Abstract

We present a careful theoretical analysis for the wavelength decorrelation of speckle intensity that occurs when plane-polarized laser illumination is propagated through an optical system consisting of a thick diffuser in cascade with the space-invariant 4F imaging system and a CCD monitoring configuration. Based on Maxwell’s equations for propagation into the right half-space, our formulation for a scalar component of the electric field is accurate well inside of the Fresnel zone and in the nonparaxial regime as well. The diffuser is described as an artificial dielectric consisting of tiny dielectric spheres embedded in a host medium and randomly spaced. We model the thick diffusers using a thin multilayer decomposition, and we write computer software describing the output speckle pattern amplitude which results from the propagation of an input plane wave. This model provides a good description for opal milk glass (OMG), and we illustrate the usefulness of this software by two applications. First, for a series of OMG diffusers of varying thickness, we present curves for the wavelength decorrelation of speckle that are found to be in good agreement with earlier experiments by George et al.[Appl. Phys. 7, 157 (1975)]. Also, these results are used to compute internal parameters of these diffusers. Second, using these values, we present some first-order statistics of the intensity for this diffuser series and show that they are in accord with the published literature.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).
  2. J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).
  3. N. George and D. C. Sinclair, eds., feature issue “Speckle in Optics,” J. Opt. Soc. Am. 66(11), (1976).
  4. N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
    [CrossRef]
  5. N. George, A. Jain, and R. D. S. Melville Jr., “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
    [CrossRef]
  6. R. D. S. Melville, “The thick diffuser,” Ph.D. thesis (California Institute of Technology, 1975).
  7. R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
    [CrossRef]
  8. T. Stone and N. George, “Wavelength performance of holographic optical elements,” Appl. Opt. 24, 3797–3810 (1985).
    [CrossRef] [PubMed]
  9. D. W. Diehl and N. George, “Holographic interference filters for infrared communications,” Appl. Opt. 42, 1203–1210 (2003).
    [CrossRef] [PubMed]
  10. Wave Train Software, MZA Associates Corp., http://www.mza.com.
  11. J. W. Goodman, “Statistical properties of laser sparkle patterns,” Tech. Rep. T. R. 2303-1 (Stanford Electronics Laboratories, 1963).
  12. N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
    [CrossRef]
  13. L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6, 765–781 (1989).
    [CrossRef]
  14. D. L. Fried, “Laser eye safety: the implications of ordinary speckle statistics and of speckled-speckle statistics,” J. Opt. Soc. Am. 71, 914–916 (1981).
    [CrossRef] [PubMed]
  15. D. J. Shertler and N. George, “Uniform scattering patterns from grating diffuser cascades for display applications,” Appl. Opt. 38, 291–303 (1999).
    [CrossRef]
  16. L. G. Shirley and P. A. Lo, “Bispectral analysis of the wavelength dependence of speckle: remote sensing of object shape,” J. Opt. Soc. Am. A 11, 1025–1046 (1994).
    [CrossRef]
  17. N. Chang and N. George, “Speckle in the 4F optical system,” Appl. Opt. 47, A13–A20 (2008).
    [CrossRef] [PubMed]
  18. K. G. Budden, The Wave-Guide Mode Theory of Wave Propagation (Logos, 1961).
  19. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).
  20. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).
  21. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  22. N. Chang, “Speckle in a thick diffuser,” Ph.D. thesis (University of Rochester, 2009).
  23. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes with Errata Sheet, 4th ed. (McGraw-Hill, 2002).

2008 (1)

2003 (1)

1999 (1)

1997 (1)

N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
[CrossRef]

1994 (1)

1989 (1)

1985 (1)

1981 (1)

1976 (1)

N. George and D. C. Sinclair, eds., feature issue “Speckle in Optics,” J. Opt. Soc. Am. 66(11), (1976).

1975 (2)

N. George, A. Jain, and R. D. S. Melville Jr., “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

1974 (1)

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

Alferness, R.

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

Budden, K. G.

K. G. Budden, The Wave-Guide Mode Theory of Wave Propagation (Logos, 1961).

Chang, N.

N. Chang and N. George, “Speckle in the 4F optical system,” Appl. Opt. 47, A13–A20 (2008).
[CrossRef] [PubMed]

N. Chang, “Speckle in a thick diffuser,” Ph.D. thesis (University of Rochester, 2009).

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

Diehl, D. W.

Fried, D. L.

George, N.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser sparkle patterns,” Tech. Rep. T. R. 2303-1 (Stanford Electronics Laboratories, 1963).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

Jain, A.

N. George, A. Jain, and R. D. S. Melville Jr., “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

Lo, P. A.

Melville, R. D. S.

N. George, A. Jain, and R. D. S. Melville Jr., “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

R. D. S. Melville, “The thick diffuser,” Ph.D. thesis (California Institute of Technology, 1975).

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes with Errata Sheet, 4th ed. (McGraw-Hill, 2002).

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes with Errata Sheet, 4th ed. (McGraw-Hill, 2002).

Shertler, D. J.

Shirley, L. G.

Sinclair, D. C.

N. George and D. C. Sinclair, eds., feature issue “Speckle in Optics,” J. Opt. Soc. Am. 66(11), (1976).

Stone, T.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

Appl. Opt. (4)

Appl. Phys. (3)

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

N. George, A. Jain, and R. D. S. Melville Jr., “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
[CrossRef]

Other (11)

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

Wave Train Software, MZA Associates Corp., http://www.mza.com.

J. W. Goodman, “Statistical properties of laser sparkle patterns,” Tech. Rep. T. R. 2303-1 (Stanford Electronics Laboratories, 1963).

R. D. S. Melville, “The thick diffuser,” Ph.D. thesis (California Institute of Technology, 1975).

K. G. Budden, The Wave-Guide Mode Theory of Wave Propagation (Logos, 1961).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

N. Chang, “Speckle in a thick diffuser,” Ph.D. thesis (University of Rochester, 2009).

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes with Errata Sheet, 4th ed. (McGraw-Hill, 2002).

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 (17)

Fig. 1
Fig. 1

(a) Decomposition of a thick diffuser with spherical particles as shown into (b) a cascade of thin screens with a small free-space region in between.

Fig. 2
Fig. 2

Plane-polarized optical wave is normally incident on the cascade-model thick diffuser and the output is imaged onto the CCD by a 4 F imager. A polarization analyzer (A) is located as the input object (IP-A). Propagation through the cascades can be computed in the amplitude plane as shown in (a). However, the actual calculation routines are based on switching between the amplitude and Fourier plane as shown in (b) to allow an all-products operation. t i is the binary maps of particle locations at each screen, which is explained in Fig. 8, and P i is the transfer function of propagating and scattering of the field.

Fig. 3
Fig. 3

(a) Section out of a cascade model where light propagates from the mth screen to the ( m + 1 ) th screen. The corresponding operation in space and Fourier domain is illustrated in (b), where a convolution expressed as a Rayleigh–Sommerfeld–Smythe integral equation in the space domain has a simpler counterpart in the Fourier domain.

Fig. 4
Fig. 4

Scattering from a small sphere, and the coordinates are illustrated. Relative index ratio n r = m 1 / m 2 , in which indices m 1 and m 2 are for sphere and medium, respectively.

Fig. 5
Fig. 5

Two particles are located a distance, d, apart. An incident wave illuminates the first particle at ( x 1 , y 1 , 0 ) , and the scattered wave subsequently illuminates the second particle, located at ( x 2 , y 2 , d ) .

Fig. 6
Fig. 6

Incident field at the second particle (a) showing the dipole moment induced at an angle ψ. In order to simplify the calculation, we only use the y component of the electric field to keep the dipole radiation symmetrical as shown in (b).

Fig. 7
Fig. 7

Scattering field at a fixed point on the ( m + 1 ) th screen is the sum of scattered fields from all particles on the mth screen.

Fig. 8
Fig. 8

This figure shows the numerical representation of the random particle positions.

Fig. 9
Fig. 9

4 F optical metrology system is shown. It consists of a Coherent Innova 70 tunable dye laser Model No. 599 (TL), polarizer (P) set to pass E y , OMG, analyzer (A), lenses of F = 200 mm , variable aperture (D) (typically 1 mm ), Atmel Camelia 4 M B/W CCD with 14 μm pixel size, digital computer (DC), and display. Illumination wavelength from the dye laser is calibrated by Coherent WaveMaster Model No. 33-2650 (W). Automatic power input adjustment is not shown. M, mirror. BS, beam splitter. F, focal length.

Fig. 10
Fig. 10

Computer simulation for the wavelength decorrelation of speckle, Eq. (13), for four thicknesses of OMG. Wavelength interval Δ λ starts at λ = 578.0 nm .

Fig. 11
Fig. 11

Computer simulation for the wavelength decorrelation as in Fig. 9 to show the variation with relative index n r for the 200 μm OMG.

Fig. 12
Fig. 12

Computer simulation for wavelength decorrelation versus Δ λ for four thicknesses.

Fig. 13
Fig. 13

Actual laser-OMG wavelength decorrelation versus Δ λ to compare to Fig. 11.

Fig. 14
Fig. 14

Speckle patterns: (a) actual laser 200 μm OMG diffuser and (b) computer simulation.

Fig. 15
Fig. 15

First-order Rician density for three OMG diffusers [2, 5].

Fig. 16
Fig. 16

First-order density of speckle intensity using computer simulation.

Fig. 17
Fig. 17

First-order density of speckle intensity using actual laser OMG for comparison to Fig. 15.

Tables (2)

Tables Icon

Table 1 Summary of the Features from the Theoretical Intensity Correlation Curves of a Series of Opal Glass Diffusers

Tables Icon

Table 2 Basic Flashed Opal Glass Properties

Equations (21)

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

E y , m + 1 ( x m + 1 , y m + 1 ; z m + 1 ) = d x m d y m E y , m ( x m , y m ; z m ) exp ( i k R ) 2 π R d m R ( i k + 1 R ) ,
V y , m + 1 ( f x , f y ; z m + 1 ) = V y , m ( f x , f y ; z m ) exp ( i k d 1 ( λ f x ) 2 ( λ f y ) 2 ) ,
x y [ exp ( i k R ) 2 π R d R ( i k + 1 R ) ] = exp ( i k d 1 ( λ f x ) 2 ( λ f y ) 2 ) , for 1 ( λ f x ) 2 ( λ f y ) 2 0 . = exp ( k d ( λ f x ) 2 + ( λ f y ) 2 1 ) , for 1 ( λ f x ) 2 ( λ f y ) 2 < 0 ,
E ψ = E ϕ ϕ ^ + E θ θ ^ = k 2 2 a 3 n r 2 1 n r 2 + 2 E inc exp ( i k 2 r ) r sin ψ ψ ^ ,
E ψ = k 2 a 3 n r 2 1 n r 2 + 2 E inc , 1 exp ( i k r ) r sin ψ ψ ^ ,
E inc , 2 y ^ = E ψ sin ψ y ^ = k 2 a 3 n r 2 1 n r 2 + 2 E inc , 1 ( ( x 2 x 1 ) 2 + d 2 ) exp ( i k r ) r 3 y ^ .
E y , m + 1 , s ( x , y ) = k 2 a 3 n r 2 1 n r 2 + 2 { [ ( x x 1 ) 2 + d 2 ] exp ( i k r 1 ) r 1 3 E inc , m ( x 1 , y 1 ) + [ ( x z 2 ) 2 + d 2 ] exp ( i k r 2 ) r 2 3 E inc , m ( z 2 , y 2 ) + + [ ( x x N m ) 2 + d 2 ] exp ( i k r N m ) r N m 3 E inc , m ( x N m , y N m ) } ,
E y , m + 1 , S ( x , y ) = k 2 a 3 n r 2 1 n r 2 + 2 [ j = 1 N m δ ( x x j , y y j ) ] E inc , m ( x , y ) × [ ( x x ) 2 + d 2 ] exp ( i k ( x x ) 2 + ( y y ) 2 + d 2 ) ( x x ) 2 + ( y y ) 2 + d 2 3 d x d y ,
E inc , m + 1 ( x , y ) = E y , m + 1 , D ( x , y ) + E y , m + 1 , S ( x , y ) = d x d y E inc , m ( x , y ) × d exp ( i k [ ( x x ) 2 + ( y y ) 2 + d 2 ] 1 / 2 ) ( x x ) 2 + ( y y ) 2 + d 2 ( i k + 1 ( x x ) 2 + ( y y ) 2 + d 2 ) + k 2 a 3 n r 2 1 n r 2 + 2 × j = 1 N m δ m ( x x j , y y j ) E inc , m ( x , y ) × [ ( x x ) 2 + d 2 ] exp ( i k ( x x ) 2 + ( y y ) 2 + d 2 ) ( x x ) 2 + ( y y ) 2 + d 2 3 d x d y .
V inc , m + 1 ( f x , f y ) = k 2 a 3 n r 2 1 n r 2 + 2 V rev , m x y [ ( x 2 + d 2 ) exp ( i k x 2 + y 2 + d 2 ) x 2 + y 2 + d 2 3 ] + V inc , m ( f x , f y ) exp ( i k d 1 ( λ f x ) 2 ( λ f y ) 2 ) ,
P ( w x ) = 1 w ¯ x ( 1 1 C R 2 ) exp [ ( w x + w ¯ x 1 C R 2 ) w ¯ x ( 1 1 C R 2 ) ] I 0 [ 2 ( w x w ¯ x 1 C R 2 ) 1 / 2 w ¯ x ( 1 1 C R 2 ) ] ,
L m = D 2 L N 3 .
M ( L 2 N D 2 ) 1 3 .
R w ( x , y , λ ; x , y , λ + Δ λ ) = λ λ + Δ λ w ( λ ) w * ( λ Δ λ ) d λ .
r π 1 s = 1 k 2 2 n = 1 i n 1 2 n + 1 n ( n + 1 ) a n ζ n ( k 2 r ) P n ( 1 ) ( cos θ ) cos ( ϕ ) ,
r π 2 s = 1 k 2 2 n = 1 i n 1 2 n + 1 n ( n + 1 ) b n ζ n ( k 2 r ) P n ( 1 ) ( cos θ ) sin ( ϕ ) ,
a n = S n ( α ) S n ( β ) n r S n ( β ) S n ( α ) ζ n ( α ) S n ( β ) n r S n ( β ) ζ n ( α ) ,
b n = n r S n ( α ) S n ( β ) S n ( β ) S n ( α ) n r ζ n ( α ) S n ( β ) S n ( β ) ζ n ( α ) ,
E ϕ = k 2 2 a 3 n r 2 1 n r 2 + 2 E inc exp ( i k 2 r ) r sin ϕ ,
E θ = k 2 2 a 3 n r 2 1 n r 2 + 2 E inc exp ( i k 2 r ) r cos θ cos ϕ ,
E ψ = E ϕ ϕ ^ + E θ θ ^ = k 2 2 a 3 n r 2 1 n r 2 + 2 E inc exp ( i k 2 r ) r sin ψ ψ ^ ,

Metrics