Abstract

Holography is reformulated by using the framework of phase-space optics. The Leith–Upatnieks off-axis reference hologram is compared with precursors, namely, single-sideband holography. The phase-space representation of complex amplitudes focuses on similarities between different holographic recording schemes and is particularly useful for investigating the degree of freedom and the space–bandwidth product of optical signals and systems. This allows one to include computer-generated holography and recent developments in digital holography in the discussion.

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
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2006 (1)

2005 (1)

1997 (1)

K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Opt. Commun. 144, 36-42 (1997).
[CrossRef]

1996 (1)

1993 (1)

1982 (1)

K.-H. Brenner and A. W. Lohmann, "Wigner distribution function display of complex 1D signals," Opt. Commun. 42, 310-314 (1982).
[CrossRef]

1979 (1)

1978 (1)

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

1969 (1)

B. R. Brown and A. W. Lohmann, "Computer-generated binary holograms," IBM J. Res. Dev. 13, 160-168 (1969).
[CrossRef]

1968 (1)

1967 (1)

1966 (2)

1963 (1)

1962 (1)

1956 (1)

A. Lohmann, "Optische Einseitenbandübertragung angewandt auf das Gabor-Mikroskop," Opt. Acta 3, 97-100 (1956).
[CrossRef]

1948 (1)

D. Gabor, "A new microscope principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

1946 (1)

D. Gabor, "Theory of communication," J. IEE 93, 429-457 (1946).

Appl. Opt. (2)

IBM J. Res. Dev. (1)

B. R. Brown and A. W. Lohmann, "Computer-generated binary holograms," IBM J. Res. Dev. 13, 160-168 (1969).
[CrossRef]

J. IEE (1)

D. Gabor, "Theory of communication," J. IEE 93, 429-457 (1946).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (4)

Nature (1)

D. Gabor, "A new microscope principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Opt. Acta (1)

A. Lohmann, "Optische Einseitenbandübertragung angewandt auf das Gabor-Mikroskop," Opt. Acta 3, 97-100 (1956).
[CrossRef]

Opt. Commun. (3)

K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Opt. Commun. 144, 36-42 (1997).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

K.-H. Brenner and A. W. Lohmann, "Wigner distribution function display of complex 1D signals," Opt. Commun. 42, 310-314 (1982).
[CrossRef]

Other (5)

M. J. Bastiaans, "Application of the Wigner distribution function in optics," in The Wigner Distribution--Theory and Applications in Signal Processing, W. Mecklenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.

A. W. Lohmann, "The space-bandwidth product, applied to spatial filtering and to holography," in Selected Papers on Phase-Space Optics, M.E.Testorf, J.Ojeda-Castañeda, and A.W.Lohmann, eds. (SPIE Press, 2006), pp. 11-32.

A. W. Lohmann, M. E. Testorf, and J. Ojeda-Castañeda, "Holography and the Wigner function," in The Art and Science of Holography, H. J. Caulfield, ed. (SPIE Press, 2004), pp. 129-144.

E. N. Leith, "A short history of the optics group at Willow Run laboratories," in Trends in Optics, A. Consortini, ed. (Academic, 1996), pp. 1-26.

P. Flandrin, Time-Frequency/Time-Scale Analysis(Academic, 1999).

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Figures (6)

Fig. 1
Fig. 1

Phase-space diagram: (a) Schematic representation of a generic complex amplitude distribution; phase-space distribution in (a) after (b) Fresnel diffraction, (c) passage through a parabolic lens, and (d) Fourier transformation. To display the change of orientation upon transformation, one corner of the schematic phase-space distribution is marked with a dot.

Fig. 2
Fig. 2

Phase-space diagram of Gabor inline holography: (a) Phase space representation of object wave O and reference plane wave R. (b) Phase-space diagram of the recorded hologram; object function O and twin signal T are overlapping with the autocorrelation functions of object wave AC [ O ] and reference AC[R].

Fig. 3
Fig. 3

Phase-space diagram of single-sideband holography: (a) The real-valued signal corresponds to a symmetric phase space distribution O. (b) Spatial filtering is used to remove one sideband of the signal. (c) The symmetry of the signal O ensures that, after recording, the superposition of the remaining sideband O SS and its twin T SS add up to the original signal O.

Fig. 4
Fig. 4

Phase-space diagram of the off-axis reference hologram. (a) Moving the reference R to a sufficiently high spatial frequency (b) separates the components of the recorded hologram in phase space.

Fig. 5
Fig. 5

Holographic recording schemes for different diffraction planes relative to the object. The phase-space distribution of the object wave O and the reference R after (a) Fourier transformation, (b) Fresnel diffraction, and (c) fractional Fourier transform is shown assuming a rectangular distribution in the object plane. The phase space representation of the detector is illustrated as a dashed rectangle D.

Fig. 6
Fig. 6

Phase-space interpretation of computer holography. (a) Three rows of a binary phase-only detour CGH. (b) PSD of the CGH; the bandwidth that can be used to encode the signal is bounded by the carrier frequency [row (I) in (a)] and the maximum change in modulation between adjacent cells [row (III) in (a)].

Equations (13)

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W ( x , ν ) = u ( x + x 2 ) u * ( x x 2 ) exp ( i 2 π ν x ) d x .
W ( x , ν ) d ν = | u ( x ) | 2 , W ( x , ν ) d x = | u ¯ ( ν ) | 2 .
W u ( x , ν ) = W g ( x , ν ) W h ( x , ν ν ) d ν = W g ( x , ν ) ν W h ( x , ν ) .
W | u | 2 ( x , ν ) = W u ( x , ν ) ν W u ( x , ν ) .
W ( x , ν ; z ) = W ( x λ z ν , ν ; 0 ) ,
W L + ( x , ν ) = W L ( x , ν + x / λ f ) ,
W 2 f ( x , ν ) = W u ( λ f ν , x / λ f ) .
u ( x ) = exp [ i 2 π ( α x 2 + β x + γ ) ] .
W u ( x , ν ) = δ ( ν α x β ) ,
W O ( x , ν ) δ [ ν 1 2 π ϕ ( x ) x ] ,
Δ x z = Δ x + λ z Δ ν .
S F = 2 [ S O + N F 1 ] ,
N F = ( λ z Δ ν 2 ) 1 .

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