Abstract

Using a phenomenological theory of diffraction gratings made by perturbing a planar waveguide allows us to deduce important properties of the sharp filtering phenomena generated by this kind of structure when the incident light excites a guided wave. It is shown that the resonance phenomenon occurring in these conditions acts on one of the two eigenvalues of the Hermitian reflection matrix only. As a consequence, we deduce a mathematical expression of the reflectivity and demonstrate that high-efficiency filtering of unpolarized light requires the simultaneous excitation of two uncoupled guided waves. Numerical examples are given.

© 2002 Optical Society of America

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References

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  1. D. Maystre, “General study of grating anomalies,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, Chichester, UK, 1982), pp. 661–724.
  2. E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  3. R. Magnusson, S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  4. A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
    [CrossRef] [PubMed]
  5. S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  6. D. Lacour, G. Granet, J. P. Plumey, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filtering,” Opt. Quantum Electron. 33, 451–470 (2001).
    [CrossRef]
  7. 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]
  8. R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
    [CrossRef]
  9. W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
    [CrossRef]

2001 (2)

D. Lacour, G. Granet, J. P. Plumey, 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]

1996 (3)

1992 (1)

R. Magnusson, 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 anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Barnes, W. L.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[CrossRef]

Botten, L. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
[CrossRef]

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
[CrossRef]

Engel, H.

Friesem, A. A.

Granet, G.

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

Iwata, K.

Kikuta, H.

Kitson, S. C.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[CrossRef]

Lacour, D.

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

Magnusson, R.

R. Magnusson, 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 anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Maystre, D.

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

D. Maystre, “General study of grating anomalies,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, Chichester, UK, 1982), pp. 661–724.

McPhedran, R. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
[CrossRef]

Mizutani, A.

Morris, G. M.

Mure-Ravaud, A.

D. Lacour, G. Granet, J. P. Plumey, 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, G. Granet, J. P. Plumey, 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 anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Preist, T. W.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[CrossRef]

Rosenblatt, D.

Sambles, J. R.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[CrossRef]

Sharon, A.

Steingrueber, R.

Wang, S.

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

Weber, H. G.

Appl. Phys. Lett. (1)

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

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

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

Opt. Lett. (1)

Opt. Quantum Electron. (1)

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

Phys. Rev. B (1)

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[CrossRef]

Other (2)

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
[CrossRef]

D. Maystre, “General study of grating anomalies,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, Chichester, UK, 1982), pp. 661–724.

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Figures (5)

Fig. 1
Fig. 1

Periodic guiding structure.

Fig. 2
Fig. 2

Examples of periodic guiding structures. (a) Classical lamellar grating that can be used in conical mounting, (b) crossed grating with circular bumps and hexagonal symmetry.

Fig. 3
Fig. 3

Two-dimensional grating with square bumps, refractive indices ns=1.5, nc=2.5, thickness ec=133 nm, bump height h=7 nm, periods dx=dy=d=930 nm, and bump width dc=465 nm. The index of the bumps is equal to nc. The incident parameters are θ=15° and ϕ=28°. (a) Reflection factor versus wavelength for both s (solid curve) and p (dashed curve) polarizations. (b) First (solid curve) and second (dashed curve) eigenvalues of the Hermitian reflection matrix versus wavelength. (c) Reflection factor versus angle of polarization δ.

Fig. 4
Fig. 4

One-dimensional lamellar grating periodic along the x axis, with refractive indices ns=1.5, nc=2.07, thickness ec=300 nm, groove depth h=87.5 nm, period dx=904 nm, and groove width c=226 nm. The incident parameters are θ=7.85°, ϕ=90°. (a) Solid arrows, scheme of the wave vectors of the guided waves of the planar waveguide; dashed arrows, direction of their associated electric field. (b) Reflection factor versus wavelength for both s (solid curve) and p (dashed curve) polarizations. (c) Reflection factor versus the angle of polarization δ at λ=1.52 μm.

Fig. 5
Fig. 5

First (solid curve) and second (dashed curve) eigenvalues versus wavelength for a two-dimensional grating with square bumps, illuminated in normal incidence, with refractive indices ns=1.448, nc=2.07, thickness ec=300 nm, bump height h=87.5 nm, period d=900 nm, and bump width dc=600 nm. The two curves are identical.

Equations (79)

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s^i+=ki+×zˆ|ki+×zˆ|,p^i+=s^i+×ki+|s^i+×ki+|
α=k sin(θ)cos(ϕ),β=k sin(θ)sin(ϕ),
γ+=(k2-α2-β2)1/2.
Ei+=Pi+exp(iαx+iβy-iγ+z),
Pi+=Pi+,ss^i++Pi+,pp^i+.
Es=n=-+m=-+Pn,m+exp(iαnx+iβmy+iγn,m+z) ifz>0,
Es=n=-+m=-+Pn,m-exp(iαnx+iβmy-iγn,m-z) ifz<-e,
αn=α+nKx,Kx=2π/dx,
βm=β+mKy,Ky=2π/dy,
γn,m+=(k2-αn2-βm2)1/2,Re(γn,m+)+Im(γn,m+)>0,
γn,m-=(k2εs-αn2-βm2)1/2,Re(γn,m-)+Im(γn,m-)>0,
 EPi+exp(iαx+iβy-iγ+z)+P0,0+exp(iαx+iβy+iγ+z) ifz,
 EP0,0-exp(iαx+iβy-iγ-z)ifz-,
s^d+=-kd+×zˆ|kd+×zˆ|,p^d+=-s^d+×kd+|s^d+×kd+|,
P0,0+=Pd+,ss^d++Pd+,pp^d+,
s^d-=kd-×zˆ|kd-×zˆ|, p^d-=s^d-×kd-|s^d-×kd-|,
P0,0-=pd-,ss^d-+Pd-,pp^d-,
s^i-=-ki-×zˆ|ki-×zˆ|,p^i-=-s^i-×ki-|s^i-×ki-|,
Ei-=Pi-exp(iαx+iβy+iγ-z),
Pi-=Pi-,ss^i-+Pi-,pp^i-,
EP-iexp(iαx+iβy+iγ-z)+P0,0-exp(iαx+iβy-iγ-z)
ifz-.
I±=(Pi±,sγ±,Pi±,pγ±),
D±=(Pd±,sγ±,Pd±,pγ±),
D+=R1I++T2I-,
D-=T1I++R2I-.
D=SI,
S=R1T2T1R2.
S*S=1,
I+, D+=I+, D+ifI-=I-=0,
I-, D-=I-, D-ifI+=I+=0,
I+, D+=I-, D- ifI-=I+=0, 
I-, D-=I+, D+,ifI+=I-=0,
I+, R1I+=I+, R1I+,
I+, R1I+=I+, t(R1)I+,
t(R1)=R1.
t(R2)=R2,
t(T2)=T1,
t(T1)=T2.
S=t(S).
E=Pg+exp(iαgx+iβgy+iγg+z),
γg+=i[(αg)2+(βg)2-(kg, plan)2]1/2.
E=Pg-exp(iαgx+iβgy-iγg-z),
γg-=i[(αg)2+(βg)2-(kg, plan)2εs]1/2,
E=n=-+m=-+Pn,mg+exp(iαngx+iβmgy+iγn,mg+z)ifz>0,
E=n=-+m=-+Pn,mg-exp(iαngx+iβmgy-iγn,mg-z)ifz<-e,
αng=αg+nKx
βmg=βg+mKy
γn,mg+=(kg, perturb)2-(αng)2-(βmg)2, Re(γn,mg+)+Im(γn,mg+)>0,
γn,mg-=(kg, perturb)2εs-(αng)2-(βmg)2,Re(γn,mg-)+Im(γn,mg-)>0.
(αng)2+(βmg)2<(|kg,perturb|)2,
EPp,qg+exp(iαpgx+iβqgy+iγp,qg+z)ifz+,
EPp,qg-exp(iαpgx+iβqgy-iγp,qg-z)ifz-.
V1S=(Pg+,s, Pg+,p, Pg-,s, Pg-,p).
kg,perturb(p)kg,planifp0.
l1T1(k, p)=l1T1(k, p)k-kg,perturb(p),
l1T1(k, 0)=l1T1(k, 0)k-kg,plan,
l1T1(k, 0)=(k-kg,plan)l1T1(k, 0)
k1T1,root(0)=kg,plan=kg,rperturb(0).
l1T1(k, p)=u(k, p) k-k1T1,root(p)k-kg,perturb(p).
S*(k¯)S(k)=1,
R1*(k¯)R1(k)+T1*(k¯)T1(k)=1,
T2*(k¯)T2(k)+R2*(k¯)R2(k)=1,
R1*(k¯)T2(k)+T1*(k¯)R2(k)=0,
T2*(k¯)R1(k)+R2*(k¯)T1(k)=0.
T1(k1T1,root)V1T1,root=0,
T2*(k¯1T1,root)R1(k1T1,root)V1T1,root=0,
T2=t(T1).
R1(k1T1,root)V1T1,root=V¯1T1,root.
ρ=D+|D+=R1I+|R1I+=R1*R1I+|I+.
R1*R1V1T1,root=V1T1,root.
ρ=l1R1*R1(k, p)|I+|V1R1*R1(k, p)|2+l2R1*R1(k, p)|I+|V2R1*R1(k, p)|2.
V1R1*R1(k, p)=[cos q,sin q exp(iϕ)],
I+=[cos(δ), sin(δ)],
ρ=l1R1*R1(k, p)+l2R1*R1(k,p)2+l1R1*R1(k, p)-l2R1*R1(k, p)2×τ cos(2δ-ψ),
τ=[cos(2q)2+sin(2q)2cos(ϕ)2]1/2,
tan(ψ)=tan(2q)cos(ϕ).
1-1-l2R1*R1(k, p)2 (1-τ)
1-1-l2R1*R1(k, p)2 (1+τ)

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