Abstract

We demonstrate theoretically a polarization-independent guided-mode resonant filter with only a one dimensional grating. A rigorous method, the modal method by Fourier expansion, is used to compute the diffracted efficiencies of the grating. Wave-vector analysis fails to correctly design a polarization-independent structure. We show that a rigorous analysis of the resonances must be employed to obtain such a device; using a pole approach, we study the effects of grating parameters on the resonances of both polarizations.

© 2003 Optical Society of America

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  1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
    [CrossRef]
  2. A. Hessel, A. A. Onliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  3. Rayleigh, “Note on the remarkable case of diffraction spectrum described by Prof. Wood,” Philos. Mag. 14, 399–416 (1907).
    [CrossRef]
  4. T. Tamir, ed., Integrated Optics (Springer-Verlag, New York, 1985).
  5. E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  6. S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
    [CrossRef] [PubMed]
  7. H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
    [CrossRef]
  8. S. Tibuleac, R. Magnusson, T. A. Maldonado, P. P. Young, T. R. Holzheimer, “Dielectric frequency-selective structures incorporating waveguide gratings,” IEEE Trans. Microwave Theory Tech. 48, 553–561 (2000).
    [CrossRef]
  9. R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  10. D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A 17, 1221–1230 (2000).
    [CrossRef]
  11. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
    [CrossRef]
  12. S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  13. S. Peng, G. M. Morris, “Experimental demonstration of resonant anomalies in diffraction from two-dimensional gratings,” Opt. Lett. 21, 549–551 (1996).
    [CrossRef] [PubMed]
  14. 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 (2000).
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  15. M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
  16. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  17. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  18. D. Lacour, J-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filter,” Opt. Quantum Electron. 33, 451–470 (2001).
    [CrossRef]
  19. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  20. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
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  21. S. M. Norton, T. Erdogan, G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1996).
    [CrossRef]
  22. R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
    [CrossRef]
  23. L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
    [CrossRef]
  24. A. L. Fehrembach, D. Maystre, A. Sentennac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19, 1136–1144 (2002).
    [CrossRef]
  25. A. L. Fehrembach, D. Maystre, A. Sentennac, “Filtering of unpolarized light by gratings,” J. Opt. A, Pure Appl. Opt. 4, 88–94 (2002).
    [CrossRef]

2002 (2)

A. L. Fehrembach, D. Maystre, A. Sentennac, “Filtering of unpolarized light by gratings,” J. Opt. A, Pure Appl. Opt. 4, 88–94 (2002).
[CrossRef]

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

2001 (1)

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

2000 (3)

1996 (5)

1994 (1)

1993 (1)

1992 (1)

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

1991 (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

1989 (1)

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

1988 (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

1987 (1)

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[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]

1981 (1)

1965 (1)

1907 (1)

Rayleigh, “Note on the remarkable case of diffraction spectrum described by Prof. Wood,” Philos. Mag. 14, 399–416 (1907).
[CrossRef]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Bendickson, J. M.

Bertoni, H. L.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Brundrett, D. L.

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Cheo, L. S.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Erdogan, T.

Fehrembach, A. L.

A. L. Fehrembach, D. Maystre, A. Sentennac, “Filtering of unpolarized light by gratings,” J. Opt. A, Pure Appl. Opt. 4, 88–94 (2002).
[CrossRef]

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

Gaylord, T. K.

Glytsis, E. N.

Granet, G.

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

Hall, D. G.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Hessel, A.

Holzheimer, T. R.

S. Tibuleac, R. Magnusson, T. A. Maldonado, P. P. Young, T. R. Holzheimer, “Dielectric frequency-selective structures incorporating waveguide gratings,” IEEE Trans. Microwave Theory Tech. 48, 553–561 (2000).
[CrossRef]

Iwata, K.

Kikuta, H.

Lacour, D.

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

Li, L.

Magnusson, R.

S. Tibuleac, R. Magnusson, T. A. Maldonado, P. P. Young, T. R. Holzheimer, “Dielectric frequency-selective structures incorporating waveguide gratings,” IEEE Trans. Microwave Theory Tech. 48, 553–561 (2000).
[CrossRef]

S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
[CrossRef] [PubMed]

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

Maldonado, T. A.

S. Tibuleac, R. Magnusson, T. A. Maldonado, P. P. Young, T. R. Holzheimer, “Dielectric frequency-selective structures incorporating waveguide gratings,” IEEE Trans. Microwave Theory Tech. 48, 553–561 (2000).
[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. Sentennac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19, 1136–1144 (2002).
[CrossRef]

A. L. Fehrembach, D. Maystre, A. Sentennac, “Filtering of unpolarized light by gratings,” J. Opt. A, Pure Appl. Opt. 4, 88–94 (2002).
[CrossRef]

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

Mizutani, A.

Moharam, M. G.

Morris, G. M.

Mure-Ravaud, A.

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

Nakajima, K.

Nevière, M.

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.

Norton, S. M.

Onliner, A. A.

Peng, S.

Plumey, J-P.

D. Lacour, J-P. Plumey, G. Granet, A. Mure-Ravaud, “Resonant waveguide grating: analysis of polarization independent filter,” 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]

Rayleigh,

Rayleigh, “Note on the remarkable case of diffraction spectrum described by Prof. Wood,” Philos. Mag. 14, 399–416 (1907).
[CrossRef]

Sambles, J. R.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Sentennac, A.

A. L. Fehrembach, D. Maystre, A. Sentennac, “Filtering of unpolarized light by gratings,” J. Opt. A, Pure Appl. Opt. 4, 88–94 (2002).
[CrossRef]

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

Tamir, T.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Tibuleac, S.

S. Tibuleac, R. Magnusson, T. A. Maldonado, P. P. Young, T. R. Holzheimer, “Dielectric frequency-selective structures incorporating waveguide gratings,” IEEE Trans. Microwave Theory Tech. 48, 553–561 (2000).
[CrossRef]

Wang, S. S.

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Young, P. P.

S. Tibuleac, R. Magnusson, T. A. Maldonado, P. P. Young, T. R. Holzheimer, “Dielectric frequency-selective structures incorporating waveguide gratings,” IEEE Trans. Microwave Theory Tech. 48, 553–561 (2000).
[CrossRef]

Zengerle, R.

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

IEEE Trans. Antennas Propag. (1)

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Tibuleac, R. Magnusson, T. A. Maldonado, P. P. Young, T. R. Holzheimer, “Dielectric frequency-selective structures incorporating waveguide gratings,” IEEE Trans. Microwave Theory Tech. 48, 553–561 (2000).
[CrossRef]

J. Lightwave Technol. (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

J. Mod. Opt. (1)

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

A. L. Fehrembach, D. Maystre, A. Sentennac, “Filtering of unpolarized light by gratings,” J. Opt. A, Pure Appl. Opt. 4, 88–94 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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. Lett. (2)

Opt. Quantum Electron. (1)

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

Philos. Mag. (2)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Rayleigh, “Note on the remarkable case of diffraction spectrum described by Prof. Wood,” Philos. Mag. 14, 399–416 (1907).
[CrossRef]

Phys. Rev. B (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Other (2)

T. Tamir, ed., Integrated Optics (Springer-Verlag, New York, 1985).

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.

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Figures (11)

Fig. 1
Fig. 1

GMRF configuration. The structure, a lamellar grating associated with a guide, rests on a semi-infinite substrate whose refractive index is equal to νs. The refractive index of the semi-infinite cover is νc. The grating is defined by its period Λ, its filling factor f, its groove depth dgrating, and its refractive index νgrating. dguide and νguide denote the depth of the guide and its refractive index. i, i, r, and t are, respectively, the incident, reflected, and transmitted-field amplitudes.

Fig. 2
Fig. 2

Coordinate system for a grating in conical mounting. The wave vector of the incident wave, k, is given by polar angle θ, azimuthal angle φ, and wave number k0. Its polarization is specified by angle δ, the angle between the (k0z) plane and the magnetic field vector H. δ=0 corresponds to a P-polarized incident wave, δ=π/2 to an S polarized incident wave.

Fig. 3
Fig. 3

Resonance reflection over the angle in normalized wave-number units.

Fig. 4
Fig. 4

Wave-vector diagram of a 1D waveguide grating periodic along the y axis. The dotted-line circles denote the circles associated with the TM0 mode, and the solid-line circles denote the circles associated with the TE0 mode. The radii of the circles are equal to the effective indices of the modes, νeff(TM0)=1.56 and νeff(TE0)=1.71. Two TM0 modes intersect at A. At this point, inside the “light disk,” two modes are simultaneously excited, and polarization independence may occur.

Fig. 5
Fig. 5

Resonant excitation of two identical guided modes in full conical mounting.

Fig. 6
Fig. 6

Angular dependence of the reflectivity of a GMRF designed with WVD approach to excite two TM0 guided modes. Parameters are dguide=0.40 μm, dgrating=0.01 μm, Λ=1.01 μm, f=0.5, νc=1, νgrating=3.48, νguide=1.98, νs=1.45, λ=1.55 μm, ϕ=0°.

Fig. 7
Fig. 7

Dependence of αp on the filling factor for both polarizations. The GMRF is designed to excite two TM0 modes. The structure parameters are the same as those for Fig. 6.

Fig. 8
Fig. 8

Dependence of αp on the filling factor for both polarizations. The GMRF is designed to excite two TM0 modes. The structure parameters are the same as those for Fig. 6.

Fig. 9
Fig. 9

Dependence of αp on grating depth for both polarizations. The GMRF is designed to excite two TM0 modes. Parameters are the same as for Fig. 6 except that f=0.25.

Fig. 10
Fig. 10

Dependence of αp on grating depth for both polarizations. The GMRF is designed to excite two TM0 modes. Parameters are the same as for Fig. 6 except that f=0.25.

Fig. 11
Fig. 11

Spectral dependence of the reflectivity of an optimized GMRF designed to excite two TM0 guided modes. Parameters are the same as for Fig. 6 except that dgrating=0.10 μm, f=0.14, and θ=18.0°.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

kk0α0k0β0k0γ0 with α0=-sin(θ)cos(ϕ)β0=-sin(θ)sin(ϕ)γ0=-cos(θ),
Ey(y, z)=m[am-exp(ik0γmz)+am+exp(-ik0γmz)]exp(-ik0βmy),
Z0Hy(y, z)=m[bm-exp(ik0γmz)+bm+exp(-ik0γmz)]exp(-ik0βmy).
γm=(ν2-α02-βm2)1/2,Re(γm)-Im(γm)<0,
βm=β0+m λΛ,
2z2 Ey(y, z)=k02α02-y1(y)y (y)-k02(y)Ey(y, z),
2z2 Hy(y, z)=-2y2+k02α02-k02(y)Hy(y, z).
Ey(y, z)=nm[am-Enmexp(ik0γEmz)+am+Enmexp(-ik0γEmz)]exp(-ik0βny),
Z0Hy(y, z)=nm[bm-Hnmexp(ik0γHmz)+bm+Hnm(-ik0γHmz)]exp(-ik0βny),
rt=Sii,
det(S-1(α0))=0.
Rn(α0)=R0 n(α0)α0-αz nα0-αp2,
α2+β2=νeff2(TM0),
α2+β2=νeff2(TE0).
α02+β021.
α2+β+m λΛ2=νeff2(TM0),
α2+β+m λΛ2=νeff2(TE0),
Λ=λ(νeff2-α02)1/2.

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