Abstract

Diffractive beam-splitting elements are typically designed for replicating beams on positions belonging to an equidistant grid in the spatial spectrum. The parameter of the output grid follows directly from the period of the beam-splitter transmission through the grating equation. Our objective is to develop design strategies allowing a more accurate positioning of the replicated beams. Issues occurring when the output grid parameter is decreased below the output beam width are discussed and shown to be avoidable. Furthermore, a design algorithm is introduced, which permits an arbitrary positioning of the replicated beams. This algorithm is constructed for high computational efficiency by utilizing fast Fourier transform operations in the major part of its iterations.

© 2002 Optical Society of America

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References

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  5. Z. Wen, P. Yeh, X. Yang, “Modified two-dimensional hamming neuronal network and its optical implementation using Dammann gratings,” Opt. Eng. 35, 2136–2144 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  22. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  23. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
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2002 (1)

2001 (2)

1998 (4)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

J. Bengtsson, “Kinoforms designed to produce different fan-out patterns for two wavelengths,” Appl. Opt. 37, 2011–2020 (1998).
[CrossRef]

F. Wyrowski, H. Aagedal, “Wave transformation by physical-optics system design,” Int. J. Optoelectron. 12, 127–143 (1998).

H. Aagedal, F. Wyrowski, “On pixel-oriented structure parametrization for design of diffractive elements,” J. Mod. Opt. 45, 1451–1464 (1998).
[CrossRef]

1997 (1)

1996 (2)

Z. Wen, P. Yeh, X. Yang, “Modified two-dimensional hamming neuronal network and its optical implementation using Dammann gratings,” Opt. Eng. 35, 2136–2144 (1996).
[CrossRef]

J. Bengtsson, N. Eriksson, A. Larsson, “Small-feature-size fan-out kinoform etched in GaAs,” Appl. Opt. 35, 801–806 (1996).
[CrossRef] [PubMed]

1995 (2)

1992 (2)

1990 (1)

1989 (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

1988 (1)

1984 (1)

1977 (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1971 (1)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Aagedal, H.

F. Wyrowski, H. Aagedal, “Wave transformation by physical-optics system design,” Int. J. Optoelectron. 12, 127–143 (1998).

H. Aagedal, F. Wyrowski, “On pixel-oriented structure parametrization for design of diffractive elements,” J. Mod. Opt. 45, 1451–1464 (1998).
[CrossRef]

H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), Chap. 6, pp. 165–188.

Bengtsson, J.

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

Brenner, K.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Cincotti, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

Dammann, H.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Dändliker, R.

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Eriksson, N.

Fabrizio, E. D.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

Fiddy, M. A.

Gale, M. T.

Gentili, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

Goodmann, J. W.

J. W. Goodmann, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gori, F.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Herzig, H. P.

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Johansson, M.

Kiryuschev, I.

Klotz, E.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Krackhardt, U.

Larsson, A.

Levi, A.

Mait, J. N.

Marom, E.

Mendlovic, D.

Noponen, E.

Ogura, Y.

Ouzieli, I.

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Prongue, D.

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

Schmid, M.

H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), Chap. 6, pp. 165–188.

Shirai, N.

Stark, H.

Streibl, N.

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
[CrossRef] [PubMed]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Testorf, M. E.

Tunida, J.

Turunen, J.

E. Noponen, J. Turunen, F. Wyrowski, “Synthesis of paraxial-domain diffractive elements by rigorous electromagnetic theory,” J. Opt. Soc. Am. A 12, 1128–1133 (1995).
[CrossRef]

J. Turunen, F. Wyrowski, “Introduction to diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), Chap. 1, pp. 1–57.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Wen, Z.

Z. Wen, P. Yeh, X. Yang, “Modified two-dimensional hamming neuronal network and its optical implementation using Dammann gratings,” Opt. Eng. 35, 2136–2144 (1996).
[CrossRef]

Wyrowski, F.

H. Aagedal, F. Wyrowski, “On pixel-oriented structure parametrization for design of diffractive elements,” J. Mod. Opt. 45, 1451–1464 (1998).
[CrossRef]

F. Wyrowski, H. Aagedal, “Wave transformation by physical-optics system design,” Int. J. Optoelectron. 12, 127–143 (1998).

E. Noponen, J. Turunen, F. Wyrowski, “Synthesis of paraxial-domain diffractive elements by rigorous electromagnetic theory,” J. Opt. Soc. Am. A 12, 1128–1133 (1995).
[CrossRef]

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

J. Turunen, F. Wyrowski, “Introduction to diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), Chap. 1, pp. 1–57.

H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), Chap. 6, pp. 165–188.

Yang, X.

Z. Wen, P. Yeh, X. Yang, “Modified two-dimensional hamming neuronal network and its optical implementation using Dammann gratings,” Opt. Eng. 35, 2136–2144 (1996).
[CrossRef]

Yeh, P.

Z. Wen, P. Yeh, X. Yang, “Modified two-dimensional hamming neuronal network and its optical implementation using Dammann gratings,” Opt. Eng. 35, 2136–2144 (1996).
[CrossRef]

Appl. Opt. (8)

Int. J. Optoelectron. (1)

F. Wyrowski, H. Aagedal, “Wave transformation by physical-optics system design,” Int. J. Optoelectron. 12, 127–143 (1998).

J. Mod. Opt. (1)

H. Aagedal, F. Wyrowski, “On pixel-oriented structure parametrization for design of diffractive elements,” J. Mod. Opt. 45, 1451–1464 (1998).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Opt. Commun. (2)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. D. Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–18 (1998).
[CrossRef]

Opt. Eng. (2)

Z. Wen, P. Yeh, X. Yang, “Modified two-dimensional hamming neuronal network and its optical implementation using Dammann gratings,” Opt. Eng. 35, 2136–2144 (1996).
[CrossRef]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Other (4)

H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), Chap. 6, pp. 165–188.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

J. Turunen, F. Wyrowski, “Introduction to diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie, Berlin, 1997), Chap. 1, pp. 1–57.

J. W. Goodmann, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (9)

Fig. 1
Fig. 1

Steps of iterative Fourier transform algorithm.

Fig. 2
Fig. 2

Steps in conventional approach for beam-splitter design.

Fig. 3
Fig. 3

(a) One period of phase-only transmission and (b) simulated output intensity distribution of the optimal solution for a regular 1×3 phase-only beam splitter.

Fig. 4
Fig. 4

(a) One period of phase-only transmission and (b) simulated output intensity distribution for an irregular 1×3 beam splitter with signal orders (spots) positioned on an equidistant grid.

Fig. 5
Fig. 5

Output intensity distribution resulting from convolution with utransf, which is four times broader than in Fig. 4(b).

Fig. 6
Fig. 6

(a) One period of phase-only transmission and (b) simulated output intensity distribution for an irregular 1×3 beam splitter with signal orders (spots) positioned on an equidistant grid, designed with a frame of size dframe=0.8a.

Fig. 7
Fig. 7

Steps in approach for designing beam splitters for arbitrary spot positions.

Fig. 8
Fig. 8

(a) Central part of phase-only transmission and (b) simulated output intensity distribution for an irregular 1×3 beam splitter with freely positioned spots.

Fig. 9
Fig. 9

Irregular 1×30 beam splitter with freely positioned spots, calculated for a phase-only transmission function with 1000×1000 sampling values. The central 100×100 pixels of the transmission are shown in (a). The simulated output intensity distribution for Gaussian illumination is shown in (b), where a normalization to the maximum intensity was applied. In (c) a normalization to 1% of the maximum intensity was applied.  

Equations (31)

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

F[u(x)](x)=-dx u(x)exp(-i2πxx).
|uout|2|F[t]*utransf|2,
t(x)=comb(x/Δx)*tp(x),
comb(x)=n=-δ(x-n),
F[t](x)=comb(x/δx)F[tp](x)
δx=1/Δx.
uniformityerror=Imax-IminImax+Imin.
width(utransf)<δx,
t(x)=exp[i arg t(x)],
Πphaset(x)=t(x)/|t(x)|.
ΠPASuout(x)
=|αusig(x)|exp{i arg[uout(x)]}ifxWsiguout(x)otherwise
α=xWsigdx|uout(x)||usig(x)|xWsigdx|usig(x)|2.
ΠPuout(x)
=usig(x)|exp{i arg[uout(x)]}ifxWsig0otherwise.
(ΠPASuout)(x)=|α usig(x)|exp{i arg[uout(x)]}ifxWsig0ifthereexistsaxWsigwith|x-x|dframe/2uout(x)otherwise.
width(utransf)<dframe/2,
t(x)=tsig(x)+tnoise(x).
Πnoiseu(x)
=0ifthereexistsanxWsigwith|x-x|dframe/2u(x)otherwise.
Π+tdt=Π[t+td]-td.
M2={t1+td: t1M1}.
t2=Π2t=Π1[t+td]-td.
d(t2, t)d(t2,t).
d(t, t)=-dx|t(x)-t(x)|2.
t2=t1-td.
t1=Π1[t+td],
t2=t1-td.
d(t2, t)=d(t1-td, t)=d(t1, t+td),
d(t2, t)=d(t1-td, t)=d(t1, t+td).
d(t1, t+td)d(t1, t+td).

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