Abstract

Based on a linear systems approach, we derive the Wigner distribution function (WDF) of a 4f imager with a volume holographic three-dimensional pupil; then we obtain the WDF of the volume hologram itself by using the shearing properties of the WDF. Two common configurations, plane and spherical wave reference volume holograms, are examined in detail. The WDF elucidates the shift variant nature of the volume holographic element in both cases.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Wigner, Phys. Rev. 40, 749 (1932).
    [CrossRef]
  2. M. J. Bastiaans, in The Wigner Distribution--Theory and Applications in Signal Processing, W.M. F.Hlawatsch, ed. (Elsevier Science, 1997).
  3. K. H. Brenner and J. Ojeda-Castañeda, Opt. Acta 31, 213 (1984).
    [CrossRef]
  4. M. J. Bastiaans, J. Opt. Soc. Am. 69, 1710 (1979).
    [CrossRef]
  5. M. J. Bastiaans and P. G. J. van de Mortel, J. Opt. Soc. Am. A 13, 1698 (1996).
    [CrossRef]
  6. V. Arrizón and J. Ojeda-Castañeda, J. Opt. Soc. Am. A 9, 1801 (1992).
    [CrossRef]
  7. W. Pan, Appl. Opt. 47, 45 (2008).
    [CrossRef]
  8. G. Barbastathis, in Photorefractive Materials and Their Applications, Vol. 3 of Springer Series in Optical Science, J.P. H. P.Günter, ed. (Springer-Verlag, 2006).
  9. G. Barbastathis and D. Psaltis, in Holographic Data Storage, 1st ed., Vol. 76 of Springer Series in Optical Sciences, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), p. xxvi, 486.
  10. A. Sinha, W. Sun, T. Shih, and G. Barbastathis, Appl. Opt. 43, 1533 (2004).
    [CrossRef] [PubMed]
  11. O. Momtahan, C. R. Hsieh, A. Adibi, and D. J. Brady, Appl. Opt. 45, 2955 (2006).
    [CrossRef] [PubMed]
  12. S. B. Oh, “Volume holographic pupils in ray, wave, statistical optics, and Wigner space,” Ph.D. dissertation (Massachusetts Institute of Technology, 2009).
  13. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2005).
  14. H. W. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

2008

2006

2004

1996

1992

1984

K. H. Brenner and J. Ojeda-Castañeda, Opt. Acta 31, 213 (1984).
[CrossRef]

1979

1969

H. W. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

1932

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Adibi, A.

Arrizón, V.

Barbastathis, G.

A. Sinha, W. Sun, T. Shih, and G. Barbastathis, Appl. Opt. 43, 1533 (2004).
[CrossRef] [PubMed]

G. Barbastathis, in Photorefractive Materials and Their Applications, Vol. 3 of Springer Series in Optical Science, J.P. H. P.Günter, ed. (Springer-Verlag, 2006).

G. Barbastathis and D. Psaltis, in Holographic Data Storage, 1st ed., Vol. 76 of Springer Series in Optical Sciences, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), p. xxvi, 486.

Bastiaans, M. J.

M. J. Bastiaans and P. G. J. van de Mortel, J. Opt. Soc. Am. A 13, 1698 (1996).
[CrossRef]

M. J. Bastiaans, J. Opt. Soc. Am. 69, 1710 (1979).
[CrossRef]

M. J. Bastiaans, in The Wigner Distribution--Theory and Applications in Signal Processing, W.M. F.Hlawatsch, ed. (Elsevier Science, 1997).

Brady, D. J.

Brenner, K. H.

K. H. Brenner and J. Ojeda-Castañeda, Opt. Acta 31, 213 (1984).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2005).

Hsieh, C. R.

Kogelnik, H. W.

H. W. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Momtahan, O.

Oh, S. B.

S. B. Oh, “Volume holographic pupils in ray, wave, statistical optics, and Wigner space,” Ph.D. dissertation (Massachusetts Institute of Technology, 2009).

Ojeda-Castañeda, J.

V. Arrizón and J. Ojeda-Castañeda, J. Opt. Soc. Am. A 9, 1801 (1992).
[CrossRef]

K. H. Brenner and J. Ojeda-Castañeda, Opt. Acta 31, 213 (1984).
[CrossRef]

Pan, W.

Psaltis, D.

G. Barbastathis and D. Psaltis, in Holographic Data Storage, 1st ed., Vol. 76 of Springer Series in Optical Sciences, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), p. xxvi, 486.

Shih, T.

Sinha, A.

Sun, W.

van de Mortel, P. G. J.

Wigner, E.

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

H. W. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

K. H. Brenner and J. Ojeda-Castañeda, Opt. Acta 31, 213 (1984).
[CrossRef]

Phys. Rev.

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Other

M. J. Bastiaans, in The Wigner Distribution--Theory and Applications in Signal Processing, W.M. F.Hlawatsch, ed. (Elsevier Science, 1997).

S. B. Oh, “Volume holographic pupils in ray, wave, statistical optics, and Wigner space,” Ph.D. dissertation (Massachusetts Institute of Technology, 2009).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2005).

G. Barbastathis, in Photorefractive Materials and Their Applications, Vol. 3 of Springer Series in Optical Science, J.P. H. P.Günter, ed. (Springer-Verlag, 2006).

G. Barbastathis and D. Psaltis, in Holographic Data Storage, 1st ed., Vol. 76 of Springer Series in Optical Sciences, H.J.Coufal, D.Psaltis, and G.T.Sincerbox, eds. (Springer, 2000), p. xxvi, 486.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Geometry of the 4 f VH imager, in which a VH is at the Fourier plane of a traditional 4 f telescope.

Fig. 2
Fig. 2

Recording geometries.

Fig. 3
Fig. 3

Wigner representation of a PRVH, where λ = 0.5 μ m , θ s = 30 ° , and L = 1   mm .

Fig. 4
Fig. 4

Wigner representation of the SRVH, where λ = 0.5 μ m , θ s = 30 ° , z f = 50   mm , and L = 1   mm .

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

W g ( x , u ) = g ( x + ξ 2 ) g ( x ξ 2 ) e i 2 π u ξ d ξ ,
E 2 ( x 2 ) = h ( x 2 ; x 1 ) E 1 ( x 1 ) d x 1 ,
W 2 ( x 2 , u 2 ) = K h ( x 2 , u 2 ; x 1 , u 1 ) W 1 ( x 1 , u 1 ) d x 1 d u 1 ,
K h ( x 2 , u 2 ; x 1 , u 1 ) = d x 2 d x 1 e i 2 π ( u 2 x 2 u 1 x 1 ) h ( x 2 + x 2 2 ; x 1 + x 1 2 ) h ( x 2 x 2 2 ; x 1 x 1 2 ) .
h 4 f VH ( x 2 ; x 1 ) = E [ x 1 + x 2 λ f , x 1 2 x 2 2 2 λ f 2 ] ,
W 2 ( x 2 , u 2 ) = K 4 f VH ( x 2 , u 2 ; x 1 , u 1 ) W 1 ( x 1 , u 1 ) d x 1 d u 1 ,
K 4 f VH ( x 2 , u 2 ; x 1 , u 1 ) = d x 1 d x 2 e i 2 π ( u 2 x 2 u 1 x 1 ) × E [ x 1 + x 1 2 + x 2 + x 2 2 λ f , ( x 1 + x 1 2 ) 2 ( x 2 + x 2 2 ) 2 2 λ f 2 ] × E [ x 1 x 1 2 + x 2 x 2 2 λ f , ( x 1 x 1 2 ) 2 ( x 2 x 2 2 ) 2 2 λ f 2 ] .
W 3 ( x 3 , u 3 ) = W 1 ( λ f 1 u 3 , L 2 f u 3 + x 3 λ f ) ,
W 4 ( x 4 , u 4 ) = W 2 ( λ f 2 u 4 , L 2 f u 4 x 4 λ f ) .
W 4 ( x 4 , u 4 ) = K VH ( x 4 , u 4 ; x 3 , u 3 ) W 3 ( x 3 , u 3 ) d x 3 d u 3 ,
K VH ( x 4 , u 4 ; x 3 , u 3 ) = K 4 f VH ( λ f u 4 , L 2 f u 4 x 4 λ f ; λ f u 3 , L 2 f u 3 + x 3 λ f ) .
ϵ ( x , z ) = exp { i 2 π λ ( θ s x θ s 2 2 z ) } rect ( z L ) ,
h ( x 2 ; x 1 ) = δ ( x 1 + x 2 f θ s ) sinc { L 2 λ f 2 ( x 1 2 x 2 2 + f 2 θ s 2 ) } .
K VH ( x 4 , u 4 ; x 3 , u 3 ) = ( 2 L θ s ) δ ( u 4 u 3 θ s λ ) Λ { 2 L θ s ( x 3 x 4 ) + λ θ s ( u 3 + u 4 ) } sinc { 2 u 3 [ L θ s | L λ ( u 3 + u 4 ) + 2 ( x 3 x 4 ) | ] } ,
W 4 ( x 4 , u 4 ) = ( 2 L θ s ) d x 3 Λ [ L λ ( u 3 + θ s 2 λ ) + ( x 3 x 4 ) L θ s ] × sinc { 2 u 3 [ L θ s | L θ s ( 2 u 3 + θ s λ ) + 2 ( x 3 x 4 ) | ] } W 3 ( x 3 , u 3 ) .
ϵ ( x , z ) = exp { i π λ x 2 z z f } exp { i 2 π λ θ s x } exp { i π λ θ s 2 z } rect ( z L ) ,
h ( x 2 ; x 1 ) = exp { i π λ z f f 2 ( x 1 + x 2 f θ s ) 2 } sinc { L λ f 2 ( x 1 + x 2 ) ( f θ s x 2 ) } .
K VH ( x 4 , u 4 ; x 3 , u 3 ) = ( λ f ) 2 d u 3 d u 4 e i 2 π { u 4 [ ( L λ / 2 ) u 4 x 4 ] u 3 [ ( L λ / 2 ) u 3 + x 3 ] } × exp [ i 2 π λ z f ( u 3 + u 4 ) ( u 3 + u 4 θ s λ ) ] × sinc [ L λ ( u 3 + u 4 + u 3 + u 4 2 ) ( u 4 + u 4 2 θ s λ ) ] sinc [ L λ ( u 3 + u 4 u 3 + u 4 2 ) ( u 4 u 4 2 θ s λ ) ] .

Metrics