Abstract

The space-bandwidth product (SBP) is a measure for the information capacity an optical system possesses. The two information processing steps in digital holography, recording, and reconstruction are analyzed with respect to the SBP. The recording setups for a Fresnel hologram, Fourier hologram, and image-plane hologram, which represent the most commonly used setup configurations in digital holography, are investigated. For the recording process, the required SBP to ensure the recording of the entire object information is calculated. This is accomplished by analyzing the recorded interference pattern in the hologram-plane. The paraxial diffraction model is used in order to simulate the light propagation from the object to hologram-plane. The SBP in the reconstruction process is represented by the product of the reconstructed field-of-view and spatial frequency bandwidth. The outcome of this analysis results in the best SBP adapted digital holographic setup.

© 2011 Optical Society of America

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References

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  1. A. W. Lohmann, “The space-bandwidth product, applied to spatial filtering and to holography,” IBM Research Paper, RJ-438 (1967).
  2. L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express 13, 2444–2552 (2005).
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    [CrossRef]
  6. A. W. Lohmann, M. E. Testorf, and J. Ojeda-Castaneda, “Holography and the Wigner function,” in The Art and Science of Holography: a Tribute to Emmett Leith and Yuri Denisyuk (SPIE Press, 2004).
  7. U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
    [CrossRef]
  11. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).
  12. X′′ matches with the object size X and can hence be replaced by it.
  13. P. Hariharan, Optical Holography (Cambridge University, 1984).
  14. A. W. Lohmann and S. Sinzinger, Optical Information Processing (Universitätsverlag Ilmenau, 2006).
  15. E. Zeidler, H. R. Schwarz, and W. Hackbusch, Taschenbuch der Mathematik (B. G. Teubner, 1996).
  16. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1986), pp. 69–97.
  17. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
    [CrossRef]
  18. A. W. Lohmann, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473(1996).
    [CrossRef]

2008 (3)

D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).
[CrossRef]

L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Numerical suppression of the twin image in in-line holography of a volume of micro-objects,” Meas. Sci. Technol. 19, 074004 (2008).
[CrossRef]

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

2007 (1)

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

1996 (1)

Ahrenberg, L.

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Asundi, A.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1986), pp. 69–97.

Callens, N.

Denis, L.

L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Numerical suppression of the twin image in in-line holography of a volume of micro-objects,” Meas. Sci. Technol. 19, 074004 (2008).
[CrossRef]

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

Dubois, F.

Ducottet, C.

L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Numerical suppression of the twin image in in-line holography of a volume of micro-objects,” Meas. Sci. Technol. 19, 074004 (2008).
[CrossRef]

Fink, H.-W.

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Fournel, T.

L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Numerical suppression of the twin image in in-line holography of a volume of micro-objects,” Meas. Sci. Technol. 19, 074004 (2008).
[CrossRef]

Fournier, C.

L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Numerical suppression of the twin image in in-line holography of a volume of micro-objects,” Meas. Sci. Technol. 19, 074004 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics2nd ed. (McGraw-Hill, 1996).

Guo, Z.

Hackbusch, W.

E. Zeidler, H. R. Schwarz, and W. Hackbusch, Taschenbuch der Mathematik (B. G. Teubner, 1996).

Hariharan, P.

P. Hariharan, Optical Holography (Cambridge University, 1984).

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).
[CrossRef]

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).
[CrossRef]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

Latychevskaia, T.

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Lohmann, A. W.

A. W. Lohmann, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473(1996).
[CrossRef]

A. W. Lohmann and S. Sinzinger, Optical Information Processing (Universitätsverlag Ilmenau, 2006).

A. W. Lohmann, M. E. Testorf, and J. Ojeda-Castaneda, “Holography and the Wigner function,” in The Art and Science of Holography: a Tribute to Emmett Leith and Yuri Denisyuk (SPIE Press, 2004).

A. W. Lohmann, “The space-bandwidth product, applied to spatial filtering and to holography,” IBM Research Paper, RJ-438 (1967).

McElhinney, C.

D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).
[CrossRef]

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Miao, J.

Naughton, T. J.

D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).
[CrossRef]

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Ojeda-Castaneda, J.

A. W. Lohmann, M. E. Testorf, and J. Ojeda-Castaneda, “Holography and the Wigner function,” in The Art and Science of Holography: a Tribute to Emmett Leith and Yuri Denisyuk (SPIE Press, 2004).

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

Peng, X.

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).

Schockaert, C.

Schwarz, H. R.

E. Zeidler, H. R. Schwarz, and W. Hackbusch, Taschenbuch der Mathematik (B. G. Teubner, 1996).

Sinzinger, S.

A. W. Lohmann and S. Sinzinger, Optical Information Processing (Universitätsverlag Ilmenau, 2006).

Testorf, M. E.

A. W. Lohmann, M. E. Testorf, and J. Ojeda-Castaneda, “Holography and the Wigner function,” in The Art and Science of Holography: a Tribute to Emmett Leith and Yuri Denisyuk (SPIE Press, 2004).

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

Xu, L.

Yourassowsky, C.

Zeidler, E.

E. Zeidler, H. R. Schwarz, and W. Hackbusch, Taschenbuch der Mathematik (B. G. Teubner, 1996).

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Numerical suppression of the twin image in in-line holography of a volume of micro-objects,” Meas. Sci. Technol. 19, 074004 (2008).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (1)

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Proc. SPIE (2)

D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).
[CrossRef]

C. McElhinney, B. M. Hennelly, L. Ahrenberg, and T. J. Naughton, “Removing the twin image in digital holography by segmented filtering of in-focus twin image,” Proc. SPIE 7072, 707208 (2008).
[CrossRef]

Other (11)

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

X′′ matches with the object size X and can hence be replaced by it.

P. Hariharan, Optical Holography (Cambridge University, 1984).

A. W. Lohmann and S. Sinzinger, Optical Information Processing (Universitätsverlag Ilmenau, 2006).

E. Zeidler, H. R. Schwarz, and W. Hackbusch, Taschenbuch der Mathematik (B. G. Teubner, 1996).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1986), pp. 69–97.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, 1989).
[CrossRef]

A. W. Lohmann, M. E. Testorf, and J. Ojeda-Castaneda, “Holography and the Wigner function,” in The Art and Science of Holography: a Tribute to Emmett Leith and Yuri Denisyuk (SPIE Press, 2004).

U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).

A. W. Lohmann, “The space-bandwidth product, applied to spatial filtering and to holography,” IBM Research Paper, RJ-438 (1967).

J. W. Goodman, Introduction to Fourier Optics2nd ed. (McGraw-Hill, 1996).

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Figures (11)

Fig. 1
Fig. 1

Adopted figures taken from [18], (a) SBP in the space-frequency domain, (b) SBP of (a) after Fresnel-transformation, (c) SBP of (a) after Fourier transformation, (d) SBP of (a) after passage through lens.

Fig. 2
Fig. 2

Nomenclature of coordinates used for the holographic recording and reconstruction process.

Fig. 3
Fig. 3

Carrier frequency introduced by inclination of plane reference wave.

Fig. 4
Fig. 4

Interference pattern caused by smallest resolvable object detail.

Fig. 5
Fig. 5

Diffracted cone of light from object coordinate x o to hologram-plane.

Fig. 6
Fig. 6

Fresnel hologram, (a) original SBP of object, (b)  SBP in-line, (c)  SBP off-axis without suppression of DC term, (d)  SBP off-axis with suppression of DC term.

Fig. 7
Fig. 7

Carrier frequency introduced by a lateral displacement of the origin of the spherical reference wave.

Fig. 8
Fig. 8

Fourier hologram, (a) original SBP of object, (b)  SBP in-line, (c)  SBP off-axis without suppression of DC term, (d)  SBP off-axis with suppression of DC term.

Fig. 9
Fig. 9

Sketch of different planes involved in the image formation.

Fig. 10
Fig. 10

Convergent object illumination to suppress quadratic phase term.

Fig. 11
Fig. 11

SBP of an image-plane hologram, (a) in-line arrangement, (b) off-axis arrangement.

Equations (71)

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SBP = X Δ ν .
Δ ν = 2 ν x _ max = 1 δ .
u ( x ) = exp ( i k d ) i λ d u ( x ) exp [ i π λ d ( x x ) 2 ] d x ,
x o 2 x o 1 = 2 δ = 1 ν o .
u o j ( x ) = A o j exp [ i π λ d ( x x o j ) 2 ] , where     j = 1 , 2..
I = | u o 1 ( x ) + u o 2 ( x ) | 2 = [ u o 1 ( x ) + u o 2 ( x ) ] · [ u o 1 ( x ) * + u o 2 ( x ) * ] = A o 1 2 + A o 2 2 + 2 A o 1 A o 2 cos [ 2 π λ d ( x o 2 x o 1 ) x ] .
I = A o 1 2 + A o 2 2 + 2 A o 1 A o 2 cos ( 2 π ν o x ) .
u o ( x ) = A o exp ( i 2 π ν o x ) exp [ i π λ d ( x x o ) 2 ] .
u r ( x ) = A r exp ( i 2 π ν r x ) ,
I ( x ) = A o A o * + A r A r * + 2 A o A r cos [ 2 π x ( ν o + ν r ) + π λ d ( x x o ) 2 ] .
x = x o + λ d ν o .
( x o , ν o ) ( x o , ν o ) = ( x o + λ d ν o , ν o ) ,
( X , Δ ν ) = ( X + λ d Δ ν , Δ ν ) .
SBP in - line = X Δ ν + λ d Δ ν 2 = SBP + N F 1 ,
( x o , ν o ) ( x o , ν o ) = ( x o + λ d ν o , [ ( ν o + ν r ) , ( ν o + ν r ) ] ) ( X , Δ ν ) = ( X + λ d Δ ν , Δ ν + 2 ν r ) .
SBP off - axis = ( X + λ d Δ ν ) ( Δ ν + 2 ν r ) .
ν r 3 X λ d = 3 2 Δ ν .
ν r X λ d = 1 2 Δ ν .
SBP off - axis = 2 ( SBP + N F 1 ) .
u r ( x ) = A r exp [ i π λ d r ( x x r ) 2 ] ,
I ( x ) = A r 2 + A o 2 + 2 A o A r cos [ π λ d E ] , where     E = 2 x ν o λ d + ( x x o ) 2 ( x x r ) 2 .
( x x o ) 2 ( x x r ) 2 = 2 x x o + x o 2 + 2 x x r x r 2 = 4 x x max .
x max = λ d 2 ν r .
I ( x ) = A r 2 + A o 2 + 2 A o A r cos [ 2 π x ( ν o + ν r ) ] .
( x o , ν o ) ( x o , ν o ) = ( λ d ν o , ν o ) ( X , Δ ν ) = ( λ d Δ ν , Δ ν ) .
SBP in - line = λ d Δ ν · Δ ν = N F 1 .
( x o , ν o ) ( x o , ν o ) = ( λ d ν o , [ ( ν o + ν r ) , ( ν o + ν r ) ] ) ( X , Δ ν ) = ( λ d Δ ν , Δ ν + 2 ν r ) .
SBP off - axis = ( λ d Δ ν ) 2 Δ ν = 2 N F 1 .
u L 1 ( x ˜ ) = u ( x ) exp [ i π ( x x ˜ ) 2 λ d 1 ] d x .
u L 2 ( x ˜ ) = u L 1 ( x ˜ ) exp ( i π x ˜ 2 λ f ) .
u ( x ) = u L 2 ( x ˜ ) exp [ i π ( x ˜ x ) 2 λ d 2 ] d x ˜ .
u ( x ) = u ( x ) exp [ i π λ { ( x x ˜ ) 2 d 1 x ˜ 2 f + ( x ˜ x ) 2 d 2 } ] d x d x ˜ = u ( x ) exp [ i π λ { x ˜ 2 ( 1 d 2 1 f + 1 d 1 ) 2 x ˜ ( x d 1 + x d 2 ) + x 2 d 1 + x 2 d 2 } ] d x d x ˜ .
u ( x ) = exp ( i π x 2 λ d 2 ) u ( x ) exp ( i π x 2 λ d 1 ) exp [ 2 i π x ˜ λ ( x d 1 + x d 2 ) ] d x ˜ d x .
u ( x ) = exp ( i π x 2 λ d 2 ) u ( x ) exp ( i π x 2 λ d 1 ) δ ( ω ^ ) .
u ( x ) = | λ d 1 | u ( x d 1 d 2 ) exp [ i π x 2 λ ( 1 d 2 + d 1 d 2 2 ) ] .
M = d 2 d 1 = x x = d 2 f f .
u ( x ) = u ( x M ) exp [ i π x 2 λ d 2 ( 1 + 1 M ) ] = u ( x M ) exp ( i π x 2 λ d 2 ) exp ( i π x 2 λ d 1 ) .
exp ( i π λ x 2 d 1 ) and exp ( i π λ x 2 d 2 ) .
I = u ( x M ) 2 + A r 2 + 2 u ( x M ) A r cos [ π λ ( x 2 d 2 x 2 d r + 2 x x r d r x r 2 d r ) ] .
I = u ( x M ) 2 + A r 2 + 2 u ( x M ) A r cos ( 2 π x ν r ) .
( x o , ν o ) ( x o · M , [ ( ν x + ν r ) , 0 , ( ν x + ν r ) ] ) .
ν x = 2 NA λ = 2 x λ M f .
| ν ( x ) | Δ ν L = 2 NA λ = 2 n sin [ arctan ( D 2 f ) ] λ D λ f .
SBP in - line = D Δ ν L = X L Δ ν L = SBP L .
ν r Δ ν L / 2.
X X L .
x Δ ν = 2 ν r + Δ ν L 2 Δ ν L .
SBP off - axis 2 X L Δ ν L = 2 SBP L .
SBP = X · 1 δ x .
φ = 2 π k Δ x ν r + π λ d ( k Δ x + X 2 ) 2 .
Δ φ Δ k φ k = 2 π Δ x ν r + 2 π λ d ( k Δ x + X 2 ) Δ x .
π | 2 π Δ x ν r + π λ d ( N Δ x + X ) Δ x | ,     N  is the pixel number.
d in - line = ( X + N Δ x ) Δ x λ .
ν r = α λ = X λ d .
d off - axis = ( 3 X + N Δ x ) Δ x λ .
X in - line = d λ Δ x N Δ x ,
X off - axis = d λ N Δ x 2 3 Δ x .
1 Δ ν ( x ) = δ ( x ) = λ d N Δ x .
1 Δ ν ( x ) in - line = δ ( x ) in - line = X + N Δ x N ,
1 Δ ν ( x ) off - axis = δ ( x ) off - axis = 3 X + N Δ x N .
SBP in - line = X · N X + N Δ x = N ( N Δ x ) 2 d λ , SBP off - axis = X · N 3 X + N Δ x = 1 3 ( N ( N Δ x ) 2 d λ ) = 1 3 · SBP in - line .
φ = π λ d ( 2 x x o + x o 2 + 2 x x r x r ) , φ = π λ d ( 2 k Δ x x o + x o 2 + 2 k Δ x x r x r ) .
π | 2 Δ x Δ k π λ d ( x o + x r ) | .
d in - line = X Δ x λ .
d off - axis = 2 X Δ x λ .
X in - line = d λ Δ x ,
X off - axis = d λ 2 Δ x .
SBP in - line = N , SBP off - axis = N 2 = 1 2 · SBP in - line .
X = N Δ x M = N Δ x f d 2 f .
| ν ( x ) | = ν ( x ) = D 1.22 λ f .
SBP = N Δ x f d 2 f ( D 1.22 λ f ) .

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