Abstract

In numerical wave optic simulations using the conventional angular spectrum method, numerical inter-periodic interference occurs as a result of the inherent periodicity of the discrete Fourier transform. In this paper, we propose an aperiodic angular spectrum representation of a scalar optical wave field without numerical inter-periodic interference in an aperiodic area. The visualization of holographic three-dimensional image light field is presented for the validity of the proposed method. It is believed that the proposed method can be broadly applied to visualization of scalar optical wave field represented by the angular spectrum plane wave bases.

© 2017 Optical Society of America

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References

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  1. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Reberts and Company Publishers, 2004).
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    [Crossref] [PubMed]
  5. Y. Lim, J. Hahn, S. Kim, J. Park, H. Kim, and B. Lee, “Plasmonic light beaming manipulation and its detection using holographic microscopy,” IEEE J. Quantum Electron. 46(3), 300–305 (2010).
    [Crossref]
  6. L. Yu and M. K. Kim, “Wavelength-scanning digital interference holography for tomographic three-dimensional imaging by use of the angular spectrum method,” Opt. Lett. 30(16), 2092–2094 (2005).
    [Crossref] [PubMed]
  7. J. Hahn, H. Kim, S.-W. Cho, and B. Lee, “Phase-shifting interferometry with genetic algorithm-based twin image noise elimination,” Appl. Opt. 47(22), 4068–4076 (2008).
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  8. E. Ye, A. H. Atabaki, N. Han, and R. J. Ram, “Miniature, sub-nanometer resolution Talbot spectrometer,” Opt. Lett. 41(11), 2434–2437 (2016).
    [Crossref] [PubMed]
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  10. T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37(19), 4128–4130 (2012).
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  11. C.-Y. Hwang, S. Oh, I.-K. Jeong, and H. Kim, “Stepwise angular spectrum method for curved surface diffraction,” Opt. Express 22(10), 12659–12667 (2014).
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  12. A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47(2-3), 317–374 (1896).
    [Crossref]
  13. J. P. Berenger, “Perfect matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antenn. Propag. 44(1), 110–117 (1996).
    [Crossref]
  14. J.-P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22(9), 1844–1849 (2005).
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  15. Y. Umul, “Modified theory of physical optics,” Opt. Express 12(20), 4959–4972 (2004).
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  16. Y. Z. Umil, “Three dimensional modified theory of physical optics,” Optik-Int. J. Light Elect. Opt. 127(2), 819–824 (2016).
    [Crossref]
  17. Y. Umul, “Diffraction by a black half plane: Modified theory of physical optics approach,” Opt. Express 13(19), 7276–7287 (2005).
    [Crossref] [PubMed]

2016 (2)

Y. Z. Umil, “Three dimensional modified theory of physical optics,” Optik-Int. J. Light Elect. Opt. 127(2), 819–824 (2016).
[Crossref]

E. Ye, A. H. Atabaki, N. Han, and R. J. Ram, “Miniature, sub-nanometer resolution Talbot spectrometer,” Opt. Lett. 41(11), 2434–2437 (2016).
[Crossref] [PubMed]

2014 (1)

2012 (1)

2011 (1)

2010 (1)

Y. Lim, J. Hahn, S. Kim, J. Park, H. Kim, and B. Lee, “Plasmonic light beaming manipulation and its detection using holographic microscopy,” IEEE J. Quantum Electron. 46(3), 300–305 (2010).
[Crossref]

2009 (1)

2008 (2)

2005 (3)

2004 (2)

1996 (1)

J. P. Berenger, “Perfect matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antenn. Propag. 44(1), 110–117 (1996).
[Crossref]

1896 (1)

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47(2-3), 317–374 (1896).
[Crossref]

Atabaki, A. H.

Barnett, S.

Berenger, J. P.

J. P. Berenger, “Perfect matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antenn. Propag. 44(1), 110–117 (1996).
[Crossref]

Brady, D. J.

Cho, S.-W.

Choi, K.

Courtial, J.

Franke-Arnold, S.

Gibson, G.

Hahn, J.

Han, N.

Horisaki, R.

Hugonin, J.-P.

Hwang, C.-Y.

Ito, T.

Jeong, I.-K.

Kakue, T.

Kim, H.

Kim, M. K.

Kim, S.

Y. Lim, J. Hahn, S. Kim, J. Park, H. Kim, and B. Lee, “Plasmonic light beaming manipulation and its detection using holographic microscopy,” IEEE J. Quantum Electron. 46(3), 300–305 (2010).
[Crossref]

Lalanne, P.

Lee, B.

Lim, S.

Lim, Y.

Y. Lim, J. Hahn, S. Kim, J. Park, H. Kim, and B. Lee, “Plasmonic light beaming manipulation and its detection using holographic microscopy,” IEEE J. Quantum Electron. 46(3), 300–305 (2010).
[Crossref]

Masuda, N.

Matsushima, K.

Nakahara, S.

Oh, S.

Padgett, M.

Park, J.

Y. Lim, J. Hahn, S. Kim, J. Park, H. Kim, and B. Lee, “Plasmonic light beaming manipulation and its detection using holographic microscopy,” IEEE J. Quantum Electron. 46(3), 300–305 (2010).
[Crossref]

Pas’ko, V.

Ram, R. J.

Shimobaba, T.

Sommerfeld, A.

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47(2-3), 317–374 (1896).
[Crossref]

Umil, Y. Z.

Y. Z. Umil, “Three dimensional modified theory of physical optics,” Optik-Int. J. Light Elect. Opt. 127(2), 819–824 (2016).
[Crossref]

Umul, Y.

Vasnetsov, M.

Ye, E.

Yu, L.

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

Y. Lim, J. Hahn, S. Kim, J. Park, H. Kim, and B. Lee, “Plasmonic light beaming manipulation and its detection using holographic microscopy,” IEEE J. Quantum Electron. 46(3), 300–305 (2010).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

J. P. Berenger, “Perfect matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antenn. Propag. 44(1), 110–117 (1996).
[Crossref]

J. Opt. Soc. Am. A (1)

Math. Ann. (1)

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47(2-3), 317–374 (1896).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Optik-Int. J. Light Elect. Opt. (1)

Y. Z. Umil, “Three dimensional modified theory of physical optics,” Optik-Int. J. Light Elect. Opt. 127(2), 819–824 (2016).
[Crossref]

Other (1)

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Reberts and Company Publishers, 2004).

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Figures (5)

Fig. 1
Fig. 1

Optical field visualization using the conventional angular spectrum method: (a) a uniform directional plane wave basis in a rectangular computation region, (b) diffraction field U( x,y,z ) for a single slit aperture contaminated by numerical inter-periodic interference, (c) a plane wave basis and (d) diffraction field distribution extended in three-period domain. In (b) and (d), each silt has a width of 5 μm , and the wavelength is set to 633 nm . In (c) and (d), the x-axis period for the unit computational domain is set to 20 μm .

Fig. 2
Fig. 2

Numerical structure of the proposed aperiodic angular spectrum representation using an MTPO wave function for a half-infinite plane: (a) diffraction wave departing from a finite bottom region with a vertical half-infinite barrier ( 0z , x= T x /2 ), (b) diffraction wave rotated by the azimuthal angle ϕ 0 , and (c) the cylindrical computation volume for the aperiodic angular spectrum representation.

Fig. 3
Fig. 3

Cross-sectional plots of U( x,y,z ) 3D field visualization of optical wave fields obtained through (a) the conventional angular spectrum method and (b) the proposed aperiodic angular spectrum method. The x-z cross-section field distributions of each method are plotted in (c) and (d) respectively. The dashed red circle at the bottom indicates the trajectory of the tangent point of the barrier wall.

Fig. 4
Fig. 4

Numerical reconstruction of diffraction fields from a binary amplitude grating pattern using (a) the conventional angular spectrum method and (b) the proposed aperiodic angular spectrum method. The period and fill factor of the grating pattern are 1um and 0.5, respectively.

Fig. 5
Fig. 5

Numerical reconstruction of a digital hologram using (a) the conventional angular spectrum method and (b) the proposed aperiodic angular spectrum method. The complex hologram pattern is placed at z=0um . The target images ‘A’ and ‘B’ are focused, respectively, at z=15um and at z=30um .

Equations (19)

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2 U( x,y,z )+ k 0 2 U( x,y,z )=0,
U( x,y,z )= 1 4 π 2 A( k x , k y ) e j ( k 0 ) 2 k x 2 k y 2 z e j( k x x+ k y y ) d k x d k y ,
U pro ( x,y,z )= 1 4 π 2 k x 2 + k y 2 k 0 2 A( k x , k y ) e j ( k 0 ) 2 k x 2 k y 2 z e j( k x x+ k y y ) d k x d k y ,
U eva ( x,y,z )= 1 4 π 2 k x 2 + k y 2 > k 0 2 A( k x , k y ) e k x 2 + k y 2 ( k 0 ) 2 z e j( k x x+ k y y ) d k x d k y .
U pro ( pΔx,lΔy,qΔz )= Δ k x Δ k y 4 π 2 n= N1 2 N1 2 m= M1 2 M1 2 A( mΔ k x ,nΔ k y ) e j ( k 0 ) 2 ( mΔ k x 2 )( nΔ k y 2 ) qΔz e j( mΔ k x pΔx+nΔ k y lΔy ) ,
U( x )= U 0 rect( x/W ).
u is ( P )= u 0 exp[ j k 0 { ( x+ T x /2 )sin θ 0 +zcos θ 0 } ]F[ ξ_ ],
F[ χ ]= e j/π4 π χ e j τ 2 dτ,
ξ = k 0 ( z 2 + ( x+ T x /2 ) 2 ) 1/2 +zcos θ 0 +( x+ T x /2 )sin θ 0 ×S[ tan 1 (x+ T x /2) z , θ 0 ].
S[ θ, θ 0 ]=sign( θ(π θ 0 ) )sign( (3π θ 0 )θ )
( x y z )=( cos ϕ 0 sin ϕ 0 0 sin ϕ 0 cos ϕ 0 0 0 0 1 )( x y z ),
u is ( P )= u 0 exp( j k 0 { x'sin θ 0 cos ϕ 0 +y'sin θ 0 sin ϕ 0 +z'cos θ 0 +( T x /2 )sin ϕ 0 } )F[ ξ_ ]
ξ_= k ( z 2 2 + ( x'cos ϕ 0 +y'sin ϕ 0 + T x /2 ) 2 ) 1/2 +z'cos θ 0 +( x'cos ϕ 0 +y'sin ϕ 0 + T x /2 )sin θ 0 ×S[ tan 1 ( ( x'cos ϕ 0 +y'sin ϕ 0 + T x /2 ) ) z' , θ 0 ].
u is ( P )=g( P )exp( jh( P ) ).
u ˜ is ( P )= e jβ ×u is ( P ).
u b ( x,y,z, k x , k y )={ u ˜ is ( x,y,z, k x , k y ), otherwise exp( j k z z ), k x = k y =0 .
u is ( x,y,z, k x , k y )= u 0 exp[ j{ k x x+ k y y+ k z z+( T x /2 ) k y k x + k y } ]F[ ξ_ ]
ξ_= k 0 ( ( z ) 2 + ( ( k x x+ k y y )/ k x + k y + T x /2 ) 2 ) 1/2 +z( k z / k 0 )+( ( k x x+ k y y )/ k 0 + T x k x + k y /( 2 k 0 ) ) . ×S[ z,( ( k x x+ k y y )/ k x + k y + T x /2 ), tan 1 ( k x + k y / k z ) ]
U( x,y,z )= 1 4 π 2 k x 2 + k y 2 k 0 2 A( k x , k y ) u b ( x,y,z, k x , k y )d k x d k y + 1 4 π 2 k x 2 + k y 2 > k 0 2 A( k x , k y ) e k x 2 + k y 2 ( k 0 ) 2 z e j( k x x+ k y y ) d k x d k y .

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