Abstract

The article deals with a method of calculation of off-axis light propagation between parallel planes using discretization of the Rayleigh-Sommerfeld integral and its implementation by fast convolution. It analyses zero-padding in case of different plane sizes. In case of memory restrictions, it suggests splitting the calculation into tiles and shows that splitting leads to a faster calculation when plane sizes are a lot different. Next, it suggests how to calculate propagation in case of different sampling rates by splitting planes into interleaved tiles and shows this to be faster than zero-padding and direct calculation. Neither the speedup nor memory-saving method decreases accuracy; the aim of the proposed method is to provide reference data that can be compared to the results of faster and less precise methods.

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (1)

2008 (1)

2007 (3)

2006 (1)

2005 (1)

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005) (Special issue on “Program Generation, Optimization, and Platform Adaptation”).
[CrossRef]

2004 (1)

J.-L. Kaiser, E. Quertemont, and R. Chevallier, “Light propagation in the pseudo-paraxial fresnel approximation,” Opt. Commun. 233(4-6), 261–269 (2004).
[CrossRef]

1998 (1)

1974 (1)

E. Sziklas and A. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62(3), 410–412 (1974).
[CrossRef]

1968 (1)

Astola, J.

Chevallier, R.

J.-L. Kaiser, E. Quertemont, and R. Chevallier, “Light propagation in the pseudo-paraxial fresnel approximation,” Opt. Commun. 233(4-6), 261–269 (2004).
[CrossRef]

Delen, N.

Egiazarian, K.

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005) (Special issue on “Program Generation, Optimization, and Platform Adaptation”).
[CrossRef]

Gotchev, A.

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech. 17(11), 1631–1646 (2007).
[CrossRef]

Hooker, B.

Johnson, S. G.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005) (Special issue on “Program Generation, Optimization, and Platform Adaptation”).
[CrossRef]

Kaiser, J.-L.

J.-L. Kaiser, E. Quertemont, and R. Chevallier, “Light propagation in the pseudo-paraxial fresnel approximation,” Opt. Commun. 233(4-6), 261–269 (2004).
[CrossRef]

Katkovnik, V.

Lalor, E.

Matsushima, K.

Muffoletto, R. P.

Nakahara, S.

Onural, L.

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24(2), 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,” IEEE Trans. Circ. Syst. Video Tech. 17(11), 1631–1646 (2007).
[CrossRef]

Osten, W.

Ozaktas, H.

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech. 17(11), 1631–1646 (2007).
[CrossRef]

Pedrini, G.

Quertemont, E.

J.-L. Kaiser, E. Quertemont, and R. Chevallier, “Light propagation in the pseudo-paraxial fresnel approximation,” Opt. Commun. 233(4-6), 261–269 (2004).
[CrossRef]

Siegman, A.

E. Sziklas and A. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62(3), 410–412 (1974).
[CrossRef]

Stoykova, E.

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech. 17(11), 1631–1646 (2007).
[CrossRef]

Sziklas, E.

E. Sziklas and A. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62(3), 410–412 (1974).
[CrossRef]

Tohline, J. E.

Tyler, J. M.

Zhang, F.

Appl. Opt. (2)

IEEE Trans. Circ. Syst. Video Tech. (1)

L. Onural, A. Gotchev, H. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech. 17(11), 1631–1646 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J.-L. Kaiser, E. Quertemont, and R. Chevallier, “Light propagation in the pseudo-paraxial fresnel approximation,” Opt. Commun. 233(4-6), 261–269 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. IEEE (2)

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005) (Special issue on “Program Generation, Optimization, and Platform Adaptation”).
[CrossRef]

E. Sziklas and A. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62(3), 410–412 (1974).
[CrossRef]

Other (1)

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

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Figures (3)

Fig. 1
Fig. 1

One-dimensional light propagation. Description of (a), (b), (c) is given in the text below.

Fig. 2
Fig. 2

The ratio of direct calculation time to (a) tiled calculation for given ratio ω of Source to Target side sizes (for Target side size N = 1024), and (b) tiled calculation for given ratio τ of Target to Source and sampling rates (for Source side size M = 1024). FFTW library was used for time measurement.

Fig. 3
Fig. 3

Speckle simulation: an example of algorithm output. A patch of size 0.1 × 0.1 mm2 with a random phase (a) and a Gaussian intensity (b) was sampled to a 1024 × 1024 array of complex amplitudes. Intensity of the off-axis propagation to the distance 5 mm is shown in the subimage (c), size of the subimage is 4.7 × 1.6 mm2. A naive approach to the convolution would require approx. 28 GiB of memory while proposed algorithm can work on a common PC.

Equations (14)

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t ( x ) = S o u r c e s ( ξ ) p ( x ξ ) d ξ
t [ n ] = m = 0 M 1 s [ m ] p [ n m ]         for   0 m M 1 ,         0 n N 1.
t [ n ] = m = 0 C 1 s [ m ] p [ ( n m )   mod   C ]         for   0 m C 1 ,         0 n C 1 ,
t = I D F T ( D F T ( s ) × D F T ( p ) ) ,
U ( x , y , z ) = 1 2 π U ( ξ , η , 0 ) z exp ( j k r ) r d ξ d η
U ( x , y , z ) = U ( x , y , 0 ) [ z 2 π ( ​ j k 1 x 2 + y 2 + z 2 ) exp ( j k x 2 + y 2 + z 2 ) x 2 + y 2 + z 2 ]
( D F T   ​ 's count ) C log C
t basic ( M , N ) = 3 ( M + N ) 2 log ( M + N ) 2 6 ( M + N ) 2 log ( M + N )
t common tiles ( M , N , T ) = ( 1 + 2 T 2 ) 2 ( M + N T ) 2 log ( M + N T )
s 1 ( ω , N , T ) = t basic ( ω N , N ) t common tiles ( ω N , N , T )
t upscaled ( M , τ ) = 6 ( τ M + τ M ) 2 log ( τ M + τ M ) 24 τ 2 M 2 log ( 2 τ M )
t interleaved tiles ( M , τ ) = ( 1 + 2 τ 2 ) 2 ( M + M ) 2 log ( M + M ) ( 1 + 2 τ 2 ) 8 M 2 log ( 2 M )
s 2 ( M , τ ) = t upscaled ( M , τ ) t interleaved tiles ( M , τ ) 3 τ 2 log ( 2 τ M ) ( 1 + 2 τ 2 ) log ( 2 M )
( M x + N x 1 ) ( M y + N y 1 ) C

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