Abstract

We report a method to generate phase-only diffractive beam splitters allowing asymmetry of the target diffracted orders, as well as providing a tailored phase difference between the diffracted orders. We apply a well-established design method that requires the determination of a set of numerical parameters, and avoids the use of image iterative algorithms. As a result, a phase lookup table is determined that can be used for any situation where a first-order (blazed) diffractive element is modified to produce higher orders with desired intensity and/or phase relation. As examples, we demonstrate the phase difference control on triplicators, as well as on other generalized diffractive elements like bifocal Fresnel lenses and phase masks for the generation of vortex beams. Results are experimentally demonstrated by encoding the calculated phase pattern onto parallel-aligned liquid crystal spatial light modulators.

© 2013 Optical Society of America

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References

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2012

2011

J. Ren and P. Guo, “Study on four-step computer-generated hologram with same diffraction efficiency of the zeroth and first-order,” Opt. Eng. 50, 08501 (2011).

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

2010

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[CrossRef]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35, 1536–1538 (2010).
[CrossRef]

A. Jesacher and M. J. Booth, “Parallel direct laser writing in three dimensions with spatially dependent aberration correction,” Opt. Express 18, 21090–21099 (2010).
[CrossRef]

2007

2006

2003

2001

1999

1998

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

1997

J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. 36, 8435–8444 (1997).
[CrossRef]

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

1996

1995

1994

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

1993

1992

1990

1985

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1971

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

1969

L. P. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Adachi, J.

Albero, J.

Aschke, L.

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

Bandres, M. A.

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Beletic, J. W.

Bengtsson, J.

Bentley, J. B.

Bizjak, T.

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

Booth, M. J.

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Campos, J.

Carcole, E.

Cincotti, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Cottrell, D. M.

Dammann, H.

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

Dändliker, R.

Davis, J. A.

Di Fabrizio, E.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Di Leonardo, R.

Dickey, F. M.

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[CrossRef]

L. A. Romero and F. M. Dickey, “Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings,” J. Opt. Soc. Am. A 24, 2280–2295 (2007).
[CrossRef]

Fernández-Pousa, C. R.

Furlan, W. D.

Gale, M. T.

Gentili, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gori, F.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Görtler, K.

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

Guo, P.

J. Ren and P. Guo, “Study on four-step computer-generated hologram with same diffraction efficiency of the zeroth and first-order,” Opt. Eng. 50, 08501 (2011).

Gutiérrez-Vega, J. C.

Herzig, H. P.

Hirsch, P. M.

L. P. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Horner, J. L.

Ianni, F.

Imgrunt, W.

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

Ivanenko, M.

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

Jesacher, A.

Jordan, J. A.

L. P. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Kachalov, D. G.

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

Krasnaberski, A.

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Leger, J. R.

Lesem, L. P.

L. P. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Lissotchenko, V. N.

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

Lissotschenko, V. N.

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

Liu, L.

Mait, J. N.

Mikhailov, A.

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

Miklyaev, V. Y.

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

Miklyaev, Yu. V.

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

Monsoriu, J. A.

Moreno, I.

Moreno, V.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

O’Shea, D. C.

Pavelyev, V. S.

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

Poutus, M.

Prongué, D.

Ren, J.

J. Ren and P. Guo, “Study on four-step computer-generated hologram with same diffraction efficiency of the zeroth and first-order,” Opt. Eng. 50, 08501 (2011).

Román, J. F.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Romero, L. A.

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[CrossRef]

L. A. Romero and F. M. Dickey, “Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings,” J. Opt. Soc. Am. A 24, 2280–2295 (2007).
[CrossRef]

Ruocco, G.

Saavedra, G.

Salgueiro, J. R.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Sand, D.

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Yuan, X. C.

Yzuel, M. J.

Zhang, N.

Zhou, C.

Am. J. Phys.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Appl. Opt.

IBM J. Res. Dev.

L. P. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

J. Opt.

J. Albero and I. Moreno, “Grating beam splitting with liquid crystal adaptive optics,” J. Opt. 14, 075704 (2012).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

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

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Opt. Eng.

J. Ren and P. Guo, “Study on four-step computer-generated hologram with same diffraction efficiency of the zeroth and first-order,” Opt. Eng. 50, 08501 (2011).

Opt. Express

Opt. Lett.

Optik

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. SPIE

V. Y. Miklyaev, W. Imgrunt, T. Bizjak, L. Aschke, V. N. Lissotschenko, V. S. Pavelyev, and D. G. Kachalov, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE 7640, 764024 (2010).
[CrossRef]

Yu. V. Miklyaev, A. Krasnaberski, M. Ivanenko, A. Mikhailov, W. Imgrunt, L. Aschke, and V. N. Lissotchenko, “Efficient diffractive optical elements from glass with continuous profiles,” Proc. SPIE 7913, 79130B (2011).
[CrossRef]

Prog. Opt.

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[CrossRef]

Other

http://support.microsoft.com/kb/214115 .

Supplementary Material (1)

» Media 1: MOV (476 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Experimental asymmetric beam splitting for duplicators (N=2, rows D1–D5), triplicators (N=3, rows T1 and T2) and quadruplicators (N=4, rows Q1 and Q2). Two periods of the encoded phase profile to obtain each result are included (vertical axis is phase from 3.5 to 3.5 rad and horizontal axis is the spatial coordinate in arbitrary units). Overall theoretical efficiencies (η) are added as well.

Fig. 2.
Fig. 2.

Three beam interference of 0th and ±1st diffracted orders, produced to demonstrate control of the relative phase among orders: (a) simulated intensity profiles when the phase difference between 0th and ±1st is 0, π/2, and π and (b) captured intensities of all three cases. Dashed lines in (b) mark the same position.

Fig. 3.
Fig. 3.

Phase profiles of diffractive lenses: (a) a standard converging lens, (b) lens in Fig. 3(a) encoded as a duplicator with 1st and 2nd order, and with an argument difference of π. Both include a transverse phase profile from the center of the image (the vertical axis denotes phase values from 3.5 to 3.5 rad, and horizontal scale is the spatial coordinate in arbitrary units), (c) schematic view of the bifocal lens focalizations and the position of the capture plane where the CCD camera is placed, and (d) experimental interference pattern of the bifocal lens produced with the duplicator lenses. The phase difference between both orders changes in steps of π/2.

Fig. 4.
Fig. 4.

Excerpt of the captured experimental output of the optimal beam splitter of N=4 at ±1st and ±3rd orders, applied to a spiral phase profile. In the media file, a phase shift Φ is added in steps of π/2, first between the ±1st (resulting in the rotation of the inner circle) and second between the ±3rd orders (resulting in the rotation of the outer circle) (media 1).

Tables (1)

Tables Icon

Table 1. Numerical Values of the Intensity of the Diffraction Orders for the Different Cases in Fig. 1. Values are Normalized to the Maximum Value in Each Case. Values for the Target Orders are Indicated with Bold Font

Equations (7)

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

f(x)=exp[iφ(x)]
ak=12πππexp[iφ(x)]exp[ikx]dx
exp[iφ(x)]=s(x)|s(x)|,
s(x)=kμkexp[iαk]exp[ikx],
ak=|ak|exp(iβk),
φ(x)=φ[ϕ(x)],
exp[iϕ(r)]=exp[iπr2λf],

Metrics