Abstract

The Maggi–Rubinowicz method (MRM) is a useful tool to compute diffraction patterns from opaque planar objects. We adapted the MRM to planar rectangles. In the first part of this study, differences between diffraction patterns, both the intensity and the phase distributions, from a tilted rectangle and from the square having the same orthogonal projection on the observation plane, have been analyzed. In the second part, we compared results obtained with the MRM to those obtained with angular spectrum theory (AST) coupled to fast Fourier transform (FFT). The main novelty of this work is the fact that MRM is particularly well suited for evaluating anti-aliasing procedures applied to AST-FFT calculations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), 7th ed.
    [Crossref]
  2. J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, 1996), 2nd ed.
  3. O. K. Ersoy, Diffraction, Fourier optics and imaging (John Wiley & Sons, 2007).
    [Crossref]
  4. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part II,” J. Opt. Soc. Am. 52, 626–636 (1962).
    [Crossref]
  5. 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. 20, 1755–1762 (2003).
    [Crossref]
  6. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
    [Crossref] [PubMed]
  7. R. L. Lucke, “Rayleigh–Sommerfeld diffraction and Poisson’s spot,” Eur. J. Phys. 27, 193–204 (2006).
    [Crossref]
  8. T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Inf. 12, 1611–1622 (2016).
    [Crossref]
  9. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [Crossref]
  10. J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [Crossref]
  11. L. Moisan, “Periodic plus smooth image decomposition,” J. Math. Imaging Vis. 39, 161–179 (2011).
    [Crossref]
  12. F. Mahmood, M. Toots, L.-G. Öfverstedt, and U. Skoglund, “2d discrete Fourier transform with simultaneous edge artifact removal for real-time applications,” in 2015 International Conference on Field Programmable Technology (FPT) (2015), pp. 236–239.
    [Crossref]

2016 (1)

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Inf. 12, 1611–1622 (2016).
[Crossref]

2011 (1)

L. Moisan, “Periodic plus smooth image decomposition,” J. Math. Imaging Vis. 39, 161–179 (2011).
[Crossref]

2009 (1)

2006 (1)

R. L. Lucke, “Rayleigh–Sommerfeld diffraction and Poisson’s spot,” Eur. J. Phys. 27, 193–204 (2006).
[Crossref]

2003 (1)

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. 20, 1755–1762 (2003).
[Crossref]

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

1965 (1)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[Crossref]

1962 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), 7th ed.
[Crossref]

Cooley, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[Crossref]

Ersoy, O. K.

O. K. Ersoy, Diffraction, Fourier optics and imaging (John Wiley & Sons, 2007).
[Crossref]

Goodman, J. W.

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

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

Ito, T.

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Inf. 12, 1611–1622 (2016).
[Crossref]

Kakue, T.

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Inf. 12, 1611–1622 (2016).
[Crossref]

Lucke, R. L.

R. L. Lucke, “Rayleigh–Sommerfeld diffraction and Poisson’s spot,” Eur. J. Phys. 27, 193–204 (2006).
[Crossref]

Mahmood, F.

F. Mahmood, M. Toots, L.-G. Öfverstedt, and U. Skoglund, “2d discrete Fourier transform with simultaneous edge artifact removal for real-time applications,” in 2015 International Conference on Field Programmable Technology (FPT) (2015), pp. 236–239.
[Crossref]

Matsushima, K.

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
[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. 20, 1755–1762 (2003).
[Crossref]

Miyamoto, K.

Moisan, L.

L. Moisan, “Periodic plus smooth image decomposition,” J. Math. Imaging Vis. 39, 161–179 (2011).
[Crossref]

Öfverstedt, L.-G.

F. Mahmood, M. Toots, L.-G. Öfverstedt, and U. Skoglund, “2d discrete Fourier transform with simultaneous edge artifact removal for real-time applications,” in 2015 International Conference on Field Programmable Technology (FPT) (2015), pp. 236–239.
[Crossref]

Schimmel, H.

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. 20, 1755–1762 (2003).
[Crossref]

Shimobaba, T.

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Inf. 12, 1611–1622 (2016).
[Crossref]

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
[Crossref] [PubMed]

Skoglund, U.

F. Mahmood, M. Toots, L.-G. Öfverstedt, and U. Skoglund, “2d discrete Fourier transform with simultaneous edge artifact removal for real-time applications,” in 2015 International Conference on Field Programmable Technology (FPT) (2015), pp. 236–239.
[Crossref]

Toots, M.

F. Mahmood, M. Toots, L.-G. Öfverstedt, and U. Skoglund, “2d discrete Fourier transform with simultaneous edge artifact removal for real-time applications,” in 2015 International Conference on Field Programmable Technology (FPT) (2015), pp. 236–239.
[Crossref]

Tukey, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[Crossref]

Wolf, E.

Wyrowski, F.

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. 20, 1755–1762 (2003).
[Crossref]

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

Eur. J. Phys. (1)

R. L. Lucke, “Rayleigh–Sommerfeld diffraction and Poisson’s spot,” Eur. J. Phys. 27, 193–204 (2006).
[Crossref]

IEEE Trans. Ind. Inf. (1)

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Inf. 12, 1611–1622 (2016).
[Crossref]

J. Math. Imaging Vis. (1)

L. Moisan, “Periodic plus smooth image decomposition,” J. Math. Imaging Vis. 39, 161–179 (2011).
[Crossref]

J. Opt. Soc. Am. (2)

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part II,” J. Opt. Soc. Am. 52, 626–636 (1962).
[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. 20, 1755–1762 (2003).
[Crossref]

Math. Comput. (1)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[Crossref]

Opt. Express (1)

Other (4)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), 7th ed.
[Crossref]

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

O. K. Ersoy, Diffraction, Fourier optics and imaging (John Wiley & Sons, 2007).
[Crossref]

F. Mahmood, M. Toots, L.-G. Öfverstedt, and U. Skoglund, “2d discrete Fourier transform with simultaneous edge artifact removal for real-time applications,” in 2015 International Conference on Field Programmable Technology (FPT) (2015), pp. 236–239.
[Crossref]

Supplementary Material (4)

NameDescription
» Visualization 1       Intensity diffraction pattern from a tilted 100 x 200 µm opaque rectangle at a distance Z = 0.5 cm from the object as a function of tilt angle.
» Visualization 2       Phase diffraction pattern from a tilted 100 x 200 µm opaque rectangle at a distance Z = 0.5 cm from the object as a function of tilt angle.
» Visualization 3       Intensity diffraction pattern from a tilted 100 x 200 µm opaque rectangle at a distance Z = 2.5 cm from the object as a function of tilt angle.
» Visualization 4       Phase diffraction pattern from a tilted 100 x 200 µm opaque rectangle at a distance Z = 2.5 cm from the object as a function of tilt angle.

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Figures (5)

Fig. 1
Fig. 1 (a) Scheme of an incident plane wave perpendicular to an opaque planar object of arbitrary shape producing a diffraction pattern at a distance Z. (b) Notations for opaque disc. (c) Notations for opaque rectangle.
Fig. 2
Fig. 2 Diffraction pattern (a) from an opaque disc with radius R at distance Z such that the Fresnel number NF = R2/() = 1 and (b) from an opaque rectangle with width 2a = b and length 2b at distance Z such that the Fresnel number NF = b2/() = 1. Due to diffraction, a bright spot —called Poisson’s spot— appears in the center of the disc shadow. The orange dashed line in the disc shadow represents the outer limit of 50 % intensity threshold. The number of bright spots in the rectangle shadow (two spots here) varies with NF.
Fig. 3
Fig. 3 (a) Intensity diffraction pattern from the 100 × 200 μm opaque rectangle at the distance Z = 2.5 cm with the tilt angle θ = 60°. (b) Difference between (a) and the intensity diffraction pattern of the corresponding opaque square of 100 × 100 μm (see text). (c) As (a) but for the phase distribution. (d) As (b) but for the difference of the phase distributions.
Fig. 4
Fig. 4 As Fig. 3 at Z = 0.5 cm.
Fig. 5
Fig. 5 (a) Diffraction image of a 200 × 400 μm opaque rectangle at Z = 2.5 cm computed with the angular spectrum theory. (b) Intensity profile along the y-axis of this image (red) compared to the one obtained with the Maggi-Rubinowicz method (blue). (c) As (b) for the x-axis. (d,e,f) As (a,b,c) with low-pass filter. The root mean square deviation (RMSD), expressed in light intensity percentage, between the MRM and the AST images is given for both comparisons.

Equations (15)

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U d ( Q ) = U g ( Q ) + U b ( Q ) ,
U g ( Q ) = { U ( Q ) if Q direct beam 0 if Q geometrical shadow ,
U b ( Q ) = 1 4 π Γ e i k k p R e i k S S ( s k ) l 1 s k d l .
U b ( Q ) = 1 4 π Γ e i k S n s S Z d l .
U b ( Q ) = 1 4 π 0 2 π e i k S ( R 2 R r cos α ) S ( S Z ) d α ,
S = R 2 + Z 2 + r 2 2 R r cos α .
U b ( Q ) = 1 4 π 0 2 π e i k S ( 1 + Z S + R r cos α r 2 S ( S Z ) ) d α .
U b ( Q ) = 1 2 e i k S ( 1 + Z S ) ,
U b ( Q ) = 1 4 π ( b b e i k S ( y Q y P ) S ( S Z ) d x P + a a e i k S ( x P x Q ) S ( S Z ) d y P + b b e i k S ( y P y Q ) S ( S Z ) d x P + a a e i k S ( x Q x P ) S ( S Z ) d y P ) ,
S = ( x Q x P ) 2 + ( y Q y P ) 2 + Z 2 .
U t ( x , y , 0 ) = t A ( x , y ) U i ( x , y , 0 ) .
t A ( x , y ) = { 1 if outside the opaque object 0 if in the opaque object .
A ( f x , f y , 0 ) = + U t ( x , y , 0 ) e i 2 π ( f x x + f y y ) d x d y .
A ( f x , f y , z ) = A ( f x , f y , 0 ) H ( f x , f y ) ,
U ( x , y , z ) = + A ( f x , f y , z ) e i 2 π ( f x x + f y y ) d f x d f y .

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