Abstract

The diffraction relation between a plane and another plane that is both tilted and translated with respect to the first one is revisited. The derivation of the result becomes easier when the impulse function over a surface is used as a tool. Such an approach converts the original 2D problem to an intermediate 3D problem and thus allows utilization of easy-to-interpret Fourier transform properties due to rotation and translation. An exact solution for the scalar monochromatic propagating waves case when the propagation direction is restricted to be in the forward direction is presented.

© 2011 Optical Society of America

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References

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    [CrossRef]
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2009 (2)

2008 (2)

2007 (1)

L. Onural and H. M. Ozaktas, “Signal processing issues in diffraction and holographic 3DTV,” Image Comm. 22, 169–177(2007).
[CrossRef]

2006 (1)

L. Onural, “Impulse functions over curves and surfaces and their applications to diffraction,” J. Math. Anal. Appl. 322, 18–27 (2006).
[CrossRef]

2003 (2)

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” Proc. SPIE 5005, 190–197 (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. A 20, 1755–1762 (2003).
[CrossRef]

1998 (1)

1993 (1)

1992 (1)

1989 (1)

1988 (1)

1978 (1)

A. W. Lohmann. “Three-dimensional properties of wave-fields,” Optik (Jena) 51, 105–117 (1978).

1968 (1)

1967 (1)

Ahrenberg, L.

Benzie, P.

Bianco, B.

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergammon, 1965).

Delen, N.

Esmer, G. B.

G. B. Esmer, “Computation of holographic patterns between tilted planes,” Master’s thesis (Bilkent University, 2004).

G. B. Esmer and L. Onural, “Simulation of scalar optical diffraction between arbitrarily oriented planes,” in Proceedings of 2004 First International Symposium on Control, Communications and Signal Processing, ISCCSP 2004 (IEEE, 2004), pp. 225–228.
[CrossRef]

G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).

Frère, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed.(Mc-Graw-Hill, 1996).

Hooker, B.

Kondoh, A.

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” Proc. SPIE 5005, 190–197 (2003).
[CrossRef]

Lalor, É.

Leseberg, D.

Lohmann, A. W.

A. W. Lohmann. “Three-dimensional properties of wave-fields,” Optik (Jena) 51, 105–117 (1978).

Magnor, M.

Matsushima, K.

K. Matsushima and S. Nakahara, “Extremely high-definition full-parallax computer-generated hologram created by the poligon-based method,” Appl. Opt. 48, H54 (2009).
[CrossRef] [PubMed]

K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to holography,” Appl. Opt. 47, D110 –D116 (2008).
[CrossRef] [PubMed]

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. A 20, 1755–1762 (2003).
[CrossRef]

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” Proc. SPIE 5005, 190–197 (2003).
[CrossRef]

K. Matsushima, “Performance of the polygon-source method for creating computer-generated holograms of surface objects,” in Proceedings of ICO Topical Meeting on Optoinformatics /Information Photonics (International Commission for Optics, 2006), pp. 99–100.

Nakahara, S.

Onural, L.

L. Onural and H. M. Ozaktas, “Signal processing issues in diffraction and holographic 3DTV,” Image Comm. 22, 169–177(2007).
[CrossRef]

L. Onural, “Impulse functions over curves and surfaces and their applications to diffraction,” J. Math. Anal. Appl. 322, 18–27 (2006).
[CrossRef]

G. B. Esmer and L. Onural, “Simulation of scalar optical diffraction between arbitrarily oriented planes,” in Proceedings of 2004 First International Symposium on Control, Communications and Signal Processing, ISCCSP 2004 (IEEE, 2004), pp. 225–228.
[CrossRef]

Ozaktas, H. M.

L. Onural and H. M. Ozaktas, “Signal processing issues in diffraction and holographic 3DTV,” Image Comm. 22, 169–177(2007).
[CrossRef]

Sakamoto, Y.

Sakata, H.

Schimmel, H.

Sherman, G. C.

Tommasi, T.

Watson, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergammon, 1965).

Wyrowski, F.

Appl. Opt. (6)

Image Comm. (1)

L. Onural and H. M. Ozaktas, “Signal processing issues in diffraction and holographic 3DTV,” Image Comm. 22, 169–177(2007).
[CrossRef]

J. Math. Anal. Appl. (1)

L. Onural, “Impulse functions over curves and surfaces and their applications to diffraction,” J. Math. Anal. Appl. 322, 18–27 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Lett. (1)

Optik (Jena) (1)

A. W. Lohmann. “Three-dimensional properties of wave-fields,” Optik (Jena) 51, 105–117 (1978).

Proc. SPIE (1)

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” Proc. SPIE 5005, 190–197 (2003).
[CrossRef]

Other (6)

K. Matsushima, “Performance of the polygon-source method for creating computer-generated holograms of surface objects,” in Proceedings of ICO Topical Meeting on Optoinformatics /Information Photonics (International Commission for Optics, 2006), pp. 99–100.

G. B. Esmer, “Computation of holographic patterns between tilted planes,” Master’s thesis (Bilkent University, 2004).

G. B. Esmer and L. Onural, “Simulation of scalar optical diffraction between arbitrarily oriented planes,” in Proceedings of 2004 First International Symposium on Control, Communications and Signal Processing, ISCCSP 2004 (IEEE, 2004), pp. 225–228.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed.(Mc-Graw-Hill, 1996).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergammon, 1965).

G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).

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Figures (2)

Fig. 1
Fig. 1

(a) Fourier transform of the 3D field as an impulse over the hemisphere, (b) Fourier transform of the rotated and translated 3D field as an impulse over the rotated hemisphere, (c) projection of the 3D Fourier transform of (b) onto ( k x , k y ) plane.

Fig. 2
Fig. 2

Planar cross sections of a propagating plane wave (a 2D propagation with 1D cross sections are shown for the sake of simplicity). (a) Typical case, (b) maximum frequency over a plane observed when the propagation direction of the plane wave is parallel to that plane.

Equations (19)

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ψ z = z 0 ( x , y ) = F 2 D 1 { F 2 D { ψ z = 0 ( x , y ) } e j z 0 k 2 k x 2 k y 2 } ,
ψ ( x , y , z ) = ψ ( x ) = k δ S ( k ) A ( k ) e j k T x d k .
δ S ( x ) , f ( x ) = R N δ S ( x ) f ( x ) d x = S f ( x ) d S .
ψ R , b ( x ) = Δ ψ ( x ) = ψ ( Rx + b ) .
Ψ R , b ( k ) = Ψ ( Rk ) e j ( Rk ) T b .
F 3 D { ψ ( x ) } = Ψ ( k ) = 8 π 3 δ S ( k ) A ( k ) .
Ψ R , b ( k ) = 8 π 3 δ S ( Rk ) A ( Rk ) e j ( Rk ) T b .
Ψ R , b ( k ) = 8 π 3 δ S R ( k ) A ( Rk ) e j ( Rk ) T b .
ψ R , b ( x ) = k δ S R ( k ) A ( Rk ) e j ( Rk ) T b e j k T x d k ,
ψ R , b ( x ) = S R A ( Rk ) e j k T ( R T b + x ) d S = i B i A ( Rk ) e j k T ( R T b + x ) d S d k x d k y d k x d k y ,
ψ R , b ( x ) = B 1 A ( Rk ) e j k T ( R T b + x ) k k 2 k x 2 k y 2 d k x d k y + B 2 A ( Rk ) e j k T ( R T b + x ) k k 2 k x 2 k y 2 d k x d k y .
4 π 2 A ( k ) k | k z | = F 2 D { ψ 0 ( x , y ) } = Ψ 0 ( k x , k y ) ,
A ( k ) = A ( k x , k y , k 2 k x 2 k y 2 ) = 1 4 π 2 | k z | k Ψ 0 ( k x , k y ) .
A ( Rk ) = A ( k x , k y , k z ) = 1 4 π 2 | k z | k Ψ 0 ( k x , k y ) .
ψ R , b ( x ) = 1 4 π 2 B 1 Ψ 0 ( k x , k y ) | k z | | k z | e j k T ( R T b ) e j k T x d k x d k y + 1 4 π 2 B 2 Ψ 0 ( k x , k y ) | k z | | k z | e j k T ( R T b ) e j k T x d k x d k y .
ψ t ( x , y ) = ψ R , b ( x ) | z = 0 = 1 4 π 2 B 1 Ψ 0 ( k x , k y ) | k z | | k z | e j k T ( R T b ) e j ( k x x + k y y ) d k x d k y + 1 4 π 2 B 2 Ψ 0 ( k x , k y ) | k z | | k z | e j k T ( R T b ) e j ( k x x + k y y ) d k x d k y = F 2 D 1 { U ( k x , k y ) } ,
U ( k x , k y ) = Δ | k z | | k z | e j k T ( R T b ) Ψ 0 ( k x , k y ) [ I ( B 1 ) + I ( B 2 ) ] ,
I ( B ) = Δ { 1 if     ( k x , k y ) B 0 else .
k x = k x r 11 + k y r 12 + k 2 k x 2 k y 2 r 13 , k y = k x r 21 + k y r 22 + k 2 k x 2 k y 2 r 23 , | k z | = k 2 k x 2 k y 2 .

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