Abstract

Band-limited angular spectrum (BLAS) methods can be used for simulating the diffractional propagation in the near field, the far field, the tilted system, and the nonparaxial system. However, it does not allow free sample interval on the output calculation window. In this paper, an improved BLAS method is proposed. This new algorithm permits a selective scaling of observation window size and sample number on the observation plane. The method is based on the linear convolution, which can be calculated by fast Fourier transform effectively.

© 2012 Optical Society of America

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References

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  1. D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
    [CrossRef]
  2. D. G. Voelz and M. C. Roggemann, “Digital simulation of scalar optical diffraction revisiting chirp function sampling criteria and consequences,” Appl. Opt. 48, 6132–6142 (2009).
    [CrossRef]
  3. T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
    [CrossRef]
  4. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh–Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
    [CrossRef]
  5. V. Nascov and P. C. Logofătu, “Fast computation algorithm for the Rayleigh–Sommerfeld diffraction formula using a type of scaled convolution,” Appl. Opt. 48, 4310–4319 (2009).
    [CrossRef]
  6. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (SPIE, 2010).
  7. C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A 23, 1616–1625 (2006).
    [CrossRef]
  8. T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17, 556–558 (1992).
    [CrossRef]
  9. T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. A 10, 299–305 (1993).
    [CrossRef]
  10. 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. A 20, 1755–1762 (2003).
    [CrossRef]
  11. D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
    [CrossRef]
  12. 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]
  13. K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation,” Opt. Express 18, 18453–18463(2010).
    [CrossRef]
  14. S. Haykin and B. Van Veen, Signals and Systems (Wiley, 2005).
  15. L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. A 24, 359–367 (2007).
    [CrossRef]
  16. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef]
  17. D. Wang, J. Zhao, F. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47, D12–D20 (2008).
    [CrossRef]
  18. C. L. Phillips, J. M. Parr, and E. Ann Riskin, Signals, Systems, and Transforms (Pearson, 2003).
  19. D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).
  20. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).
  21. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef]
  22. J. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
    [CrossRef]
  23. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

2010

2009

2008

2007

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. A 24, 359–367 (2007).
[CrossRef]

2006

2003

J. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (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. A 20, 1755–1762 (2003).
[CrossRef]

2002

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

1997

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

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

1993

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. A 10, 299–305 (1993).
[CrossRef]

1992

1975

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]

Ann Riskin, E.

C. L. Phillips, J. M. Parr, and E. Ann Riskin, Signals, Systems, and Transforms (Pearson, 2003).

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Bengtsson, J.

Bianco, B.

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. A 10, 299–305 (1993).
[CrossRef]

T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17, 556–558 (1992).
[CrossRef]

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Curtis, J.

J. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

Franke-Arnold, S.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Grier, D. G.

J. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

Haykin, S.

S. Haykin and B. Van Veen, Signals and Systems (Wiley, 2005).

Hodgson, N.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

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]

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

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]

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Logofatu, P. C.

Matsushima, K.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Nascov, V.

Onural, L.

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. A 24, 359–367 (2007).
[CrossRef]

Osten, W.

Padgett, M. J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Parr, J. M.

C. L. Phillips, J. M. Parr, and E. Ann Riskin, Signals, Systems, and Transforms (Pearson, 2003).

Pedrini, G.

Phillips, C. L.

C. L. Phillips, J. M. Parr, and E. Ann Riskin, Signals, Systems, and Transforms (Pearson, 2003).

Roggemann, M. C.

Rydberg, C.

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. A 20, 1755–1762 (2003).
[CrossRef]

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (SPIE, 2010).

Shen, F.

Shimobaba, T.

Siegman, A. E.

Sziklas, E. A.

Tommasi, T.

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. A 10, 299–305 (1993).
[CrossRef]

T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17, 556–558 (1992).
[CrossRef]

Van Veen, B.

S. Haykin and B. Van Veen, Signals and Systems (Wiley, 2005).

Voelz, D. G.

Wang, A.

Wang, D.

Weber, H.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

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. A 20, 1755–1762 (2003).
[CrossRef]

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Zhang, F.

Zhao, J.

Appl. Opt.

J. Mod. Opt.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

J. Opt. Soc. A

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. A 10, 299–305 (1993).
[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. A 20, 1755–1762 (2003).
[CrossRef]

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. A 24, 359–367 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

J. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

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

C. L. Phillips, J. M. Parr, and E. Ann Riskin, Signals, Systems, and Transforms (Pearson, 2003).

D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (SPIE, 2010).

S. Haykin and B. Van Veen, Signals and Systems (Wiley, 2005).

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Figures (6)

Fig. 1.
Fig. 1.

Coordinate systems and the geometry of the model.

Fig. 2.
Fig. 2.

One-dimensional geometric parameters.

Fig. 3.
Fig. 3.

Replicas of the physical window in the input plane.

Fig. 4.
Fig. 4.

Aliasing error from the replicas of the physical window.

Fig. 5.
Fig. 5.

Setup for (a) numerical simulation of the scaling BLAS method, (b) the amplitude distribution of the input field, and (c) the phase distribution of the input field.

Fig. 6.
Fig. 6.

Simulation result for the normal BLAS method and the scaling BLAS method.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

U(x,y,z)=ΣU(x1,y1,0)exp(ikr)zr(1rik)dx1dy1,
A(u,v,z)=A(u,v,0)G(u,v,z),
A(u,v,0)=U(x1,y1,0)exp[i2π(ux1+vy1)]dx1dy1,
G(u,v,z)=exp[i2πz(λ2u2v2)1/2].
U(x,y,z)=A(u,v,z)exp(i2π(ux+vy))dudv.
A(un,z)=A1(un,0)exp[i2πz(λ2un2)1/2],
U(xm,z)=n=N/2N/21A(un,z)exp(i2πunxm)·Δu,forxm=mΔxandm=M2,,M21.
exp(i2πunxm)=exp[i2πα(αunxm)]=exp[iπα(αunxm)2]exp{iπα[(αun)2+xm2]}.
U(xm,z)=exp(iπαxm2)n=N/2N/211αA(un,z)exp[iπα(αun)2]exp[iπα(αunxm)2]·(α·Δu)=exp(iπαxm2)·Δwn=N/2N/21B(wn)exp[iπα(wnxm)2]=exp(iπαxm2)·Δw[B(wn)*f(wn)],
Lx=Wx+Sx.
Lx=hWx,h=2or3.
Lx=max(Wx+Sx,hWx).
uneed=1λ[(2z/(Wx+Sx))2+1]1/2.
umax1λ[(2zΔu)2+1]1/2,
umaxuneed.
G(u,z)=exp[i2πz(λ2u2)1/2]rect(u/(2uneed)).
C=O(N2log2N)+O((M+N1)2log2(M+N1))+O((M+N1)2)+O(M2)+O(N2).
A1=A0rexp(r2/2a2)exp(iθ),

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