Abstract

A hybrid optical/digital processing scheme for measuring phase distributions is described and demonstrated. It is intended to be an alternative to interferometric methods of measuring optical path length changes in flow diagnostics and can also be used as a flow visualization technique. The processing scheme enables one to make accurate measurements of phase at arbitrary points in the image plane. The system is based on a simple coherent optical Fourier processor but incorporates three separate measurements and postdetection digital processing to eliminate extraneous parts of the signal. The addition of a holographic filter to the system enables one to measure deformation or displacement of diffusely reflecting opaque objects. The technique is demonstrated by using it to visualize the flow of an expanding compressible gas jet and to measure the optical path length through a heated plume of air.

© 1983 Optical Society of America

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References

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  1. R. Dändliker, Prog. Opt. 17, 1 (1980).
    [CrossRef]
  2. J. B. Schemm, C. M. Vest, Appl. Opt. 22, 2850 (1983).
    [CrossRef] [PubMed]
  3. R. Dändliker, R. Thalmann, N. Brown, Opt. Commun. 42, 301 (1982).
    [CrossRef]
  4. P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
    [CrossRef]
  5. L. M. Frantz, A. A. Sawchuk, W. von der Ohe, Appl. Opt. 18, 3301 (1979).
    [CrossRef] [PubMed]
  6. R. A. Sprague, B. J. Thompson, Appl. Opt. 11, 1469 (1972).
    [CrossRef] [PubMed]
  7. R. L. Cody, “A Comparison of Various Coherent Optical Filtering Operations,” M.S. Thesis, U. Tennessee (1971).
  8. D. Kermisch, J. Op. Soc. Am. 65, 887 (1975).
    [CrossRef]
  9. B. A. Horwitz, Opt. Commun. 17, 231 (1976).
    [CrossRef]
  10. S. H. Lee, in Optical Information Processing Fundamentals, S. H. Lee, Ed. (Springer, Berlin, 1981), pp. 45–50.
  11. I. Prikryl, Opt. Acta 21, 675 (1974).
    [CrossRef]

1983 (1)

1982 (2)

R. Dändliker, R. Thalmann, N. Brown, Opt. Commun. 42, 301 (1982).
[CrossRef]

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

1980 (1)

R. Dändliker, Prog. Opt. 17, 1 (1980).
[CrossRef]

1979 (1)

1976 (1)

B. A. Horwitz, Opt. Commun. 17, 231 (1976).
[CrossRef]

1975 (1)

D. Kermisch, J. Op. Soc. Am. 65, 887 (1975).
[CrossRef]

1974 (1)

I. Prikryl, Opt. Acta 21, 675 (1974).
[CrossRef]

1972 (1)

Brown, N.

R. Dändliker, R. Thalmann, N. Brown, Opt. Commun. 42, 301 (1982).
[CrossRef]

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

Cody, R. L.

R. L. Cody, “A Comparison of Various Coherent Optical Filtering Operations,” M.S. Thesis, U. Tennessee (1971).

Dändliker, R.

R. Dändliker, R. Thalmann, N. Brown, Opt. Commun. 42, 301 (1982).
[CrossRef]

R. Dändliker, Prog. Opt. 17, 1 (1980).
[CrossRef]

Frantz, L. M.

Hariharan, P.

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

Horwitz, B. A.

B. A. Horwitz, Opt. Commun. 17, 231 (1976).
[CrossRef]

Kermisch, D.

D. Kermisch, J. Op. Soc. Am. 65, 887 (1975).
[CrossRef]

Lee, S. H.

S. H. Lee, in Optical Information Processing Fundamentals, S. H. Lee, Ed. (Springer, Berlin, 1981), pp. 45–50.

Oreb, B. F.

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

Prikryl, I.

I. Prikryl, Opt. Acta 21, 675 (1974).
[CrossRef]

Sawchuk, A. A.

Schemm, J. B.

Sprague, R. A.

Thalmann, R.

R. Dändliker, R. Thalmann, N. Brown, Opt. Commun. 42, 301 (1982).
[CrossRef]

Thompson, B. J.

Vest, C. M.

von der Ohe, W.

Appl. Opt. (3)

J. Op. Soc. Am. (1)

D. Kermisch, J. Op. Soc. Am. 65, 887 (1975).
[CrossRef]

Opt. Acta (1)

I. Prikryl, Opt. Acta 21, 675 (1974).
[CrossRef]

Opt. Commun. (3)

B. A. Horwitz, Opt. Commun. 17, 231 (1976).
[CrossRef]

R. Dändliker, R. Thalmann, N. Brown, Opt. Commun. 42, 301 (1982).
[CrossRef]

P. Hariharan, B. F. Oreb, N. Brown, Opt. Commun. 41, 393 (1982).
[CrossRef]

Prog. Opt. (1)

R. Dändliker, Prog. Opt. 17, 1 (1980).
[CrossRef]

Other (2)

S. H. Lee, in Optical Information Processing Fundamentals, S. H. Lee, Ed. (Springer, Berlin, 1981), pp. 45–50.

R. L. Cody, “A Comparison of Various Coherent Optical Filtering Operations,” M.S. Thesis, U. Tennessee (1971).

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Figures (5)

Fig. 1
Fig. 1

Optical processor. The filter is placed in the back focal plane of the transforming lens. In general, the output lens is aligned to form an image of the test region in the detector plane.

Fig. 2
Fig. 2

Processed image of a nozzle with no flow.

Fig. 3
Fig. 3

Processed image of a jet of nitrogen gas flowing from a nozzle. A composite grating filter is used to differentiate approximately in the horizontal direction.

Fig. 4
Fig. 4

Measured optical path length distributions across a heated plume of air. The dashed curve includes the effect of wave front tilt. The solid curve is the corrected distribution after wave front tilt has been removed.

Fig. 5
Fig. 5

Holographic system used when the object wave is diffused. The lens images the object onto the hologram plane.

Equations (36)

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u 0 ( x , y ) = g ( x , y ) exp [ i ϕ ( x , y ) ]
u ( X , Y ) = g ( X , Y ) exp { i [ ϕ ( X , Y ) + X ] }
d u ( X , Y ) = { d g ( X , Y ) + i g ( X , Y ) [ d ϕ ( X , Y ) + d X ] } exp [ i ϕ ( X , Y ) ] ,
d g ( X , Y ) = g ( X , Y ) X + g ( X , Y ) Y d Y ,
d ϕ ( X , Y ) = ϕ ( X , Y ) X d X + ϕ ( X , Y ) Y d Y .
I ( X , Y ) = [ d g ( X , Y ) ] 2 + g 2 ( X , Y ) [ d ϕ ( X , Y ) + d X ] 2
d ϕ ( X , Y ) = 1 2 ( 0 d X ) I 2 ( X , Y ) I 1 ( X , Y ) I 1 ( X , Y ) + I 2 ( X , Y ) 2 I 0 ( X , Y ) .
ϕ ( P ) = P 0 P d ϕ ( P ) + ϕ ( P 0 ) .
{ f ( X , Y ) / X } = iuF ( u , υ ) , { f ( X , Y ) / Y } = i υ F ( u , υ ) , }
F [ ( k ξ / f ) , ( k η / f ) ] ,
T ( k ξ / f , k η / f ) = C 2 ( k ξ / f + k η / f ) 2
θ = { 0 for ( k / f ) ( ξ + η ) 0 , π for ( k / f ) ( ξ + η ) < 0 ,
C 2 ( k ξ / f + k η / f ) max 2 = 1 .
d u ( X , Y ) i C [ u ( X , Y ) / X + u ( X , Y ) / Y ] , d X i C , d Y i C .
( d X ) 2 = ( d Y ) 2 = ( dXdY ) = C 2 .
Δ X = λ f ( Δ T ) / T 2 ,
δ ( X + Δ X , Y ) δ ( X , Y ) ,
u ( X + Δ X , Y ) u ( X , Y ) .
( u / X ) Δ X .
Δ u ( X , Y ) = { [ g ( X , Y ) + g X Δ X ] × exp [ [ i ( ϕ X Δ X + Δ X ) ] g ( X , Y ) } × exp { i [ ϕ ( X , Y ) ] } .
I ( X , Y ) = g 2 ( X , Y ) + [ g ( X , Y ) + g X Δ X ] 2 2 g ( X , Y ) [ g ( X , Y ) + g X Δ X ] × cos ( ϕ X Δ X + Δ X ) .
tan ( ϕ X Δ X ) = 1 cos ( 0 Δ X ) sin ( 0 Δ X ) I 2 ( X , Y ) I 1 ( X , Y ) I 1 ( X , Y ) + I 2 ( X , Y ) 2 I 0 ( X , Y ) .
ϕ ( X , Y ) X Δ X 1 2 ( 0 Δ X ) I 2 ( X , Y ) I 1 ( X , Y ) I 1 ( X , Y ) + I 2 ( X , Y ) 2 I 0 ( X , Y ) ,
tan [ ( ϕ / X ) Δ X ] ( Φ / X ) Δ X
π / 2 < ϕ X Δ X < π / 2 .
Δ X ( π / 2 ) ( | ϕ X | max ) 1
0 < 0 Δ X < π .
0 = | ϕ / X | max .
δ ( X + Δ X , Y ) + δ ( X , Y + Δ Y ) 2 δ ( X , Y ) .
sin ( R / f ) R / f > > λ / [ 2 ( Δ X 2 + Δ Y 2 ) 1 / 2 ] ,
u 0 ( x , y ) = a 0 exp [ i Φ ( x , y ) ] .
u R ( x , y ) = a R exp [ i ( α x + γ z ] .
t a = t a ( x , y ) + β ( exp [ i Φ ( x , y ) α x ] + exp { i [ Φ ( x , y ) α x ] } ) .
u 0 ( x , y ) = a 0 exp { i [ Φ ( x , y ) + ϕ ( x , y ) ] } ,
u 0 ( x , y ) t a ( x , y ) = a 0 exp [ i ( Φ + ϕ ) ] + β exp [ i ( 2 Φ + ϕ α x ) ] + β exp [ i ( ϕ + α x ) ] .
x = γ x / k , y = y .

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