Abstract

In computational Fourier optics, computer generated holography, etc., coherent light propagation calculation between parallel planes is the essential task. A proper calculation discretization in the off-axis case leads to big memory demands in order to avoid aliasing errors. The proposed method typically cuts down the memory demands one hundred times. The principle of the method is based on the observation that there is a close correspondence between the reconstruction process (opposite of the sampling process) and prefiltering of the convolution kernel.

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004), 3rd ed.
  2. D. G. Voelz, Computational Fourier Optics: A Matlab Tutorial, Tutorial texts in optical engineering (SPIE Press, 2011).
  3. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).
  4. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Applied Optics44, 4607–4614 (2005).
    [CrossRef] [PubMed]
  5. I. Hanák, M. Janda, and V. Skala, “Detail-driven digital hologram generation,” Visual Computer26, 83–96 (2010).
    [CrossRef]
  6. Y. Sakamoto, M. Takase, and Y. Aoki, “Hidden surface removal using z-buffer for computer-generated hologram,” Practical Holography XVII and Holographic Materials IX5005, 276–283 (2003).
    [CrossRef]
  7. M. Yamaguchi, “Ray-based and wavefront-based holographic displays for high-density light-field reproduction,” Three-Dimensional Imaging, Visualization, and Display 20118043, 804306 (2011).
    [CrossRef]
  8. P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T. C. Poon, “Modern methods for fast generation of digital holograms,” 3D Research1, 11–18–18 (2010).
    [CrossRef]
  9. N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast fourier transform approach,” J. Opt. Soc. Am. A15, 857–867 (1998).
    [CrossRef]
  10. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A20, 1755–1762 (2003).
    [CrossRef]
  11. L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A28, 290–295 (2011).
    [CrossRef]
  12. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am.58, 1235–1237 (1968).
    [CrossRef]
  13. H. M. Ozaktas, S. O. Arik, and T. Coşkun, “Fundamental structure of fresnel diffraction: natural sampling grid and the fractional fourier transform,” Opt. Lett.36, 2524–2526 (2011).
    [CrossRef] [PubMed]
  14. K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation,” Opt. Express18, 18453–18463 (2010).
    [CrossRef] [PubMed]
  15. L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A24, 359–367 (2007).
    [CrossRef]
  16. L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” Circuits and Systems for Video Technology, IEEE Transactions on17, 1631 –1646 (2007).
    [CrossRef]
  17. P. Lobaz, “Reference calculation of light propagation between parallel planes of different sizes and sampling rates,” Opt. Express19, 32–39 (2011).
    [CrossRef] [PubMed]
  18. V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wavefield distributions,” Appl. Opt.47, 3481–3493 (2008).
    [CrossRef] [PubMed]
  19. E. Steward, Fourier Optics: An Introduction, Ellis Horwood Series in Physics (Dover Publications, 2004), 2nd ed.
  20. K. Turkowski, “Filters for common resampling tasks,” in “Graphics gems,”, A. S. Glassner, ed. (Academic Press Professional, Inc., San Diego, CA, USA, 1990).
  21. D. P. Mitchell and A. N. Netravali, “Reconstruction filters in computer-graphics,” SIGGRAPH Comput. Graph.22, 221–228 (1988).
    [CrossRef]

2011 (4)

2010 (3)

K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation,” Opt. Express18, 18453–18463 (2010).
[CrossRef] [PubMed]

P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T. C. Poon, “Modern methods for fast generation of digital holograms,” 3D Research1, 11–18–18 (2010).
[CrossRef]

I. Hanák, M. Janda, and V. Skala, “Detail-driven digital hologram generation,” Visual Computer26, 83–96 (2010).
[CrossRef]

2008 (1)

2007 (2)

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A24, 359–367 (2007).
[CrossRef]

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” Circuits and Systems for Video Technology, IEEE Transactions on17, 1631 –1646 (2007).
[CrossRef]

2005 (1)

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Applied Optics44, 4607–4614 (2005).
[CrossRef] [PubMed]

2003 (2)

Y. Sakamoto, M. Takase, and Y. Aoki, “Hidden surface removal using z-buffer for computer-generated hologram,” Practical Holography XVII and Holographic Materials IX5005, 276–283 (2003).
[CrossRef]

K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A20, 1755–1762 (2003).
[CrossRef]

1998 (1)

1988 (1)

D. P. Mitchell and A. N. Netravali, “Reconstruction filters in computer-graphics,” SIGGRAPH Comput. Graph.22, 221–228 (1988).
[CrossRef]

1968 (1)

Aoki, Y.

Y. Sakamoto, M. Takase, and Y. Aoki, “Hidden surface removal using z-buffer for computer-generated hologram,” Practical Holography XVII and Holographic Materials IX5005, 276–283 (2003).
[CrossRef]

Arik, S. O.

Astola, J.

Cheung, K. W. K.

P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T. C. Poon, “Modern methods for fast generation of digital holograms,” 3D Research1, 11–18–18 (2010).
[CrossRef]

Coskun, T.

Delen, N.

Egiazarian, K.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004), 3rd ed.

Gotchev, A.

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” Circuits and Systems for Video Technology, IEEE Transactions on17, 1631 –1646 (2007).
[CrossRef]

Hanák, I.

I. Hanák, M. Janda, and V. Skala, “Detail-driven digital hologram generation,” Visual Computer26, 83–96 (2010).
[CrossRef]

Hooker, B.

Janda, M.

I. Hanák, M. Janda, and V. Skala, “Detail-driven digital hologram generation,” Visual Computer26, 83–96 (2010).
[CrossRef]

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).

Katkovnik, V.

Lalor, E.

Liu, J. P.

P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T. C. Poon, “Modern methods for fast generation of digital holograms,” 3D Research1, 11–18–18 (2010).
[CrossRef]

Lobaz, P.

Matsushima, K.

Mitchell, D. P.

D. P. Mitchell and A. N. Netravali, “Reconstruction filters in computer-graphics,” SIGGRAPH Comput. Graph.22, 221–228 (1988).
[CrossRef]

Netravali, A. N.

D. P. Mitchell and A. N. Netravali, “Reconstruction filters in computer-graphics,” SIGGRAPH Comput. Graph.22, 221–228 (1988).
[CrossRef]

Onural, L.

L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A28, 290–295 (2011).
[CrossRef]

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A24, 359–367 (2007).
[CrossRef]

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” Circuits and Systems for Video Technology, IEEE Transactions on17, 1631 –1646 (2007).
[CrossRef]

Ozaktas, H.

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” Circuits and Systems for Video Technology, IEEE Transactions on17, 1631 –1646 (2007).
[CrossRef]

Ozaktas, H. M.

Poon, T. C.

P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T. C. Poon, “Modern methods for fast generation of digital holograms,” 3D Research1, 11–18–18 (2010).
[CrossRef]

Sakamoto, Y.

Y. Sakamoto, M. Takase, and Y. Aoki, “Hidden surface removal using z-buffer for computer-generated hologram,” Practical Holography XVII and Holographic Materials IX5005, 276–283 (2003).
[CrossRef]

Schimmel, H.

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).

Skala, V.

I. Hanák, M. Janda, and V. Skala, “Detail-driven digital hologram generation,” Visual Computer26, 83–96 (2010).
[CrossRef]

Steward, E.

E. Steward, Fourier Optics: An Introduction, Ellis Horwood Series in Physics (Dover Publications, 2004), 2nd ed.

Stoykova, E.

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” Circuits and Systems for Video Technology, IEEE Transactions on17, 1631 –1646 (2007).
[CrossRef]

Takase, M.

Y. Sakamoto, M. Takase, and Y. Aoki, “Hidden surface removal using z-buffer for computer-generated hologram,” Practical Holography XVII and Holographic Materials IX5005, 276–283 (2003).
[CrossRef]

Tsang, P. W. M.

P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T. C. Poon, “Modern methods for fast generation of digital holograms,” 3D Research1, 11–18–18 (2010).
[CrossRef]

Turkowski, K.

K. Turkowski, “Filters for common resampling tasks,” in “Graphics gems,”, A. S. Glassner, ed. (Academic Press Professional, Inc., San Diego, CA, USA, 1990).

Voelz, D. G.

D. G. Voelz, Computational Fourier Optics: A Matlab Tutorial, Tutorial texts in optical engineering (SPIE Press, 2011).

Wyrowski, F.

Yamaguchi, M.

M. Yamaguchi, “Ray-based and wavefront-based holographic displays for high-density light-field reproduction,” Three-Dimensional Imaging, Visualization, and Display 20118043, 804306 (2011).
[CrossRef]

3D Research (1)

P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T. C. Poon, “Modern methods for fast generation of digital holograms,” 3D Research1, 11–18–18 (2010).
[CrossRef]

Appl. Opt. (1)

Applied Optics (1)

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Applied Optics44, 4607–4614 (2005).
[CrossRef] [PubMed]

Circuits and Systems for Video Technology, IEEE Transactions on (1)

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” Circuits and Systems for Video Technology, IEEE Transactions on17, 1631 –1646 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Express (2)

Opt. Lett. (1)

Practical Holography XVII and Holographic Materials IX (1)

Y. Sakamoto, M. Takase, and Y. Aoki, “Hidden surface removal using z-buffer for computer-generated hologram,” Practical Holography XVII and Holographic Materials IX5005, 276–283 (2003).
[CrossRef]

SIGGRAPH Comput. Graph. (1)

D. P. Mitchell and A. N. Netravali, “Reconstruction filters in computer-graphics,” SIGGRAPH Comput. Graph.22, 221–228 (1988).
[CrossRef]

Three-Dimensional Imaging, Visualization, and Display 2011 (1)

M. Yamaguchi, “Ray-based and wavefront-based holographic displays for high-density light-field reproduction,” Three-Dimensional Imaging, Visualization, and Display 20118043, 804306 (2011).
[CrossRef]

Visual Computer (1)

I. Hanák, M. Janda, and V. Skala, “Detail-driven digital hologram generation,” Visual Computer26, 83–96 (2010).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004), 3rd ed.

D. G. Voelz, Computational Fourier Optics: A Matlab Tutorial, Tutorial texts in optical engineering (SPIE Press, 2011).

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).

E. Steward, Fourier Optics: An Introduction, Ellis Horwood Series in Physics (Dover Publications, 2004), 2nd ed.

K. Turkowski, “Filters for common resampling tasks,” in “Graphics gems,”, A. S. Glassner, ed. (Academic Press Professional, Inc., San Diego, CA, USA, 1990).

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Figures (5)

Fig. 1
Fig. 1

Light diffraction at sine grating 5 × 5 mm2 of period 50 cycles/mm illuminated perpendicularly by a plane wave with λ = 650 nm at a distance z0 = 0.5 m. a) Discretization using Δ = 10 μm. b) Discretization using Δ = 10/12 μm = 0.83 μm.

Fig. 2
Fig. 2

Geometry of the simplified 1D case. Samples with index 0 are depicted as full circles, the others as empty circles.

Fig. 3
Fig. 3

Examples of interpolation convolution kernels (filters) for ups = 3. The windowed sinc filter shown is the normalized Lanczos filter for a = 2.

Fig. 4
Fig. 4

Examples of diffraction by grating with vertical strips; grating size 5 × 5 mm2, sampling period 10 μm, samples in each row 1, 0, 1, 0, ... (i.e. 50 slits/mm). Propagation distance 500 mm, illumination at normal incidence, λ = 650 nm. The left images show right halves of the diffraction patterns (compare with Fig. 1); the graphs to the right show the intensity relative to the central intensity. The interpolation used is a) none, b) rectangular filter, c) triangular filter, d) Lanczos filter, a = 2, e) Lanczos filter, a = 3.

Fig. 5
Fig. 5

Time of the calculation comparison. The graphs to the left and in the middle show the dependency of the time of propagation calculation for N = M = 500 on the upsampling factor ups; the vertical scale used defines the time of the basic method for ups = 1 to be 1. Besides the complete time of the calculation, the times of the FFTs and the propagation kernel calculation times are shown. The rightmost graph shows the ratio of the proposed and the basic method calculation times; e.g. a value of 4 means that the proposed method is 4× faster for a given ups.

Equations (12)

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U ( x , y , z 0 ) = 1 2 π Σ U ( ξ , η , 0 ) z exp ( j k r ) r d ξ d η
target [ j ] = U ( j Δ + x 0 , z 0 ) = Δ i = U ( i Δ , 0 ) h ( ( j i ) Δ + x 0 , 0 , z 0 ) = = Δ i = source [ i ] h x 0 , z 0 , 1 [ j i ] = = Δ ( source [ ] h x 0 , z 0 , 1 [ ] ) [ j ]
h x 0 , z 0 , ups [ j ] = h ( j Δ / ups + x 0 , 0 , z 0 )
h ( x , y , z ) = 1 2 π z exp ( j k r ) r = z 2 π ( j k 1 r ) exp ( j k r ) r 2 , r = x 2 + y 2 + z 2
source ups [ j ] = { source [ j / ups ] if ( j / ups ) Z 0 otherwise
target ups [ ] = ( source ups [ ] filter [ ] ) h x 0 , z 0 , ups [ ] = source ups [ ] ( filter [ ] h x 0 , z 0 , ups [ ] ) = = source ups [ ] h x 0 , z 0 , ups fin [ ]
target ups [ j ] = i = source ups [ j i ] h x 0 , z 0 , ups fin [ i ] = = i = source ups [ j i ] k = f w h f w h filter [ k ] h x 0 , z 0 , ups [ i k ]
target [ j ] = target ups [ ups × j ] = i = source ups [ ups × ( j i ) ] h x 0 , z 0 , ups fin [ ups × i ] = = i = source [ j i ] h x 0 , z 0 , 1 fin [ i ]
h x 0 , z 0 fin [ i ] = k = f w h f w h filter [ k ] h x 0 , z 0 , ups [ ups × i k ]
target [ j ] = i = 0 C 1 source [ i mod C ] h x 0 , z 0 , 1 fin [ ( j i ) mod C ]
h x 0 , z 0 , 1 fin [ i ] = { k = fwh fwh filter [ k ] h x 0 , z 0 , ups [ ups × i k ] if  0 i < N k = fwh fwh filter [ k ] h x 0 , z 0 , ups [ ups × ( i C ) k ] if  N i < C
target [ , ] = IFFT ( FFT ( source [ , ] ) FFT ( h x 0 , y 0 , z 0 , 1 fin [ , ] )

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