Abstract

A novel method is proposed for simulating free-space propagation from an input source field to a destination sampling window laterally shifted from that in the source field. This off-axis type numerical propagation is realized using the shifted-Fresnel method (Shift-FR) and is very useful for calculating non-paraxial and large-scale fields. However, the Shift-FR is prone to a serious problem, in that it causes strong aliasing errors in short distance propagation. The proposed method, based on the angular spectrum method, resolves this problem. Numerical examples as well as the formulation are presented.

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References

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2009

2008

2007

2006

2005

2004

2003

1998

1997

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

1995

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[CrossRef]

Adams, M.

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Alfieri, D.

De Nicola, S.

Delen, N.

Ferraro, P.

Finizio, A.

Hong, C. K.

Hooker, B.

Jeong, S. J.

Jüptner, W. P. O.

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Kim, M. K.

Kreis, T. M.

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Matsushima, K.

Muffoletto, R. P.

Nakahara, S.

Osten, W.

Pedrini, G.

Pierattini, G.

Schimmel, H.

Shimobaba, T.

Sypek, M.

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[CrossRef]

Tohline, J. E.

Tyler, J. M.

Wang, D.

Wyrowski, F.

Yamaguchi, I.

Yaroslavsky, L. P.

Yu, L.

Zhang, F.

Zhao, J.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), chap. 3.10.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), chap. 2.2.

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Figures (8)

Fig. 1
Fig. 1

Calculation of the diffracted field using (a) conventional methods and (b) methods for off-axis propagation, where the region of interest is apart from the optical axis in the destination pane.

Fig. 2
Fig. 2

The coordinate system used for formulation.

Fig. 3
Fig. 3

Example of a sampled transfer function. Only the real part of H ^ ( u ; z 0 ) is depicted in the sampling interval Δ u = ( 2 S x ) 1 = ( 2 N x Δ x ) 1 , where N x = 1024 , Δ x = 2 λ , x 0 = + S x / 2 , and z 0 = 20 S x .

Fig. 4
Fig. 4

Schematic illustrations of frequency regions to avoid aliasing errors of the sampled transfer function. Here, x 0 0 and y 0 = 0 .

Fig. 5
Fig. 5

Schematic illustrating the approximated rectangular region for the band-limit of a two-dimensional wave field. Here, the region is depicted with x 0 > + S x and y 0 < S y as an example.

Fig. 6
Fig. 6

Setup for (a) numerical simulation of off-axis propagation, and (b) the sampled source field used in the simulation.

Fig. 7
Fig. 7

Amplitude images of the destination fields calculated for different propagation and shift distances using three different methods.

Fig. 8
Fig. 8

Model for discussing the highest and lowest spatial frequencies required for off-axis numerical propagation.

Tables (1)

Tables Icon

Table 1 Constants used for the band-limit to avoid aliasing errors

Equations (26)

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x ^ = x x 0 y ^ = y y 0
g ^ ( x ^ , y ^ , z 0 ) = g ^ ( x , y , z 0 ) = g ( x , y , z 0 ) = g ( x , y , 0 ) h ( x x , y y , z 0 ) d x d y
G ( u , v ; z 0 ) = G ( u , v ; 0 ) H ( u , v ; z 0 ) ,
G ( u , v ; 0 ) = g ( x , y , 0 ) exp [ i 2 π ( u x + v y ) ] d x d y = F { g ( x , y , 0 ) } .
H ( u , v ; z 0 ) = exp [ i 2 π w z 0 ] , w = w ( u , v ) = { ( λ 2 u 2 v 2 ) 1 / 2   u 2 + v 2 λ 2 0  otherwise
g ^ ( x ^ , y ^ , z 0 ) = g ( x ^ + x 0 , y ^ + y 0 , z 0 ) = g ( x , y , 0 ) h ( x ^ + x 0 x , y ^ + y 0 y , z 0 ) d x d y .
G ^ ( u , v ; z 0 ) = G ( u , v ; 0 ) H ^ ( u , v ; z 0 ) , H ^ ( u , v ; z 0 ) = H ( u , v ; z 0 ) exp [ i 2 π ( x 0 u + y 0 v ) ] = exp [ i 2 π ( x 0 u + y 0 v + z 0 w ) ]
g ^ ( x ^ , y ^ , z 0 ) = F 1 { G ^ ( u , v ; z 0 ) } .
H ^ ( u ; z 0 ) = exp [ i ϕ ( u ; z 0 ) ] , ϕ ( u ; z 0 ) = 2 π [ x 0 u + z 0 ( λ 2 u 2 ) 1 / 2 ] .
f u = 1 2 π ϕ u = x 0 u ( λ 2 u 2 ) 1 / 2 z 0 ,
Δ u 1 2 | f u | .
( x 0 1 2 Δ u ) 1 z 0 u ( λ 2 u 2 ) 1 / 2 ( x 0 + 1 2 Δ u ) 1 z 0
{ u limit ( ) < u < u limit ( + )  if S x < x 0 u limit ( ) < u < u limit ( + )  if S x x 0 < S x u limit ( ) < u < u limit ( + )  if x 0 S x
u limit ( ± ) [ ( x 0 ± 1 2 Δ u ) 2 z 0 2 + 1 ] 1 / 2 λ 1 .
H ^ ( u ; z 0 ) = H ^ ( u ; z 0 ) rect ( u u 0 u width ) ,
f u = 1 2 π ϕ u = x 0 u w z 0 ,       f v = 1 2 π ϕ v = y 0 v w z 0 ,
Δ u 1 2 | f u | ,
Δ v 1 2 | f v | ,
u 0   and   u 2 [ u limit ( ) ] 2 + v 2 λ 2 1   and   u 2 [ u limit ( + ) ] 2 + v 2 λ 2 1 ,
( u 2 [ u limit ( ) ] 2 + v 2 λ 2 1   and   u 0 )   or   ( u 2 [ u limit ( + ) ] 2 + v 2 λ 2 1   and   u > 0 ) ,
u 0   and   u 2 [ u limit ( + ) ] 2 + v 2 λ 2 1   and   u 2 [ u limit ( ) ] 2 + v 2 λ 2 1.
u 2 u limit 2 + v 2 λ 2 1 ,      u 2 λ 2 + v 2 v limit 2 1.
H ^ ( u , v ; z 0 ) = H ^ ( u , v ; z 0 ) rect ( u u 0 u width ) rect ( v v 0 v width ) ,
v limit ( ± ) [ ( y 0 ± 1 2 Δ v ) 2 z 0 2 + 1 ] 1 / 2 λ 1 .
sin θ max = ( x 0 + S x ) / [ z 0 + ( x 0 + S x ) 2 ] 1 / 2 , sin θ min = ( x 0 S x ) / [ z 0 + ( x 0 S x ) 2 ] 1 / 2 .
u high = sin θ max / λ = u limit ( + ) , u low = sin θ min / λ = u limit ( ) .

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