Abstract

Improved diffraction efficiencies can be obtained in the paraxial domain of diffractive optics by considering light explicitly as an electromagnetic rather than a scalar field because of the extra freedoms provided by the state of polarization. For example, diffractive beam splitters with 100% efficiency are made possible by means of space-variant subwavelength-carrier surface-relief elements. Some aspects of the general design theory of polarization-modulating elements for vector fields, including design freedoms and constraints, are presented. Upper bounds of diffraction efficiency are derived and compared with those for the scalar case. Iterative design algorithms are developed. Several design examples with different constraints are presented, and the effects of replacing continuous-fringe structures by pixel structures containing locally linear gratings are evaluated.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  9. O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, Chap. 1.
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    [CrossRef]
  11. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
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    [CrossRef]
  14. L. Pajewski, R. Borghi, G. Schettini, F. Frezza, M. Santarsiero, “Design of a binary grating with subwavelength features that acts as a polarizing beam splitter,” Appl. Opt. 40, 5898–5905 (2001).
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  17. C. R. Fernández-Pousa, I. Moreno, J. A. Davis, J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. 26, 1651–1653 (2001).
    [CrossRef]

2002 (1)

2001 (4)

2000 (2)

J. Tervo, J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[CrossRef]

M. Honkanen, V. Kettunen, J. Tervo, J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).
[CrossRef]

1999 (1)

1995 (2)

1993 (1)

1992 (1)

1991 (3)

F. Wyrowski, “Upper bound of the efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1817 (1991).
[CrossRef] [PubMed]

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings, Opt. Lett. 89, 1921–1923 (1991).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Adachi, J.

Biener, G.

Bomzon, Z.

Borghi, R.

Bräuer, R.

Bryngdahl, O.

M. Schmitz, R. Bräuer, O. Bryngdahl, “Phase gratings with subwavelength structures,” J. Opt. Soc. Am. A 12, 2458–2462 (1995).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, Chap. 1.

Davis, J. A.

Fainman, Y.

Farn, M. W.

Fernández-Pousa, C. R.

Frezza, F.

Gori, F.

Haidner, H.

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings, Opt. Lett. 89, 1921–1923 (1991).
[CrossRef]

Hasman, E.

Honkanen, M.

M. Honkanen, V. Kettunen, J. Tervo, J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).
[CrossRef]

Kettunen, V.

M. Honkanen, V. Kettunen, J. Tervo, J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).
[CrossRef]

Kipfer, P.

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings, Opt. Lett. 89, 1921–1923 (1991).
[CrossRef]

Kleiner, V.

Kuittinen, M.

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, Sec. 5.7.

Moreno, I.

Pajewski, L.

Richter, I.

Santarsiero, M.

Schettini, G.

Schmitz, M.

Stork, W.

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings, Opt. Lett. 89, 1921–1923 (1991).
[CrossRef]

Streibl, N.

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings, Opt. Lett. 89, 1921–1923 (1991).
[CrossRef]

Sun, P.-C.

Tervo, J.

J. Tervo, J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
[CrossRef]

M. Honkanen, V. Kettunen, J. Tervo, J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).
[CrossRef]

J. Tervo, J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[CrossRef]

Turunen, J.

J. Tervo, J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
[CrossRef]

J. Tervo, J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[CrossRef]

M. Honkanen, V. Kettunen, J. Tervo, J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).
[CrossRef]

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, Sec. 5.7.

Wyrowski, F.

F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

F. Wyrowski, “Upper bound of the efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1817 (1991).
[CrossRef] [PubMed]

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, Sec. 5.7.

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, Chap. 1.

Xu, F.

Appl. Opt. (3)

J. Mod. Opt. (1)

M. Honkanen, V. Kettunen, J. Tervo, J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. 47, 2351–2359 (2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. Tervo, J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
[CrossRef]

Opt. Lett. (7)

Rep. Prog. Phys. (1)

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other (2)

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, Sec. 5.7.

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, Chap. 1.

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Figures (2)

Fig. 1
Fig. 1

Discretization of a polarization grating into a structure with K=16 slices in each of which the fringe orientation is constant.

Fig. 2
Fig. 2

Efficiency of a 1:2 beam splitter as a function of the number of slices K. The dashed line corresponds to the scalar upper bound.

Tables (1)

Tables Icon

Table 1 Diffraction Efficiencies of Beam Splitters for Vector and Scalar Fields under Different Constraints (see text)

Equations (24)

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T(x, y)=A(x, y)cos2 ϕ(x, y)+B(x, y)sin2 ϕ(x, y)[A(x, y)-B(x, y)]sin ϕ(x, y)cos ϕ(x, y)[A(x, y)-B(x, y)]sin ϕ(x, y)cos ϕ(x, y)A(x, y)sin2 ϕ(x, y)+B(x, y)cos2 ϕ(x, y),
T(x, y)E=14[A(x, y)-B(x, y)](Ex+iEy)1i×exp[-i2ϕ(x, y)]+14[A(x, y)-B(x, y)]×(Ex-iEy)1-iexp[i2ϕ(x, y)]+12[A(x, y)+B(x, y)]ExEy.
E(x, y)=m=-n=-Dmn exp[i2π(mx+ny)],
Dmn=0101T(x, y)E exp[-i2π(mx+ny)]dxdy
ηmn=Dmn|Dmn.
ηmn=0101T(x, y)E exp[-i2π(mx+ny)]dxdy2=14|A-B|20101 cos[2ϕ(x, y)]×exp[-i2π(mx+ny)]dxdy2.
ηu=0101ES(x, y)dxdy2,
ES(x, y)=m,nSDxmn exp[i2π(mx+ny)]2+m,nSDymn exp[i2π(mx+ny)]21/2.
E(x, 0)2=m=14n=14Dm|Dnexp[i2π(m-n)x].
D1|D4=D1|D3+D2|D4=D1|D2+D2|D3+D3|D4=0.
D1|D4=D2|D4=D1|D2=0.
D11|D12+D21|D22=D11|D22=D21|D12=D11|D21+D12|D22=0.
D11|D12=D21|D12=D11|D21=0
D1|D6=D1|D5+D2|D6=D2|D5+D3|D6=D1|D3+D3|D5=D1|D2+D2|D3+D5|D6=0.
14(A-B){(c-i)(Ex+iEy)exp[-i2ϕ(x, y)]+(c+i)(Ex-iEy)exp[i2ϕ(x, y)]} +12(A+B)(cEx-Ey)=0.
(A-B)(c-i)(Ex+iEy)=(A-B)(c+i)(Ex-iEy)=(A+B)(cEx-Ey)=0,
E(x, y)=A(x, y)EA+B(x, y)EB=A(x, y)Ex cos2 ϕ+Ey sin ϕ cos ϕEx sin ϕ cos ϕ+Ey sin2 ϕ+B(x, y)Ex sin2 ϕ-Ey sin ϕ cos ϕ-Ex sin ϕ cos ϕ+Ey cos2 ϕ.
A(x, y)=1,
B(x, y)=exp(iθ),
|Dmn|2=η/N,
|Dx,mn|=1-|Dy,mn|2,
arg Dx,mn=arg Dy,mn,
ηm=14|A+B|2δm,0+18sinc2(m/P)|A-B|2×s[(1+2|ExEy|sin ϑ)δm,sP-1+(1-2|ExEy|sin ϑ)δm,1-sP],
Dmn=1PQexp[iπ(m/P+n/Q)]sinc(m/P)sinc(n/Q)×p=1Pq=1Q exp[-i2π(mp/P+nq/Q)]×12(Apq+Bpq)ExEy+14(Apq-Bpq)(Ex+iEy)1iexp(-i2ϕpq)+14(Apq-Bpq)(Ex-iEy)1-iexp(-i2ϕpq).

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