Abstract

The range of validity of the scalar diffraction analysis is quantified for the case of two-dimensionally-periodic diffractive optical elements (crossed gratings). Three canonical classes of two-dimensionally-periodic grating structures are analyzed by using the rigorous coupled-wave analysis as well as the scalar diffraction analysis. In all cases the scalar-analysis diffraction efficiencies are compared with the exact diffraction efficiencies. The error in using the scalar analysis is then determined as a function of the grating-period(s)-to-wavelength ratio(s), the minimum feature size, the grating depth, the refractive index of the grating, the incident polarization, and the number of phase levels. The three classes of two-dimensional (2-D) unit cells are as follows: (1) a rectangular pillar, (2) an elliptical pillar, and (3) an arbitrarily pixellated multilevel 2-D unit cell that is representative of more complicated diffractive optical elements such as computer-generated holograms. In all cases a normally incident electromagnetic plane wave is considered. It is shown that the error of the scalar diffraction analysis in the case of two-dimensionally-periodic diffractive optical elements is greater than that for the corresponding one-dimensionally-periodic counterparts. In addition, the accuracy of the scalar diffraction analysis degrades with increasing refractive index, grating thickness, and asymmetry of the 2-D unit cell and with decreasing grating-period-to-wavelength ratio and feature size.

© 2002 Optical Society of America

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References

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    [CrossRef]
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2000

1999

1998

1997

1996

1995

1994

1993

1992

1990

1988

1987

1982

1978

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

Bagieu, M.

Bagnoud, V.

Becker, M. F.

Bendickson, J. M.

Bengtsson, J.

Dändliker, R.

Ehbets, P.

Gale, M. T.

Gallagher, N. C.

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, 1996), Chaps. 3 and 4.

Grann, E. B.

Gremaux, D. A.

Han, S. T.

Harris, J. B.

Herzig, H. P.

Hirayama, K.

Ichikawa, H.

Jaakkola, T.

Jahns, J.

Kok, Y.-L.

Körner, T. O.

Kuisma, S.

Lalanne, P.

Leger, J. R.

Li, L.

Mainguy, S.

Mait, J. N.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

Maystre, D.

Miller, J. M.

Mirotznik, M. S.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Nevière, M.

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

Noponen, E.

Peng, S.

Petit, R.

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

Pommet, D. A.

Prather, D. W.

Preist, T. W.

Prongué, D.

Rossi, M.

Sambles, J. R.

Sandström, S. E.

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

Schütz, H.

Schwider, J.

Sheridan, J. T.

Shi, S.

Swanson, G. J.

Taghizadeh, M. R.

Tayeb, G.

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

Thorpe, R. N.

Tsao, Y.-L.

Turunen, J.

Vasara, A.

Veldkamp, W. B.

Walker, S. J.

Wasler, R. M.

Watts, R. A.

Westerholm, J.

Appl. Opt.

J. Electromagn. Waves Appl.

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

J. Opt. (Paris)

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

T. O. Körner, J. T. Sheridan, J. Schwider, “Interferometric resolution examined by means of electromagnetic theory,” J. Opt. Soc. Am. A 12, 752–760 (1995).
[CrossRef]

S. J. Walker, J. Jahns, “Array generation with multilevel phase gratings,” J. Opt. Soc. Am. A 7, 1509–1513 (1990).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
[CrossRef]

D. W. Prather, S. Shi, “Electromagnetic analysis of axially symmetric diffractive lenses with the method of moments,” J. Opt. Soc. Am. A 17, 729–739 (2000).
[CrossRef]

J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
[CrossRef]

Y.-L. Kok, N. C. Gallagher, “Relative phases of electromagnetic waves diffracted by a perfectly conducting rectangular-groove grating,” J. Opt. Soc. Am. A 5, 65–73 (1988).
[CrossRef] [PubMed]

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

V. Bagnoud, S. Mainguy, “Diffraction of electromagnetic waves by dielectric crossed gratings: a three-dimensional Rayleigh–Fourier solution,” J. Opt. Soc. Am. A 16, 1277–1285 (1999).
[CrossRef]

M. Bagieu, D. Maystre, “Regularized Waterman and Rayleigh methods: extension to two-dimensional gratings,” J. Opt. Soc. Am. A 16, 284–292 (1999).
[CrossRef]

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[CrossRef]

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

Other

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, 1996), Chaps. 3 and 4.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

(a) Perspective view of the unit cell of an arbitrarily pixellated two-dimensionally-periodic grating. Λx and Λy are the x- and y-direction periodicities, and nc and ns are the refractive indices of the cover and the substrate, respectively. For simplicity the z axis is not shown, but its direction is implied from the right-handed xyz coordinate system. Variable-thickness pixels are permitted as shown in order to represent multilevel gratings. The (l1, l2)th pixel of thickness dl1,l2 is also shown. (b) Top view of the unit cell of the arbitrarily pixellated grating shown in (a). Lx and Ly are the pixel dimensions along the x and y directions, respectively. (c) Top view of the unit cell of a rectangular two-dimensionally-periodic grating. The filling factors are Fx and Fy along the x and y directions, respectively. (d) Top view of the unit cell of an elliptical two-dimensionally-periodic grating. The ellipse semiaxes are a and b along the x and y directions, respectively.

Fig. 2
Fig. 2

Exact and scalar diffraction efficiencies shown as functions of the normalized grating thickness for Λ/λ0=10, 5, 2.5, and 1 (or s/λ0=5, 2.5, 1.25, and 0.5, respectively). The two-dimensionally-periodic grating has a square pillar unit cell with Fx=Fy=0.50. The polarization of the normally incident plane wave is E=Eyyˆ. (a) (0, 0)th forward diffraction efficiency, (b) (1, 0)th forward diffraction efficiency, (c) (0, 0)th backward diffraction efficiency.

Fig. 3
Fig. 3

(1, 0)th forward exact and scalar diffraction efficiencies shown as functions of the normalized grating thickness for Λ/λ0=10, 5, 2.5, and 1 (or s/λ0=5, 2.5, 1.25, and 0.5, respectively). The two-dimensionally-periodic grating has a square pillar unit cell with Fx=Fy=0.50. The polarization of the normally incident plane wave is E=Exxˆ.

Fig. 4
Fig. 4

Errors of the scalar diffraction efficiencies shown as functions of the normalized grating thickness for Λ/λ0=10, 5, 2.5, and 1 (or s/λ0=5, 2.5, 1.25, and 0.5, respectively). The two-dimensionally-periodic grating has a square pillar unit cell with Fx=Fy=0.50. The polarization of the normally incident plane wave is E=Eyyˆ (solid curves) or E=Exxˆ (dashed curves). (a) Error η0,0f of the (0, 0)th forward diffraction efficiency (no polarization dependence), (b) error η1,0f of the (1, 0)th forward diffraction efficiency, (c) error η0,0b of the (0, 0)th backward diffraction efficiency (no polarization dependence).

Fig. 5
Fig. 5

Comparison of the error in the scalar diffraction analysis between a two-dimensionally-periodic square pillar unit-cell grating (with Fx=Fy=0.50 and Λx=Λy=Λ) and a one-dimensionally-periodic lamellar grating of the same feature size (F=0.50) and period Λ for Λ/λ0=10 (or s/λ0=5). The errors η1,0f of the (1, 0)th forward-diffracted order of the 2-D grating and η1f of the +1-th forward-diffracted order of the 1-D grating are shown as functions of the normalized grating thickness for both orthogonal polarizations for a normally incident plane wave.

Fig. 6
Fig. 6

Exact and scalar diffraction efficiencies shown as functions of the normalized grating thickness for Λ/λ0=10, 5, 2.5, and 1 (or s/λ0=2.5, 1.25, 0.675, and 0.25, respectively). The two-dimensionally-periodic grating has a rectangular pillar unit cell with Fx=0.25 and Fy=0.75. The polarization of the normally incident plane wave is E=Eyyˆ. (a) (0, 0)th forward diffraction efficiency, (b) (1, 0)th forward diffraction efficiency, (c) (0, 0)th backward diffraction efficiency.

Fig. 7
Fig. 7

Errors of the scalar diffraction efficiencies shown as functions of the normalized grating thickness for Λ/λ0=10, 5, 2.5, and 1 (or s/λ0=2.5, 1.25, 0.675, and 0.25, respectively). The two-dimensionally-periodic grating has a rectangular pillar unit cell with Fx=0.25 and Fy=0.75. The polarization of the normally incident plane wave is E=Eyyˆ (solid curves) or E=Exxˆ (dashed curves). (a) Error η0,0f of the (0, 0)th forward diffraction efficiency, (b) error η1,0f of the (1, 0)th forward diffraction efficiency, (c) error η0,0b of the (0, 0)th backward diffraction efficiency.

Fig. 8
Fig. 8

Errors of the scalar diffraction efficiencies shown as functions of the normalized grating thickness for Λ/λ0=10, 5, and 2.5 (or s/λ0=5, 2.5, and 1.25, respectively). The two-dimensionally-periodic grating has a square pillar unit cell with Fx=Fy=0.50. The polarization of the normally incident plane wave is E=Eyyˆ (solid curves) or E=Exxˆ (dashed curves). For the (0, 0)th forward- and backward-diffracted orders, there is no polarization dependence, owing to the symmetry of the grating. The substrate refractive index is ns=2.0. (a) Error η0,0f of the (0, 0)th forward diffraction efficiency, (b) error η1,0f of the (1, 0)th forward diffraction efficiency, (c) error η0,0b of the (0, 0)th backward diffraction efficiency.

Fig. 9
Fig. 9

Similar to Fig. 8 but for ns=4.0. Because of the more oscillatory nature of the curves in this case, only the range of d/λ0 between 0 and 1 is shown.

Fig. 10
Fig. 10

Exact and scalar diffraction efficiencies shown as functions of the normalized grating thickness for Λ/λ0=10, 5, and 2.5 (or s/λ0=5, 2.5, and 1.25, respectively). The two-dimensionally-periodic grating has a circular pillar unit cell with 2a/Λ=2b/Λ=0.50. (a) (1, 0)th forward diffraction efficiency and (b) (1, 1)th forward diffraction efficiency.

Fig. 11
Fig. 11

Exact and scalar diffraction efficiencies shown as functions of the normalized grating thickness for Λ/λ0=10, 5, and 2.5 (or s/λ0=3.333, 1.667, and 0.833, respectively). The two-dimensionally-periodic grating has an elliptical pillar unit cell with 2a/Λ=1/3 and 2b/Λ=2/3. The polarization of the normally incident plane wave is E=Eyyˆ. (a) (1, 0)th forward diffraction efficiency and (b) (1, 1)th forward diffraction efficiency.

Fig. 12
Fig. 12

Unit cell of a multilevel arbitrarily pixellated two-dimensionally-periodic grating. This unit cell can be represented by 16 (Mx×My, where Mx=My=4) pixels per grating and a stack of three cascaded gratings. This unit cell was selected to show the generality of the arbitrarily pixellated 2-D unit-cell gratings.

Tables (4)

Tables Icon

Table 1 Rigorous and Scalar Diffraction Efficiencies of the (1,0)th Forward-Diffracted Order of a Square Pillar 2-D Unit Cell for d/λ0=1, Λx/λ0=Λy/λ0=10

Tables Icon

Table 2 Rigorous and Scalar Diffraction Efficiencies of Selected Forward-Diffracted Orders of the Arbitrarily Pixellated Unit-Cell Grating Shown in Fig. 12

Tables Icon

Table 3 Convergence of Rigorous Diffraction Efficiencies of the Square Pillar 2-D Unit Cell for d/λ0=1, Fx=Fy=0.50, ns=1.5, and E=Eyyˆ

Tables Icon

Table 4 Convergence of Rigorous Diffraction Efficiencies of the Square Pillar 2-D Unit Cell for d/λ0=1, Fx=Fy=0.50, Λx/λ0=Λy/λ0=10, and E=Eyyˆ

Equations (24)

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ddz S˜xS˜yU˜xU˜y=jA˜S˜xS˜yU˜xU˜y,
A˜=0˜0˜-K˜xE˜-1K˜yK˜xE˜-1K˜x-I˜0˜0˜I˜-K˜yE˜-1K˜yK˜yE˜-1K˜xK˜xK˜yαE˜+(1-α)A˜-1-K˜xK˜x0˜0˜K˜yK˜y-αA˜-1+(1-α)E˜-K˜yK˜x0˜0˜.
DER,i1,i2b=|Ri1,i2|2 Re(-kz,i1,i2I)kinc,z,
DER,i1,i2f=|Ti1,i2|2 Re(kz,i1,i2III)kinc,z,
t(x, y)=t(kinc)tuc(x, y)**×h1h2δ(x-h1Kx)δ(y-h2Ky),
r(x, y)=r(kinc)ruc(x, y)**×h1h2δ(x-h1Kx)δ(y-h2Ky),
Ut(kx, ky)=t(kinc)h1h2Tuch1,h2(2π)2×δ(kx+kinc,x-h2Kx)×δ(ky+kinc,y-h2Ky),
Ur(kx, ky)=r(kinc)h1h2Ruch1,h2(2π)2×δ(kx+kinc,x-h1Kx)×δ(ky+kinc,y-h2Ky),
DES,i1,i2b=|r(kinc)|2|Ruci1,i2|2,
DES,i1,i2f=nsnc|t(kinc)|2|Tuci1,i2|2=[1-|r(kinc)|2]|Tuci1,i2|2.
Error %=ηi1,i2u=DER,i1,i2u-DES,i1,i2uDER,i1,i2u×100,
ϵh1,h2=(ns2-nc2) sin(πh1Fx)πh1 sin(πh2Fy)πh2+nc2 sin(πh1)πh1 sinh(πh2)πh2.
ϵh1,h2=ns2-nc2Λx1Λy1 abu0h1Λx12+h2Λy121/2J12πabu0h1Λx12+h2Λy121/2ifh1,h20(ns2-nc2) πabΛxΛy+nc2ifh1=h2=0.
ϵh1,h2=l1l2ϵh1,h2l1,l2 exp[-j2π(h1axl1,l2+h2ayl1,l2)],
tuc(x, y)=rectxFxΛx, yFyΛy[exp(-jϕt)-1]+rectxΛx, yΛy,
ruc(x, y)=rectxFxΛx, yFyΛy[1-exp(-jϕr)]+exp(-jϕr)rectxΛx, yΛy,
Tuch1,h2=[exp(-jϕt)-1] sin(πh1Fx)πh1 sin(πh2Fy)πh2+sin(πh1)πh1 sin(πh2)πh2,
Ruch1,h2=[1-exp(-jϕr)] sin(πh1Fx)πh1 sin(πh2Fy)πh2+exp(-jϕr) sin(πh1)πh1 sin(πh2)πh2.
Tuch1,h2=exp(-jϕt)-1ΛxΛy u0h1Λx12+h2Λy121/2×J12πabu0h1Λx12+h2Λy121/2+sin(πh1)πh1 sin(πh2)πh2,
Ruch1,h2=1-exp(-jϕr)ΛxΛy u0h1Λx12+h2Λy121/2×J12πabu0h1Λx12+h2Λy121/2+exp(-jϕr) sin(πh1)πh1 sin(πh2)πh2.
tuc(x, y)=l1l2rectx-axl1,l2Fxl1,l2Λx, y-ayl1,l2Fyl1,l2Λy×[exp(-jϕtl1,l2)-1]+rectxΛx, yΛy,
ruc(x, y)=l1l2rectx-axl1,l2Fxl1,l2Λx, y-ayl1,l2Fyl1,l2Λy×[1-exp(-jϕrl1,l2)]+exp(-jϕrl1,l2)rectxΛx, yΛy,
Tuch1,h2=l1l2Tuc,l1,l2h1,h2 exp[-j2π(h1axl1,l2+h2ayl1,l2)],
Ruch1,h2=l1l2Ruc,l1,l2h1,h2 exp[-j2π(h1axl1,l2+h2ayl1,l2)],

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