Abstract

An efficient algorithm for calculating nonparaxial scalar field distributions in the focal region of a lens is discussed. The algorithm is based on fast Fourier transform implementations of the first Rayleigh–Sommerfeld diffraction integral and assumes that the input field at the pupil plane has a larger extent than the field in the focal region. A sampling grid is defined over a finite region in the output plane and referred to as a tile. The input field is divided into multiple separate spatial regions of the size of the output tile. Finally, the input tiles are added coherently to form a summed tile, which is propagated to the output plane. Since only a single tile is propagated, there are significant reductions of computational load and memory requirements. This method is combined either with a subpixel sampling technique or with a chirp z-transform to realize smaller sampling intervals in the output plane than in the input plane. For a given example the resulting methods enable a speedup of approximately 800× in comparison to the normal angular spectrum method, while the memory requirements are reduced by more than 99%.

© 2014 Optical Society of America

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References

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2014

2013

P. Lobaz, “Memory-efficient reference calculation of light propagation using the convolution method,” Opt. Express 21, 2795–2806 (2013).
[CrossRef]

S. van Haver and A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Pub. 8, 13044 (2013).
[CrossRef]

2012

2011

2010

2009

2008

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

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]

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

2006

1998

1997

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

1981

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

1973

1970

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

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

1969

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

1961

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]

Asoubar, D.

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).

Corle, T. R.

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

Dainty, J. C.

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]

Eide, H. A.

Ersoy, O. K.

O. K. Ersoy, Diffraction, Fourier Optics and Imaging, Vol. 30 of Wiley Series in Pure and Applied Optics (Wiley-Interscience, 2007).

Goodman, J. W.

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

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Graulig, C.

Hao, P.

Hennelly, B. M.

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.

Heurtley, J. C.

Hooker, B.

Hopkins, H. H.

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[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]

Janssen, A. J. E. M.

S. van Haver and A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Pub. 8, 13044 (2013).
[CrossRef]

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.

Kanka, M.

Kelly, D. P.

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

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.

Kino, G. S.

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

Koike, C.

Koike, T.

Kou, S. S.

Kozacki, T.

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

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]

Kreuzer, H. J.

Kuhn, M.

Lin, J.

Lobaz, P.

Logofatu, P. C.

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.

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.

Odate, S.

Osten, W.

Osterberg, H.

Otaki, K.

Pandey, N.

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.

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

Riesenberg, R.

Rodríguez-Herrera, O. G.

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]

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).

Shen, F. B.

Sheppard, C. J. R.

Sheridan, J. T.

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.

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.

Site, Z.

Smith, L. W.

Stamnes, J. J.

J. J. Stamnes and H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

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

Sugaya, A.

Sugisaki, K.

Toba, H.

Uchikawa, K.

van Haver, S.

S. van Haver and A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Pub. 8, 13044 (2013).
[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]

Voelz, D. G.

Wang, A. B.

Wei, W.

Wilson, T.

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy, 2nd ed. (Academic, 1985).

Wuttig, A.

Wyrowski, F.

Xiahui, T.

Yingxiong, Q.

Yu, X.

Yuan, X. C.

Yzuel, M. J.

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Zhang, F.

Appl. Math. Comput.

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

Appl. Opt.

Bell Syst. Tech. J.

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.

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

J. Eur. Opt. Soc. Rapid Pub.

S. van Haver and A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Pub. 8, 13044 (2013).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Opt. Commun.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

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

Opt. Eng.

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]

Opt. Express

Opt. Lett.

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]

Prog. Opt.

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]

Other

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

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

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

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

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.

M. Kanka, “Bildrekonstruktion in der digitalen inline-holographischen Mikrsokopie,” Ph.D. thesis (Technische Universität Ilmenau, 2011).

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy, 2nd ed. (Academic, 1985).

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

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.

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

O. K. Ersoy, Diffraction, Fourier Optics and Imaging, Vol. 30 of Wiley Series in Pure and Applied Optics (Wiley-Interscience, 2007).

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Figures (8)

Fig. 1.
Fig. 1.

Schematic of the SFM, the untiled AS, and of an alternative method for calculating the summed field at the output plane.

Fig. 2.
Fig. 2.

Error analysis of the SFM for the AS propagation of a spherical wave to out-of-focus planes with defocus distances 50μm and 250μm. Comparison between the SFM implementation of the AS and an untiled AS. The fields show the magnitude in the output plane and are normalized with respect to the maximum magnitude in this plane for the SFM cases [(c) and (g)].

Fig. 3.
Fig. 3.

Errors of the SFM from the perspective of aliasing between neighboring replicas. During the AS propagation two zeros have been inserted in the x-direction between each value of F{uΣ0}HΣz before performing the inverse DFT to increase the output field size in the x-direction [42]. The fields show the magnitude in the output plane and are normalized with respect to the maximum magnitude in this plane.

Fig. 4.
Fig. 4.

Irradiance in the output plane with sampling intervals of (a) 400 nm and (b) 25 nm. The field (b) was calculated using 322 tiles and 162 subpixels. The fields are normalized with respect to the maximum irradiance in the output plane.

Fig. 5.
Fig. 5.

Irradiance 10 μm in front of the geometrical focus calculated with three different methods: DNI, AS + SFM + SST (322 tiles, 162 subpixels), and RSC + SFM + SST (322 tiles, 162 subpixels).

Fig. 6.
Fig. 6.

Relative errors of the AS + SFM + SST and AS + SFM + CZT methods with respect to the DNI results for different numbers of tiles.

Fig. 7.
Fig. 7.

Relative error of the AS + SFM + SST and RSC + SFM + SST methods with respect to the DNI calculation.

Fig. 8.
Fig. 8.

Relative calculation times of the different steps required for the AS + SFM + SST and the AS + SFM + CZT methods. The calculation times are evaluated for different numbers of tiles and normalized with respect to the AS + SST method (162 subpixels, without tiling).

Tables (1)

Tables Icon

Table 1. Comparison of the Calculation Speed for the Different Propagation Methods Discussed in this Papera

Equations (24)

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

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{hz(x0,y0)}},
hz(x0,y0)=12πz[exp(jkR)R]
B=F{b}=bexp[j2π(νxx+νyy)]dxdy,
b=F1{B}=Bexp[j2π(νxx+νyy)]dνxdνy.
Hz(νx,νy)=F{hz}={exp[j2πz1λ2νx2νy2]forνx2+νy21λ2,exp[2πzνx2+νy21λ2]forνx2+νy2>1λ2.
uz(x,y)=F1{F{u0(x0,y0)}Hz(νx,νy)}.
B=F{b}dft{b}δxδy,
b=F1{B}idft{B}/(δxδy),
dft{bmn}=m=0N1n=0N1bmnexp[2πjmv+nwN],
idft{Bvw}=1N2v=0N1w=0N1Bvwexp[2πjmv+nwN].
1/N2=δx·δy·δνx·δνy.
uz={idft{shift{Hz}dft{u0}}forASidft{dft{shift{hz}}dft{u0}}δxδyforRSC.
BTs,Tσ=r=0R1ρ=0R1t=0T1τ=0T1btR+r,τR+ρ×exp[2πjTs(tR+r)+Tσ(τR+ρ)RT].
Ts(tR+r)+Tσ(τR+ρ)RT=st+στ+sr+σρR,
BTs,Tσ=r=0R1ρ=0R1[t=0T1τ=0T1btR+r,τR+ρ]exp[2πjsr+σρR].
b(xsx,ysy)=F1{B(νx,νy)exp[j2π(νxsx+νysy)]}.
bmn=CZT(Bvw)=δωxδωyCmn(BvwEvwαxαyDvw),
Cmn=exp[jπ(x2/αx+y2/αy)],
Dvw=exp[jπ(ωx2/αx+ωy2/αy)],
Evw=exp[jπ(ωx2/αx+ωy2/αy)].
u0={exp[jkRf]Rfforx2+y2rmax,0forx2+y2rmax,
Rf=x2+y2+zf2.

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