Abstract

Correlation properties of speckle fields at the output of quadratic phase systems with hard square and circular apertures are examined. Using the linear canonical transform and ABCD ray matrix techniques to describe these general optical systems, we first derive analytical formulas for determining axial and lateral speckle sizes. Then using a numerical technique, we extend the analysis so that the correlation properties of nonaxial speckles can also be considered. Using some simple optical systems as examples, we demonstrate how this approach may be conveniently applied. The results of this analysis apply broadly both to the design of metrology systems and to speckle control schemes.

© 2009 Optical Society of America

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References

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  1. P. K. Rastogi, “Techniques of displacement and deformation Measurements in speckle metrology,” in Speckle Metrology, R.S.Sirohi, ed. (Marcel Dekker, 1993), pp. xx-xx.
  2. H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271-274 (1972).
    [CrossRef]
  3. J. T. Sheridan and R. Patten, “Holographic interferometry and the fractional Fourier transformation,” Opt. Lett. 25, 448-450 (2000).
    [CrossRef]
  4. D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. 44, 2720-2727 (2005).
    [CrossRef] [PubMed]
  5. B. M. Hennelly, D. P. Kelly, J. E. Ward, R. Patten, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
    [CrossRef]
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  14. H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16, 1402-1412 (1999).
    [CrossRef]
  15. D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394-3396 (2007).
    [CrossRef] [PubMed]
  16. J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. XIV, E.Wolf, ed. (North-Holland, Amsterdam, 1976).
  17. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).
  18. B. R. A. Nijboer, “The diffraction theory of optical aberrations: II. Diffraction pattern in the presence of small aberrations,” Physica (Utrecht) 13, 605-620 (1947).
    [CrossRef]
  19. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  20. T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A 10, 324-328 (1993).
    [CrossRef]
  21. E. W. Weisstein, “ANOVA,” in MathWorld, A Wolfram Web Resource; http://mathworld.wolfram.com/ANOVA.html (July 2009).

2007

2006

B. M. Hennelly, D. P. Kelly, J. E. Ward, R. Patten, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

2005

2003

2000

1999

1997

1995

1994

S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 9, 1801-1803 (1994).
[CrossRef]

1993

1990

1972

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271-274 (1972).
[CrossRef]

1970

1965

1947

B. R. A. Nijboer, “The diffraction theory of optical aberrations: II. Diffraction pattern in the presence of small aberrations,” Physica (Utrecht) 13, 605-620 (1947).
[CrossRef]

Abe, S.

S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 9, 1801-1803 (1994).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Collins, S. A.

Dainty, J. C.

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. XIV, E.Wolf, ed. (North-Holland, Amsterdam, 1976).

Fricke-Begemann, T.

Goldfischer, L. I.

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Gopinathan, U.

D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394-3396 (2007).
[CrossRef] [PubMed]

B. M. Hennelly, D. P. Kelly, J. E. Ward, R. Patten, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

Hansen, R. S.

Hanson, S. G.

Hennelly, B. M.

B. M. Hennelly, D. P. Kelly, J. E. Ward, R. Patten, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. 44, 2720-2727 (2005).
[CrossRef] [PubMed]

Iwamoto, S.

Kelly, D. P.

Kirchner, M.

Leushacke, L.

Liyan, Z.

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of optical aberrations: II. Diffraction pattern in the presence of small aberrations,” Physica (Utrecht) 13, 605-620 (1947).
[CrossRef]

O'Neill, F. T.

B. M. Hennelly, D. P. Kelly, J. E. Ward, R. Patten, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

Patten, R.

B. M. Hennelly, D. P. Kelly, J. E. Ward, R. Patten, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

J. T. Sheridan and R. Patten, “Holographic interferometry and the fractional Fourier transformation,” Opt. Lett. 25, 448-450 (2000).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, “Techniques of displacement and deformation Measurements in speckle metrology,” in Speckle Metrology, R.S.Sirohi, ed. (Marcel Dekker, 1993), pp. xx-xx.

Rose, B.

Sheridan, J. T.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Tiziani, H.

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271-274 (1972).
[CrossRef]

Ward, J. E.

D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394-3396 (2007).
[CrossRef] [PubMed]

B. M. Hennelly, D. P. Kelly, J. E. Ward, R. Patten, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

Weisstein, E. W.

E. W. Weisstein, “ANOVA,” in MathWorld, A Wolfram Web Resource; http://mathworld.wolfram.com/ANOVA.html (July 2009).

Yamahai, K.

Yoshimura, T.

Yura, H. T.

Zhou, M.

Appl. Opt.

J. Mod. Opt.

B. M. Hennelly, D. P. Kelly, J. E. Ward, R. Patten, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271-274 (1972).
[CrossRef]

Opt. Lett.

Physica (Utrecht)

B. R. A. Nijboer, “The diffraction theory of optical aberrations: II. Diffraction pattern in the presence of small aberrations,” Physica (Utrecht) 13, 605-620 (1947).
[CrossRef]

Other

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

P. K. Rastogi, “Techniques of displacement and deformation Measurements in speckle metrology,” in Speckle Metrology, R.S.Sirohi, ed. (Marcel Dekker, 1993), pp. xx-xx.

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. XIV, E.Wolf, ed. (North-Holland, Amsterdam, 1976).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

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

E. W. Weisstein, “ANOVA,” in MathWorld, A Wolfram Web Resource; http://mathworld.wolfram.com/ANOVA.html (July 2009).

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Figures (7)

Fig. 1
Fig. 1

Contour plot of μ I ( A B C D 1 , A B C D 2 ) for the square aperture case, where the BOC marks the location of the first minima parameterized in terms of a and b.

Fig. 2
Fig. 2

Plot of the BOC for the square aperture case, showing the range of a values for which each segment of the BOC curve is valid.

Fig. 3
Fig. 3

Schematic representation of (a) a free space optical system, (b) a single-lens system.

Fig. 4
Fig. 4

Contour plot of μ I ( A B C D 1 , A B C D 2 ) parameterized in terms of p and q, showing the BOC for the circular aperture case. Dashes in an oval mark the half-maximum contour.

Fig. 5
Fig. 5

BOC for the circular aperture case, showing the range of p values for which each segment of the BOC is valid.

Fig. 6
Fig. 6

Longitudinal speckle size ϵ z as a function of displacement x from the optical axis for a square aperture FST and a single-lens system.

Fig. 7
Fig. 7

Longitudinal speckle size ϵ z as a function of displacement r from the optical axis for a circular aperture FST and a single lens optical system.

Tables (4)

Tables Icon

Table 1 Equations Describing the Fits to the Various Segments of the BOC, with the Associated Ranges and Mean-Square-Error Values

Tables Icon

Table 2 a and b Values for Some Commonly Used Optical Systems That Are Laterally Stationary a

Tables Icon

Table 3 Equations Describing the Fits to the Various Segments of the BOC, with the Associated Ranges and Mean-Square-Error Values for the Circular Aperture Case

Tables Icon

Table 4 p and q Values for Some Commonly Used Optical Systems That Are Laterally Stationary a

Equations (52)

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

u ( x , y ) = LCT { u ( x 0 , y 0 ) } ( x , y ) = 1 j λ B u ( x 0 , y 0 ) p ( x 0 , y 0 ) exp [ j π λ B ( D x 2 2 x x 0 + A x 0 2 ) ] × exp [ j π λ B ( D y 2 2 y y 0 + A y 0 2 ) ] d x 0 d y 0 ,
[ 1 z 0 1 ] ,
[ 1 0 1 f 1 ] ,
[ 0 1 1 0 ] ,
R ( x , x ̃ , A B C D 1 , A B C D 2 ) = I A B C D 1 ( x ) I A B C D 2 ( x ̃ ) + | J A ( x , x ̃ , A B C D 1 , A B C D 2 ) | 2 ,
J A ( x , x ̃ , A B C D 1 , A B C D 2 ) = u A B C D 1 ( x ) u ABCD 2 * ( x ̃ ) ,
1 j λ B 1 B 2 p 2 ( x 0 ) u ( x 0 ) u * ( x ̃ 0 ) exp [ j π λ B 1 ( D 1 x 2 2 x x 0 + A 1 x 0 2 ) ] , × exp [ j π λ B 2 ( D 2 x ̃ 2 2 x ̃ x ̃ 0 + A 2 x ̃ 0 2 ) ] d x 0 d x ̃ 0 ,
C exp [ j π λ ( D 1 B 1 x 2 D 2 B 2 x ̃ 2 ) ] j λ B 1 B 2 p 2 ( x 0 ) exp { j π λ [ ( x ̃ B 2 x B 1 ) 2 x 0 + ( A 1 B 1 A 2 B 2 ) x 0 2 ] } d x 0 .
μ I ( x , x ̃ , A B C D 1 , A B C D 2 ) = | J A ( x , x ̃ , A B C D 1 , A B C D 2 ) J A ( x , x , A B C D 1 , A B C D 1 ) J A ( x ̃ , x ̃ , A B C D 2 , A B C D 2 ) | 2 .
| = p 2 ( x 0 ) exp [ j ( α x 0 + τ x 0 2 ) ] d x 0 = p 2 ( x 0 ) d x 0 | 2 ,
α = 2 π λ ( x ̃ B 2 x B 1 ) , τ = π λ ( A 1 B 1 A 2 B 2 ) .
p ( x 0 ) = { 1 | x 0 | L 2 0 , otherwise } ,
μ I ( x , x ̃ , A B C D 1 , A B C D 2 ) = | 1 L L 2 L 2 exp [ j ( α x 0 + τ x 0 2 ) ] d x 0 | 2 .
μ I ( x , x ̃ ) = | 1 L L 2 L 2 exp ( j α x 0 ) d x 0 | 2 = | 2 sin ( L α 2 ) L α | 2 .
L 2 2 π λ B ( x ̃ x ) = π .
ϵ x = x ̃ x = λ B L .
μ I ( A B C D 1 , A B C D 2 ) = | 1 L π 2 τ exp [ j ( α 2 4 τ ) ] α 2 τ π L τ 2 π α 2 τ π + L τ 2 π exp ( j π t 2 2 ) d t | 2 .
| 1 2 b exp [ j ( π a 2 2 ) ] [ C ( a + b ) C ( a b ) + j S ( a + b ) j S ( a b ) ] | 2 ,
1 | 2 b | 2 { [ C ( a + b ) C ( a b ) ] 2 + [ S ( a + b ) S ( a b ) ] 2 } .
C 2 ( b ) + S 2 ( b ) | b 2 | ,
A 1 B 1 A 2 B 2 = 7.31 λ L 2 .
p ( r 0 ) = { 1 , | r 0 | D 0 2 0 , otherwise } ,
μ I ( x , x ̃ , y , y ̃ , A B C D 1 , A B C D 2 ) = | p 2 ( x 0 , y 0 ) exp { j [ ( α x x 0 + α y y 0 ) + τ ( x 0 2 + y 0 2 ) ] } d x 0 d y 0 p 2 ( x 0 , y 0 ) e d x 0 d y 0 | 2 ,
α x = 2 π λ ( x ̃ B 2 x B 1 ) , α y = 2 π λ ( y ̃ B 2 y B 1 ) .
r 0 = x 0 2 + y 0 2 , Ω = α x 2 + α y 2 = 2 π λ ( r ̃ B 2 r B 1 ) ,
μ I ( r , r ̃ , A B C D 1 , A B C D 2 ) = | 0 0 2 π p 2 ( r 0 ) exp { j [ Ω r 0 ( cos θ cos ϕ + sin θ sin ϕ ) + τ r 0 2 ] } r 0 d θ d r 0 0 0 2 π p 2 ( r 0 ) r 0 d θ d r | 2 .
| 0 D 0 2 0 2 π exp [ j Ω r 0 cos ( θ ϕ ) ] exp [ j τ r 0 2 ] r 0 d θ d r 0 0 D 0 2 0 2 π r 0 d θ d r | 2 .
| 1 π 0 1 0 2 π exp [ j Ω D 0 r N 2 cos ( θ ϕ ) ] exp [ j τ D 0 2 r N 2 4 ] r N d θ d r N | 2 = | 0 1 J 0 ( q r N ) exp [ j p r N 2 ] r N d r N | 2 ,
μ I ( r , r ̃ ) = | 2 0 1 J 0 ( q r N ) r N d r N | 2 = | 2 J 1 ( q ) q | 2 ,
D 0 π λ B ( r ̃ r ) = 3.382 .
ϵ r = r ̃ r = 1.22 λ B D 0 .
μ I ( A B C D 1 , A B C D 2 ) = | 2 exp ( j p 2 ) π p n = 0 ( 2 n + 1 ) ( j ) n J n + 1 2 ( p 2 ) J 2 n + 1 ( q ) q | 2 .
lim q 0 [ J 2 n + 1 ( q ) q ] = { 1 2 , n = 0 0 , n > 0 } .
| π p J 1 2 ( p 2 ) | 2 = | sinc ( p 2 ) | 2 ,
A 1 B 1 A 2 B 2 = 8 λ D 0 2 .
a = x 2 ( λ B 1 B 2 ) ( B 1 B 2 ) B 2 A 1 A 2 B 1 ,
b = L 1 2 λ ( A 1 B 1 A 2 B 2 ) .
p = D 0 2 π 4 λ ( A 2 B 2 A 1 B 1 ) ,
q = D 0 π r λ ( 1 B 2 1 B 1 ) .
[ A 2 B 2 C 2 D 2 ] = [ 1 z + Δ z 0 1 ] .
ϵ x = λ z L .
Δ z z ( z + Δ z ) = 7.31 λ L 2 .
ϵ z = Δ z = 7.31 λ ( z L ) 2 .
ϵ r = 1.22 λ z D 0 .
Δ z z ( z + Δ z ) = 8 λ D 0 2 ,
ϵ z = Δ z = 8 λ ( z D 0 ) 2 ,
A B C D 1 = [ 1 z 2 f z 2 + z 1 ( 1 z 2 f ) 1 f 1 z 1 f ] ,
A B C D 2 = [ 1 z 2 + Δ z f z 2 + Δ z + z 1 ( 1 z 2 + Δ z f ) 1 f 1 z 1 f ] .
ϵ x = λ L [ z 2 + z 1 ( 1 z 2 f ) ]
ϵ z = Δ z = 7.31 λ [ f ( z 1 + z 2 ) z 1 z 2 f L ] 2 .
ϵ r = 1.22 λ D 0 [ z 2 + z 1 ( 1 z 2 f ) ]
ϵ z = Δ z = 8 λ [ f ( z 1 + z 2 ) z 1 z 2 D 0 f ] 2 .

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