Abstract

A technique to eliminate the lateral dispersion in the correlation signal from a holographic Vander Lugt filter is described. Both spatially coherent, and spatially noncoherent, object illumination are considered; and expressions for the color-corrected correlation intensity are written in each case. Experimental results of the correlation plane intensity are shown using laser and spatially noncoherent white-light illumination. The latter is seen to be useful to search automatically for object scale.

© 1980 Optical Society of America

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References

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  1. C. B. Burchardt, Bell Syst. Tech. J. 45, 1841 (1966).
  2. D. J. DeBitetto, Appl. Phys. Lett. 9, 417 (1966).
    [CrossRef]
  3. R. H. Katyl, Appl. Opt. 11, 1241 (1972).
    [CrossRef] [PubMed]
  4. J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
    [CrossRef]
  5. S. P. Almeida, S. K. Case, W. J. Dallas, Appl. Opt. 18, 4025 (1979).
    [CrossRef] [PubMed]
  6. G. M. Morris, N. George, Opt. Lett. 5, 202 (1980).
    [CrossRef] [PubMed]
  7. A. Lacourt, Opt. Commun. 27, 47 (1978).
    [CrossRef]
  8. H. O. Bartelt, Opt. Commun. 29, 37 (1979).
    [CrossRef]
  9. A. W. Lohmann, Appl. Opt. 7, 561 (1968).
    [CrossRef]
  10. A. W. Lohmann, H. W. Werlich, Appl. Opt. 10, 670 (1971).
    [CrossRef] [PubMed]
  11. S. Lowenthal, A. Werts, C. R. Acad. Sci. Ser. B: 266, 542 (1968).
  12. B. Watrasiewicz, Opt. Acta 16, 321 (1969).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, (Pergamon, New York, 1975), Chap. 10.
  14. E. Wolf, W. H. Carter, J. Opt. Soc. Am. 68, 953 (1978).
    [CrossRef]
  15. W. B. Davenport, W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 6, p. 106.
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  18. B. D. Guenther, C. R. Christensen, J. Upatnieks, IEEE J. Quantum Electron. QE-15, 1348 (1979).
    [CrossRef]
  19. A. Vander Lugt, Appl. Opt. 5, 1760 (1966).
    [CrossRef]
  20. R. E. Williams, IEEE Trans. Inf. Theory IT-10, 227 (1964).
    [CrossRef]
  21. K. Hinch, Opt. Acta 24, 1027 (1977).
    [CrossRef]
  22. R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience, New York, 1953), p. 49.

1980

1979

H. O. Bartelt, Opt. Commun. 29, 37 (1979).
[CrossRef]

S. P. Almeida, S. K. Case, W. J. Dallas, Appl. Opt. 18, 4025 (1979).
[CrossRef] [PubMed]

B. D. Guenther, C. R. Christensen, J. Upatnieks, IEEE J. Quantum Electron. QE-15, 1348 (1979).
[CrossRef]

1978

E. Wolf, W. H. Carter, J. Opt. Soc. Am. 68, 953 (1978).
[CrossRef]

A. Lacourt, Opt. Commun. 27, 47 (1978).
[CrossRef]

J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
[CrossRef]

1977

K. Hinch, Opt. Acta 24, 1027 (1977).
[CrossRef]

1972

1971

1969

B. Watrasiewicz, Opt. Acta 16, 321 (1969).
[CrossRef]

1968

S. Lowenthal, A. Werts, C. R. Acad. Sci. Ser. B: 266, 542 (1968).

A. W. Lohmann, Appl. Opt. 7, 561 (1968).
[CrossRef]

1966

C. B. Burchardt, Bell Syst. Tech. J. 45, 1841 (1966).

D. J. DeBitetto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

A. Vander Lugt, Appl. Opt. 5, 1760 (1966).
[CrossRef]

1964

R. E. Williams, IEEE Trans. Inf. Theory IT-10, 227 (1964).
[CrossRef]

Almeida, S. P.

Bartelt, H. O.

H. O. Bartelt, Opt. Commun. 29, 37 (1979).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, (Pergamon, New York, 1975), Chap. 10.

Burchardt, C. B.

C. B. Burchardt, Bell Syst. Tech. J. 45, 1841 (1966).

Carter, W. H.

Case, S. K.

Christensen, C. R.

B. D. Guenther, C. R. Christensen, J. Upatnieks, IEEE J. Quantum Electron. QE-15, 1348 (1979).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience, New York, 1953), p. 49.

Dallas, W. J.

Davenport, W. B.

W. B. Davenport, W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 6, p. 106.

DeBitetto, D. J.

D. J. DeBitetto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

Gazeu, R.

J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
[CrossRef]

George, N.

Goedgebuer, J. P.

J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Guenther, B. D.

B. D. Guenther, C. R. Christensen, J. Upatnieks, IEEE J. Quantum Electron. QE-15, 1348 (1979).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience, New York, 1953), p. 49.

Hinch, K.

K. Hinch, Opt. Acta 24, 1027 (1977).
[CrossRef]

Katyl, R. H.

Lacourt, A.

A. Lacourt, Opt. Commun. 27, 47 (1978).
[CrossRef]

Lohmann, A. W.

Lowenthal, S.

S. Lowenthal, A. Werts, C. R. Acad. Sci. Ser. B: 266, 542 (1968).

Morris, G. M.

Root, W. L.

W. B. Davenport, W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 6, p. 106.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Upatnieks, J.

B. D. Guenther, C. R. Christensen, J. Upatnieks, IEEE J. Quantum Electron. QE-15, 1348 (1979).
[CrossRef]

Vander Lugt, A.

Watrasiewicz, B.

B. Watrasiewicz, Opt. Acta 16, 321 (1969).
[CrossRef]

Werlich, H. W.

Werts, A.

S. Lowenthal, A. Werts, C. R. Acad. Sci. Ser. B: 266, 542 (1968).

Williams, R. E.

R. E. Williams, IEEE Trans. Inf. Theory IT-10, 227 (1964).
[CrossRef]

Wolf, E.

E. Wolf, W. H. Carter, J. Opt. Soc. Am. 68, 953 (1978).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, (Pergamon, New York, 1975), Chap. 10.

Appl. Opt.

Appl. Phys. Lett.

D. J. DeBitetto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

Bell Syst. Tech. J.

C. B. Burchardt, Bell Syst. Tech. J. 45, 1841 (1966).

C. R. Acad. Sci. Ser. B

S. Lowenthal, A. Werts, C. R. Acad. Sci. Ser. B: 266, 542 (1968).

IEEE J. Quantum Electron.

B. D. Guenther, C. R. Christensen, J. Upatnieks, IEEE J. Quantum Electron. QE-15, 1348 (1979).
[CrossRef]

IEEE Trans. Inf. Theory

R. E. Williams, IEEE Trans. Inf. Theory IT-10, 227 (1964).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

B. Watrasiewicz, Opt. Acta 16, 321 (1969).
[CrossRef]

K. Hinch, Opt. Acta 24, 1027 (1977).
[CrossRef]

Opt. Commun.

J. P. Goedgebuer, R. Gazeu, Opt. Commun. 27, 53 (1978).
[CrossRef]

A. Lacourt, Opt. Commun. 27, 47 (1978).
[CrossRef]

H. O. Bartelt, Opt. Commun. 29, 37 (1979).
[CrossRef]

Opt. Lett.

Other

M. Born, E. Wolf, Principles of Optics, (Pergamon, New York, 1975), Chap. 10.

W. B. Davenport, W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 6, p. 106.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience, New York, 1953), p. 49.

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Figures (5)

Fig. 1
Fig. 1

(a) Concept of the dispersion-compensation system; (b) matched filter setup consisting of object plane, L1 transform lens, matched filter plane, L2-L3 imaging lens pair, limiting aperture to block direct beam, compensation grating, L4 lens, and output or correlation plane.

Fig. 2
Fig. 2

Intensity of the correlation peak vs wavelength with λ0 = 514.5 nm. Correlation is for currency recognition using collimated illumination from an argon laser.

Fig. 3
Fig. 3

Correlation peak for a two-dollar transparency bill with its corresponding matched filter: (a) with object scale M = 1.0, and (b) an enlarged object magnified by M = 1.15.

Fig. 4
Fig. 4

Intensity of the correlation signal as a function of wavelength when the illumination is spectrally broad. Effect of an input object that is larger than that used to make the matched filter is shown, i.e., the correlation signal shifts to a longer wavelength.

Fig. 5
Fig. 5

Correlation output using spectrally broad illumination with a low spatial coherence length of 14 μm and the same matched filter as that for Fig. 3. Peaks of different color are seen for (a) object scale M = 1.0 and (b) a magnification factor M = 1.15. Color shift is as shown in Fig. 4.

Equations (40)

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E III ( x 3 , y 3 ; ν ) = i [ exp ( i 4 π F / λ ) / ( λ F ) ] d x 2 d y 2 E II ( x 2 , y 2 ; ν ) × exp [ i 2 π ( x 3 x 2 + y 3 y 2 ) / ( λ F ) ] ,
S ( x 2 , y 2 ; λ ) = i [ exp ( i 4 π F / λ ) / ( λ F ) ] d x 1 d y 1 s ( x 1 , y 1 ) × exp [ i 2 π ( x 2 x 1 + y 2 y 1 ) / ( λ F ) ] .
T * ( x 2 , y 2 ; λ 0 ) = i [ exp ( i 4 π F / λ 0 ) / ( λ 0 F ) ] d x 1 d y 1 t * ( x 1 , y 1 ) × exp [ + i 2 π ( x 2 x 1 + y 2 y 1 ) / ( λ 0 F ) ] .
E II ( x 2 , y 2 ; ν ) = exp ( i 2 π x 2 sin θ 0 / λ 0 ) S ( x 2 , y 2 , λ ) T * ( x 2 , y 2 , λ 0 ) .
E III ( x 3 , y 3 ; ν ) = i { exp [ i 4 π F ( 2 / λ 1 / λ 0 ) ] / ( λ 0 F ) } × d x 1 d y 1 t * ( x 1 , y 1 ) × s [ x 1 λ / λ 0 + F sin θ 0 ( λ / λ 0 ) x 3 ; y 1 λ / λ 0 y 3 ] .
G ( x 4 , y 4 0 ) = exp ( i 2 π x 4 sin θ 0 / λ 0 ) .
E IV ( x 4 , y 4 ; ν ) = exp ( i 8 π F / λ ) S ( x 4 , y 4 ; λ ) T * ( x 4 , y 4 ; λ 0 ) .
E V ( x 5 , y 5 ; ν ) = ( i / λ 0 F ) exp [ i 4 π F ( 4 / λ 1 / λ 0 ) ] × d x 1 d y 1 t * ( x 1 , y 1 ) × s ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ) .
E V ( x 5 , y 5 ; ν ) = d x 1 d y 1 s ( x 1 , y 1 ) h ( x 5 , y 5 ; x 1 , y 1 ) ,
h ( x 5 , y 5 ; x 1 , y 1 ) = [ i λ 0 / ( λ 2 F ) ] e [ i 4 π F ( 4 / λ 1 / λ 0 ) ] × t * [ ( λ 0 / λ ) ( x 1 x 5 ) , ( λ 0 / λ ) ( y 1 y 5 ) ] .
U V ( x 5 , y 5 ; ν ) = | d x 1 d y 1 s ( x 1 , y 1 ) h ( x 5 , y 5 ; x 1 , y 1 ) | 2 ;
U V ( x 5 , y 5 ; ν ) = [ 1 / ( λ 0 F ) 2 ] | d x 1 d y 1 t * ( x 1 , y 1 ) × s ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ) | 2 .
E I ( x , y ; ν ) = T T d t e I ( x , y , t ) exp ( + i 2 π ν t ) .
e I ( x , y , t ) rect ( t T ) = d v E I ( x , y ; ν ) exp ( i 2 π ν t ) .
Γ 12 ( τ ) = e I ( x , y , t ) e * I ( x , y , t τ ) s
U 12 ( ν ) = d τ Γ 12 ( τ ) exp ( i 2 π ν τ ) ,
Γ 12 ( τ ) = d ν U 12 ( ν ) exp ( i 2 π ν τ ) .
U 12 ( ν ) = lim T E I ( x , y ; ν ) E * I ( x , y ; ν ) s 2 T ,
U 12 ( ν ) = E 1 ( x , y ; ν ) E * I ( x , y ; ν ) ,
E 1 ( x 1 , y 1 ; ν ) = E in ( x 1 , y 1 ; ν ) s ( x 1 , y 1 ) .
E V ( x 5 , y 5 ; ν ) = [ i / ( λ 0 F ) ] exp [ i 4 π F ( 4 / λ 1 / λ 0 ) ] × d x 1 d y 1 t * ( x 1 , y 1 ) × s ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ) × E in ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ; ν ) .
U V ( x 5 , y 5 ; ν ) = [ 1 / ( λ 0 F ) 2 ] −∞ × d x 1 d y 1 d x 1 d y 1 t * ( x 1 , y 1 ) t ( x 1 , y 1 ) × s ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ) × s * ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ) × E in ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ; ν ) × E * in ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ; ν ) .
E in ( x 1 , y 1 ; ν ) E * in ( x 1 , y 1 ; ν ) = κ U in ( x 1 , y 1 ; ν ) δ ( x 1 x 1 , y 1 y 1 ) ,
U V ( x 5 , y 5 ; ν ) = κ [ 1 / ( λ F ) 2 ] d x 1 d y 1 | t ( x 1 , y 1 ) | 2 × | E I ( x 1 λ / λ 0 + x 5 , y 1 λ / λ 0 + y 5 ; ν ) | 2 ,
U V ( x 5 , y 5 ; ν ) = κ [ λ 0 / ( λ 2 F ) ] 2 d ξ d η × | t [ ( ξ x 5 ) λ 0 / λ , ( η y 5 ) λ 0 / λ ] | 2 | E I ( ξ, η ; ν ) | 2 .
ϒ i ( x 5 , y 5 ; x 1 , y 1 ) = κ | h ( x 5 , y 5 ; x 1 , y 1 ) | 2 ,
ϒ i ( x 5 , y 5 ; ξ, η ) = k [ λ 0 / ( λ 2 F ) ] 2 | t [ ( ξ x 5 ) λ 0 / λ , ( η y 5 ) λ 0 / λ ] | 2 .
E in ( x 1 , y 1 ; ν ) E * in ( x 1 , y 1 ; ν ) = | E in ( x 1 , y 1 ; ν ) | 2 W ( Δ x 1 , Δ y 1 ; ν ) .
W ( Δ x 1 , Δ y 1 ; ν ) = [ 2 J 1 ( χ ) ] / χ
Δ x 1 = x 1 x 1 , Δ y 1 = y 1 y 1 , χ = [ π d / ( λ z 0 ) ] ( Δ x 1 2 + Δ y 1 2 ) 1 / 2 .
U V ( x 5 , y 5 ; v ) [ 1 / ( λ 0 F ) 2 ] d x 1 d y 1 t * ( x 1 , y 1 ) × s ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ) × | E i n ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ; ν ) | 2 × d Δ x 1 d Δ y 1 t ( x 1 + Δ x 1 , y 1 + Δ y 1 ) × s * [ λ ( x 1 + Δ x 1 ) / λ 0 + x 5 , λ ( y 1 + Δ y 1 ) / λ 0 + y 5 ] × W ( λ Δ x 1 / λ 0 , λ Δ y 1 / λ 0 ; ν ) .
t ( x 1 + Δ x 1 , y 1 + Δ y 1 ) t ( x 1 , y 1 )
s * [ λ ( x 1 + Δ x 1 ) / λ 0 + x 5 , λ ( y 1 + Δ y 1 ) / λ 0 + y 5 ) s * ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 )
U V ( x 5 , y 5 ; ν ) [ 1 / ( λ F ) 2 ] d x 1 d y 1 | t ( x 1 , y 1 ) | 2 × | E I ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ; ν ) | 2 × d Δ x 1 d Δ y 1 W ( Δ x 1 , Δ y 1 ; ν )
U V ( x 5 , y 5 ; ν ) 2 π ( z 0 F d ) 2 { 1 J 0 [ π d 0 / ( λ z 0 ) ] } × d x 1 d y 1 | t ( x 1 , y 1 ) | 2 × | E I ( λ x 1 / λ 0 + x 5 , λ y 1 / λ 0 + y 5 ; ν ) | 2
κ = ( 2 / π ) ( λ z 0 / d ) 2 { 1 J 0 [ π d 0 / ( λ z 0 ) ] } ,
κ = 1.46 0 2 , with 0 = 0.32 λ z 0 / d .
U V ( 0,0 ; ν ) = [ 1 / ( λ 0 F ) 2 ] | d x 1 d y 1 t * ( x 1 , y 1 ) t ( x 1 λ / λ 0 , y 1 λ / λ 0 ) | 2 .
U V ( 0,0 ; ν ) 2 π ( z 0 F d ) 2 { 1 J 0 [ π d 0 / ( λ z 0 ) ] } × d x 1 d y 1 | t ( x 1 , y 1 ) | 2 | t [ x 1 λ / ( λ 0 M ) , y 1 λ / ( λ 0 M ) ] | 2 .
d x 1 d y 1 | t ( x 1 , y 1 ) | 2 | t [ x 1 λ / ( λ 0 M ) , y 1 λ / ( λ 0 M ) ] | 2 ( M λ 0 / λ ) d x 1 d y 1 | t ( x 1 , y 1 ) | 4 ,

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