Abstract

Coherent-light diffraction on random phase screens with fractal properties leads to the formation of speckle patterns with peculiarities in correlation characteristics in the small-scale region. Such peculiarities are manifested in asymptotic behavior in intensity autocorrelation and structure functions in the vicinity of the zero values of their arguments. Intensity fluctuations in the far and the near diffraction zones are also characterized by values of fractal (Hausdorff–Besicovitch) dimensions D HB, differing from the corresponding Euclidean dimension. Relationships between the exponential factors of the structure functions of boundary field phase and scattered-light intensity fluctuations as well as between values of D HB have been obtained as a result of speckle-formation analysis for different conditions. Their dependencies on the illumination and observation conditions obtained in experiments with fractallike scatterers (rough glass plates) are in satisfactory agreement with theoretical results.

© 1996 Optical Society of America

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References

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  13. V. V. Tuchin, D. A. Zimnyakov, G. G. Akchurin, A. A. Mishin, I. L. Kon, “Coherence-domain optical methods for cell and tissue structure and function monitoring,” in Laser Chemistry, Biophysics, and Biomedicine, V. N. Zadkov, ed., Proc. SPIE2802, 152–163 (1996).
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    [CrossRef]

1994 (2)

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of statistical properties of partially developed speckles for diagnostics of human skin structural changes,” Opt. Spectrosc. USSR 76, 838–844 (1994).

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 33, 3178–3188 (1994).
[CrossRef]

1993 (1)

1988 (1)

1987 (2)

1986 (2)

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

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

1985 (1)

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct comparison of mechanical and optical measurements of precision machined surfaces,” Opt. Eng. 24, 388–396 (1985).

1982 (1)

1979 (2)

Akchurin, G. G.

V. V. Tuchin, D. A. Zimnyakov, G. G. Akchurin, A. A. Mishin, I. L. Kon, “Coherence-domain optical methods for cell and tissue structure and function monitoring,” in Laser Chemistry, Biophysics, and Biomedicine, V. N. Zadkov, ed., Proc. SPIE2802, 152–163 (1996).

Allain, C.

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

Angelsky, O. V.

Bennett, J. M.

Benzoni, J. F.

Berry, M. V.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

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

Church, E. L.

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
[CrossRef] [PubMed]

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct comparison of mechanical and optical measurements of precision machined surfaces,” Opt. Eng. 24, 388–396 (1985).

Cloitre, M.

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

Elson, J. M.

Jaggard, D. L.

Jakeman, E.

Kim, Y.

Kon, I. L.

V. V. Tuchin, D. A. Zimnyakov, G. G. Akchurin, A. A. Mishin, I. L. Kon, “Coherence-domain optical methods for cell and tissue structure and function monitoring,” in Laser Chemistry, Biophysics, and Biomedicine, V. N. Zadkov, ed., Proc. SPIE2802, 152–163 (1996).

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics. Part 2. Random Fields (Nauka, Moscow, 1978), p. 464 (in Russian).

Maksimyak, P. P.

Mishin, A. A.

V. V. Tuchin, D. A. Zimnyakov, G. G. Akchurin, A. A. Mishin, I. L. Kon, “Coherence-domain optical methods for cell and tissue structure and function monitoring,” in Laser Chemistry, Biophysics, and Biomedicine, V. N. Zadkov, ed., Proc. SPIE2802, 152–163 (1996).

Percival, I. C.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Perun, T. O.

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics. Part 2. Random Fields (Nauka, Moscow, 1978), p. 464 (in Russian).

Sarkar, S.

Sherrington, D.

Tatarsky, V. I.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics. Part 2. Random Fields (Nauka, Moscow, 1978), p. 464 (in Russian).

Tuchin, V. V.

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of statistical properties of partially developed speckles for diagnostics of human skin structural changes,” Opt. Spectrosc. USSR 76, 838–844 (1994).

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 33, 3178–3188 (1994).
[CrossRef]

V. V. Tuchin, D. A. Zimnyakov, G. G. Akchurin, A. A. Mishin, I. L. Kon, “Coherence-domain optical methods for cell and tissue structure and function monitoring,” in Laser Chemistry, Biophysics, and Biomedicine, V. N. Zadkov, ed., Proc. SPIE2802, 152–163 (1996).

Utz, S. R.

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of statistical properties of partially developed speckles for diagnostics of human skin structural changes,” Opt. Spectrosc. USSR 76, 838–844 (1994).

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 33, 3178–3188 (1994).
[CrossRef]

Vorburger, T. V.

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct comparison of mechanical and optical measurements of precision machined surfaces,” Opt. Eng. 24, 388–396 (1985).

Wyant, J. C.

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct comparison of mechanical and optical measurements of precision machined surfaces,” Opt. Eng. 24, 388–396 (1985).

Yaroslavsky, I. V.

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 33, 3178–3188 (1994).
[CrossRef]

Zimnyakov, D. A.

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of statistical properties of partially developed speckles for diagnostics of human skin structural changes,” Opt. Spectrosc. USSR 76, 838–844 (1994).

V. V. Tuchin, D. A. Zimnyakov, G. G. Akchurin, A. A. Mishin, I. L. Kon, “Coherence-domain optical methods for cell and tissue structure and function monitoring,” in Laser Chemistry, Biophysics, and Biomedicine, V. N. Zadkov, ed., Proc. SPIE2802, 152–163 (1996).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Phys. A (1)

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

Opt. Acta (1)

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Opt. Eng. (2)

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 33, 3178–3188 (1994).
[CrossRef]

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct comparison of mechanical and optical measurements of precision machined surfaces,” Opt. Eng. 24, 388–396 (1985).

Opt. Spectrosc. USSR (1)

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of statistical properties of partially developed speckles for diagnostics of human skin structural changes,” Opt. Spectrosc. USSR 76, 838–844 (1994).

Phys. Rev. B (1)

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

Other (2)

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics. Part 2. Random Fields (Nauka, Moscow, 1978), p. 464 (in Russian).

V. V. Tuchin, D. A. Zimnyakov, G. G. Akchurin, A. A. Mishin, I. L. Kon, “Coherence-domain optical methods for cell and tissue structure and function monitoring,” in Laser Chemistry, Biophysics, and Biomedicine, V. N. Zadkov, ed., Proc. SPIE2802, 152–163 (1996).

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Figures (11)

Fig. 1
Fig. 1

Near-zone observation of paraxial-region intensity fluctuations: H1, illuminating beam-waist plane; RPS, random phase screen; H2, observation plane; O, reimaging lens; H3, detector position plane.

Fig. 2
Fig. 2

Optical scheme of the scanning differential microinterferometer: 1, single-mode He–Ne laser; 2, 3, telescopic system as the beam expander; 4–6, Michelson interferometer unit; 7, beam splitter; 8, focused micro-objective; 9, sample being studied; 10, micro-objective; 11, 12, photodetectors (photomultiplier tubes); 13, piezoelectric phase-shift modulator; 14, 2D scanning device; 15, interface board.

Fig. 3
Fig. 3

Empirical structure functions for rough glass plates (samples 1–6).

Fig. 4
Fig. 4

Angular dependence of the normalized mean value of the scattered light intensity for sample N7; light source, single-mode He–Ne laser.

Fig. 5
Fig. 5

Experimental setup for the speckle intensity fluctuation analysis: 1, single-mode He–Ne laser; 2, telescopic system; 3, mirror; 4, focusing micro-objective; 5, 2D scanning device; 6, sample being studied; 7, image-transforming system (lens or free space); 8, photodetector (photomultiplier tube); 9, interface board.

Fig. 6
Fig. 6

Dependencies of the exponential factor of the intensity fluctuations for different samples of rough glass plates (1, sample N1; 2, sample N4; 3, sample N6) in the case of focused (R ≈ 20 μm) Gaussian-beam illumination.

Fig. 7
Fig. 7

Relationships between the exponential factors of the boundary field phase and the scattered light intensity fluctuations in the case of broad Gaussian-beam illumination. Objects being studied—samples NN1–6.

Fig. 8
Fig. 8

Relationships between the values of ν I b and ν I n for samples NN1–6; ν I n have been estimated for a sharply focused (R ≈ 3.2 μm) illuminating beam.

Fig. 9
Fig. 9

Dependencies of the exponential factor ν I on wave parameter Θ in the Fresnel diffraction zone for different samples (■, sample N2; □, sample N4; Δ, sample N6).

Fig. 10
Fig. 10

Dependencies of ν I (+, ■) and D HB (--, …)on defocusing parameter Δz in the near diffraction zone for sample N7 (1, L = 1.5 mm, R ≈ 3.2 μm; 2, L = 2.5 mm, R ≈ 50 μm).

Fig. 11
Fig. 11

Relationships between exponential factor ν I (■, +, *) and Hausdorff–Besicovitch dimension D HB (□) of the near-zone intensity fluctuations and the Ω parameter for sample N7 (Ω N =1.0 μm−1; +, *, R ≈ 3.2 μm, L = 1.5 mm; ■, R ≈ 50 μm, L = 2.0 mm).

Tables (1)

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Table 1 Parameters of Phase Distributions for Scatterers being Studied

Equations (22)

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D ϕ ( τ x , τ y ) = [ ϕ ( x + τ x , y + τ y ) ϕ ( x , y ) ] 2 ,
ϕ ˜ ( x , y ) = ϕ ( x , y ) ϕ ( x , y ) , D ϕ ( τ x , τ y ) = 2 R ϕ ( 0 , 0 ) 2 R ϕ ( τ x , τ y ) .
D ϕ ( | r ¯ | ) L T 2 v | r ¯ | ν ,
ν = 2 ( 2 D HB ) .
S ( f x ) = K α f x α
S ( f ¯ ) = K α f x α 1 [ 0 . 5 Γ ( α + 0 . 5 ) Γ 1 ( 0 . 5 ) Γ 1 ( α / 2 ) ]
U ( ξ , η ) = + d y + Φ ( x ξ , y η ) A ( x , y ) d x ,
F I ( ω ξ , ω η ) = [ F Φ ( ω x , ω y ) F A ( ω x , ω y ) ] [ F Φ ( ω x , ω y ) F A ( ω x , ω y ) ] * ,
F I ( ω ξ , ω η ) = [ F Φ ( ω x , ω y ) δ ( ω x , ω y ) ] [ F Φ ( ω x , ω y ) δ ( ω x , ω y ) ] * .
Γ υ ( ρ ¯ ) = exp [ 0 . 5 D ϕ ( ρ ¯ ) ] ,
R I ( ρ ¯ ) = | Γ υ ( ρ ¯ ) | 2 | υ 0 | 4 ,
D I ( ρ ¯ ) = 2 [ | Γ υ ( 0 ) | 2 | | Γ υ ( ρ ¯ ) | 2 ] .
D I ( ρ ¯ ) | | ρ ¯ | 0 2 D ϕ ( ρ ¯ ) .
ν I b = ν ϕ ,
F I ( ω ξ , ω η ) = [ F Φ ( ω x , ω y ) ] [ F Φ ( ω x , ω y ) ] * .
ν I n = 2 ν ϕ 2 .
ν I n = 2 ν I b 2 .
D HB ( I ) = 2 D HB ( ϕ ) 1 .
A ( x , y ) = exp { [ w 2 ( Δ z ) + j k / 2 R ˜ ( Δ z ) + j k / 2 ( L Δ z ) ] ( x 2 + y 2 ) } ,
w ( Δ z ) w 0 Δ z / R
| F A ( ω ξ , ω η ) | = k 1 [ R 2 ( Δ z ) 4 + ( L Δ z ) 2 ] 0 . 5 × exp { 0 . 5 ( ω ξ 2 + ω η 2 ) [ R k ( Δ z ) 2 + R 1 k ( Δ z ) 2 ( L Δ z ) 2 ] 1 } .
Ω = [ R k ( Δ z ) 2 + R 1 k ( Δ z ) 2 ( L Δ z ) 2 ] 0 . 5 .

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