Abstract

In this paper, we present sampling conditions for fast-Fourier-transform-based field propagations. The input field and the propagation kernel are analyzed in a combined manner to derive sampling criteria that guarantee accurate calculation results in the output plane. These sampling criteria are also applicable to the propagation of general fields. For focal field calculations, geometrical optics is used to obtain a priori knowledge about the input and output fields. This a priori knowledge is used to determine an optimum balance between computational load and calculation accuracy. In a numerical example, correct results are obtained even though both the input field and the propagation kernel are sampled below the Nyquist rate. Finally, we show how chirp z-transform-based zoom-algorithms may be analyzed using the same techniques.

© 2014 Optical Society of America

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2014 (2)

2013 (1)

2012 (3)

2011 (1)

2009 (5)

2008 (3)

L. F. Shampine, “MATLAB program for quadrature in 2D,” Appl. Math. Comput. 202, 266–274 (2008).
[CrossRef]

T. Kozacki, “Numerical errors of diffraction computing using plane wave spectrum decomposition,” Opt. Commun. 281, 4219–4223 (2008).
[CrossRef]

J. J. Braat, S. van Haver, A. J. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” Prog. Opt. 51, 349–468 (2008).
[CrossRef]

2007 (1)

2006 (5)

F. B. Shen and A. B. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh–Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
[CrossRef]

D. P. Kelly, W. T. Rhodes, J. T. Sheridan, and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

F. Zhang, G. Pedrini, and W. Osten, “Reconstruction algorithm for high-numerical-aperture holograms with diffraction-limited resolution,” Opt. Lett. 31, 1633–1635 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part I: 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part II: 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

2004 (1)

2003 (1)

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

2000 (1)

1999 (1)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

1998 (1)

1997 (3)

1996 (1)

1970 (1)

L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

1969 (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Syst. Tech. J. 48, 1249–1292 (1969).
[CrossRef]

1967 (1)

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]

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Bluestein, L.

L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

Braat, J. J.

J. J. Braat, S. van Haver, A. J. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” Prog. Opt. 51, 349–468 (2008).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and Its Applications, Prentice-Hall Signal Processing Series (Prentice Hall, 1988).

Delen, N.

Dirksen, P.

J. J. Braat, S. van Haver, A. J. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” Prog. Opt. 51, 349–468 (2008).
[CrossRef]

Dorsch, R. G.

Falaggis, K.

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Hao, P.

Hennelly, B. M.

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

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part II: 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

D. P. Kelly, W. T. Rhodes, J. T. Sheridan, and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom algorithms for digital holography,” in Information Optics and Photonics, B. Javidi and T. Fournel, eds. (Springer, 2010), pp. 187–204.

Hernández, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Hillenbrand, M.

Hoffmann, A.

Hooker, B.

Illueca, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Ito, T.

Janssen, A. J.

J. J. Braat, S. van Haver, A. J. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” Prog. Opt. 51, 349–468 (2008).
[CrossRef]

Javidi, B.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (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]

Kakue, T.

Kelly, D. P.

M. Hillenbrand, A. Hoffmann, D. P. Kelly, and S. Sinzinger, “Fast nonparaxial scalar focal field calculations,” J. Opt. Soc. Am. A 31, 1206–1214 (2014).
[CrossRef]

D. P. Kelly, “Numerical calculation of the Fresnel transform,” J. Opt. Soc. Am. A 31, 755–764 (2014).
[CrossRef]

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

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part II: 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part I: 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

D. P. Kelly, W. T. Rhodes, J. T. Sheridan, and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2011), paper DTuC5.

B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom algorithms for digital holography,” in Information Optics and Photonics, B. Javidi and T. Fournel, eds. (Springer, 2010), pp. 187–204.

Koike, C.

Koike, T.

Kozacki, T.

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

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]

Kujawinska, M.

Li, J.-C.

Logofatu, P. C.

Lohmann, A. W.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 2008).

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Masuda, N.

Matsushima, K.

Meinecke, T.

D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2011), paper DTuC5.

Mendlovic, D.

Miret, J. J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Monaghan, D. S.

B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom algorithms for digital holography,” in Information Optics and Photonics, B. Javidi and T. Fournel, eds. (Springer, 2010), pp. 187–204.

Nascov, V.

Naughton, T. J.

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

Odate, S.

Onural, L.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. (Pearson, 2010).

Osten, W.

Otaki, K.

Pandey, N.

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

B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom algorithms for digital holography,” in Information Optics and Photonics, B. Javidi and T. Fournel, eds. (Springer, 2010), pp. 187–204.

Pedrini, G.

Peng, Z.-J.

Pérez, J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Picart, P.

Rabiner, L. R.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Syst. Tech. J. 48, 1249–1292 (1969).
[CrossRef]

Rader, C. M.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Syst. Tech. J. 48, 1249–1292 (1969).
[CrossRef]

Rhodes, W. T.

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

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part II: 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part I: 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

D. P. Kelly, W. T. Rhodes, J. T. Sheridan, and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Roggemann, M. C.

Sabitov, N.

D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2011), paper DTuC5.

Schafer, R. W.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Syst. Tech. J. 48, 1249–1292 (1969).
[CrossRef]

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. (Pearson, 2010).

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).

Shampine, L. F.

L. F. Shampine, “MATLAB program for quadrature in 2D,” Appl. Math. Comput. 202, 266–274 (2008).
[CrossRef]

Shen, F. B.

Sheridan, J. T.

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part I: 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination. part II: 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

D. P. Kelly, W. T. Rhodes, J. T. Sheridan, and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Sherman, G. C.

Shimobaba, T.

Sinzinger, S.

M. Hillenbrand, A. Hoffmann, D. P. Kelly, and S. Sinzinger, “Fast nonparaxial scalar focal field calculations,” J. Opt. Soc. Am. A 31, 1206–1214 (2014).
[CrossRef]

D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2011), paper DTuC5.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves, Adam Hilger Series on Optics and Optoelectronics (Hilger, 1986).

Stern, A.

Sugaya, A.

Sugisaki, K.

Tankam, P.

Toba, H.

Uchikawa, K.

van Haver, S.

J. J. Braat, S. van Haver, A. J. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” Prog. Opt. 51, 349–468 (2008).
[CrossRef]

Vázquez, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Voelz, D. G.

Wang, A. B.

Wei, W.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 2008).

Xiahui, T.

Yingxiong, Q.

Yu, X.

Zalevsky, Z.

Zhang, F.

Appl. Math. Comput. (1)

L. F. Shampine, “MATLAB program for quadrature in 2D,” Appl. Math. Comput. 202, 266–274 (2008).
[CrossRef]

Appl. Opt. (6)

Bell Syst. Tech. J. (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Syst. Tech. J. 48, 1249–1292 (1969).
[CrossRef]

IEEE Trans. Audio Electroacoust. (1)

L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

D. Mendlovic and A. W. Lohmann, “Space–bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997).
[CrossRef]

D. Mendlovic, A. W. Lohmann, and Z. Zalevsky, “Space–bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
[CrossRef]

D. P. Kelly, “Numerical calculation of the Fresnel transform,” J. Opt. Soc. Am. A 31, 755–764 (2014).
[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

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

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Figures (10)

Fig. 1.
Fig. 1.

Propagation of a focusing field u0 from the input plane z=0 to the output plane z. The discrete sampling leads to SRs and may also cause SAs. The effective extents of u0 and of the output field uz are given by LE(u0) and LE(uz).

Fig. 2.
Fig. 2.

Schematic of the ASM for the propagation of continuous and discretely sampled fields. (a) Spatial distribution of the input field, (b) spatial frequency distribution of the input field and of the propagation kernel, and (c) spatial distribution of the propagated field.

Fig. 3.
Fig. 3.

Schematic of the RSC for the propagation of continuous and discretely sampled fields. (a) Spatial distribution of the input field and the propagation kernel, (b) spatial frequency distribution of the input field and of the propagation kernel, and (c) spatial distribution of the propagated field.

Fig. 4.
Fig. 4.

SAs in the RSC output plane due to aliasing in the spatial frequency domain. (a) In the case of a paraxial propagation, the SAs are evenly spaced and represent shifted replicas of the correct output field. (b) For a nonparaxial propagation, the SAs are distorted. (c) Cross sections of the fields shown in (a) and (b) along the line y=0.

Fig. 5.
Fig. 5.

Schematic for avoiding the appearance of SAs within the region LE(uz).

Fig. 6.
Fig. 6.

Illustration of Eqs. (24)–(26). SR(u˜z) given by the cyclic convolution of u˜0 and h˜ have a finite extent LN(SR)2LN(u˜z). Thus, areas of size 2LN(u˜z)LN(SR) are free of spatial aliasing. These areas should be larger than LE(u˜z).

Fig. 7.
Fig. 7.

Spatial frequency extent of a spherical wave (blue) in comparison with the geometric optics approximation (red). Simulation parameters: λ=500nm, NA=0.30, and zf=2.0mm. The geometric estimation is given by νx,max=6e5m1.

Fig. 8.
Fig. 8.

Comparison of FFT-based field calculations using the ASM and the RSC. The simulations are discussed as cases (a)–(e) in the main body of the paper. All simulations are performed with λ=500nm, NA=0.30, and rExP=0.79mm. zf is the distance to the geometric focus while z is the propagation distance. To make three adjacent Fourier replicas visible, zeros were inserted between existing space domain values of u˜0 and h˜ [41]. Subfigures (c) and (d) show aliasing between neighboring replicas of U˜0 and H˜. We added plots of the central replica, calculated from densely sampled space domain fields.

Fig. 9.
Fig. 9.

Radial intensity distribution and error of the ASM and RSC compared with a direct numerical integration. Simulation parameters: LN(u˜0)=(2mm)2, zf=2mm, z=1.95mm, and rExP=0.63mm. The ASM is performed with 163842 sampling points (Δx=Δy=0.1μm). The RSC is performed for two different sets of sampling points: 163842 (Δx=Δy=0.1μm) and 5122 (Δx=Δy=3.2μm). In case of the 5122 sampling points, the propagation has been performed several times for slightly shifted output fields to produce output values at the spacing of the ASM calculation.

Fig. 10.
Fig. 10.

Example of a RSC calculation with additional CZT for a converging spherical wave. (a) Normalized magnitude of the field U˜ωz, (b) normalized magnitude of u˜ωz, (c) cross section of the field U˜ωz at ωy=0, and (d) cross section of the output field u˜ωz at y=0. Simulation parameters: LN(u˜0)=(1.6mm)2, zf=2mm, z=1.95mm, rExP=0.63mm, Nx=Ny=4096, and Δx=Δy=0.4μm. The final CZT uses the same number of sampling points and provides a zoom factor of 20×. Subfigure (d) also contains the results of an ASM calculation and a DNI, which were both performed with the same parameters.

Equations (44)

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uz(x,y)=12πu0(x0,y0)z[exp(jkR)R]dx0dy0.
R=(xx0)2+(yy0)2+z2.
uz(x,y)=F1{F{u0(x0,y0)}F{h(x0,y0)}}
h(x0,y0)=12πz[exp(jkR)R]=zexp[jkR]2πR2(jk1R),
B=F{b}=bexp[j2π(νxx+νyy)]dxdy
b=F1{B}=Bexp[j2π(νxx+νyy)]dνxdνy.
uz(x,y)=F1{F{u0(x0,y0)}Hz(νx,νy)}.
H(νx,νy)=F{h}={exp[j2πz1λ2νx2νy2]forνx2+νy21λ2,exp[2πzνx2+νy21λ2]forνx2+νy2>1λ2.
x=nxΔx,y=nyΔy,νx=nνxΔνx,νy=nνyΔνy
nx,nνx=1Nx,andny,nνy=1Ny.
LN(b˜)=LN(b˜,nx)×LN(b˜,ny)
LN(b˜,nx)=Nx×Δx,andLN(b˜,ny)=Ny×Δy.
LN(B˜)=LN(B˜,nνx)×LN(B˜,ny)
LN(B˜,nνx)=Nx×Δνx,LN(B˜,nνy)=Ny×Δνy.
1/Nx=ΔxΔνx,1/Ny=ΔyΔνy.
LE(b˜)=LE(b˜,nx)×LE(b˜,ny).
LE(B˜)=LE(B˜,nνx)×LE(B˜,nνy)
1/Δx=LN(B˜,nνx)and1/Δy=LN(B˜,nνy).
1/Δνx=LN(b˜,nx)and1/Δνy=LN(b˜,ny).
LE(U˜0)<LN(U˜0)
λzΔx>LE(u˜z,nx),λzΔy>LE(u˜z,ny).
LE(U˜0)<LE(H˜).
LE(u˜z)<LN(u˜z).
LN(SR)=LN(SR,nx)×LN(SR,ny)
LN(SR,x)=LN(u˜0,nx)+LN(h˜,nx),LN(SR,y)=LN(u˜0,ny)+LN(h˜,ny).
LE(u˜z,x)<2LN(u˜z,nx)LN(SR,nx)=LN(u˜z,nx)LE(u˜0,nx),LE(u˜z,y)<2LN(u˜z,ny)LN(SR,ny)=LN(u˜z,ny)LE(u˜0,ny)
ASM-1:LE(U˜0)<LN(U˜0),
ASM-2:LE(u˜z)<LN(u˜z).
RSC-1:LE(u˜z,nx)<λz/Δx,LE(u˜z,ny)<λz/Δy,
RSC-2:LE(u˜0,nx)+LE(u˜z,nx)<LN(u˜0,nx),LE(u˜0,ny)+LE(u˜z,ny)<LN(u˜0,ny).
νx=12πxϕ(x,y)andνy=12πyϕ(x,y).
α=λνx,β=λνy,γ=1(λνx)2(λνy)2.
LE(B˜,nνx)LN(B˜,nνx)=1/Δx,LE(B˜,nνy)LN(B˜,nνy)=1/Δy.
Δx|ϕhx0|maxπandΔy|ϕhy0|maxπ.
Δνx|ϕHνx|maxπandΔνy|ϕHνy|maxπ.
|νx|max=|νy|max=NA/λ=6×105m1
u˜z=CZT(U˜z)=ΔωxΔωyC(U˜z·EαxαyD)
C=exp[jπ(x2/αx+y2/αy)],
D=exp[jπ(ωx2/αx+ωy2/αy)],
E=exp[jπ(ωx2/αx+ωy2/αy)].
CZT-1:LE(U˜ωz)<LN(U˜ωz).
CZT-2:LN(U˜ωz,nωx)>LE(U˜ωz,nωx)+LE(u˜ωz,nωx),LN(U˜ωz,nωy)>LE(U˜ωz,nωy)+LE(u˜ωz,nωy).
LE(FFT(U˜ωz))<LN(FFT(U˜ωz)),LE(FFT(D))<LN(FFT(D))
LN(FFT(U˜ωz))=LN(FFT(D))=1/δωx×1/δωy.

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