Abstract

We derive closed-form expressions for the effective index of subwavelength gratings up to the fourth and the second order for TE and TM polarization, respectively. These expressions are valid for arbitrary grating structures and are a generalization of previous results obtained for lamellar gratings with one groove per period (a structure often called a two-component layered medium). The effective-medium-theory predictions are carefully validated with exact electromagnetic theories for slanted and unslanted sinusoidally modulated volume gratings and for classical mounting. It is shown that, even for large period-to-wavelength ratios near the cutoff value, the form birefringence is accurately predicted at any angle of incidence.

© 1998 Optical Society of America

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References

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    [CrossRef]
  2. R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
    [CrossRef] [PubMed]
  3. T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
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  4. Y. Ono, Y. Kimura, Y. Otha, N. Nishida, “Antireflection effects in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
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  5. M. E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in silicon using binary optics technology,” Appl. Opt. 31, 4371–4376 (1992).
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  6. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef] [PubMed]
  7. L. H. Cescato, E. Gluch, N. Streibl, “Holographic quarterwave plates,” Appl. Opt. 29, 3286–3290 (1990).
    [CrossRef] [PubMed]
  8. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
    [CrossRef]
  9. P. Yeh, “A new optical model for wire grid polarizers,” Opt. Commun. 26, 289–292 (1978).
    [CrossRef]
  10. W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
    [CrossRef] [PubMed]
  11. F. T. Chen, H. G. Craighead, “Diffractive phase elements based on two-dimensional artificial dielectrics,” Opt. Lett. 20, 121–123 (1995).
    [CrossRef] [PubMed]
  12. Ph. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
    [CrossRef]
  13. G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
    [CrossRef]
  14. R. C. McPhedran, L. C. Botten, M. S. Craig, N. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
    [CrossRef]
  15. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  16. C. Gu, P. Yeh, “Form birefringence dispersion in periodic layered media,” Opt. Lett. 21, 504–506 (1996).
    [CrossRef] [PubMed]
  17. J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Acta 29, 1475–1489 (1982).
    [CrossRef]
  18. Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
    [CrossRef]
  19. G. Campbell, R. K. Kostuk, “Effective-medium theory of sinusoidally modulated volume holograms,” J. Opt. Soc. Am. A 12, 1113–1117 (1995).
    [CrossRef]
  20. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 6.
  21. T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by grating,” Proc. IEEE 73, 894–936 (1985).
    [CrossRef]
  22. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1984), p. 62.

1997

1996

C. Gu, P. Yeh, “Form birefringence dispersion in periodic layered media,” Opt. Lett. 21, 504–506 (1996).
[CrossRef] [PubMed]

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

1995

1993

1992

1991

1990

1987

1986

1985

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by grating,” Proc. IEEE 73, 894–936 (1985).
[CrossRef]

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

1983

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

R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
[CrossRef] [PubMed]

1982

R. C. McPhedran, L. C. Botten, M. S. Craig, N. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

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]

1978

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

1956

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

Baird, W. E.

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, Principles of Optics, 6th ed. (Pergamon, New York, 1984), p. 62.

Botten, L. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, N. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Bouchitté, G.

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

Campbell, G.

Case, S. K.

Cescato, L. H.

Chen, F. T.

Craig, M. S.

R. C. McPhedran, L. C. Botten, M. S. Craig, N. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Craighead, 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.

Flanders, D. C.

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

Gaylord, T. K.

Gluch, E.

Gu, C.

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]

Kimura, Y.

Kipfer, P.

Kostuk, R. K.

Lalanne, Ph.

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

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

Lemercier-Lalanne, D.

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

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

Maystre, D.

R. C. McPhedran, L. C. Botten, M. S. Craig, N. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, N. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (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]

Moharam, M. G.

Morris, G. M.

Motamedi, M. E.

Nevière, N.

R. C. McPhedran, L. C. Botten, M. S. Craig, N. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Nishida, N.

Ono, Y.

Otha, Y.

Petit, R.

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

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.

Stork, W.

Streibl, N.

Wilson, S. J.

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

Wolf, E.

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

Yeh, P.

C. Gu, P. Yeh, “Form birefringence dispersion in periodic layered media,” Opt. Lett. 21, 504–506 (1996).
[CrossRef] [PubMed]

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

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 6.

Appl. Opt.

Appl. Phys. Lett.

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

Electromagnetics

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

J. Mod. Opt.

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

Opt. Acta

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]

R. C. McPhedran, L. C. Botten, M. S. Craig, N. Nevière, D. Maystre, “Lossy lamellar gratings in the quasi-static limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Opt. Commun.

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

Opt. Lett.

Proc. IEEE

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by grating,” Proc. IEEE 73, 894–936 (1985).
[CrossRef]

Sov. Phys. JETP

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

Other

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 6.

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

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Figures (7)

Fig. 1
Fig. 1

(a) Geometry for the nonconical grating diffraction problem analyzed in the paper. The relative permittivity is assumed to be independent of the z direction, and an unslanted grating is considered in the figure ( depends only on the x coordinate). (b) Corresponding periodic structure with an infinite spatial extent in the z direction. β and γ are the normalized x and z components of the wave vector along the x and the z direction, respectively.

Fig. 2
Fig. 2

Effective relative permittivity as a function of the angle of incidence for TE polarization and for different period-to-wavelength ratios: comparison between EMT predictions (dotted and solid curves) and Bloch-wave computation (circles). Dotted and solid curves are obtained with second-order and fourth-order EMT of Eq. (17), respectively.

Fig. 3
Fig. 3

Effective relative permittivity as a function of the angle of incidence for TM polarization and for different period-to-wavelength ratios: comparison between the second-order EMT predictions (dotted curves) of Eq. (18) and Bloch-wave computation (circles).

Fig. 4
Fig. 4

Effective relative permittivity as a function of the angle of incidence for TE polarization and for different period-to-wavelength ratios: comparison between EMT predictions (dotted and solid curves) and results (circles) obtained by minimizing the error function of Eq. (19) by use of RCWA. Dotted and solid curves are obtained with second-order and fourth-order EMT of Eq. (17), respectively.

Fig. 5
Fig. 5

Effective relative permittivity as a function of the angle of incidence for TM polarization and for different period-to-wavelength ratios: comparison between second-order EMT predictions (dotted curves) and results (circles) obtained by minimizing the error function of Eq. (19) by use of RCWA.

Fig. 6
Fig. 6

TM–TE phase shift predicted by EMT (dotted curves) and RCWA (solid curves) for the dichromated gelatin grating. (a) and (b) are obtained for Λ=(1/2)(λ/1.36) and (1/4)(λ/1.36), respectively. They can be directly compared with those obtained with a zero-order EMT by Campbell and Kostuk [see Figs. 4(a) and 4(b) in Ref. 20].

Fig. 7
Fig. 7

TM–TE phase shift predicted by EMT (dotted curves) and RCWA (solid curves) for a slanted-fringe dichromated gelatin grating for several period-to-wavelengths, Λ/λ=κ/5, κ/3, 2κ/3, and κ. The slant angle is Φ=45°.

Equations (27)

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neff=n0+n1Λ/λ+n2(Λ/λ)2+.
2Eyz2+2Eyx2+k2Ey=0.
2Hyz2+x1Hyx+k2Hy=0,
2Uz2+α x1αUx+k2U=0,
U(x, z)=exp(jkγz)V1(x/Λ).
1αλ2πΛdV1dx,
α dV2dx+2πΛλ(-γ2)V1=0.
ddxV1V2=2πΛλ0a(x)b(x)0V1V2
ddxV=2πΛλMV.
ddxv=2πΛλ(M-jβI)v,
C=I+2πΛλC1(1)+2πΛλ2C2(1)+2πΛλ3C3(1)+O2πΛλ4,
C1(1)+(2πΛ/λ)C2(1)+(2πΛ/λ)2C3(1)
+O(2πΛ/λ)3=0.
a*(x)=a(x)-0α,
b*(x)=b(x)-γ2a0α+01-α,
A(x)=0xa*(x)dx,
B(x)=0xb*(x)dx,
0α(γ2a0α-01-α)+β2=(2πΛ/λ)2R,
R=02αB|B+β402αA|A+2β2B|A-20αB|b*A+2 β20αA|a*B-a*|B2.
β=n1 sin(θ).
eff=β2+γ2,
γ2=f(β)=10αa0α0-β2+2πΛλ2R(β).
eff=0+Λλ2p0 p-pp2+Λλ4×(4β2-0)p0 p-pp4+p0,k0 pk-p-kp2k2.
TE:eff=0+(Δ)22Λλ2+2β2(Δ)2Λλ4.
TM:eff=10a00-β2+0a0β2+Λλ2β4Δ2202+β2Δ1-β202a1a0+Δa220a0+0a021+β402-2β200n0ann2+2Δn>0 anan+1n(n+1).
e=|r(nRCWA)-r|+|t(nRCWA)-t|,
(γ sin Φ-β cos Φ)2=f(γ cos Φ+β sin Φ),

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