Abstract

The well-sampling conditions for the digital calculations of the scalar diffraction, including the Huygens convolution method (HCM), the angular spectrum method (ASM), and the Fresnel diffraction integral (FDI), were discussed. We found the aliasing always occurs unless proper zero-padding—that is, to pad zero-value pixels around the initial field—is applied prior to the simulation of the diffraction. From the aspect of well-sampling, the ASM is applicable to a short propagation distance, while the HCM is applicable to a long propagation distance. However, we found that the free-space point spread function in the HCM is low-pass filtered when the propagation distance is long. As a result, it is recommended to always use the ASM in conjunction with sufficient zero-padding for the digital calculation of the diffraction field. The FDI can be directly applied to a long-distance propagation without the necessity of the zero-padding, provided only the intensity is of interest. If the phase of the diffraction field is important, the zero-padding is necessary and the propagation distance is severely restricted.

© 2012 Optical Society of America

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References

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  1. T.-C. Poon, ed., Digital Holography and Three-Dimensional Display (Springer, 2006).
  2. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
    [CrossRef]
  3. L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929–5935 (2000).
    [CrossRef]
  4. T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
    [CrossRef]
  5. T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839(2002).
    [CrossRef]
  6. C.-S. Guo, L. Zhang, Z.-Y. Rong, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
    [CrossRef]
  7. A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
    [CrossRef]
  8. L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359–367(2007).
    [CrossRef]
  9. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25, 1744–1761 (2008).
    [CrossRef]
  10. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
    [CrossRef]
  11. S. G. Chamberlain and D. H. Harper, “MTF simulation including transmittance effects and experimental results of charge-coupled imagers,” IEEE Trans. Electron Devices 25, 145–154 (1978).
    [CrossRef]
  12. J. C. Feltz and M. A. Karim, “Modulation transfer function of charge-coupled devices,” Appl. Opt. 29, 717–722 (1990).
    [CrossRef]
  13. M. Marchywka and D. G. Socker, “Modulation transfer function measurement technique for small-pixel detectors,” Appl. Opt. 31, 7198–7213 (1992).
    [CrossRef]
  14. 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]
  15. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distanceand wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29, 854–856 (2004).
    [CrossRef]
  16. T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, 2007).
  17. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).
  18. J.-P. Liu and T.-C. Poon, “Two-step-only quadrature phase-shifting digital holography,” Opt. Lett. 34, 250–252 (2009).
    [CrossRef]
  19. J.-P. Liu, T.-C. Poon, G.-S. Jhou, and P.-J. Chen, “Comparison of two-, three-, and four-exposure quadrature phase-shifting holography,” Appl. Opt. 50, 2443–2450 (2011).
    [CrossRef]

2011 (1)

2009 (3)

2008 (1)

2007 (1)

2004 (2)

2003 (1)

C.-S. Guo, L. Zhang, Z.-Y. Rong, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
[CrossRef]

2002 (2)

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839(2002).
[CrossRef]

2000 (1)

1997 (1)

1992 (1)

1990 (1)

1978 (1)

S. G. Chamberlain and D. H. Harper, “MTF simulation including transmittance effects and experimental results of charge-coupled imagers,” IEEE Trans. Electron Devices 25, 145–154 (1978).
[CrossRef]

Alfieri, D.

Chamberlain, S. G.

S. G. Chamberlain and D. H. Harper, “MTF simulation including transmittance effects and experimental results of charge-coupled imagers,” IEEE Trans. Electron Devices 25, 145–154 (1978).
[CrossRef]

Chen, P.-J.

Coppola, G.

De Nicola, S.

Feltz, J. C.

Ferraro, P.

Finizio, A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).

Guo, C.-S.

C.-S. Guo, L. Zhang, Z.-Y. Rong, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
[CrossRef]

Harper, D. H.

S. G. Chamberlain and D. H. Harper, “MTF simulation including transmittance effects and experimental results of charge-coupled imagers,” IEEE Trans. Electron Devices 25, 145–154 (1978).
[CrossRef]

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Javidi, B.

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Jhou, G.-S.

Karim, M. A.

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Kreis, T. M.

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839(2002).
[CrossRef]

Lalanne, P.

Leval, J.

Liu, J.-P.

Marchywka, M.

Naughton, T. J.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Onural, L.

Pandey, N.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Picart, P.

Pierattini, G.

Poon, T.-C.

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Roggemann, M. C.

Rong, Z.-Y.

C.-S. Guo, L. Zhang, Z.-Y. Rong, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
[CrossRef]

Socker, D. G.

Stern, A.

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Voelz, D. G.

Wang, H.-T.

C.-S. Guo, L. Zhang, Z.-Y. Rong, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
[CrossRef]

Zhang, L.

C.-S. Guo, L. Zhang, Z.-Y. Rong, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Electron Devices (1)

S. G. Chamberlain and D. H. Harper, “MTF simulation including transmittance effects and experimental results of charge-coupled imagers,” IEEE Trans. Electron Devices 25, 145–154 (1978).
[CrossRef]

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

Opt. Eng. (5)

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839(2002).
[CrossRef]

C.-S. Guo, L. Zhang, Z.-Y. Rong, and H.-T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the papers ‘Frequency analysis of digital holography’ and ‘Frequency analysis of digital holography with reconstruction by convolution’,” Opt. Eng. 42, 2768–2771 (2003).
[CrossRef]

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Opt. Lett. (2)

Other (3)

T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, 2007).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).

T.-C. Poon, ed., Digital Holography and Three-Dimensional Display (Springer, 2006).

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Figures (13)

Fig. 1.
Fig. 1.

(a) Geometry for the scalar diffraction, and (b) the corresponding parameters in the digital calculation.

Fig. 2.
Fig. 2.

Critical propagation distance as a function of the sampling period for the ASM.

Fig. 3.
Fig. 3.

Modulus of the diffraction field calculated using the HCM (left column) and using the ASM (right column). The propagation distances are 1.24 cm [(a) and (b)], 3.24 cm [(c) and (d)], and 7.24 cm [(e) and (f)], respectively.

Fig. 4.
Fig. 4.

Modulus of the diffraction field at z=7.24cm, by using (a) the truncated transfer function and with normal zero-padding (μ0=2), and (b) the normal transfer function and with proper zero-padding (μ0=4).

Fig. 5.
Fig. 5.

Cross section of Fig. 4(b) and the corresponding result obtained from the numerical integration.

Fig. 6.
Fig. 6.

Modulus of the diffraction field calculated using the ASM at z=3.24cm: (a) without zero-padding, and (b) without zero-padding but with double sampling density (Δ0=5λ).

Fig. 7.
Fig. 7.

Geometry for the cascaded free-space propagation (a), and the correct procedure in digital calculation (b).

Fig. 8.
Fig. 8.

Modulus of the diffraction field calculated by the cascaded free-space propagation. (a) The zero-padding is performed before each propagation; (b) The zero-padding is only performed before the first propagation.

Fig. 9.
Fig. 9.

Simulations of the FDI. (a) The original object; (b)–(d) modulus of the diffraction field at z=1cm (b), z=6.14cm (c), and z=20cm (d), respectively.

Fig. 10.
Fig. 10.

Modulus of the diffraction field at z=1cm calculated by the FDI (a) with zero-padding (μ0=4), and (b) without zero-padding but with four times sampling density (Δ0=2.5λ).

Fig. 11.
Fig. 11.

Cross section of Fig. 9(c) and the corresponding result obtained from the numerical integration.

Fig. 12.
Fig. 12.

Real part of selected region of the diffraction field at z=6.14cm (a) without zero-padding, and (b) with zero-padding (μ0=4).

Fig. 13.
Fig. 13.

Intensity of the reconstructed images by using (a) FDI; (b) HCM (μ0=2); (c) ASM (μ0=1), and (d) ASM (μ0=4). For the methods with zero-padding, only the central 512×512 pixels of the reconstructed image are shown for comparison.

Tables (1)

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Table 1. Simulation Conditions for Achieving Minimized Aliasing

Equations (30)

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u(x,y;z)=u0(x,y)h(x,y;z),
F{u(x,y;z)}=F{u0(x,y)}×H(fx,fy;z),
h(x,y;z)=zjλ(x2+y2+z2)exp(j2πλx2+y2+z2),
H(fx,fy;z)=exp[j2πz(1/λ)2fr2]forfr1/λ=0otherwise,
u(x,y;z)=F1{F{u0(x,y)}×F{h(x,y;z)}},
u(x,y;z)=F1{F{u0(x,y)}×H(fx,fy;z)},
uz(m,n)=Fd1{Fd{u0(m,n)}×Fd{hz(m,n)}},
uz(m,n)=Fd1{Fd{u0(m,n)}×Hz(k,l)},
Fd{}=m=M/2M/21n=M/2M/21exp[j2π(mk+nl)/M],
Fd1{}=1M2k=M/2M/21l=M/2M/21exp[j2π(mk+nl)/M].
hz(m,n)=zjλ(m2Δ02+n2Δ02+z2)exp(j2πλm2Δ02+n2Δ02+z2),
Hz(k,l)=exp[j2πz(1λ)2(kMΔ0)2(lMΔ0)2],
h(x,y;z)ej2πz/λjλzexp(jπx2+y2λz).
u(x,y;z)=ej2πz/λjλzejπx2+y2λzu0(x0,y0)ejπx02+y02λzej2π(xx0+yy0)/λzdx0dy0=ej2πz/λjλzejπx2+y2λzF{u0(x0,y0)ejπx02+y02λz}fx=x/λzfy=y/λz.
Δ=λzMΔ0.
uz(k,l)=ej2πz/λjλzϕ2(k,l)×Fd{u0(m,n)×ϕ1(m,n)},
ϕ1(m,n)=exp[jπ(mΔ0)2+(nΔ0)2λz]andϕ2(k,l)=exp{jπλz[(kMΔ0)2+(lMΔ0)2]}.
|12πddr[2πλr2+z2]max|12Δ0,
z[r2(4Δ02λ21)]max.
zM0Δ024Δ02λ21.
M0Δ02+M0Δ0212Δf,
M0M/2.
M0Δ02+|12πddfr[2πz(1/λ)2fr2]max|12Δf,
zM0Δ02(μ01)4Δ02λ21.
μf=[λ2z2Δ04M02(μ01)2+λ24Δ02]1/2,
M0Δ02λz+14Δ012Δ0,
z2M0Δ02λ.
12μ0Δ+MwΔ2λz12Δ.
zμwM0Δ02λ(μ01),
μw(μ01)2.

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