Abstract

Understanding speckle behavior is very important in speckle metrology application. The contrast of a polychromatic speckle depends not only on surface roughness and the coherence length of a light source, as shown in previous works, but also on optical geometry. We applied the Fresnel approach of diffraction theory for the free-space geometry and derived a simple analytical relationship between contrast, coherence length, size of illuminated spot, and distances between source, object, and observation plane. The effect of contrast reduction is found to be significant for low-coherence light sources.

© 2006 Optical Society of America

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References

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  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Application, Version 5, August 2005 (work in progress available at http://www-ee.stanford.edu/~goodman).
  2. H. M. Pedersen, 'On the contrast of polychromatic speckle pattern and its dependence on surface roughness,' Opt. Acta 22, 15-24 (1975).
    [CrossRef]
  3. H. M. Pedersen, 'Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependences of speckle,' Opt. Acta 22, 523-535 (1975).
    [CrossRef]
  4. G. Parry, 'Speckle pattern in partially coherent light,' in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), pp. 77-122.
  5. T. Iwai, N. Takai, and T. Asakura, 'Space-time correlation function of the dynamic polychromatic laser speckle,' Opt. Acta 30, 759-776 (1983).
    [CrossRef]
  6. T. McKechnie, 'Image-plane speckle in partially coherent illumination,' Opt. Quantum Electron. 8, 61-67 (1976).
    [CrossRef]
  7. Y.-Q. Hu, 'Dependence of polychromatic-speckle-pattern contrast on imaging and illumination directions,' Appl. Opt. 33, 2707-2714 (1994).
    [CrossRef] [PubMed]
  8. N. George and A. Jain, 'Space and wavelength dependence of speckle intensity,' Appl. Phys. 4, 201-212 (1974).
    [CrossRef]
  9. J. M. Huntley, 'Simple model for image-plane polychromatic speckle contrast,' Appl. Opt. 38, 2212-2215 (1999).
    [CrossRef]
  10. C. Rodrigues and J. Pinto, 'Contrast of polychromatic speckle patterns and its dependence to surface height distribution,' Opt. Eng. (Bellingham) 42, 1699-1703 (2003).
    [CrossRef]
  11. T. Iwai, N. Takai, and T. Asakura, 'Statistical properties of the dynamic dichromatic laser speckle,' Opt. Commun. 44, 13-18 (1982).
    [CrossRef]
  12. S. S. Ul'yanov, D. A. Zimnyakov, and V. V. Tuchin, 'Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,' Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
    [CrossRef]
  13. T. S. Tkaczyk, K. W. Gossage, and J. K. Barton, 'Speckle image properties in optical coherence tomography,' Proc. SPIE 4619, 59-70 (2002).
    [CrossRef]

2003

C. Rodrigues and J. Pinto, 'Contrast of polychromatic speckle patterns and its dependence to surface height distribution,' Opt. Eng. (Bellingham) 42, 1699-1703 (2003).
[CrossRef]

2002

T. S. Tkaczyk, K. W. Gossage, and J. K. Barton, 'Speckle image properties in optical coherence tomography,' Proc. SPIE 4619, 59-70 (2002).
[CrossRef]

1999

1994

S. S. Ul'yanov, D. A. Zimnyakov, and V. V. Tuchin, 'Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,' Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

Y.-Q. Hu, 'Dependence of polychromatic-speckle-pattern contrast on imaging and illumination directions,' Appl. Opt. 33, 2707-2714 (1994).
[CrossRef] [PubMed]

1983

T. Iwai, N. Takai, and T. Asakura, 'Space-time correlation function of the dynamic polychromatic laser speckle,' Opt. Acta 30, 759-776 (1983).
[CrossRef]

1982

T. Iwai, N. Takai, and T. Asakura, 'Statistical properties of the dynamic dichromatic laser speckle,' Opt. Commun. 44, 13-18 (1982).
[CrossRef]

1976

T. McKechnie, 'Image-plane speckle in partially coherent illumination,' Opt. Quantum Electron. 8, 61-67 (1976).
[CrossRef]

1975

H. M. Pedersen, 'On the contrast of polychromatic speckle pattern and its dependence on surface roughness,' Opt. Acta 22, 15-24 (1975).
[CrossRef]

H. M. Pedersen, 'Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependences of speckle,' Opt. Acta 22, 523-535 (1975).
[CrossRef]

1974

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

Asakura, T.

T. Iwai, N. Takai, and T. Asakura, 'Space-time correlation function of the dynamic polychromatic laser speckle,' Opt. Acta 30, 759-776 (1983).
[CrossRef]

T. Iwai, N. Takai, and T. Asakura, 'Statistical properties of the dynamic dichromatic laser speckle,' Opt. Commun. 44, 13-18 (1982).
[CrossRef]

Barton, J. K.

T. S. Tkaczyk, K. W. Gossage, and J. K. Barton, 'Speckle image properties in optical coherence tomography,' Proc. SPIE 4619, 59-70 (2002).
[CrossRef]

George, N.

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

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Application, Version 5, August 2005 (work in progress available at http://www-ee.stanford.edu/~goodman).

Gossage, K. W.

T. S. Tkaczyk, K. W. Gossage, and J. K. Barton, 'Speckle image properties in optical coherence tomography,' Proc. SPIE 4619, 59-70 (2002).
[CrossRef]

Hu, Y.-Q.

Huntley, J. M.

Iwai, T.

T. Iwai, N. Takai, and T. Asakura, 'Space-time correlation function of the dynamic polychromatic laser speckle,' Opt. Acta 30, 759-776 (1983).
[CrossRef]

T. Iwai, N. Takai, and T. Asakura, 'Statistical properties of the dynamic dichromatic laser speckle,' Opt. Commun. 44, 13-18 (1982).
[CrossRef]

Jain, A.

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

McKechnie, T.

T. McKechnie, 'Image-plane speckle in partially coherent illumination,' Opt. Quantum Electron. 8, 61-67 (1976).
[CrossRef]

Parry, G.

G. Parry, 'Speckle pattern in partially coherent light,' in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), pp. 77-122.

Pedersen, H. M.

H. M. Pedersen, 'On the contrast of polychromatic speckle pattern and its dependence on surface roughness,' Opt. Acta 22, 15-24 (1975).
[CrossRef]

H. M. Pedersen, 'Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependences of speckle,' Opt. Acta 22, 523-535 (1975).
[CrossRef]

Pinto, J.

C. Rodrigues and J. Pinto, 'Contrast of polychromatic speckle patterns and its dependence to surface height distribution,' Opt. Eng. (Bellingham) 42, 1699-1703 (2003).
[CrossRef]

Rodrigues, C.

C. Rodrigues and J. Pinto, 'Contrast of polychromatic speckle patterns and its dependence to surface height distribution,' Opt. Eng. (Bellingham) 42, 1699-1703 (2003).
[CrossRef]

Takai, N.

T. Iwai, N. Takai, and T. Asakura, 'Space-time correlation function of the dynamic polychromatic laser speckle,' Opt. Acta 30, 759-776 (1983).
[CrossRef]

T. Iwai, N. Takai, and T. Asakura, 'Statistical properties of the dynamic dichromatic laser speckle,' Opt. Commun. 44, 13-18 (1982).
[CrossRef]

Tkaczyk, T. S.

T. S. Tkaczyk, K. W. Gossage, and J. K. Barton, 'Speckle image properties in optical coherence tomography,' Proc. SPIE 4619, 59-70 (2002).
[CrossRef]

Tuchin, V. V.

S. S. Ul'yanov, D. A. Zimnyakov, and V. V. Tuchin, 'Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,' Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

Ul'yanov, S. S.

S. S. Ul'yanov, D. A. Zimnyakov, and V. V. Tuchin, 'Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,' Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

Zimnyakov, D. A.

S. S. Ul'yanov, D. A. Zimnyakov, and V. V. Tuchin, 'Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,' Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

Appl. Opt.

Appl. Phys.

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

Opt. Acta

H. M. Pedersen, 'On the contrast of polychromatic speckle pattern and its dependence on surface roughness,' Opt. Acta 22, 15-24 (1975).
[CrossRef]

H. M. Pedersen, 'Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependences of speckle,' Opt. Acta 22, 523-535 (1975).
[CrossRef]

T. Iwai, N. Takai, and T. Asakura, 'Space-time correlation function of the dynamic polychromatic laser speckle,' Opt. Acta 30, 759-776 (1983).
[CrossRef]

Opt. Commun.

T. Iwai, N. Takai, and T. Asakura, 'Statistical properties of the dynamic dichromatic laser speckle,' Opt. Commun. 44, 13-18 (1982).
[CrossRef]

Opt. Eng. (Bellingham)

S. S. Ul'yanov, D. A. Zimnyakov, and V. V. Tuchin, 'Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,' Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

C. Rodrigues and J. Pinto, 'Contrast of polychromatic speckle patterns and its dependence to surface height distribution,' Opt. Eng. (Bellingham) 42, 1699-1703 (2003).
[CrossRef]

Opt. Quantum Electron.

T. McKechnie, 'Image-plane speckle in partially coherent illumination,' Opt. Quantum Electron. 8, 61-67 (1976).
[CrossRef]

Proc. SPIE

T. S. Tkaczyk, K. W. Gossage, and J. K. Barton, 'Speckle image properties in optical coherence tomography,' Proc. SPIE 4619, 59-70 (2002).
[CrossRef]

Other

J. W. Goodman, Speckle Phenomena in Optics: Theory and Application, Version 5, August 2005 (work in progress available at http://www-ee.stanford.edu/~goodman).

G. Parry, 'Speckle pattern in partially coherent light,' in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), pp. 77-122.

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Figures (2)

Fig. 1
Fig. 1

Optical setup.

Fig. 2
Fig. 2

Theoretically calculated contrast C versus z eff for the SLD. The solid curve represents the approximation by Eq. (30) and the dashed curve is the numerical true integration from Eq. (22).

Equations (39)

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C = σ I I ,
σ I 2 = I 2 I 2 .
I ( x ) = 0 F ( k ) I ( x , k ) d k ,
I 2 ( x ) = 0 0 F ( k 1 ) F ( k 2 ) I ( x , k 1 ) I ( x , k 2 ) d k 1 d k 2 .
σ I 2 = I 2 I 2 = 0 0 F ( k 1 ) F ( k 2 ) [ I ( x , k 1 ) I ( x , k 2 ) I ( x , k 1 ) I ( x , k 2 ) ] d k 1 d k 2 .
J A ( x , k 1 , x , k 2 ) 2 = I ( x , k 1 ) I ( x , k 2 ) I ( x , k 1 ) I ( x , k 2 ) .
A ( x , k ) = exp ( i k R ) exp [ i φ ( ζ ) ] d S ,
J A ( x , k 1 , x , k 2 ) = A ( x , k 1 ) A * ( x , k 2 ) = exp [ i ( k 1 R 1 k 2 R 2 ) ] exp { i [ φ ( ζ 1 ) φ ( ζ 2 ) ] } d S 1 d S 2 .
exp { i [ φ ( ζ 1 ) φ ( ζ 2 ) ] } = exp { i [ k 1 Δ h ( ζ 1 ) k 2 Δ h ( ζ 2 ) ] } = exp [ i ( k 1 k 2 ) Δ h ( ζ 2 ) ] δ ( ζ 1 ζ 2 ) ,
J A ( x , k 1 , x , k 2 ) = exp [ i ( k 1 k 2 ) R ] exp [ i ( k 1 k 2 ) Δ h ( ζ ) ] d S .
exp [ i ( k 1 k 2 ) Δ h ( ζ ) ] = M Δ h ( k 1 k 2 ) = M Δ h ( Δ k ) ,
σ I 2 = I 2 I 2 = 0 0 F ( k 1 ) F ( k 2 ) M Δ h ( k 1 k 2 ) 2 exp [ i ( k 1 k 2 ) R ] d S 2 d k 1 d k 2 .
F ( k ) = F 0 exp [ ( k k 0 ) 2 2 σ k 2 ] ,
F ( k 1 ) F ( k 2 ) = F 0 2 exp [ ( k k 0 ) 2 σ k 2 Δ k 2 4 σ k 2 ] ,
σ I 2 = Δ k 2 Δ k 2 F 0 2 exp [ ( k k 0 ) 2 σ k 2 Δ k 2 4 σ k 2 ] M Δ h ( Δ k ) 2 e i Δ k R d S 2 d k d Δ k .
σ I 2 = F 0 2 π 1 2 σ k e ( Δ k 2 4 σ k 2 ) M Δ h ( Δ k ) 2 e i Δ k R d S 2 d Δ k .
M Δ h ( w ) = e ( σ Δ 2 w 2 2 ) ,
σ I 2 = F 0 2 π 1 2 σ k exp [ ( 1 4 σ k 2 + σ Δ 2 ) Δ k 2 ] e i Δ k ( R 2 R 1 ) d Δ k d S 1 d S 2 .
σ I 2 = 2 π F 0 2 σ k 2 1 + ( 2 σ k σ Δ ) 2 exp [ Δ R 2 4 ( 1 4 σ k 2 + σ Δ 2 ) ] d S 1 d S 2 .
I ( x , k ) = A ( x , k ) A * ( x , k ) = exp [ i ( k 1 k 2 ) R ] exp { i [ φ ( ζ 1 ) φ ( ζ 2 ) ] } d S 1 d S 2 = d S = π q 2 .
I ( x ) = 0 F ( k ) I ( x , k ) d k = π q 2 0 F 0 exp [ ( k k 0 ) 2 2 σ k 2 ] d k = π q 2 F 0 ( 2 π ) 1 2 σ k ,
I ( x ) 2 = 2 π 3 q 4 F 0 2 σ k 2 ,
C 2 = σ I 2 I 2 = 1 ( π q 2 ) 2 1 + ( 2 σ k σ Δ ) 2 exp ( Δ R 2 1 σ k 2 + 4 σ Δ 2 ) d S 1 d S 2 .
C = [ 1 + ( 2 σ k σ Δ ) 2 ] 1 4 .
C 2 = 1 ( π q 2 ) 2 1 + ( 2 σ k σ Δ ) 2 exp ( Δ R 2 1 σ k 2 + 4 σ Δ 2 ) r 1 r 2 d φ 1 d φ 2 d r 1 d r 2 = ( 2 π ) 2 ( π q 2 ) 2 1 + ( 2 σ k σ Δ ) 2 exp ( Δ R 2 1 σ k 2 + 4 σ Δ 2 ) r 1 r 2 d r 1 d r 2 .
C 2 = ( 2 π ) 2 ( π q 2 ) 2 1 + ( 2 σ k σ Δ ) 2 exp ( Δ R 2 1 σ k 2 + 4 σ Δ 2 ) ( ρ 2 ε 2 4 ) d ε d ρ .
Δ R 2 ε 2 ρ 2 ( 1 z + 1 z 0 ) 2 ,
C 2 = ( 2 π ) 2 ( π q 2 ) 2 1 + ( 2 σ k σ Δ ) 2 exp [ ε 2 ρ 2 ( z + z 0 ) 2 ( 1 σ k 2 + 4 σ 0 2 ) z 2 z 0 2 ] ( ρ 2 ε 2 4 ) d ε d ρ .
C 2 = 2 π 1 2 z 0 z q 2 σ k ( z 0 + z ) .
C = 1 N .
C 2 = π 1 2 ( 2 ln 2 ) 1 2 2 z 0 z L c q 2 ( z 0 + z ) = 0.664 2 z eff L c q 2 .
R z 1 = ( r 1 2 + z 2 ) 1 2 ,
R z 2 = ( r 2 2 + z 2 ) 1 2 ,
R 01 = ( r 1 2 + z 0 2 ) 1 2 ,
R 02 = ( r 2 2 + z 0 2 ) 1 2 .
( r 2 + z 2 ) 1 2 = z ( 1 + r 2 2 z 2 r 4 8 z 4 + ) .
( R z 2 R z 1 ) 2 = r 1 4 4 z 2 r 1 2 r 2 2 2 z 2 + r 2 4 4 z 2 + = 1 4 z 2 ( r 2 2 r 1 2 ) 2 + = ε 2 ρ 2 z 2 + ,
( R 02 R 01 ) 2 = r 1 4 4 z 0 2 r 1 2 r 2 2 2 z 0 2 + r 2 4 4 z 0 2 + = 1 4 z 0 2 ( r 2 2 r 1 2 ) 2 + = ε 2 ρ 2 z 0 2 + .
Δ R 2 = ε 2 ρ 2 z 2 + 2 ε ρ z ε ρ z 0 + ε 2 ρ 2 z 0 2 + ε 2 ρ 2 ( 1 z + 1 z 0 ) 2 .

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