Abstract

A method for producing Fourier transform holograms in spatially coherent but temporally broadband light is presented. The technique employs a broadband Fourier transforming system incorporated into an achromatic grating interferometer. The theory has been experimentally verified by producing a Fourier transform hologram in temporally broadband light from a high pressure Hg arc lamp.

© 1981 Optical Society of America

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References

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  1. E. N. Leith, J. A. Roth, Appl. Opt. 16, 2565 (1977).
    [CrossRef] [PubMed]
  2. E. N. Leith, J. A. Roth, Appl. Opt. 18, 2803 (1979).
    [CrossRef] [PubMed]
  3. J. A. Roth, Ph.D. Thesis, Michigan (1981), available from University Microfilms.
  4. E. N. Leith, G. J. Swanson, Appl. Opt. 19, 638 (1980).
    [CrossRef] [PubMed]
  5. G. M. Morris, N. George, Opt. Lett. 5, 446 (1980).
    [CrossRef] [PubMed]
  6. G. M. Morris, N. George, Opt. Lett. 5, 446 (1980).
    [CrossRef] [PubMed]
  7. G. M. Morris, N. George, Appl. Opt. 19, 3843 (1980).
    [CrossRef] [PubMed]
  8. R. H. Katyl, Appl. Opt. 4, 1241, 1248, 1255 (1972).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 7.
  10. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 207.
  11. A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
    [CrossRef]
  12. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 323.
  13. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 200.
  14. E. N. Leith, U. Michigan; private communication.
  15. J. Upatnieks, Environmental Research Institute of Michigan; private communication.
  16. J. R. Fienup, C. D. Leonard, Appl. Opt. 18, 631 (1979).
    [CrossRef] [PubMed]

1980 (4)

1979 (2)

1977 (1)

1972 (1)

R. H. Katyl, Appl. Opt. 4, 1241, 1248, 1255 (1972).
[CrossRef]

1964 (1)

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 207.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 207.

Fienup, J. R.

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 7.

Katyl, R. H.

R. H. Katyl, Appl. Opt. 4, 1241, 1248, 1255 (1972).
[CrossRef]

Leith, E. N.

Leonard, C. D.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 207.

Morris, G. M.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 323.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 200.

Roth, J. A.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 323.

Swanson, G. J.

Upatnieks, J.

J. Upatnieks, Environmental Research Institute of Michigan; private communication.

Vander Lugt, A.

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Inf. Theory (1)

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

Opt. Lett. (2)

Other (7)

J. A. Roth, Ph.D. Thesis, Michigan (1981), available from University Microfilms.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 7.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 207.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 323.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 200.

E. N. Leith, U. Michigan; private communication.

J. Upatnieks, Environmental Research Institute of Michigan; private communication.

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Figures (8)

Fig. 1
Fig. 1

AFT is an achromatic Fourier transforming system. Input light is derived from a polychromatic point source. Õ(·) is defined to be the Fourier transform of O(·).

Fig. 2
Fig. 2

Exosystems for Fourier transform holography in monochromatic light: (a) grating and (b) prism.

Fig. 3
Fig. 3

VanderLugt method of producing a Fourier transform hologram.

Fig. 4
Fig. 4

Chirp z transform method of producing a Fourier transform. λ0 and FI are constants. Impulse response of the linear filter is exp(jπx20FI).

Fig. 5
Fig. 5

Standard coherent optical processor configuration.

Fig. 6
Fig. 6

Conceptual system used for analysis, G1, G2, and G3 are linear gratings of spatial frequency fc. L1 and L2 are achromats of focal length F. R is a reference plane in the reference beam path. O is the object plane containing the signal input S(x) exp(−jπx20FI). I is the output (image) plane. H ˜ (x′,λ) is the ideal filter function.

Fig. 7
Fig. 7

Actual system configuration employed for the experiment. G2 now denotes an off-axis zone plate (see text). tHOE(x′) represents a holographic optical element. All other elements are as before (Fig. 6).

Fig. 8
Fig. 8

Reconstructed image.

Equations (67)

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H ( x ) = exp ( j π x 2 λ 0 F I ) ,
H ˜ ( f x = x / λ F ) = exp ( - j π λ 0 F I f x 2 ) = exp [ - j π ( x ) 2 λ F E ( λ ) ] ,
t HOE ( x ) = exp [ j π ( x ) 2 λ 0 F ZP ] ,
t ACH ( x , λ ) = exp [ j π ( x ) 2 λ F A ] ,
t D ( x , λ ) = t HOE ( x ) t ACH ( x , λ ) = exp [ j π ( x ) 2 λ 0 F ZP ] exp [ j π ( x ) 2 λ F A ] .
t D ( x , λ ) H ˜ * ( x , λ ) = exp [ j ϕ E ( x , λ ) ] = exp [ j π ( x ) 2 λ 0 F ZP ψ ( μ = λ λ 0 ) ] ,
ψ ( μ ) = 1 μ 2 ( μ 2 + F ZP F A μ + F ZP F I F 2 ) .
t D ( x , λ ) = H ˜ ( x , λ ) exp [ j ϕ E ( x , λ ) ] .
( 1 ) ψ ( μ = 1 ) = 0 ,
( 2 ) d ψ d μ | μ = 1 = 0.
( 1 ) F 2 F ZP = F I ,
( 2 ) F 2 F A = - 2 F I ,
( 3 ) ψ ( μ ) = 1 μ 2 ( μ - 1 ) 2 , ϕ E ( x , μ ) = π ( x ) 2 λ 0 ( F 2 F I ) 1 μ 2 ( μ - 1 ) 2 .
( 1 )             F ZP = F = F I ;
( 2 )             F A = - F I 2 ;
( 3 )             ϕ E ( x , μ ) = π ( x ) 2 λ 0 F I 1 μ 2 ( μ - 1 ) 2 .
Δ λ 2 λ - λ 0 ;
ϕ E ( x , Δ λ ) = π ( x ) 2 λ 0 F I ( 1 + 2 λ 0 Δ λ ) 2 .
x max | 1 - 2 λ 0 Δ λ max | ( λ 0 F I ) 1 / 2 2 ,
( 1 + 2 λ 0 Δ λ ) 2 ,
| 1 - 2 λ 0 Δ λ | < | 1 + 2 λ 0 Δ λ | .
SBW = 2 x max Δ x min ,
Δ x min 1.22 F λ 0 D ,
SBW = ( 2 x max ) 2 1.22 F I λ 0 0.82 | 1 - 2 λ 0 Δ λ max | 2 .
Δ λ max λ 0 = 0.083.
Δ λ max λ 0 ,
t D ( x , λ ) = exp [ j π ( x ) 2 λ 0 F ZP ] exp [ j π ( x ) 2 λ F A ] ,
t D ( x , λ ) = exp [ j π ( x ) 2 λ 0 F I ( 1 - 2 λ 0 λ ) ] = exp [ j π ( x ) 2 λ 0 F I ( 1 - 2 μ ) ] = t D ( x , μ ) .
t D ( f x , μ ) = exp [ j π λ 0 F I f x 2 μ 2 ( 1 - 2 μ ) ] .
H A ( x , μ ) = exp { j π x 2 λ 0 F I [ 1 - Δ ( μ ) ] } ,
Δ ( μ ) = ( μ - 1 ) 2 μ ( μ - 2 ) ,
I ( x , μ ) = exp [ - j π x 2 Δ ( μ ) λ 0 F I ] [ S { x λ 0 F I [ 1 - Δ ( μ ) ] } * exp { j π x 2 λ 0 F I [ 1 - 1 Δ ( μ ) ] 2 } ] ,
exp [ - j π x 2 Δ ( μ ) λ 0 F I ] ;
x λ 0 F I [ 1 - Δ ( μ ) ] of S ˜ ( ) ;
exp { - j π x 2 λ 0 F I [ 1 - 1 Δ ( μ ) ] 2 } .
t ACH ( x , λ ) = exp [ j π ( x ) 2 λ F A ] = exp [ - j π λ ( 2 F I ) f x 2 ] = t ACH ( f x , λ )
H ˜ F S ( z , λ , f x ) = exp ( - j π λ z f x 2 ) .
exp ( - j π x 2 λ 0 F I ) = exp [ - j π x 2 λ F ( λ ) ] ,
H ( x ) = exp ( j π x 2 λ 0 F I ) .
S ( x ) exp ( - j π x 2 λ 0 F I )
S ˜ ( x λ 0 F I ) ,
S ˜ ( x λ 0 F I ) exp ( j π x 2 λ 0 F I ) ,
exp ( j 2 π f c x ) exp ( - j 2 π f c x ) exp ( - j π λ d 1 f c 2 ) = exp ( - j π λ d 1 f c 2 ) .
exp ( - j π λ d 1 f c 2 ) S ( x ) exp ( - j π x 2 λ 0 F I ) .
[ exp ( - j π λ d 1 f c 2 ) S ( x ) exp ( - j π x 2 λ 0 F I ) ] * exp ( j π x 2 λ 0 F I ) = exp ( - j π λ d 1 f c 2 ) S ˜ ( x λ 0 F I ) exp ( j π x 2 λ 0 F I ) ,
S ( x ) exp ( - j π x 2 λ 0 F I )
exp ( - j π λ d 1 f c 2 ) [ exp ( j 2 π f c x ) + S ˜ ( x λ 0 F I ) exp ( j π x 2 λ 0 F I ) ] ,
1 + | S ˜ ( x λ 0 F I ) | 2 + S ˜ ( x λ 0 F I ) exp ( - j 2 π f c x ) exp ( + j π x 2 λ 0 F I ) + S ˜ ( x λ 0 F I ) exp ( - j 2 π f c x ) exp ( - j π x 2 λ 0 F I ) .
ϕ E ( x , Δ λ ) = π ( x ) 2 λ 0 F I ( 1 + 2 λ 0 Δ λ ) 2 .
1 + | [ S ( x ) exp ( - j π x 2 λ 0 F I ) ] * H A ( x , λ ) | 2 + exp ( j 2 π f c x ) { [ S ( x ) exp ( - j π x 2 λ 0 F I ) ] * H A ( x , λ ) } * + exp ( - j 2 π f c x ) { [ S ( x ) exp ( - j π x 2 λ 0 F I ) ] * H A ( x , λ ) } .
λ exp ( - j 2 π f c x ) { [ S ( x ) exp ( - j π x 2 λ 0 F I ) ] * H A ( x , λ ) } d λ = exp ( - j 2 π f c x ) { [ S ( x ) exp ( - j π x 2 λ 0 F I ) ] * λ H A ( x , λ ) d λ } .
H A ( x , λ ) = exp { j π x 2 λ 0 F I [ 1 - Δ ( μ ) ] } ,
Δ ( μ ) = ( μ - 1 ) 2 μ ( μ - 2 ) = [ 1 - ( 2 λ 0 Δ λ ) 2 ] - 1 .
( Δ λ 2 λ 0 ) 2 1 , Δ ( ) - ( Δ λ 2 λ 0 ) 2
H A ( x , λ ) exp { j π x 2 λ 0 F I [ 1 + ( Δ λ 2 λ 0 ) 2 ] } .
λ 0 - Δ λ max 2 λ 0 + Δ λ max 2 H A ( x , λ ) d λ exp ( j π x 2 λ 0 F I ) Δ λ max * n = 0 α n [ j π x 2 4 λ 0 F I ( Δ λ max λ 0 ) 2 ] n ,
α n = [ n ! ( 2 n + 1 ) ] - 1 .
λ 0 - Δ λ max 2 λ 0 + Δ λ max 2 H A ( x , λ ) d λ exp ( j π x 2 λ 0 F I ) Δ λ max [ 1 + j π x 2 12 λ 0 F I ( Δ λ max λ 0 ) 2 ] Δ λ max exp ( j π x 2 λ 0 F I ) exp [ j π x 2 12 λ 0 F I ( Δ λ max λ 0 ) 2 ] = Δ λ max exp { j π x 2 λ 0 F I [ 1 + 1 12 ( Δ λ max λ 0 ) 2 ] } .
1 12 ( Δ λ max λ 0 ) 2
exp ( - j 2 π f c x ) exp { j π x 2 λ 0 F I [ 1 + 1 12 ( Δ λ max λ 0 ) 2 ] } ( S ˜ { x λ 0 F I [ 1 + 1 12 ( Δ λ max λ 0 ) 2 ] } * exp { j π x 2 λ 0 F I [ 1 + 12 ( λ 0 Δ λ max ) 2 ] 2 } ) ,
exp ( - j 2 π f c x ) exp [ j π x 2 λ 0 F I ( 1 + Δ λ max λ 0 ) 2 ] exp [ j π x 0 2 12 λ 0 F I ( Δ λ max λ 0 ) 2 ] exp { - j 2 π x x 0 λ 0 F I [ 1 + 1 12 ( Δ λ max λ 0 ) 2 ] } ~ exp ( - j 2 π f c x ) exp ( j π x 2 λ 0 F I ) exp ( - j 2 π x x 0 λ 0 F I ) ,
F I F I [ 1 + 1 12 ( Δ λ max λ 0 ) 2 ] - 1 ,
exp ( - j 2 π f c x ) exp ( j π x 2 λ 0 F I ) exp ( - j 2 π x x 0 λ 0 F I ) .
S ( x ) exp ( - j π x 2 λ 0 F I )
1 2 [ 1 + 1 2 exp ( j 2 π f c x ) exp ( j π 2 λ 0 F I ) + 1 2 exp ( - j 2 π f c x ) exp ( - j π x 2 λ 0 F I ) ]
exp ( - j π x 2 λ 0 F I ) .
exp ( - j π x 2 λ 0 F I ) S ( x ) ,

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