F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
    [CrossRef]
  18. S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
    [CrossRef]
  19. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  20. A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transform,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
    [CrossRef]
  21. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  22. J. Feder, Fractals (Plenum, New York, 1988), Chap. 14, pp. 227–237.

1996 (3)

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

T. Alieva, “Fractional Fourier transform as a tool for investigation of fractal objects,” J. Opt. Soc. Am. A 13, 1189–1192 (1996).
[CrossRef]

1995 (3)

J. Uozumi, Y. Sakurada, T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995), and references therein.
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[CrossRef]

1994 (3)

A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transform,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
[CrossRef]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

1993 (2)

1992 (2)

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[CrossRef]

1991 (1)

1987 (1)

A. Lakhtakia, N. S. Holter, V. K. Varadan, V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Trans. Antennas Propag. AP-35, 236–239 (1987).
[CrossRef]

1986 (1)

C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

1982 (1)

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Abe, S.

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[CrossRef]

Agullo-Lopez, F.

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Alieva, T.

T. Alieva, “Fractional Fourier transform as a tool for investigation of fractal objects,” J. Opt. Soc. Am. A 13, 1189–1192 (1996).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Allain, C.

C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

Almeida, L. B.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Asakura, T.

J. Uozumi, Y. Sakurada, T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995), and references therein.
[CrossRef]

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Cloitre, M.

C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

Feder, J.

J. Feder, Fractals (Plenum, New York, 1988), Chap. 14, pp. 227–237.

Gomez-Reino, C.

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

Grebel, H.

Holter, N. S.

A. Lakhtakia, N. S. Holter, V. K. Varadan, V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Trans. Antennas Propag. AP-35, 236–239 (1987).
[CrossRef]

Jaggard, D. L.

Kim, Y.

Klein, S.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, N. S. Holter, V. K. Varadan, V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Trans. Antennas Propag. AP-35, 236–239 (1987).
[CrossRef]

Lohmann, A. W.

Lopez, V.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, Calif., 1966), Chap. 4, pp. 217–268.

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature, 2nd ed. (Freeman, New York, 1982), Chap. 1, pp. 1–5.

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Mendlovic, D.

Nazarathy, M.

Ozaktas, H. M.

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Sakurada, Y.

J. Uozumi, Y. Sakurada, T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995), and references therein.
[CrossRef]

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[CrossRef]

Shamir, J.

Sheridan, J. T.

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[CrossRef]

Soffer, B. H.

Uozumi, J.

J. Uozumi, Y. Sakurada, T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995), and references therein.
[CrossRef]

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[CrossRef]

Varadan, V. K.

A. Lakhtakia, N. S. Holter, V. K. Varadan, V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Trans. Antennas Propag. AP-35, 236–239 (1987).
[CrossRef]

Varadan, V. V.

A. Lakhtakia, N. S. Holter, V. K. Varadan, V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Trans. Antennas Propag. AP-35, 236–239 (1987).
[CrossRef]

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), pp. 381–393.

IEEE Trans. Antennas Propag. (1)

A. Lakhtakia, N. S. Holter, V. K. Varadan, V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Trans. Antennas Propag. AP-35, 236–239 (1987).
[CrossRef]

Int. J. Optoelectron. (1)

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

J. Mod. Opt. (3)

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

J. Uozumi, Y. Sakurada, T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995), and references therein.
[CrossRef]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (4)

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun. 114, 161–169 (1995).
[CrossRef]

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Phys. Rev. B (1)

C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[CrossRef]

Other (4)

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, Calif., 1966), Chap. 4, pp. 217–268.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), pp. 381–393.

B. B. Mandelbrot, The Fractal Geometry of Nature, 2nd ed. (Freeman, New York, 1982), Chap. 1, pp. 1–5.

J. Feder, Fractals (Plenum, New York, 1988), Chap. 14, pp. 227–237.

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Equations (50)

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[RMf(x)](u)=FM(u)=-KM(x, u)f(x)dx
KM(x, u)=(iB)-1/2 expiπB(Ax2+Du2-2xu).
uku=ABCDxkx,
Wf(x, kx)=-fx+X2f*x-X2×exp(-i2πXkx)dX,
f(x)=1f*(0)-Wf(x/2, kx)exp(i2πxkx)dkx.
Wf(x, kx)=-Gf(x, X)exp(-i2πXkx)dX,
Gf(x, X)=f(x+X/2)f*(x-X/2)
WFM(u, ku)=Wf(uD-kuB,-Cu+kuA).
|FM(u)|2=-WFM(u, ku)dku=-Wf(uD-kuB,-Cu+kuA)dku.
f(λ1x1, , λixi, , λnxn)
=λ1H1λiHiλnHnf(x1, , xi, xn),
W0(λx, γkx)=λHλγHγW0(x, kx).
I(λx)=W0(λx, kx)dkx=λHλI(x),
|Fπ/2(γkx)|2=W0(x, γkx)dx=γHγ|Fπ/2(kx)|2.
f(λx)=1f*(0)-Wf(λx/2, kx)exp(i2πλxkx)dkx
=λ-1f*(0)-Wf(λx/2, λ-1kx)exp(i2πxkx)dkx
=λHf*(0)-Wf(x/2, kx)exp(i2πxkx)dkx=λHf(x),
G0(λx, σX)=λHλσHσG0(x, X),
W0(λx, σ-1kx)=λHλσHσ+1W0(x, kx).
|FM1(u)|2=-W0(uD1-kuB1,-uC1+kuA1)dku=λ-Hλσ-Hσ-1-W0(λ(uD1-kuB1),σ-1(-uC1+kuA1))dku.
r=u(λA2D1-σ-1B2C1)+ku(-λA2B1+σ-1B2A1),
p=u(λC2D1-σ-1D2C1)+ku(D2A1σ-1-λC2B1),
dku=(D2A1σ-1-λC2B1)-1 dp=P dp,
|FM1(u)|2=λ-Hλσ-Hσ-1P-W0(rD2-pB2,-rC2+pA2)dp.
A1B2=λσA2B1,
|FM1(u)|2=λ-Hλσ-Hσ(A2/A1)|FM2(uσ-1B2/B1)|2
|FM1(u)|2=σHλ-Hσ(B2/B1)-Hλ(A2/A1)1+Hλ×|FM2(uσ-1B2/B1)|2.
λσZ1=Z2
λσ tan α1=tan α2
I(λx)=λHλI(x),
|Fπ/2(λ-1kx)|2=λHσ+1|Fπ/2(kx)|2.
|FM1(u)|2=A2B1A1B2(Hλ+Hσ)/2A2A1×FM2uA2B2A1B12.
A2B2=A1B1,
|FM1(u)|2=(B1/B2)Hλ+Hσ+1|FM2(u)|2.
|Fα(u)|2=(tan α)Hλ+Hσ+1|Fπ/2-α(u)|2.
I(λx)=λHλI(x),
|Fπ/2(λkx)|2=λ-Hσ-1|Fπ/2(kx)|2.
|FM(λu)|2=λHλ-Hσ|FM(u)|2.
|FM1(u)|2=(A2B1/A1B2)Hλ(A2/A1)|FM2(uB2/B1)|2.
|FZ1(u)|2=(Z1/Z2)Hλ|FZ2(uZ2/Z1)|2,
|Fα(u)|2=(tan α)2Hλ+1|Fπ/2-α(u cot α)|2.
|FM1(u)|2=(A2B1/A1B2)Hσ(A2/A1)|FM2(uA1/A2)|2.
|FZ1(u)|2=(Z1/Z2)Hσ|FZ2(u)|2,
I(σZ, u)=σHσI(Z, u).
|FM1(u)|2=(A2/A1)Hλ+1(B1/B2)Hσ|FM2(u)|2.
|Fα1(u)|2=cos α2cos α1Hλ+1sin α1sin α2Hσ|Fα2(u)|2.
W0(λxx, σx-1kx, λyy, σy-1ky)
=λxHλxσxHσx+1λyHλyσyHσy+1W0(x, kx, y, ky).
|FM1x, y(u, v)|2=λx-Hλxσx-Hσxλy-Hλyσy-Hσy(A2xA2y/A1xA1y)×|FM2x, y(uσx-1B2x/B1x, vσy-1B2y/B1y)|2,
A1x, yB2x, y=λx, yσx, yA2x, yB1x, y.

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