Abstract

A mathematical model is derived, and numerical simulation is analyzed for laser beam shaping by using multilevel phase-only diffractive optical elements (DOEs). We used the simulated annealing algorithm to design the beam shapers. The result has an essential effect on the diffractive pattern quality caused by the spatial frequency composition of target patterns for the same incident Gaussian beam size and target pattern area. The root mean square error between the diffractive and target patterns is smaller for the target patterns with lower spatial frequencies. Moreover, the effect of spatial frequency composition can be relaxed for the cases of larger incident Gaussian beam size. In addition, finer quality control of a diffraction pattern can be obtained by increasing the number of quantization levels at the DOEs.

© 2012 Optical Society of America

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References

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  1. J. Turunen and F. Wyrowski eds., Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, 1997).
  2. M. B. Stern, “Binary optics fabrication,” in Micro-optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, 1997), pp. 53–86.
  3. J. R. Leger, D. Chen, and Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19, 108–110 (1994).
    [CrossRef]
  4. F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
    [CrossRef]
  5. N. J. Jenness, R. T. Hill, A. Hucknall, A. Chilkoti, and R. L. Clark, “A versatile diffractive maskless lithography for single-shot and serial microfabrication,” Opt. Express 18, 11754–11762 (2010).
    [CrossRef]
  6. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef]
  7. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. 7, 961–969 (1990).
    [CrossRef]
  8. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  9. W.-F. Hsu, “Backward iterative quantization methods for designs of multilevel diffractive optical elements,” Opt. Express 13, 5052–5063 (2005).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics, 3rd Ed.(Roberts and Company, 2005).
  11. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef]
  12. N. Yoshikawa and T. Yatagai, “Phase optimization of a Kinoform by simulated annealing,” Appl. Opt. 33, 863–868 (1994).
    [CrossRef]
  13. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  14. P.J.M. van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Applications (Kluwer Academic, 1987).
  15. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” IBM Research Report RC 9355 (1982).

2010

2005

1996

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[CrossRef]

1995

1994

1990

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. 7, 961–969 (1990).
[CrossRef]

1989

1983

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

1953

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Aarts, E. H. L.

P.J.M. van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Applications (Kluwer Academic, 1987).

Chen, D.

Chilkoti, A.

Clark, R. L.

Dickey, F. M.

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” IBM Research Report RC 9355 (1982).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd Ed.(Roberts and Company, 2005).

Hill, R. T.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[CrossRef]

Hsu, W.-F.

Hucknall, A.

Jenness, N. J.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” IBM Research Report RC 9355 (1982).

Leger, J. R.

Mait, J. N.

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Stern, M. B.

M. B. Stern, “Binary optics fabrication,” in Micro-optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, 1997), pp. 53–86.

Teller, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

van Laarhoven, P.J.M.

P.J.M. van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Applications (Kluwer Academic, 1987).

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” IBM Research Report RC 9355 (1982).

Wang, Z.

Wyrowski, F.

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. 7, 961–969 (1990).
[CrossRef]

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
[CrossRef]

Yatagai, T.

Yoshikawa, N.

Appl. Opt.

J. Chem. Phys.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Opt. Soc. Am.

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. 7, 961–969 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

Science

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Other

P.J.M. van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Applications (Kluwer Academic, 1987).

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” IBM Research Report RC 9355 (1982).

J. W. Goodman, Introduction to Fourier Optics, 3rd Ed.(Roberts and Company, 2005).

J. Turunen and F. Wyrowski eds., Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, 1997).

M. B. Stern, “Binary optics fabrication,” in Micro-optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, 1997), pp. 53–86.

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Figures (8)

Fig. 1.
Fig. 1.

Flow chart of the simulated annealing algorithm.

Fig. 2.
Fig. 2.

Target patterns comprising squares with various frequency compositions.

Fig. 3.
Fig. 3.

RMSE vs. numbers of squares.

Fig. 4.
Fig. 4.

Target patterns of (a) stripes and (b) squared frames.

Fig. 5.
Fig. 5.

Target patterns of two-dimensional sinusoidal functions with different frequencies.

Fig. 6.
Fig. 6.

The four-level beam shapers designed for the target patterns (a) A1, (b) A3, and (c) A5 by using an R=0.25mm incident Gaussian beam.

Fig. 7.
Fig. 7.

The diffractive patterns of the four-level beam shapers by using (a) R=0.25mm and (b) R=0.50mm incident Gaussian beams.

Fig. 8.
Fig. 8.

Diffractive patterns of the designed (a) two-level, (b) four-level, (c) eight-level, and (d) 16-level beam shapers by using an R=0.50mm incident Gaussian beam.

Tables (3)

Tables Icon

Table 1. Performance of the B-and C-Groups

Tables Icon

Table 2. Performance of the D-Group

Tables Icon

Table 3. Nonuniformity for Various Numbers of Quantization Levels

Equations (23)

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gg(u,v)=exp(u2+v22w2)·circ(u2+v24w2),
ge=mnrect(umaa,vnaa)·exp(jϕmn)·rect(uD,vD),m=l2,l2+1,,l2,n=l2,l2+1,,l2.
t(u,v)=gg(u,v)·ge(u,v)=exp(u2+v22w2)·circ(u2+v24w2)·mnrect(umaa,vnaa)·exp(jϕmn)·rect(uD,vD)=mnexp(u2+v22w2)·rect(umaa,vnaa)·exp(jϕmn).
U(x,y)=t(x,y)h(x,y)=t(u,v)·h(xu,yv)dudv,
h(x,y)=exp(j2πλz)jλz·exp[jπλz(x2+y2)],
H(fx,fy)=exp(j2πλz)·exp[jπλz(fx2+fy2)].
U(x,y)=exp(j2πλz)·exp[jπλz(x2+y2)]·2πw2·exp[2π2w2λ2z2(x2+y2)]a2sinc(aλzx,aλzy)exp[jπλz(x2+y2)]·mnexp[j2πλza(mx+ny)]·exp(jϕmn)=2πw2·exp[jπλz(x2+y2)]·exp[2π2w2λ2z2(x2+y2)]mna2sinc(aλzx,aλzy)·exp[j2πλza(mx+ny)]·exp[j(ϕmn+2πλz)]exp[jπλz(x2+y2)],
t(u,v)=mnexp[(ma)2+(na)22w2]·rect(umaa,vnaa)·exp(jϕmn).
U(x,y)=exp(j2πλz)jλz·exp[jπλz(x2+y2)]·{mnexp[(ma)2+(na)22w2]·rect(umaa,vnaa)}·exp(jϕmn)·exp[jπλz(u2+v2)]·exp[j2πλz(ux+vy)]dudv=exp{j[2πλz+πλz(x2+y2)]}jλz·mnexp[(ma)2+(na)22w2]·exp{jπλz[(ma)2+(na)2]}·exp(jϕmn)·rect(umaa,vnaa)·exp(j2πλz(ux+vy))dudv=exp{j[2πλz+πλz(x2+y2)]}jλz·a2sinc(aλzx,aλzy)·mnexp[(ma)2+(na)22w2]·exp{jπλz[(ma)2+(na)2]+jϕmn}·exp[j2πλza(mx+ny)]=exp{j[2πλz+πλz(x2+y2)]}jλz·a2sinc(aλzx,aλzy)·mnexp[(ma)2+(na)22w2]·exp{j[2πaλz(mx+ny)πa2(m2+n2)λzϕmn]}.
RMSE=[(ItId)2/It2]1/2,
EFF=PsPtotal,
Cost=|ItαId|2,
α=It/Id.
Pa=exp(ΔCT(kMC)).
T(kMC)=TikMC+1.
Ti=ΔCaveln(Pinitial),
F(x,y)=rect(xxd,yyd).
f(x,y)=sinc(xxi,yyi),
F(x,y)=xiyi·rect(xiλzx,yiλzy).
ID1(x,y)=[cos2(x)·cos2(y)]1/2.
ID2(x,y)=[cos2(x)·cos2(10y)]1/2,
ID3(x,y)=[cos2(10x)·cos2(10y)]1/2.
σ=Is,maxIs,minIs,max+Is,min,

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