Abstract

Optical configurations for performing holography with light of limited coherence are analyzed. Such configurations, which may employ mirrors and gratings in preference to lenses and prisms, are space invariant in that the optical pathlength of a ray between object and hologram recording planes depends only on its initial direction, not on its location in the object plane. Holograms of arbitrary size made from object transparencies of arbitrary size and containing an arbitrary number of fringes may be produced with light of limited coherence.

© 1973 Optical Society of America

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References

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  1. R. E. Brooks, L. O. Heflinger, R. F. Wuerker, IEEE J. Quantum Electron. QE-2, 275 (1966).
    [Crossref]
  2. J. M. Burch, J. W. Gates, R. G. N. Hill, L. H. Tanner, Nature 212, 1347 (1966).
    [Crossref]
  3. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  4. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 57, 975 (1967).
    [Crossref]
  5. R. H. Katyl, Appl. Opt. 11, 1241, 1248, 1255 (1972).
    [Crossref] [PubMed]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36227 (1959).
    [Crossref]

1972 (1)

R. H. Katyl, Appl. Opt. 11, 1241, 1248, 1255 (1972).
[Crossref] [PubMed]

1967 (1)

1966 (2)

R. E. Brooks, L. O. Heflinger, R. F. Wuerker, IEEE J. Quantum Electron. QE-2, 275 (1966).
[Crossref]

J. M. Burch, J. W. Gates, R. G. N. Hill, L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

1962 (1)

1959 (1)

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36227 (1959).
[Crossref]

Brooks, R. E.

R. E. Brooks, L. O. Heflinger, R. F. Wuerker, IEEE J. Quantum Electron. QE-2, 275 (1966).
[Crossref]

Burch, J. M.

J. M. Burch, J. W. Gates, R. G. N. Hill, L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Gates, J. W.

J. M. Burch, J. W. Gates, R. G. N. Hill, L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Goodman, J. W.

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

Heflinger, L. O.

R. E. Brooks, L. O. Heflinger, R. F. Wuerker, IEEE J. Quantum Electron. QE-2, 275 (1966).
[Crossref]

Hill, R. G. N.

J. M. Burch, J. W. Gates, R. G. N. Hill, L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Katyl, R. H.

R. H. Katyl, Appl. Opt. 11, 1241, 1248, 1255 (1972).
[Crossref] [PubMed]

Leith, E. N.

Tanner, L. H.

J. M. Burch, J. W. Gates, R. G. N. Hill, L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Upatnieks, J.

Weinberg, F. J.

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36227 (1959).
[Crossref]

Wood, N. B.

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36227 (1959).
[Crossref]

Wuerker, R. F.

R. E. Brooks, L. O. Heflinger, R. F. Wuerker, IEEE J. Quantum Electron. QE-2, 275 (1966).
[Crossref]

Appl. Opt. (1)

R. H. Katyl, Appl. Opt. 11, 1241, 1248, 1255 (1972).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

R. E. Brooks, L. O. Heflinger, R. F. Wuerker, IEEE J. Quantum Electron. QE-2, 275 (1966).
[Crossref]

J. Opt. Soc. Am. (2)

J. Sci. Instrum. (1)

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36227 (1959).
[Crossref]

Nature (1)

J. M. Burch, J. W. Gates, R. G. N. Hill, L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Model for system analysis. S is the source, OS is a system for illuminating the grating G. O is the object, H is the recording plate. One diffracted order is used for the object beam, and one diffracted order is used for the reference beam, which travels a path (not shown) around the object, G′ is the grating image formed in the reference illumination, but projected back through the object beam path; x0 is a lateral displacement of the reference beam relative to the object beam.

Fig. 2
Fig. 2

Plot of Eq. (9). The system passband consists of those values of β for which the parabola lies between the horizontally dashed lines. The dotted line shows an ideally located parabola.

Fig. 3
Fig. 3

Plot of Eq. (15) as a function of β. The system passband is from β1 to β2.

Fig. 4
Fig. 4

Graphs of Eqs. (25) and (27) showing improved transfer characteristics when angularly sensitive filter is used. For convenience, we have chosen α0 = 0.

Fig. 5
Fig. 5

All-grating interferometer after Weinberg and Wood. A and B are gratings, O is the object, and C is the recording plane.

Fig. 6
Fig. 6

Fringes produced in the interferometer of Fig. 5. Left, using point source; right, using extended source. Grating A has 50 lines/mm, and grating B has 100 lines/mm. Also, da = db = 55 cm.

Fig. 7
Fig. 7

Use of extended source illumination in a space-invariant holographic system (Fig. 5) to reduce artifact noise. Top, using point source; bottom, using extended source (diameter about 2 mm). Both pictures were read out using a point source. Each arrow was about 2.5 mm across. Also, da = db = 55 cm, and d2 = 20 cm.

Tables (1)

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Table I Summary of Configurations

Equations (29)

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u = ( a 0 + a 1 cos α 0 x ) exp ( j p x ) ,
u 1 = c r exp ( j 2 π d 3 / λ ) exp ( - j λ d 3 α r 2 / 4 π ) × exp [ j α r ( x - x o ) ] ,
u 2 = c s exp [ j 2 π ( d 1 - d 2 ) / λ ] × exp [ - j λ ( d 1 - d 2 ) α s 2 / 4 π ] exp ( j α s x ) ,
u 3 = c s exp ( j π d 1 / λ ) exp [ - j λ ( d 1 - d 2 ) α s 2 / 4 π ] × exp [ - j λ d 2 ( α s + β ) 2 / 4 π ] exp [ j ( α s + β ) x ] .
Re u 1 * u 3 = 2 c r c s Re exp j [ ( α s - α r + β ) x + ϕ ] ,
ϕ = λ 4 π [ d 3 ( p + δ r α 0 ) 2 - ( d 1 - d 2 ) ( p + δ s α 0 ) 2 - d 2 ( p + δ s α 0 + β ) 2 ] - 2 π λ ( d 3 - d 1 ) + ( p + δ r α 0 ) x 0 .
λ 0 - Δ λ λ λ 0 + Δ λ .
< Re u 1 u 3 * > α sinc ( K Δ λ / 4 π 2 ) ,
K = d 3 ( δ r α 0 ) 2 - ( d 1 - d 2 ) ( δ s α 0 ) 2 - d 2 ( δ s α 0 + β ) 2 + 8 π 2 λ 0 2 ( d 3 - d 1 )
( λ 0 + Δ λ ) - 1 ( 1 - Δ λ / λ 0 ) / λ 0 .
β = β 0 = - δ s α 0 , K = K 0 = d 3 ( δ r α 0 ) 2 - ( d 1 - d 2 ) ( δ s α 0 ) 2 + 8 π 2 λ 0 ( d 3 - d 1 ) ,
Re u 1 u 3 * ¯ α sinc λ 0 M / 4 π 2 ,
M = 2 p 0 α 0 d 1 ( δ r - δ s ) - 2 d 2 p 0 β + 4 π p 0 x 0 / λ 0 .
( Δ λ / 4 π ) K ( π / 2 ) .
s ( x ) = 1 2 π S ( β ) exp ( j β x ) d β , S ( β ) = S exp ( j ϕ s ) .
v ( λ ) u ( x ) = 1 2 π v ( λ ) S ( β ) exp ( j β x ) exp ( - j λ d 2 β 2 / 4 π ) d β .
v ( λ ) u 0 = v ( λ ) c r exp ( - j d 3 α 0 2 / 4 π ) exp ( j α 0 x ) .
d I = v ( λ ) u 0 + v ( λ ) u 2 = v ( λ ) 2 u 0 2 + 2 ( c r S / 2 π ) cos [ ( α 0 - β ) x - ϕ s + λ η / 4 π ] d β ,
I = c r 2 v ( λ ) 2 d λ + 2 λ β v ( λ ) 2 c r c s F ( λ , β ) × cos [ ( α 0 - β ) x + λ 4 π η ] d β d λ .
F ( λ , β ) = rect [ λ - λ 0 2 Δ λ ( β ) ] .
I = c r 2 v 0 2 λ 0 - Δ λ λ 0 + Δ λ d λ + 2 v 0 2 β c r c s λ 0 - Δ λ λ 0 + Δ λ × cos [ ( α 0 - β ) x + λ 4 π η ] d λ d β
= K + 2 K c s C r cos [ ( α 0 - β ) x - λ 0 η / 4 π ] × sinc ( η Δ λ / 4 π 2 ) d β ,
β c = ± ( 2 π 2 / d 2 Δ λ + α 0 2 d 3 / d 2 ) 1 / 2 .
η Δ λ / 4 π 2 / < ½ .
I = K + 2 K c s c r Δ λ Δ λ cos [ ( α 0 - β ) x - λ 0 η ] sinc ( η Δ λ / 4 π 2 ) d β .
H ( β ) = 2 K ( c s / c r ) sinc ( η Δ λ / 4 π 2 ) ,
H ( β ) = 2 K ( c s / c r ) ( Δ λ / Δ λ ) sinc ( η Δ λ / 4 π 2 ) .
H ( β ) = 2 K ( c s / c r ) [ 4 π / ( d 2 β 2 - d 3 α 0 2 ) ] .
sinc ( d 2 θ 0 β / π ) sinc [ Δ λ ( 2 d 2 α 0 β + d 2 β 2 ) / 4 π 2 ] ,

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