Abstract

Effective-medium theory is often helpful for understanding the behavior of subwavelength gratings with small period-to-wavelength ratios. By analytically solving Maxwell’s equations in the small-depth limit, we show that the effective properties of subwavelength gratings strongly depend on the grating depth. Moreover, the effective properties are shown to depend not only on the grating structure but also on the optical indices of the surrounding media. A simple expression for the effective indices of one-dimensional (1-D) and two-dimensional (2-D) gratings with arbitrary depths is proposed. Comparison with rigorous computations shows that for TE polarization of 1-D gratings the depth dependence of the effective index prediction is accurate. For TM polarization of 1-D volume gratings and for 2-D volume gratings a slight deviation between rigorous computation and the prediction is observed for approximately quarter-wave depths. For TM polarization of 1-D surface-relief gratings and for 2-D surface-relief gratings, no closed-form expression for the depth dependence of the effective indices is found, but a very sharp variation of the effective index at small thicknesses is predicted. This prediction is confirmed by rigorous computation.

© 1997 Optical Society of America

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References

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    [CrossRef]
  13. J. L. Jackson, S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
    [CrossRef]
  14. E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary grating,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
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  15. R. Bräuer, A. Bryngdahl, “Design of antireflection gratings with approximate and rigorous methods,” Appl. Opt. 33, 7875–7882 (1994).
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  17. M. Born, E. Wolf, Principle of Optics, 6th ed. (Pergamon, New York, 1984), p. 62.
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    [CrossRef]
  19. Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
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  20. A. Wirgin, “On the reduction of reflection between two transparent media by interface roughening,” Opt. Commun. 37, 321–325 (1981).
    [CrossRef]
  21. G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
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1996 (2)

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

Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

1995 (4)

1994 (2)

1993 (1)

1992 (2)

1991 (1)

1990 (1)

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

1985 (1)

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).

1983 (1)

1982 (2)

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

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

1981 (1)

A. Wirgin, “On the reduction of reflection between two transparent media by interface roughening,” Opt. Commun. 37, 321–325 (1981).
[CrossRef]

1978 (1)

P. Yeh, “A new optical model for wire grid polarizers,” Opt. Commun. 26, 289–292 (1978).
[CrossRef]

1968 (1)

J. L. Jackson, S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
[CrossRef]

1956 (1)

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

Bagby, J. S.

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

Bell, J. M.

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principle of Optics, 6th ed. (Pergamon, New York, 1984), p. 62.

Bouchitté, G.

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).

Bräuer, R.

Bryngdahl, A.

Case, S. K.

Chen, F. T.

Coriell, S. R.

J. L. Jackson, S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
[CrossRef]

Craighhead, H. G.

Derrick, G. H.

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

Enger, R. C.

Farn, W. M.

Gaylord, T. K.

Grann, E. B.

Gunning, W. J.

Haidner, H.

Hutley, M. C.

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

Jackson, J. L.

J. L. Jackson, S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
[CrossRef]

Kipfer, P.

Lalanne, Ph.

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

Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

Lemercier-Lalanne, D.

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

Magnusson, R.

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

McPhedran, R. C.

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Motamedi, M. E.

Peng, S.

Petit, R.

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).

Pommet, D. A.

Raguin, D. H.

Rytov, S. M.

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

Southwell, W. H.

Sreibl, N.

Stork, W.

Wang, S. S.

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

Wilson, S. J.

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

Wirgin, A.

A. Wirgin, “On the reduction of reflection between two transparent media by interface roughening,” Opt. Commun. 37, 321–325 (1981).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principle of Optics, 6th ed. (Pergamon, New York, 1984), p. 62.

Yeh, P.

P. Yeh, “A new optical model for wire grid polarizers,” Opt. Commun. 26, 289–292 (1978).
[CrossRef]

Appl. Opt. (5)

Electromagnetics (1)

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).

J. Appl. Phys. (1)

J. L. Jackson, S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (6)

Opt. Acta (2)

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

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
[CrossRef]

Opt. Commun. (2)

A. Wirgin, “On the reduction of reflection between two transparent media by interface roughening,” Opt. Commun. 37, 321–325 (1981).
[CrossRef]

P. Yeh, “A new optical model for wire grid polarizers,” Opt. Commun. 26, 289–292 (1978).
[CrossRef]

Opt. Lett. (2)

Sov. Phys. JETP (1)

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

Other (1)

M. Born, E. Wolf, Principle of Optics, 6th ed. (Pergamon, New York, 1984), p. 62.

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Figures (6)

Fig. 1
Fig. 1

General diffraction problem investigated in this paper (1-D case).

Fig. 2
Fig. 2

Dependence of the effective relative permittivity with the substrate optical index. Crosses are obtained with rigorous computations, and the solid curve corresponds to the η̃TM prediction of Eq. (14b). h=0.001λ, α=10, and n1=1.

Fig. 3
Fig. 3

TE polarization of a sinusoidally modulated grating (0=5, Δ=3, n1=1, and n3=3). Crosses, plus signs, and circles are rigorous computation results obtained for α=20, 6, and 4, respectively. Solid curves are obtained from Eq. (16) with nL given by Eq. (18a).

Fig. 4
Fig. 4

Same as Fig. 3 (for TM polarization), except that the effective index nL in the large-depth limit is given by Eq. (18).

Fig. 5
Fig. 5

Comparison between the EMT prediction of Eq. (16) and rigorous computation results for the 2-D grating whose relative permittivity is given by Eq. (19). Crosses, plus signs, and circles are defined as for Fig. 3.

Fig. 6
Fig. 6

EMT of lamellar gratings: (a) TE polarization of a 1-D grating, (b) TM polarization of a 1-D grating, and (c) 2-D grating. In (a) the solid curve corresponds to Eq. (16), and in (b) and (c) the solid lines correspond to the EMT prediction of Ref. 12 in the large-depth limit.

Equations (33)

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E1,y=exp[jn1k(z-h)]+i Ri×exp[-j(n12k2-i2K2)1/2(z-h)]exp(jiKx),
E3,y=i Ti exp[j(n32k2-i2K2)1/2z]exp(jiKx),
Ey=m Sm(z)exp(jKmx),
Hx=j 1μ0cm Um(z)exp(jKmx).
Si=-kUi,
Ui=-ki2α2Si-p i-pSp,
Si(h)=Si(0)-hkUi(0)+(hk)22i2α2Si(0)-p i-pSp(0)+O(h3k3),
Ui(h)=Ui(0)-hki2α2Si(0)-p i-pSp(0)+(hk)22i2α2Ui(0)-p i-pUp(0)+O(h3k3).
Ri=Ri(0)+hkRi(1)+(hk)22Ri(2)+O(h3k3),
Ti=Ti(0)+hkTi(1)+(hk)22Ti(2)+O(h3k3).
R0=n1-n3n1+n31+j2n1 0-n32n32-n12hk+4n1n32-n12×j p0 p-p(p2α2-n32)1/2+(p2α2-n12)1/2+(n1n3+0)(0-n32)n1+n3 (hk)22+O(h3k3),
T0=2n1n1+n31-j 0+n1n3n1+n3hk+0-2jn1+n3×p0 p-p(p2α2-n32)1/2+(p2α2-n12)1/2-2 (n1n3+0)2(n1+n3)2 (hk)22+O(h3k3).
η=η0+hkη1+O(h2k2).
Rfilm=n1-n3n1+n31+j2n1 η0-n32n32-n12hk+4n1n32-n12×jη1+(n1n3+η0)(η0-n32)n1+n3 (hk)22+O(h3k3),
Tfilm=2n1n1+n31-j η0+n1n3n1+n3hk+η0-2jη1n1+n3-2 (n1n3+η0)2(n1+n3)2 (hk)22+O(h3k3).
p0 p-p(p2α2-n32)1/2+(p2α2-n12)1/2,
ηTE=0+p0 p-p(p2α2-n32)1/2+(p2α2-n12)1/2hk+O(h2k2).
ηTM=0-p0 p-pn32(p2α2-n32)-1/2+n12(p2α2-n12)-1/2×hk+O(h2k2).
ηx=0,0-(p,q)(0,0) -p,-qp,qCp,qAp,q2-Cp,qCp,qhk+O(h2k2),
ηy=0,0-(p,q)(0,0) -p,-qp,qCp,qAp,q2-Cp,qCp,qhk+O(h2k2).
Cp,q=N3p,q+N1p,q-q2α2(1/N1p,q+1/N3p,q),
Cp,q=N3p,q+N1p,q-p2ρ2α2(1/N1p,q+1/N3p,q),
Ap,q=pqρα2(1/N1p,q+1/N3p,q),
η˜TE=0+p0 p-p2|p|αhk+O(h2k2),
η˜TM=0-p0 |p|αp-pn32+n12hk+O(h2k2);
η˜x=0,0-p0,q0 (p2ρ2+q2)3/2αp,q-p,-q2q2(n12+n32)+p0 |p|ραp,0-p,0n12+n32-q0 0,q0,-q2|q|αhk+O(h2k2),
η˜y=0,0-p0,q0 (p2ρ2+q2)3/2αp,q-p,-q2p2ρ2(n12+n32)+q0 |q|α0,q0,-qn12+n32-p0 p,0-p,02|p|ραhk+O(h2k2).
r=0+Δ cos(Kx).
n2(h)=η0+2π(nL2-η0)arctanπ2 η1nL2-η0hλ.
e=|r(nRCWA)-r|2+|t(nRCWA)-t|2,
TE:nL2=0+12Δ2α-2+O(α-4),
TM:nL2=1a0+a1a03Δa22+20a1α-2+Oα-4.
r=0+Δ2cos(Kx)+Δ2cos(Ky).

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