Abstract

We present a scalar model to overcome the computation time and sampling interval limitations of the traditional Rayleigh–Sommerfeld (RS) formula and angular spectrum method in computing wide-angle diffraction in the far-field. Numerical and experimental results show that our proposed method based on an accurate nonparaxial diffraction step onto a hemisphere and a projection onto a plane accurately predicts the observed nonparaxial far-field diffraction pattern, while its calculation time is much lower than the more rigorous RS integral. The results enable a fast and efficient way to compute far-field nonparaxial diffraction when the conventional Fraunhofer pattern fails to predict correctly.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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  22. M. V. Kessels, M. El Bouz, R. Pagan, and K. Heggarty, “Versatile stepper based maskless microlithography using a liquid crystal display for direct write of binary and multilevel microstructures,” J. Micro/Nanolithogr. MEMS MOEMS 6, 033002 (2007).
    [CrossRef]

2012 (1)

2010 (1)

2009 (2)

2008 (2)

2007 (1)

M. V. Kessels, M. El Bouz, R. Pagan, and K. Heggarty, “Versatile stepper based maskless microlithography using a liquid crystal display for direct write of binary and multilevel microstructures,” J. Micro/Nanolithogr. MEMS MOEMS 6, 033002 (2007).
[CrossRef]

2006 (1)

2004 (1)

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

2003 (1)

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]

1995 (1)

1992 (1)

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

1989 (1)

1979 (1)

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

1967 (1)

Barouch, E.

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Beck, W. A.

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]

Bogunovic, D.

Cole, D. C.

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

El Bouz, M.

M. V. Kessels, M. El Bouz, R. Pagan, and K. Heggarty, “Versatile stepper based maskless microlithography using a liquid crystal display for direct write of binary and multilevel microstructures,” J. Micro/Nanolithogr. MEMS MOEMS 6, 033002 (2007).
[CrossRef]

Falaggis, K.

Feng, D.

Feng, L.-S.

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]

Flury, M.

Gao, X.

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]

Gaylord, T. K.

Gérard, P.

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice-Hall, 2007).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Grann, E. B.

Harvey, J. E.

J. E. Harvey, D. Bogunovic, and A. Krywonos, “Aberrations of diffracted wave fields: distortion,” Appl. Opt. 42, 1167–1174 (2003).
[CrossRef]

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

Heggarty, K.

M. V. Kessels, M. El Bouz, R. Pagan, and K. Heggarty, “Versatile stepper based maskless microlithography using a liquid crystal display for direct write of binary and multilevel microstructures,” J. Micro/Nanolithogr. MEMS MOEMS 6, 033002 (2007).
[CrossRef]

Herzig, H. P.

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

Hollerbach, U.

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Hu, S.-L.

Kessels, M. V.

M. V. Kessels, M. El Bouz, R. Pagan, and K. Heggarty, “Versatile stepper based maskless microlithography using a liquid crystal display for direct write of binary and multilevel microstructures,” J. Micro/Nanolithogr. MEMS MOEMS 6, 033002 (2007).
[CrossRef]

Kettunen, V.

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

Kozacki, T.

Kress, B.

Kress, B. C.

B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-optics to Nanophotonics (Wiley, 2009).

Krywonos, A.

Kujawinska, M.

Mait, J. N.

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

Matsushima, K.

Meyrueis, P.

Mirotznik, M. S.

Moharam, M. G.

Orszag, S. A.

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Ou, P.

Pagan, R.

M. V. Kessels, M. El Bouz, R. Pagan, and K. Heggarty, “Versatile stepper based maskless microlithography using a liquid crystal display for direct write of binary and multilevel microstructures,” J. Micro/Nanolithogr. MEMS MOEMS 6, 033002 (2007).
[CrossRef]

Pommet, D. A.

Prather, D. W.

Raulot, V.

Ripoll, O.

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

Roggemann, M. C.

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010).

Serio, B.

Shen, F.

Sherman, G. C.

Shi, S.

Shimobaba, T.

Voelz, D. G.

Wang, A.

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice-Hall, 2007).

Wyrowski, F.

Zhang, C.-X.

Am. J. Phys. (1)

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

Appl. Opt. (6)

J. Micro/Nanolithogr. MEMS MOEMS (1)

M. V. Kessels, M. El Bouz, R. Pagan, and K. Heggarty, “Versatile stepper based maskless microlithography using a liquid crystal display for direct write of binary and multilevel microstructures,” J. Micro/Nanolithogr. MEMS MOEMS 6, 033002 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Opt. Commun. (2)

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

Opt. Eng. (1)

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

Opt. Express (3)

Other (5)

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010).

D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011), Vol. TT89.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice-Hall, 2007).

B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-optics to Nanophotonics (Wiley, 2009).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (7)

Fig. 1.
Fig. 1.

Diffraction geometry.

Fig. 2.
Fig. 2.

(a) Harvey model and the diverging effect from the hemisphere to the observation plane. (b) Multiradii approach.

Fig. 3.
Fig. 3.

(a) Target image, which is a 256×256 pixel image at a diffraction distance of 18 cm and with a spot separation of about 3.5 cm. (b) Fourier DOE designed by an iterative Fourier transform algorithm, where a period is 8 μm.

Fig. 4.
Fig. 4.

Ratio of difference to the RS calculated diffraction pattern for the observation plane diffraction patterns calculated using (a) the paraxial approximation in angular coordinates or (b) the Harvey method plus plane wave projection. (c) Harvey method plus spherical wave projection.

Fig. 5.
Fig. 5.

Comparison between the simulated output patterns of the test DOE obtained by different methods. (a) RS integral. (b) RS convolution. (c) ASM. (d) Repeated convolution. (e) Fraunhofer approximation. (f) Multiradii approach. Because of the sampling constraint, the RS convolution and the ASM only obtained a small area around the optical axis.

Fig. 6.
Fig. 6.

Experimental verification. (a) Optical interferometric microscope image of the fabricated DOE, where a period is about 8 μm. (b) Optical setup. (c) Superposition of the simulated and experimental diffraction patterns. Green spots are those predicted by the multiradii Harvey calculation, whereas the red spots are those of the experimentally observed pattern. Yellow regions show where they overlap.

Fig. 7.
Fig. 7.

Effect of exposure time on the etch profile and pixel linewidth.

Tables (2)

Tables Icon

Table 1. Simulated Diffraction Angles and Efficiencies of Sample Diffraction Orders Calculated by Different Methodsa

Tables Icon

Table 2. Simulated and Experimentally Observed Diffraction Efficiencies for Sample Diffraction Orders of the Test DOE without and with Fabrication Errors

Equations (22)

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

U(x2,y2;z)=z2πU(x1,y1;0)g(x2x1,y2y1;z)dx1dy1,
g(x,y;z)=(jk1r)exp(jkr)r2
r=z2+(x2x1)2+(y2y1)2.
U(xm,yn;z)=z2πi=1Nk=1NU(xi,yk;0)g(xmxi,ynyk;z)δ12,
U(x2,y2;z)=z2πIFT{FT[U(x1,y1;0)]·FT[g(x1,y1;z)]},
tan45°=O2P2maxzmaxzmax=Nδ2222.9mm,
U(x2,y2;z)=A(fx,fy;0)G(fx,fy;z)exp[j2π(fxx2+fyy2)]dfxdfy,
U(x2,y2;z)=IFT{FT[U(x1,y1;0)]·G(fx,fy;z)}
δf|ϕGf|maxπ,whereϕG(f)=kz1λ2f2.
zNδ12λ1λ24δ12,withδ1λ/2,
U(α,β;R)γexp(jkR)jλRU(x1,y1;0)exp(jkW)(1+ϵ)2exp[j2πλ(αx1+βy1)]dx1dy1,
α=xR,β=yR,γ=zR,withR=x2+y2+z2.
U(α,β;R)γexp(jkR)jλRFT{U(x1,y1;0)}.
R=Rγ,x2=Rαγ,y2=Rβγ,
U(α,β;z)=U(α,β;R)=exp[jk(RR)]U(α,β;R),
U(α,β;R)=Rexp(jkR)U(α,β;R)exp(jkR)R=RRexp[jk(RR)]U(α,β;R).
U(α,β;R)γexp(jkR)jλRFT{U(x1,y1;0)},
U(mδ2,nδ2;z)U(pδα,qδβ;R),
Pspotarea|U|2γδ22,
U(α,β;z)U(α,β;R)exp(jkz)jλzFT{U(x1,y1;0)}.
U(x2,y2;z)exp(jkz)jλzFT{U(x1,y1;0)},withx2Rαδ2Rδα=zλNδ1
Ratio of difference=|U||URS||U|+|URS|,

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