Abstract

The dispersion relation of eigenmodes of two-dimensional waveguide gratings is studied with a perturbative model. The analytic expression of the complex wavelength of the modes permits us to predict the shape of the anomalies in the grating reflectivity with respect to the wavelength and the polarization of the incident plane wave. The simultaneous excitation of two independent modes is necessary for obtaining high-efficiency filtering of unpolarized light. We show how this requirement can be met.

© 2003 Optical Society of America

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References

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  1. P. Vincent, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 17, 239–248 (1978).
    [CrossRef]
  2. E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  3. R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  4. A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
    [CrossRef]
  5. A. Sharon, D. Rosenblatt, A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997).
    [CrossRef]
  6. D. Jacob, S. Dunn, M. G. Moharam, “Design considerations for narrow-band dielectric resonant grating reflection filters of finite length,” J. Opt. Soc. Am. A 17, 1241–1249 (2000).
    [CrossRef]
  7. F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide grat-ings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
    [CrossRef]
  8. S. Peng, M. G. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  9. D. Lacour, J.-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filtering,” Opt. Quantum Electron. 33, 451–470 (2001).
    [CrossRef]
  10. A. Mizutani, H. Kikuta, K. Nakajima, K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
    [CrossRef]
  11. A.-L. Fehrembach, D. Maystre, A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19, 1136–1144 (2002).
    [CrossRef]
  12. N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders, New York, 1976).
  13. K. Sakoda, “Symmetry, degeneracy and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982–7986 (1995).
    [CrossRef]
  14. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  15. A.-L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” Pure Appl. Opt. 4, S88–S94 (2002).
    [CrossRef]
  16. L. Tsang, J.-A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

2002 (2)

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” Pure Appl. Opt. 4, S88–S94 (2002).
[CrossRef]

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19, 1136–1144 (2002).
[CrossRef]

2001 (2)

D. Lacour, J.-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filtering,” Opt. Quantum Electron. 33, 451–470 (2001).
[CrossRef]

A. Mizutani, H. Kikuta, K. Nakajima, K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
[CrossRef]

2000 (1)

1999 (1)

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide grat-ings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

1997 (2)

1996 (2)

S. Peng, M. G. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

1995 (1)

K. Sakoda, “Symmetry, degeneracy and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982–7986 (1995).
[CrossRef]

1992 (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1978 (1)

P. Vincent, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders, New York, 1976).

Cambril, E.

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide grat-ings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Dunn, S.

Fehrembach, A.-L.

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19, 1136–1144 (2002).
[CrossRef]

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” Pure Appl. Opt. 4, S88–S94 (2002).
[CrossRef]

Friesem, A. A.

A. Sharon, D. Rosenblatt, A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

Giovannini, H.

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide grat-ings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Granet, G.

D. Lacour, J.-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filtering,” Opt. Quantum Electron. 33, 451–470 (2001).
[CrossRef]

Iwata, K.

Jacob, D.

Kikuta, H.

Kong, J.-A.

L. Tsang, J.-A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

Lacour, D.

D. Lacour, J.-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filtering,” Opt. Quantum Electron. 33, 451–470 (2001).
[CrossRef]

Lemarchand, F.

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide grat-ings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Li, L.

Magnusson, R.

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Maystre, D.

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” Pure Appl. Opt. 4, S88–S94 (2002).
[CrossRef]

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19, 1136–1144 (2002).
[CrossRef]

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Mermin, N. D.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders, New York, 1976).

Mizutani, A.

Moharam, M. G.

Morris, M. G.

Mure-Ravaud, A.

D. Lacour, J.-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filtering,” Opt. Quantum Electron. 33, 451–470 (2001).
[CrossRef]

Nakajima, K.

Peng, S.

Plumey, J.-P.

D. Lacour, J.-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filtering,” Opt. Quantum Electron. 33, 451–470 (2001).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Rosenblatt, D.

A. Sharon, D. Rosenblatt, A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

Sakoda, K.

K. Sakoda, “Symmetry, degeneracy and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982–7986 (1995).
[CrossRef]

Sentenac, A.

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19, 1136–1144 (2002).
[CrossRef]

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” Pure Appl. Opt. 4, S88–S94 (2002).
[CrossRef]

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide grat-ings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Sharon, A.

A. Sharon, D. Rosenblatt, A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

Shin, R. T.

L. Tsang, J.-A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

Tsang, L.

L. Tsang, J.-A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

Vincent, P.

P. Vincent, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

Wang, S. S.

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Appl. Phys. (1)

P. Vincent, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

Appl. Phys. Lett. (2)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

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

Opt. Acta (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Quantum Electron. (1)

D. Lacour, J.-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filtering,” Opt. Quantum Electron. 33, 451–470 (2001).
[CrossRef]

Phys. Rev. B (1)

K. Sakoda, “Symmetry, degeneracy and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982–7986 (1995).
[CrossRef]

Pure Appl. Opt. (2)

A.-L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” Pure Appl. Opt. 4, S88–S94 (2002).
[CrossRef]

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide grat-ings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Other (2)

L. Tsang, J.-A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley Interscience, New York, 1985).

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders, New York, 1976).

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Figures (5)

Fig. 1
Fig. 1

Notation. (a) Illumination configuration, (b) geometry of the perturbed planar waveguide.

Fig. 2
Fig. 2

Top view of the gratings. (a) Lamellar grating periodic along one direction; (b) grating periodic along two orthogonal directions, square bumps; (c) grating periodic along two directions, triangular lattice, circular bumps; (d) Description of the basis in the direct space, (xˆ, yˆ), and reciprocal space, (kx, ky). The bisector of xˆ and yˆ belongs to a vertical plane of symmetry of the structure. The planar incident wave vector k is chosen along the bisector of (Ox) and (Oy).

Fig. 3
Fig. 3

Comparison of the resonant wavelengths obtained with the rigorous (solid line) and the perturbative (dotted line) methods for both symmetrical and antisymmetrical eigenmodes. The structure is a lamellar grating periodic along one direction with period d=864 nm and filling factor f=0.75, deposited on the planar waveguide described in Fig. 1(b); e=130 nm, a=1.0, l=9.0, s=2.25, |k|=3.5 μm-1. 25 orders (from -12 to +12) are used in the Fourier modal method to calculate the resonant wavelength with enough accuracy. (a) Square root of the imaginary part of the resonant wavelengths versus the grating height h, (b) real part of the resonant wavelengths versus h.

Fig. 4
Fig. 4

(a) Difference between the real parts of the symmetrical and the antisymmetrical resonant wavelengths versus the grating Fourier coefficient -1,1 calculated with the rigorous method. The grating is a square lattice of circular holes of depth h=10 nm, with period d=984.3 nm. -1,1 is modified by changing the radii of the holes. The holes are drilled in a layer, l=4, e=230 nm, deposited on a substrate, s=2.25, a=1. |k|=3/2 μm-1. The square cell is discretized into 256×256 square pixels to describe the motif of the grating. [-3, 3]×[-3, 3] orders along kx and ky are taken in the Fourier modal method. (b) Reflectivity versus wavelength for p and s incident polarizations of the resonant grating described in (a) when the real part of λAS-λS=0. -1,1=0.0096 (rA=123 nm, rB=38.5 nm, and rC=196 nm). The plane of incidence is the bisector of xˆ and yˆ, and the angle of incidence with respect to (Oz) is 39.5°.

Fig. 5
Fig. 5

(a) Top view of the structure whose reflectivity is plotted in (b). The grating is ruled on a multilayer stack that behaves as an antireflection film outside the resonance. The planar waveguide is the superposition of a substrate (dielectric constant 2.097), a first layer (dielectric constant 4.285, thickness 79.1 nm), a second layer (dielectric constant 2.161, thickness 263.5 nm), and a third layer (dielectric constant 4.285, thickness 404.3 nm). The grating consists of circular holes drilled in the third layer (radius 200 nm, depth 30 nm) along a nonregular triangular lattice, with period d=953.1 nm. The diamond-shaped cell is discretized into 256×256 pixels. [-3, 3]×[-3, 3] orders are taken in the Fourier modal method. (b) Reflectivity of the structure versus wavelength for both s (solid curve) and p (dotted curve) polarizations. The plane of incidence is the bisector of xˆ and yˆ, and the angle of incidence is 13.5°.

Equations (30)

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kx·xˆ=2π/dx,kx·yˆ=0,
E(r)=f(r)exp(ik·r)=f(r+mdxxˆ+ndyyˆ)exp(ik·r),
××E(r)-(r)2πλ2E(r)=0,
l1T(λinc)=u(λinc)λinc-λT,rootλinc-λ.
λref(|k+Ku,v|)=λref(|k+Kp,q|)=λ(0).
××E(r)-ref(z)k02E(r)=k02per(r)E(r),
××G¯¯(r-r, z, z)-ref(z)k02G¯¯(r-r, z, z)=δ(r-r)I¯¯,
E(r)=k020hper(r)G¯¯(r-r, z, z)E(r)drdz.
E(r)=mnEm,n(z)exp[i(k+Km,n)·r].
per(r)=mnm,n exp(iKm,n·r).
Em,n(z)=k02jlm-j,n-l×0hg¯¯(Km,n+k, z, z)Ej,l(z)dz,
g¯¯(Km,n+k, z, z)=G¯¯(r-r, z, z)×exp(-i(Km,n+k)·(r-r))dr.
λ-λ(0)=O(h).
Em,n(z)=Em,n(0)(z)+hEm,n(1)(z)+O(h2),
Em,n=hk02g¯¯m,n(k, λ)jlm-j,n-lEj,l(0)+O(h2g¯¯m,n),
g¯¯m,n(k, λ)=g¯¯(Km,n+k, 0, 0),Em,n=Em,n(0),
Ej,l(0)=Ej,l(0)(0).
g¯¯m,n(k, λ)=gm,ns(k, λ)000gm,nk(k, λ)gm,nkz(k, λ)0gm,nzk(k, λ)gm,nz(k, λ).
gp,qs(k, λ)=Ap,q(k, λ)λ-λ(0),gu,vs(k, λ)=Au,v(k, λ)λ-λ(0),
Ep,qs(0)=hk02gp,qs(k, λ)[0,0Ep,qs(0)+p-u,q-vEu,vs(0)×cos(ψp,q-ψu,v)]+O(h),
Eu,vs(0)=hk02gu,vs(k, λ)[0,0Eu,vs(0)+u-p,v-qEp,qs(0)×cos(ψu,v-ψp,q)]+O(h).
λ(k)=λ(0)+2πλ(0)2hAp,q(k, λ(0))×[0,0-σp-q,q-p cos(2ψp,q)]+O(h2).
F[λ(k)]=h22πλ(0)4Ap,q(k, λ(0))|p,q|2×[g0,0s(k, λ(0))cos2(ψp,q)(1-σ)+g0,0k(k, λ(0))×sin2(ψp,q)(1+σ)]+O(h3).
gm,ns(k, λ)=iπγm,na(1+Rm,ns),
gm,nk(k, λ)=iπγm,naak02(1-Rm,np),
gm,nkz(k, λ)=iπkm,nak02(-1-Rm,np),
gm,nzk(k, λ)=iπkm,nak02(-1+Rm,np),
gm,nz(k, λ)=iπkm,n2ak02γm,na(1+Rm,np)-2πak02h,
Rm,ns=ra,l+rl,s exp(i2γm,nle)1+ra,lrl,s exp(i2γm,nle),
rμ,ν=γm,nμ-γm,nνγm,nμ+γm,ngn

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