Abstract

We discuss the schlieren imaging of quasi-sinusoidal phase objects. We demonstrate that, when the zero-order (Fourier) spatial component of the input image is not blocked by the schlieren-knife at the Fourier plane, the intensity distribution on the reconstructed image is a linear function of the phase amplitude. In contrast, if the zero order is completely blocked (i.e., dark Schlieren processing), the intensity distribution on the output image becomes essentially a quadratic function of the phase, and thus a direct phase retrieval is not possible. We discuss the possibility of contrast enhancement and present validation experiments.

© 2007 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. 8.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. L. Joannes, F. Dubois, and J.-C. Legros, "Phase-shifting schlieren: high resolution quantitative schlieren that uses the phase-shifting technique principle," Appl. Opt. 42, 5046-5053 (2003).
    [CrossRef] [PubMed]
  7. E. Garbusi, J. A. Ferrari, and C. D. Perciante, "Harmonic suppression and defect enhancement using schlieren processing," Appl. Opt. 44, 2963-2969 (2005).
    [CrossRef] [PubMed]
  8. J. Glückstad and P. C. Mogensen, "Optimal phase contrast in common-path interferometry," Appl. Opt. 40, 268-282 (2001).
    [CrossRef]
  9. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Academic, 2005).

2005

2004

2003

2001

1989

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, Tutorials in Fourier Optics (SPIE Press , 1989).

1967

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Academic, 2005).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. 8.

Brackenridge, J. B.

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, Tutorials in Fourier Optics (SPIE Press , 1989).

Dubois, F.

Ferrari, J. A.

Garbusi, E.

Glückstad, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Joannes, L.

Legros, J.-C.

Mogensen, P. C.

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, Tutorials in Fourier Optics (SPIE Press , 1989).

Perciante, C. D.

Peterka, J.

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, Tutorials in Fourier Optics (SPIE Press , 1989).

Stricker, J.

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, Tutorials in Fourier Optics (SPIE Press , 1989).

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Academic, 2005).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. 8.

Zakharin, B.

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Figures (4)

Fig. 1
Fig. 1

Schlieren setup with the knife border orthogonal to the x direction. PO, phase object to be tested; K, knife; C, CCD camera; and L 1,2 , Fourier lenses of focal length f.

Fig. 2
Fig. 2

Experimental setup to observe an ultrasonic pulse. AOM is an acousto-optic modulator that flashes the laser light when an electrical pulse is sent to the ultrasonic transducer.

Fig. 3
Fig. 3

Optical image of the ultrasonic pulse captured 16 μ s after its emission. (a) Bright-field schlieren method is used, (b) the same pulse is shown with the dark-field schlieren method.

Fig. 4
Fig. 4

Relative light intensity distributions along the direction of propagation of the ultrasonic pulse. (a) Bright-field schlieren imaging with maximum modulation intensity of | a | 0.3 . (b) Dark-field schlieren imaging with maximum modulation intensity of ∼0.1.

Equations (18)

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ϕ ( x , y ) = a ( x , y ) sin ( 2 π x / d ) ,
E ( x , y ) = exp [ i ϕ ( x , y ) ] = 1 + i ϕ ( x , y ) ϕ 2 ( x , y ) 2 + O ( ϕ 3 ) ,
E ( x , y ) = ( 1 a 2 ( x , y ) 4 ) + a ( x , y ) 2 exp ( i 2 π x / d ) + a 2 ( x , y ) 8 exp ( i 4 π x / d ) a ( x , y ) 2 exp ( i 2 π x / d ) + a 2 ( x , y ) 8 exp ( i 4 π x / d ) + O ( a 3 ) .
  E image ( x , y ) = ( 1 ε ) ( 1 a 2 ( x , y ) 4 ) + a ( x , y ) 2   exp ( i 2 π x / d ) + a 2 ( x , y ) 8   exp ( i 4 π x / d ) + O ( a 3 ) .
I ( x , y ) = ( 1 ε ) 2 ( 1 a 2 ( x , y ) 4 ) 2 + a 2 ( x , y ) 4 + ( 1 - ε ) a ( x , y ) cos ( 2 π x / d ) + ( 1 - ε ) a 2 ( x , y ) 4   cos ( 4 π x / d ) + a 3 ( x , y ) 8   cos ( 2 π x / d ) + ( 1 ε ) O ( a 3 ) .
I ( x , y ) 1 + a ( x , y ) cos ( 2 π x / d ) .
I ( x , y ) a 2 ( x , y ) 4 + a 3 ( x , y ) 8   cos ( 2 π x / d ) .
I ( x , y ) a 2 ( x 0 , y 0 ) 4 + a 2 ( x , y ) 4 + a ( x 0 , y 0 ) a ( x , y ) cos ( 2 π x / d ) 2 .
V = 2 a ( x 0 , y 0 ) a ( x , y ) a 2 ( x 0 , y 0 ) + a 2 ( x , y ) 1
E ( x , y ) = E 0   exp [ i a ( x , y ) sin ( 2 π x / d ) ] .
exp [ i a ( x , y ) sin ( 2 π x / d ) ] n = J n [ a ( x , y ) ] × exp [ i ( 2 π n x / d ) ] ,
E image ( x , y ) ( 1 ε ) J 0 [ a ( x , y ) ] + n = 1 J n [ a ( x , y ) ] exp [ i ( 2 π n x / d ) ] .
I ( x , y ) ( 1 ε ) 2 J 0 2 ( a ) + 2 ( 1 ε ) J 0 ( a ) × n = 1 J n ( a ) cos ( 2 π n x / d ) + n = 1 J n 2 ( a ) + 2 p = 1 [ n = 1 J n ( a ) J n + p ( a ) ] cos ( 2 π p x / d ) .
n = 1 J n 2 = 1 J 0 2 2 ,
I ( x , y ) 1 / 2 + [ ( 1 ε ) 2 ( 1 / 2 ) ] J 0 2 ( a ) + 2 ( 1 ε ) J 0 ( a ) n = 1 J n ( a )   cos ( 2 π n x / d ) + 2 p = 1 [ n = 1 J n ( a ) J n + p ( a ) ] cos ( 2 π p x / d ) .
J 0 ( a ) 1 a 2 / 4 , J 1 ( a ) a / 2 a 3 / 16 ,
J 2 ( a ) a 2 / 8 , J 3 ( a ) a 3 / 48 ,
J n ( a ) 0 for  n > 3.

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