Abstract

An efficient way to numerically calculate diffraction from large volume holograms is developed using the first-order Born approximation. For this, everything except the propagating part of the Green’s function is neglected, and the fact that the gratings have a slowly varying envelope is used. The results of the new method are compared with analytical solutions of plane-wave diffraction with absorption, with phase-conjugated readout of a hologram recorded with a point source, and with numerical simulations of shift multiplexing with high-numerical-aperture microscope objectives. We show that the new method gives correct results in all cases and is several orders of magnitudes faster than FFT-based integration.

© 2009 Optical Society of America

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References

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  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1969).
  2. S. Q. Tao, B. Wang, G. W. Burr, and J. B. Chen, “Diffraction efficiency of volume gratings with finite size: corrected analytical solution,” J. Mod. Opt. 51, 1115-1122 (2004).
  3. G. Barbastathis, M. Levene, and D. Psaltis, “Shift multiplexing with spherical reference waves,” Appl. Opt. 35, 2403-2417 (1996).
    [Crossref] [PubMed]
  4. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592-1598 (1997).
    [Crossref]
  5. P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax (R),” Opt. Express 16, 7416-7524 (2008).
    [Crossref]
  6. B. Gombkoto, P. Koppa, A. Suto, and E. Lorincz, “Computer simulation of reflective volume gratings holographic data storage,” J. Opt. Soc. Am. A 24, 2075-2081 (2007).
    [Crossref]
  7. H. F. Arnoldus and J. T. Foley, “Traveling and evanescent parts of the electromagnetic Green's tensor,” J. Opt. Soc. Am. A 19, 1701-1711 (2002).
    [Crossref]
  8. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173-178 (1984).
    [Crossref]
  9. G. Gao and C. Torres-Verdin, “High-order generalized extended Born approximation for electromagnetic scattering,” IEEE Trans. Antennas Propag. 54, 1243-1256 (2006).
    [Crossref]
  10. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).
  11. H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. d. Phys. 365, 481-500 (1919).
    [Crossref]

2008 (1)

P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax (R),” Opt. Express 16, 7416-7524 (2008).
[Crossref]

2007 (1)

2006 (1)

G. Gao and C. Torres-Verdin, “High-order generalized extended Born approximation for electromagnetic scattering,” IEEE Trans. Antennas Propag. 54, 1243-1256 (2006).
[Crossref]

2004 (1)

S. Q. Tao, B. Wang, G. W. Burr, and J. B. Chen, “Diffraction efficiency of volume gratings with finite size: corrected analytical solution,” J. Mod. Opt. 51, 1115-1122 (2004).

2002 (1)

1997 (1)

1996 (1)

1984 (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173-178 (1984).
[Crossref]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1969).

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. d. Phys. 365, 481-500 (1919).
[Crossref]

Arnoldus, H. F.

Barbastathis, G.

P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax (R),” Opt. Express 16, 7416-7524 (2008).
[Crossref]

G. Barbastathis, M. Levene, and D. Psaltis, “Shift multiplexing with spherical reference waves,” Appl. Opt. 35, 2403-2417 (1996).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Burr, G. W.

S. Q. Tao, B. Wang, G. W. Burr, and J. B. Chen, “Diffraction efficiency of volume gratings with finite size: corrected analytical solution,” J. Mod. Opt. 51, 1115-1122 (2004).

Chen, J. B.

S. Q. Tao, B. Wang, G. W. Burr, and J. B. Chen, “Diffraction efficiency of volume gratings with finite size: corrected analytical solution,” J. Mod. Opt. 51, 1115-1122 (2004).

Foley, J. T.

Gao, G.

G. Gao and C. Torres-Verdin, “High-order generalized extended Born approximation for electromagnetic scattering,” IEEE Trans. Antennas Propag. 54, 1243-1256 (2006).
[Crossref]

Gombkoto, B.

Guibelalde, E.

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173-178 (1984).
[Crossref]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1969).

Koppa, P.

Lalanne, P.

Levene, M.

Lorincz, E.

Oh, S. B.

P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax (R),” Opt. Express 16, 7416-7524 (2008).
[Crossref]

Psaltis, D.

Suto, A.

Tao, S. Q.

S. Q. Tao, B. Wang, G. W. Burr, and J. B. Chen, “Diffraction efficiency of volume gratings with finite size: corrected analytical solution,” J. Mod. Opt. 51, 1115-1122 (2004).

Torres-Verdin, C.

G. Gao and C. Torres-Verdin, “High-order generalized extended Born approximation for electromagnetic scattering,” IEEE Trans. Antennas Propag. 54, 1243-1256 (2006).
[Crossref]

Wang, B.

S. Q. Tao, B. Wang, G. W. Burr, and J. B. Chen, “Diffraction efficiency of volume gratings with finite size: corrected analytical solution,” J. Mod. Opt. 51, 1115-1122 (2004).

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. d. Phys. 365, 481-500 (1919).
[Crossref]

Wissmann, P.

P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax (R),” Opt. Express 16, 7416-7524 (2008).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Ann. d. Phys. (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. d. Phys. 365, 481-500 (1919).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1969).

IEEE Trans. Antennas Propag. (1)

G. Gao and C. Torres-Verdin, “High-order generalized extended Born approximation for electromagnetic scattering,” IEEE Trans. Antennas Propag. 54, 1243-1256 (2006).
[Crossref]

J. Mod. Opt. (1)

S. Q. Tao, B. Wang, G. W. Burr, and J. B. Chen, “Diffraction efficiency of volume gratings with finite size: corrected analytical solution,” J. Mod. Opt. 51, 1115-1122 (2004).

J. Opt. Soc. Am. A (3)

Opt. Express (1)

P. Wissmann, S. B. Oh, and G. Barbastathis, “Simulation and optimization of volume holographic imaging systems in Zemax (R),” Opt. Express 16, 7416-7524 (2008).
[Crossref]

Opt. Quantum Electron. (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173-178 (1984).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

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Figures (5)

Fig. 1
Fig. 1

Scheme of the novel approach to simulate diffraction on large volume holograms: The angular distribution S ( θ , φ ) of the diffracted wave E d is shown by the black arrows. This function is calculated by sectioning the volume into several subvolumes and assuming that a uniform grating can be found in each subvolume. The gratings are shown by the black lines. Each subvolume diffracts light into certain angular spectra, which are indicated by white arrows for two subvolumes. The angular spectra are summed to obtain S ( θ , φ ) . The angle φ is shown in the coordinate system.

Fig. 2
Fig. 2

Scheme of the three test cases. The reference beam R, the signal beam S, the phase-conjugated reference beam R * , and the signal beam S * are shown in the three cases during recording and readout of the hologram.

Fig. 3
Fig. 3

Angular selectivity curve for a 1 m m × 1 m m × 1 m m hologram recorded by two plane waves. The solid curves are computed with the method described in this article, and the dashed curves are analytical solutions [2]. The absorption coefficients are (from top to bottom) α = 0 , α = 500 m 1 , α = 1000 m 1 .

Fig. 4
Fig. 4

Phase-conjugated light at the exit surface of a 20 μ m × 20 μ m × 20 μ m hologram. The signal wave is a spherical wave, where the focus is f z = 2 μ m behind the exit surface. The reference wave is a plane wave entering the volume at a 90° angle: (a) was obtained with the new method described in this paper and (b) was obtained by direct FFT-based integration. Here (c) shows a cross section through the center of the phase-conjugated beam. The solid curve is from (a), the points are from (b).

Fig. 5
Fig. 5

Shift multiplexing curve of a large volume hologram. The solid curves are simulations based on the method described in this article, and the black points are extracted from Fig. 2 in [6]. The parameters in both simulations are identical as far as we know.

Equations (10)

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2 | E r ( r ) E s ( r ) | cos [ K g ( r ) r ] | E r ( r ) | 2 + | E s ( r ) | 2 ,
ε ( r ) = ε ( 0 ) + ε ( 1 ) ( r ) = ε ( 0 ) + 2 | E r ( r ) E s ( r ) | | E r ( r ) | 2 + | E s ( r ) | 2 ( 2 ε ( 0 ) Δ n max cos [ K g ( r ) r ] + i 2 ε ( 0 ) λ 0 2 π Δ α max 2 cos [ K g ( r ) r + γ ] ) = ε ( 0 ) + 2 | E r ( r ) E s ( r ) | | E r ( r ) | 2 + | E s ( r ) | 2 ( κ + exp [ i K g ( r ) r ] + κ exp [ i K g ( r ) r ] ) ,
E d ( r ) = k 2 ε ( 0 ) d 3 r e i k r r 4 π r r | ε ( 1 ) ( r ) E p ( r ) .
e i k r r r r = i k 2 π π π d θ π 2 π 2 d φ cos ( φ ) e i k s ( θ , φ ) ( r r ) ,
s ( θ , φ ) = ( cos φ cos θ cos φ sin θ sin φ ) .
E d ( r ) = i k 3 8 π 2 ε ( 0 ) π π d θ π 2 π 2 d φ cos ( φ ) e i k s ( θ , φ ) r d 3 r ε ( 1 ) ( r ) E p ( r ) e i k s ( θ , φ ) r = i k 3 8 π 2 ε ( 0 ) π π d θ π 2 π 2 d φ cos ( φ ) e i k s ( θ , φ ) r S ( θ , φ ) ,
S ( θ , φ ) = d 3 r ε ( 1 ) ( r ) E p ( r ) e i k s ( θ , φ ) r .
S ( θ , φ ) = d 3 r ε ̂ ( 1 ) ( r ) E ̂ p ( r ) e Im ( k ) s ( θ , φ ) r e i D ( r ) r .
S ( θ , φ ) = m Q m d 3 r ε ̂ m ( 1 ) E ̂ p , m e Im ( k ) s ( θ , φ ) r m e i D m r = m W x , m W y , m W z , m e Im ( k ) s ( θ , φ ) r m e i D m r m ε ̂ m ( 1 ) E ̂ p , m sinc ( 1 2 D x , m W x , m ) sinc ( 1 2 D y , m W y , m ) sinc ( 1 2 D z , m W z , m ) ,
S ( θ , φ ) sinc ( 1 2 Re ( k ) cos ( φ ) cos ( θ ) W x ) sinc ( 1 2 Re ( k ) cos ( φ ) sin ( θ ) W y ) sinc ( 1 2 Re ( k ) [ 1 sin ( φ ) ] W z ) .

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