Abstract

Effective medium theory is useful for designing optical elements with a form-birefringent subwavelength structure. We describe a way to determine the effective refractive indices and the directions of their principal axes for two-dimensional (2-D) subwavelength gratings in the normal incidence case. The effective indices and the directions are calculated from finite coupled-coefficient equations that are used in 2-D rigorous coupled-wave analysis. We use the effective medium theory to calculate transmittance of several kinds of 2-D periodic subwavelength gratings. These results are compared with results of rigorous grating analysis to confirm the predictions of the effective medium theory for 2-D subwavelength gratings.

© 1998 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  6. H. Kikuta, Y. Ohira, K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566–1572 (1997).
    [CrossRef] [PubMed]
  7. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  8. M. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. P. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
    [CrossRef]
  22. M. G. Moharam, “Coupled wave analysis of twodimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Chindrich, ed., Proc. SPIE883, 8–11 (1988).For details of the calculation procedure, see M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  23. Z. Zhang, S. Satpathy, “Electromagnetic wave propaga-tion in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
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  24. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
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1997 (3)

1996 (2)

F. Xu, R. Tyan, P. Sun, Y. Fainman, C. Cheng, A. Scherer, “Form-birefringent computer-generated holograms,” Opt. Lett. 21, 1513–1515 (1996).
[CrossRef] [PubMed]

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

1995 (5)

1994 (2)

1993 (2)

1991 (1)

1990 (1)

Z. Zhang, S. Satpathy, “Electromagnetic wave propaga-tion in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

1983 (1)

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

1982 (1)

S. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

1981 (1)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

1978 (2)

1956 (1)

S. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Adams, J. L.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J. R.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Aoyama, S.

S. Aoyama, T. Yamashita, “Grating beam splitting polarizer using multi-layer resist method in International Conference on the Application and Theory of Periodic Structures,” J. M. Lerner, W. R. Mckinney, eds., Proc. SPIE1545, 241–250 (1991).
[CrossRef]

Botten, I. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Bräuer, R.

Bryngdahl, O.

Chen, F. T.

Cheng, C.

Craig, M. S.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Craighead, H. G.

Fainman, Y.

Flanders, D. C.

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

Gaylord, T. K.

Grann, E.

Grann, E. B.

Haggans, C. W.

Hutley, M. C.

S. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Iwata, K.

H. Kikuta, Y. Ohira, K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566–1572 (1997).
[CrossRef] [PubMed]

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Kikuta, H.

H. Kikuta, Y. Ohira, K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566–1572 (1997).
[CrossRef] [PubMed]

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Knop, K.

Kostuk, R. K.

Lalanne, P.

P. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
[CrossRef]

P. Lalanne, G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8, 53–56 (1997).
[CrossRef]

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Lemercier-Lalanne, D.

P. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
[CrossRef]

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Li, L.

Magnusson, R.

McPhedran, R. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Moharam, M.

Moharam, M. G.

E. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
[CrossRef]

M. G. Moharam, “Coupled wave analysis of twodimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Chindrich, ed., Proc. SPIE883, 8–11 (1988).For details of the calculation procedure, see M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

Morris, G. H.

Morris, G. M.

P. Lalanne, G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8, 53–56 (1997).
[CrossRef]

Ohira, Y.

Pommet, D. A.

Raguin, D. H.

Richter, I.

Rytov, S.

S. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Satpathy, S.

Z. Zhang, S. Satpathy, “Electromagnetic wave propaga-tion in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Scherer, A.

Smith, R. E.

Southwell, W.

Sun, P.

Tyan, R.

Vawter, G. A.

Warren, M. E.

Wendt, J. R.

Wilson, S.

S. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Xu, F.

Yamashita, T.

S. Aoyama, T. Yamashita, “Grating beam splitting polarizer using multi-layer resist method in International Conference on the Application and Theory of Periodic Structures,” J. M. Lerner, W. R. Mckinney, eds., Proc. SPIE1545, 241–250 (1991).
[CrossRef]

Yariv, A.

A. Yariv, P. Yhe, Optical Waves in Crystals (Wiley, New York, 1984), pp. 165–174.

Yhe, P.

A. Yariv, P. Yhe, Optical Waves in Crystals (Wiley, New York, 1984), pp. 165–174.

Yoshida, H.

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Zhang, Z.

Z. Zhang, S. Satpathy, “Electromagnetic wave propaga-tion in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

J. Mod. Opt. (1)

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Nanotechnology (1)

P. Lalanne, G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8, 53–56 (1997).
[CrossRef]

Opt. Acta (2)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

S. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Opt. Lett. (3)

Opt. Rev. (1)

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

Z. Zhang, S. Satpathy, “Electromagnetic wave propaga-tion in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

S. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (3)

A. Yariv, P. Yhe, Optical Waves in Crystals (Wiley, New York, 1984), pp. 165–174.

S. Aoyama, T. Yamashita, “Grating beam splitting polarizer using multi-layer resist method in International Conference on the Application and Theory of Periodic Structures,” J. M. Lerner, W. R. Mckinney, eds., Proc. SPIE1545, 241–250 (1991).
[CrossRef]

M. G. Moharam, “Coupled wave analysis of twodimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Chindrich, ed., Proc. SPIE883, 8–11 (1988).For details of the calculation procedure, see M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Two-dimensional (2-D) subwavelength structure. Pillars are arranged with periods Λx and Λy in the x and y directions.

Fig. 2
Fig. 2

2-D square subwavelength grating on a substrate. The subwavelength structure in region 2 is regarded as a thin film that is isotropic to the z direction. The relative dielectrics of region 1 and the groove in region 2, ε1 and εg, respectively, are both 1.0; the dielectrics of the substrate (region 3) and the ridge in region 2, ε3 and εr, respectively, are 4.0. The filling factor f is defined by the ratio of the ridge width to period Λ.

Fig. 3
Fig. 3

Convergence of the effective refractive index with respect to the maximum order N of the Fourier coefficients of dielectrics. The calculated grating has period Λ=0.4λ0 and filling factor f=0.5. The effective refractive index is independent of groove depth d.

Fig. 4
Fig. 4

Transmittance of the 2-D subwavelength grating as a function of groove depth at a fixed period Λ=0.4λ0 and filling factor f=0.5. (a) Transmittance power, (b) phase delay. Circles show results estimated by the effective medium theory with effective refractive index 1.220. The solid curve is the result from RCWA.

Fig. 5
Fig. 5

Effective refractive index as a function of the period of the 2-D grating. The filling factor f is fixed at 0.5.

Fig. 6
Fig. 6

2-D rectangular subwavelength grating on a substrate. The dielectrics are ε1=εg=1.0 and εr=ε3=4.0.

Fig. 7
Fig. 7

(a) Transmittance and (b) phase delay for the rectangular subwavelength grating as a function of groove depth d. The periods and filling factors are Λx=0.4λ0, Λy=0.2λ0 and fx=fy=0.707, respectively. The crosses show results estimated by the effective medium theory for x polarization with effective index 1.405; circles show y polarization with effective index 1.465. Thin and thick solid curves show the results from RCWA for x and y polarization, respectively.

Fig. 8
Fig. 8

Effective refractive indices for x polarization with various periods Λx and Λy and filling factors fx and fy: (a) indices with various periods at filling factor fx=fy=0.5, (b) indices with various filling factors at period Λx=Λy=0.4λ0.

Fig. 9
Fig. 9

2-D triangle subwavelength grating on a substrate.

Fig. 10
Fig. 10

Transmittance of the 45° polarized wave as a function of groove depth d for the triangle subwavelength grating: (a) transmittance power, (b) phase delay. The dielectrics are ε1=εg=1.0 and εr=ε3=4.0. Periods Λx and Λy are both 0.4λ0, and fx=fy=0.5.

Fig. 11
Fig. 11

Effective refractive indices and the directions of the principal axes for the triangle subwavelength structure as a function of filling factor fy at fixed fx=0.5 and Λx=Λy=0.4λ0.

Fig. 12
Fig. 12

Change of calculated effective refractive indices obtained from three determination methods with respect to the Fourier expansion size of the dielectrics for an isotropic case. Size N means that Fourier coefficients from -N to +N order for each direction are used in the calculation of effective refractive indices. Bräuer’s method is independent of Fourier expansion size.

Equations (21)

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

ε(x, y)=m,nε˜m,n exp[j(mKxx+nKyy)],
1/ε(x, y)=m,nξ˜m,n exp[j(mKxx+nKyy)],
Eg(x, y, z)=p,qSipq(z)exp[-j(kxpx+kyqy)],
i=x, y,
kxp=kx0-pKx,kyq=ky0-qKy,
S¨xpq=kxpm,nξ˜p-m,q-nr,tε˜m-r,n-t(kxmSxrt-kynSyrt)-kyq(kxpSypq-kyqSxpq)-k02m,nε˜p-m,q-nSxmn,
S¨ypq=kyqm,nξ˜p-m,q-nr,tε˜m-r,n-t(kxmSxrt-kynSyrt)+kxp(kxpSypq-kyqSxpq)-k02m,nε˜p-m,q-nSymn,
S¨ipq=2Sipqz2.
s¨xs¨y=Msxsy,
si=si-1si0si1,sip=Sip-1Sip0Sip1,
i=x, y; p=-N, ,-1, 0, 1,, N,
M=Ky2+(KxXKx-I)AKx(XKyA-Ky)Ky(XKxA-Kx)Kx2+(KyXKy-I)A.
E˜lx(z)=exp(±γl1/2z)p,qwlxpq exp[-j(kxpx+kyqy)],
E˜ly(z)=exp(±γl1/2z)p,qwlypq exp[-j(kxpx+kyqy)],
neff=λ02π(kx02+ky02-γl)1/2,
ψ=tan-1wly00wlx00.
wx1=[wx10-1wx100wx101]T=[0.0150.9290.015]T,
wy1=[wy10-1wy100wy101]T=[0.0000.0000.000]T,
wx2=[wx20-1wx200wx201]T=[0.0000.0000.000]T,
wy2=[wy20-1wy200wy201]T=[-0.2460.870-0.246]T.
ψ1=tan-10.0000.929=0,ψ2=tan-10.8700.000=π2.

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