Abstract

We analyze the shape of Bragg diffraction images from plane wave reference volume holograms in a 4-f geometry. When the volume hologram is probed by out-of-plane probe beams, the diffraction images become curved lines. Exploiting the k-sphere formulation and Fourier optics analysis, we present both geometrical and analytical solutions of the curved shape, which are distorted ellipses. Parameters and conditions related to the curvature are characterized, and experimental evidence is presented.

© 2009 Optical Society of America

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  1. H. Coufal, D. Psaltis, and G. T. Sincerbox, Holographic Data Storage (Springer, 2000).
  2. A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533-1551 (2004).
    [CrossRef] [PubMed]
  3. O. Momtahan, C. R. Hsieh, A. Karbaschi, A. Adibi, M. E. Sullivan, and D. J. Brady, “Spherical beam volume holograms for spectroscopic applications: modeling and implementation,” Appl. Opt. 43, 6557-6567 (2004).
    [CrossRef]
  4. O. Momtahan, C. R. Hsieh, A. Adibi, and D. J. Brady, “Analysis of slitless holographic spectrometers implemented by spherical beam volume holograms,” Appl. Opt. 45, 2955-2964 (2006).
    [CrossRef] [PubMed]
  5. G. Barbastathis and D. Psaltis, “Volume holographic multiplexing methods,” in Holographic Data Storage, Vol. 76 of Springer Series in Optical Sciences, 1st ed., H. J. Coufal, D. Psaltis, and G. T. Sincerbox, eds. (Springer, 2000), pp. xxvi and 486.
  6. G. Barbastathis, “The transfer function of volume holographic optical systems,” in Photorefractive Materials and Their Applications, Vol. 3 of Springer Series in Optical Science, J.P. H.Gu¨nter, ed. (Springer-Verlag, 2006).
  7. P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax,” Opt. Express 16, 7516-7524 (2008).
    [CrossRef] [PubMed]
  8. J. M. Watson, “Evaluation of spatial-spectral filtering in non-paraxial volume holographic imaging systems,” M.S.thesis (Massachusetts Institute of Technology, 2008).
  9. G. Barbastathis, “Imaging properties of three-dimensional pupils,” in Computational Optical Sensing and Imaging (COSI) (Optical Society of America, 2005), paper CMC4.
  10. M. Abramowitz and I. A. Stegun, “Spherical Bessel functions,” in Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1972), pp. 437-442.
  11. W. Sun and G. Barbastathis, “Rainbow volume holographic imaging,” Opt. Lett. 30, 976-978 (2005).
    [CrossRef] [PubMed]
  12. W. Sun, A. Sinha, G. Barbastathis, and M. A. Neifeld, “High-resolution volume holographic profilometry using the Viterbi algorithm,” Opt. Lett. 30, 1297-1299 (2005).
    [CrossRef] [PubMed]
  13. Y. Luo, P. J. Gelsinger-Austin, J. M. Watson, G. Barbastathis, J. K. Barton, and R. K. Kostuk, “Laser induced fluorescence imaging of subsurface tissue structures with a volume holographic spatial-spectral imaging system,” Opt. Lett. 33, 2098-2100 (2008).
    [CrossRef] [PubMed]
  14. A. Sinha and G. Barbastathis, “Volume holographic telescope,” Opt. Lett. 27, 1690-1692 (2002).
    [CrossRef]
  15. W. Liu, G. Barbastathis, and D. Psaltis, “Volume holographic hyperspectral imaging,” Appl. Opt. 43, 3581-3599 (2004).
    [CrossRef] [PubMed]
  16. W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27, 854-856(2002).
    [CrossRef]

2008 (2)

2006 (1)

2005 (2)

2004 (3)

2002 (2)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, “Spherical Bessel functions,” in Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1972), pp. 437-442.

Adibi, A.

Barbastathis, G.

P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax,” Opt. Express 16, 7516-7524 (2008).
[CrossRef] [PubMed]

Y. Luo, P. J. Gelsinger-Austin, J. M. Watson, G. Barbastathis, J. K. Barton, and R. K. Kostuk, “Laser induced fluorescence imaging of subsurface tissue structures with a volume holographic spatial-spectral imaging system,” Opt. Lett. 33, 2098-2100 (2008).
[CrossRef] [PubMed]

W. Sun and G. Barbastathis, “Rainbow volume holographic imaging,” Opt. Lett. 30, 976-978 (2005).
[CrossRef] [PubMed]

W. Sun, A. Sinha, G. Barbastathis, and M. A. Neifeld, “High-resolution volume holographic profilometry using the Viterbi algorithm,” Opt. Lett. 30, 1297-1299 (2005).
[CrossRef] [PubMed]

A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533-1551 (2004).
[CrossRef] [PubMed]

W. Liu, G. Barbastathis, and D. Psaltis, “Volume holographic hyperspectral imaging,” Appl. Opt. 43, 3581-3599 (2004).
[CrossRef] [PubMed]

W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27, 854-856(2002).
[CrossRef]

A. Sinha and G. Barbastathis, “Volume holographic telescope,” Opt. Lett. 27, 1690-1692 (2002).
[CrossRef]

G. Barbastathis, “Imaging properties of three-dimensional pupils,” in Computational Optical Sensing and Imaging (COSI) (Optical Society of America, 2005), paper CMC4.

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

G. Barbastathis, “The transfer function of volume holographic optical systems,” in Photorefractive Materials and Their Applications, Vol. 3 of Springer Series in Optical Science, J.P. H.Gu¨nter, ed. (Springer-Verlag, 2006).

Barton, J. K.

Brady, D. J.

Coufal, H.

H. Coufal, D. Psaltis, and G. T. Sincerbox, Holographic Data Storage (Springer, 2000).

Gelsinger-Austin, P. J.

Hsieh, C. R.

Karbaschi, A.

Kostuk, R. K.

Liu, W.

Luo, Y.

Momtahan, O.

Neifeld, M. A.

Oh, S. B.

Psaltis, D.

W. Liu, G. Barbastathis, and D. Psaltis, “Volume holographic hyperspectral imaging,” Appl. Opt. 43, 3581-3599 (2004).
[CrossRef] [PubMed]

W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27, 854-856(2002).
[CrossRef]

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

H. Coufal, D. Psaltis, and G. T. Sincerbox, Holographic Data Storage (Springer, 2000).

Shih, T.

Sincerbox, G. T.

H. Coufal, D. Psaltis, and G. T. Sincerbox, Holographic Data Storage (Springer, 2000).

Sinha, A.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, “Spherical Bessel functions,” in Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1972), pp. 437-442.

Sullivan, M. E.

Sun, W.

Watson, J. M.

Wissmann, P.

Appl. Opt. (4)

Opt. Express (1)

Opt. Lett. (5)

Other (6)

H. Coufal, D. Psaltis, and G. T. Sincerbox, Holographic Data Storage (Springer, 2000).

J. M. Watson, “Evaluation of spatial-spectral filtering in non-paraxial volume holographic imaging systems,” M.S.thesis (Massachusetts Institute of Technology, 2008).

G. Barbastathis, “Imaging properties of three-dimensional pupils,” in Computational Optical Sensing and Imaging (COSI) (Optical Society of America, 2005), paper CMC4.

M. Abramowitz and I. A. Stegun, “Spherical Bessel functions,” in Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1972), pp. 437-442.

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

G. Barbastathis, “The transfer function of volume holographic optical systems,” in Photorefractive Materials and Their Applications, Vol. 3 of Springer Series in Optical Science, J.P. H.Gu¨nter, ed. (Springer-Verlag, 2006).

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Figures (17)

Fig. 1
Fig. 1

Recording and probing geometries and their k-sphere representations, where k f , k s , k p , and k d are wave vectors of the reference, signal, probe, and diffracted beams, respectively, and θ f , θ s , θ p , and θ d are the angles of the reference, signal, probe, and diffracted beams with respect to the z axis, respectively. The wavelengths of the reference and probe beams are λ f and λ p , respectively. f 1 and f 2 are the focal lengths of the two lenses, and x and x are Cartesian coordinates of the object and image planes, respectively. The x - z coordinates are defined by the signal beam k s , where the maximum Bragg diffraction is generated by on-axis probe beams with the same wavelength.

Fig. 2
Fig. 2

k-sphere representation in 3D space, where the grating vector K g is constructed by k s and k f .

Fig. 3
Fig. 3

Definition of ϕ x and ϕ y . k d is the projection of k d onto the y - z plane. In (b), gray wave vectors are projections of k d and k d onto the x - z plane. ϕ x is the angle of the projected k d onto the x - z plane with respect to the z axis, and ϕ y is the angle between k d and k d .

Fig. 4
Fig. 4

Geometry used to find the maximal Bragg diffraction. (a) Bragg matching from out-of-plane probe beam and (b) the coordinate transform from ϕ x and ϕ y to θ x and θ y .

Fig. 5
Fig. 5

Trajectory of the maximum intensity of Bragg diffraction on the image plane for various θ s and θ f values when λ p = λ f . Note that the scales of horizontal and vertical axes are different.

Fig. 6
Fig. 6

Trajectory of the maximum Bragg diffraction on the image plane of unslanted holograms ( θ f = θ s ).

Fig. 7
Fig. 7

Maximum k dy of a Bragg matched diffracted beam. (a) Cutoff condition of k dy and (b) the k-sphere for an unslanted reflection hologram.

Fig. 8
Fig. 8

k-spheres for three different wavelengths. Degenerate readout occurs when vector K g is Bragg matched to k-spheres of different radii.

Fig. 9
Fig. 9

Trajectories of maximum Bragg diffracted intensity at the image plane, where the wavelength of the reference beam is 532 nm . In (a), they appear shifted due to the wavelength degeneracy. For comparison of the curvatures, they are replotted shifted and centered at the origin in (b). The hologram is unslanted with θ s = θ f = 40 ° .

Fig. 10
Fig. 10

Evaluated values of θ x b for various θ s and θ f . The diagonal line corresponds to unslanted holograms.

Fig. 11
Fig. 11

Computed field of view in object plane [in (a) and (b)] and trajectory of the maximum Bragg diffraction in image planes [in (c) and (d)]. The colors of the lines represent different θ s .

Fig. 12
Fig. 12

Crescent shape of the Bragg diffraction image. The experimental images are obtained from a multiplexed hologram. The curve at the center in (a) is the Bragg diffraction image of a plane wave reference hologram ( θ s = θ f = 48.1 ° , λ p = 650 nm , L = 2.11 mm , NA = 0.55 , f 2 = 20 mm , CCD pixel size is 16 μm , and total number of the pixels is 512 × 512 [8]).

Fig. 13
Fig. 13

Assuming a fixed value for θ p x , the coefficients C and D are plotted on a grid of ( x , y ) for θ p y increasing with the y coordinate. These values are shown for θ f = 40 ° .

Fig. 14
Fig. 14

Using varying θ p y and θ p x , two curves are shown, each with θ s = θ f . Note that these curves match those shown in Fig. 6.

Fig. 15
Fig. 15

Center of Bragg slit as a function of position in the image plane for the weak diffraction solution; θ f = 20 ° , 40 ° . This result was generated numerically by finding the locations of maximum focal plane intensity over a grid of incident point sources.

Fig. 16
Fig. 16

Reproduction of Fig. 12 using the weak diffraction formulation and the error with respect to the exact solution presented in Section 2. Also the error of the analytical solution of the k-sphere is plotted together. Across the y direction, both models of Sections 3, 4 yield less than 0.4 mrad , which corresponds to less than 1 pixel offset at the image plane.

Fig. 17
Fig. 17

Curves of maximum diffraction efficiency for Bragg matched input at three wavelengths. The hologram was recorded at λ = 532 nm and was unslanted with θ f = 40 ° .

Equations (48)

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| K g | = 2 2 π λ f sin ( θ s θ f 2 ) = 2 2 π λ p cos ( ϕ y ) sin ( ϕ x + ( θ s θ f 2 ) ) .
ϕ x = sin 1 [ λ p sin ( θ s θ f 2 ) λ f cos ( ϕ y ) ] ( θ s θ f 2 ) .
sin θ x = cos ϕ y sin ϕ x ,
sin θ y = sin ϕ y .
K g = k s k f = 2 π λ [ x ^ ( sin θ s sin θ f ) + z ^ ( cos θ s cos θ f ) ] ,
k p = 2 π λ ( x ^ cos α x + y ^ cos α y + z ^ cos α z ) ,
cos α x sin θ p x ,
cos α y sin θ p y .
k d x = 2 π λ ( sin θ s sin θ f + sin θ p x ) ,
k d y = 2 π λ sin θ p y ,
k d z = ( 2 π λ ) [ 1 k d x 2 k d y 2 ] 1 / 2 = ( 2 π λ ) [ 1 ( sin θ s sin θ f + sin θ p x ) 2 sin 2 θ p y ] 1 / 2 .
θ p x = θ f + θ x ,
θ p y = θ y ,
sin θ p x = sin ( θ f + θ x ) sin θ f + θ x cos θ f ,
sin θ p y θ y .
k d z = 2 π λ [ 1 ( sin θ f + θ x cos θ f ) 2 θ y 2 ] 1 / 2 = 2 π λ [ 1 sin 2 θ s θ x 2 cos 2 θ f 2 θ x sin θ s cos θ f θ y 2 ] 1 / 2 = 2 π λ [ cos 2 θ s 2 θ x sin θ s cos θ f ( θ y 2 cos 2 θ f θ x 2 + ) ] 1 / 2 = 2 π λ cos θ s [ 1 2 sin θ s cos θ f cos 2 θ s θ x cos 2 θ f θ x 2 + θ y 2 cos 2 θ s ] 1 / 2 .
f ( ϵ 1 , ϵ 2 ) = 1 + α 1 ϵ 1 + β 1 ϵ 1 2 + β 2 ϵ 2 2 .
f ( ϵ 1 , ϵ 2 ) 1 + α 1 2 ϵ 1 + 4 β 1 α 1 2 8 ϵ 1 2 + β 2 2 ϵ 2 2 .
k d z 2 π λ cos θ s [ 1 sin θ s cos θ f cos 2 θ s θ x 1 2 ( cos 2 θ f cos 2 θ s + sin 2 θ s cos 2 θ f cos 4 θ s ) θ x 2 1 2 1 cos 2 θ s θ y 2 ] .
Δ k d z = ( 2 π λ ) ( cos θ s cos θ f + cos α z ) k d z .
cos α z = 1 sin 2 θ p x sin 2 θ p y 1 ( sin θ f + θ x cos θ f ) 2 θ y 2 cos θ f [ 1 sin θ f cos θ f cos 2 θ f θ x 1 2 ( 1 + sin 2 θ f cos 2 θ f ) θ x 2 1 2 θ x 2 cos 2 θ f ] .
Δ k d z = ( 2 π λ ) [ ( sin θ s cos θ f cos θ s sin θ f ) θ x + 1 2 ( 1 cos θ s 1 cos θ f ) θ y 2 + 1 2 ( cos 2 θ f cos θ s + sin 2 θ s cos 2 θ f cos 3 θ s cos θ f sin 2 θ f cos θ f ) θ x 2 ] .
( θ x θ x b ) 2 X 2 + θ y 2 Y 2 = 1 ,
X 2 = cos 4 θ s cos 2 θ f sin ( θ s θ f ) ( cos θ f cos θ s ) 2 ( cos 2 θ f + cos θ f cos θ s + cos 2 θ s ) 2 ,
Y 2 = cos 2 θ s cos 2 θ f sin 2 ( θ s θ f ) ( cos θ f cos θ s ) 2 ( cos 2 θ f + cos θ f cos θ s + cos 2 θ s ) ,
θ x b = cos 2 θ s cos θ f sin ( θ s θ f ) ( cos θ f cos θ s ) ( cos 2 θ f + cos θ f cos θ s + cos 2 θ s ) .
ϵ = Y 2 X 2 = cos 2 θ f + cos θ f cos θ s + cos 2 θ s cos 2 θ s 0 ;
( x z ) = ( cos θ s sin θ s sin θ s cos θ s ) ( x z ) .
g ( x , y , z ) = ϵ ( x , y , z ) × P ( x , y , z ) × a ( x , y , z ) ,
q ( x 0 , y 0 , z 0 ) = g ( x , y , z ) h ( x 0 x , y 0 y , z 0 z ) d x d y d z ,
h ( x 0 x , y 0 y , z 0 z ) = exp [ i 2 π λ p ( z 0 z ) ] exp [ i π ( x 0 x ) 2 + ( y 0 y ) 2 λ p ( z 0 z ) ] .
q ( x , y ) = F 2 D [ g ( x , y , z ) h ( x 0 x , y 0 y , f 2 z , ) d x d y d z ] ( x 0 , y 0 ) ( x λ p f 2 , y λ p f 2 ) .
h ( x 0 x , y 0 y , f 2 z ) = exp [ i 2 π λ p ( f 2 x sin θ s z cos θ s ) ] × exp [ i π ( x 0 x cos θ s + z sin θ s ) 2 + ( y 0 y ) 2 λ p ( f 2 x sin θ s z cos θ s ) ] .
| q ( x , y ) | = | B × g ( x , y , z ) × exp [ i 2 π λ p f 2 ( A x + y y ) ] × exp [ i 2 π B λ p 2 ( 1 1 2 f 2 2 ( x 2 + y 2 ) ) ] d x d y d z | ,
A = x cos θ s + z sin θ s ,
B = λ p ( f 2 x sin θ s z cos θ s ) .
g ( x , y , z ) = rect ( x A x ) rect ( y A y ) rect ( z L ) × exp [ i ( z { 2 π λ f ( cos θ s cos θ f ) + 2 π λ p cos θ p } + x { 2 π λ f ( sin θ s sin θ f ) + 2 π λ p sin θ p x } + y 2 π λ p sin θ p y ) ] ,
θ p = sin 1 sin 2 ( θ p x ) + sin 2 ( θ p y ) .
( λ p ( f 2 x sin θ s ) L / 2 L / 2 exp [ i 2 C z ] d z λ p cos θ s L / 2 L / 2 z exp [ i 2 C z ] d z ) d x d y ,
= ( λ p ( f 2 x sin θ s ) ( L sinc ( CL π ) ) i λ p cos θ s L 2 C ( cos ( CL ) sinc ( CL π ) ) ) d x d y ,
C = π λ p ( λ p λ f ( cos θ s cos θ f ) + cos θ p + x sin θ s f 2 cos θ s ( 1 x 2 + y 2 2 f 2 2 ) ) .
A y / 2 A y / 2 exp [ i 2 π λ p ( sin θ p y + y f 2 ) ] d x d y ,
= A y sinc ( A y λ p ( sin θ p y + y f 2 ) ) d x .
d x = [ f 2 L sinc ( C L π ) i cos θ s L 2 C { cos ( C L ) sinc ( C L π ) } ] × A x / 2 A x / 2 exp [ i 2 D x ] d x + L sin θ s sinc ( C L π ) × A x / 2 A x / 2 x exp [ i 2 D x ] d x = [ f 2 L sinc ( C L π ) + i cos θ s L 2 C { cos ( C L ) sinc ( C L π ) } ] { A x sinc ( A x D π ) } L sin θ s sinc ( C L π ) i A x 2 D { cos ( A x D ) sinc ( A x D π ) } ,
D = 1 2 ( 2 π λ f ( sin θ s sin θ f ) + 2 π λ p sin θ p x 2 π λ p cos θ s ( x f 2 ) 2 π λ p sin θ s ( 1 x 2 + y 2 2 f 2 2 ) ) .
| q ( x , y ) | = λ p | [ A y sinc ( A y λ p ( sinc θ p y + y f 2 ) ) × ( f 2 L sinc ( C L π ) i L 2 C cos θ s { cos ( C L ) sinc ( C L π ) } ) × { A x sinc ( A x D π ) } L sin θ s sinc ( C L π ) × i A x 2 D { cos ( D A x ) sinc ( D A x π ) } ] | .
| q ( x , y ) | = λ p L A x A y | sinc ( A y λ p ( sin θ p y + y f 2 ) ) | × | f 2 sinc ( C L π ) sinc ( D A x π ) + i ( 1 2 c cos θ s sinc ( D A x π ) [ cos ( C L ) sinc ( C L π ) ] + 1 2 D sin θ s sinc ( C L π ) [ cos ( D A x ) sinc ( D A x π ) ] ) | .
| q ( x , y ) | = | j 0 ( π A y λ p ( sin θ p y ) + y f 2 ) | × | f 2 j 0 ( C L ) j 0 ( D A x ) i 2 ( L cos θ s j 0 ( D A x ) j 1 ( C L ) + A y sin θ s j 0 ( C L ) j 1 ( D A x ) ) | .

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