Abstract

The diffraction characteristics of finite-number-of-periods (FNP) gratings are determined by using rigorous electromagnetic analysis based on the two-region formulation of the exact boundary element method. Gratings with 5, 9, 13, and 17 periods are analyzed and compared with infinite-number-of-periods (INP) gratings in terms of diffraction efficiencies and field patterns. Both dielectric transmission diffractive devices and metallic reflection diffractive devices are treated for TE and TM normally incident light. Furthermore, both 2-level and 8-level grating structures are treated. The diffraction properties of FNP gratings are shown to approach smoothly those of INP gratings as the number of periods increases.

© 1997 Optical Society of America

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References

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  7. M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
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1996

1995

1994

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

1993

1991

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

1989

1986

1985

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1982

Brundrett, D. L.

E. N. Glytsis, T. K. Gaylord, D. L. Brundrett, “Rigorous coupled-wave analysis and applications of grating diffraction,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of Critical Reviews (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 3–31.

Gallagher, N. C.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Gaylord, T. K.

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

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, D. L. Brundrett, “Rigorous coupled-wave analysis and applications of grating diffraction,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of Critical Reviews (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 3–31.

Glytsis, E. N.

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

E. N. Glytsis, T. K. Gaylord, D. L. Brundrett, “Rigorous coupled-wave analysis and applications of grating diffraction,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of Critical Reviews (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 3–31.

Hirayama, K.

Huang, A.

Ido, J.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

Jahns, J.

Kojima, T.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

Kok, Y.-L.

Koshiba, M.

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

Lichtenberg, B.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Mait, J. N.

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

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

Manara, G.

Mirotznik, M. S.

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modelling 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.

Pelosi, G.

Prather, D. W.

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

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

Toso, G.

Wilson, D. W.

Appl. Opt.

Electron. Commun. Jpn. Pt. 2

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

J. Mod. Opt.

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Proc. IEEE

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other

E. N. Glytsis, T. K. Gaylord, D. L. Brundrett, “Rigorous coupled-wave analysis and applications of grating diffraction,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of Critical Reviews (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 3–31.

P. K. Banerjee, R. Butterfield, eds., Developments in Boundary Element Methods (Applied Science Publishers, London, 1979).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C., 1972).

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modelling 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 (11)

Fig. 1
Fig. 1

Configuration of the FNP (N=9) gratings investigated. Both 2-level and 8-level gratings are shown for dielectric (diffraction primarily from air to grating into glass) and metallic (diffraction primarily from glass to grating and back to glass) structures. The w(x) beam-shaping aperture, as well as the incident plane wave, is shown.

Fig. 2
Fig. 2

Representation of space filled with two materials of refractive indices n1 and n2, respectively (where n2 can be complex). One representative groove is shown in order to represent the scattering problem. Contours C1 and C2 are used in the derivation8 of Eqs. (1) and (2).

Fig. 3
Fig. 3

Resulting field patterns of a FNP (N=5) 2-level dielectric (n1=1.0, n2=1.5) grating with a normally incident TE-polarized plane wave. The instantaneous electric field contours are shown in (a), while the intensity of the total field contours is shown in (b). The grating groove depth is d=0.63 µm.

Fig. 4
Fig. 4

Same as Fig. 3, but for an 8-level dielectric FNP grating and grating groove depth d=1.09 µm.

Fig. 5
Fig. 5

Normalized diffracted power density [|ϕ2s(r2)|2r2/2η2]/Pinc as a function of the diffraction angle φ in region 2 for 2-level dielectric FNP gratings with N=5, 9, 13, and 17. In (a) the normally incident plane wave is TE polarized, and the groove depth is d=0.63 µm. In (b) the normally incident plane wave is TM polarized, and the groove depth is d=1.10 µm.

Fig. 6
Fig. 6

Normalized diffracted power density [|ϕ1s(r1)|2r1/2η1]/Pinc as a function of the diffraction angle φ in region 1 for 8-level metallic FNP gratings with N=5, 9, 13, and 17. In (a) the normally incident plane wave is TE polarized, and the groove depth is d=1.09 µm. In (b) the normally incident plane wave is TM polarized, and the groove depth is d=1.47 µm.

Fig. 7
Fig. 7

Diffraction efficiency of the +1 diffracted order as a function of the groove depth for a 2-level dielectric FNP grating with N =5, 9, 13, and 17. The RCWA results correspond to an otherwise identical INP grating. Plot (a) shows the efficiency for a TE-polarized normally incident plane wave, while plot (b) shows the efficiency for a TM-polarized normally incident plane wave.

Fig. 8
Fig. 8

Same as Fig. 7, but for an 8-level dielectric FNP grating.

Fig. 9
Fig. 9

Diffraction efficiencies of the +1, -1, and 0 diffracted orders as a function of the groove depth for a 2-level metallic FNP grating with N=5, 9, 13, and 17. The RCWA results correspond to an otherwise identical INP grating. The sum of all diffraction efficiencies is also shown. Plot (a) shows the efficiency for a TE-polarized normally incident plane wave, while plot (b) shows the efficiency for a TM-polarized normally incident plane wave.

Fig. 10
Fig. 10

Same as Fig. 9, but for an 8-level metallic FNP grating.

Fig. 11
Fig. 11

Diffraction efficiency of the +1 diffracted order as a function of the groove depth for a FNP grating with N=5 for a TM-polarized normally incident plane wave. The effect of the element length (le) on the diffraction efficiency is shown. For the le<1/6 µm case shown, an element length of le<1/4 µm was used on the y=0 planar boundary surrounding the grating. Plot (a) is for a 2-level dielectric grating, and plot (b) is for an 8-level one.

Tables (2)

Tables Icon

Table 1 Diffraction Efficiency for 2-Level Dielectric Grating with 17 Periods for TM Polarization with Groove Depth d=1.10µm

Tables Icon

Table 2 Diffraction Efficiency for 8-Level Dielectric Grating with 17 Periods for TM Polarization with Groove Depth d=1.47µm

Equations (19)

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-ϕ1t(r1)+Γ[ϕΓ(rΓ)nˆ·G1(r1, rΓ)
-p1G1(r1, rΓ)ψΓ(rΓ)]dl=-ϕinc(r1),r1S1,
ϕ2t(r2)+Γ[ϕΓ(rΓ)nˆ·G2(r2, rΓ)
-p2G2(r2, rΓ)ψΓ(rΓ)]dl=0,r2S2,
θΓ2π-1ϕΓ(rΓ)+Γ[ϕΓ(rΓ)nˆ·G1(rΓ, rΓ)
-p1G1(rΓ, rΓ)ψΓ(rΓ)]dl=-ϕinc(rΓ),
θΓ2πϕΓ(rΓ)+Γ[ϕΓ(rΓ)nˆ·G2(rΓ, rΓ)
-p2G2(rΓ, rΓ)ψΓ(rΓ)]dl=0,
ϕis(ri)=±Γ [ϕΓ(r)nˆ·Gi(ri, r)-piGi(ri, r)ψΓ(r)]dl,
Ezis(x, yi)=-+ ai(ρ)exp[-j(ρx±βiyi)]dρ,
Hxis(x, yi)=±1ηi-+βikiai(ρ)exp[-j(ρx±βiyi)]dρ,
ai(ρ)exp(jβiyi)=12π-+Ezis exp(jρx)dx,
Ps=Re±12-+Ezis(x, yi)[Hxis(x, yi)]* dx=2π2ηi-kikiβiki|ai(ρ)|2 dρ=2πki2ηiφ1φ2 |ai(ki cos φ)|2 sin2 φ dφ,
Ezis(x, yi)=n=-M/2M/2-1 Ai(ρn)exp(-jρnx),
Ps=L2ηin βinki|Ai(ρn)|2=12ηikiL22πn |Ai(ki cos φn)|2(sin2 φn)Δφn,
ϕis(ri)±18πkiri exp-jkiri-π4×Γ[ki(nˆi·rˆi)ϕΓ(r)+jpiψΓ(r)]×exp[jki(r·rˆi)]dl,
Ps12ηiφ1φ2 |ϕis(ri)|2ri dφ,
w(x)=1,0|x|D/2-lcos2|x|-D/2+l4lπ,D/2-l|x|D/2+l0,D/2+l|x|<,
Pinc=(D-l/2)/2η1η1(D-l/2)/2forTEpolarizationforTMpolarization,

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