Abstract

We are concerned with optimal design of guided-mode grating resonant structures (GMGRs). A typical structure is the integration of a zeroth-order grating and a planar waveguide. Our approach has two main steps. The first is to find the resonant wavelength. For any fixed grating structure the resonant wavelength is found by solving a nonlinear eigenvalue problem. The second step is to develop a Newton-type local optimization method. A crucial step is to determine an appropriate initial guess of the design parameters. Numerical design examples for both TE and TM polarization are presented. The design algorithm is expected to provide systematic guidance in engineering design of GMGRs.

© 2005 Optical Society of America

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    [CrossRef]
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  9. R. Magnusson, D. Shin, Z. S. Liu, “Guided-mode resonance Brewster filter,” Opt. Lett. 23, 612–614 (1998).
    [CrossRef]
  10. Z. S. Liu, S. Tibuleac, D. Shin, P. P. Young, R. Magnusson, “High-efficiency guided-mode resonance filter,” Opt. Lett. 23, 1556–1558 (1998).
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    [CrossRef]
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    [CrossRef]
  13. S. T. Thurman, M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. 42, 3225–3233 (2003).
    [CrossRef] [PubMed]
  14. F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide gratings: increasing the angular tolerance of guided-mode filters,” J. Opt. A, Pure Appl. Opt. 1, 545–551 (1999).
    [CrossRef]
  15. V. M. Fitio, Y. V. Bobitski, “Resonance effects in a dielectric grating; total absorption of electromagnetic waves by a dielectric grating on metal system,” J. Opt. A Pure Appl. Opt. 6, 943–951 (2004).
    [CrossRef]
  16. A. L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” J. Opt. A Pure Appl. Opt. 4, 588–594 (2002).
    [CrossRef]
  17. G. Bao, L. Cowsar, and W. Masters, eds. Mathematical Modeling in Optical Sciences, SIAM Frontiers in Applied Mathematics (Society of Industrial and Applied Mathematics, Philadelphia, Pa., 2001).
    [CrossRef]
  18. G. Bao, D. Dobson, J. A. Cox, “Mathematical studies of rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
    [CrossRef]
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    [CrossRef]
  22. G. Bao, K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 16, 755–760 (2002).
    [CrossRef]
  23. K. Huang, “Optimal design of diffractive optics,” Ph.D. thesis (Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1024, 2002).
  24. G. Bao, K. Huang, “Optimal design of guided-mode grating resonance filters,” IEEE Photonics Technol. Lett. 16, 141–144 (2004).
    [CrossRef]
  25. J. Elschner, G. Schmidt, “Diffraction in periodic structures and optimal design of binary gratings. I. Direct problems and gradient formulas,” Math. Methods Appl. Sci. 21, 1297–1342 (1998).
    [CrossRef]

2004 (2)

V. M. Fitio, Y. V. Bobitski, “Resonance effects in a dielectric grating; total absorption of electromagnetic waves by a dielectric grating on metal system,” J. Opt. A Pure Appl. Opt. 6, 943–951 (2004).
[CrossRef]

G. Bao, K. Huang, “Optimal design of guided-mode grating resonance filters,” IEEE Photonics Technol. Lett. 16, 141–144 (2004).
[CrossRef]

2003 (2)

P. S. Priambodo, T. A. Maldonado, R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83, 3248–3250 (2003).
[CrossRef]

S. T. Thurman, M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. 42, 3225–3233 (2003).
[CrossRef] [PubMed]

2002 (2)

G. Bao, K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 16, 755–760 (2002).
[CrossRef]

A. L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” J. Opt. A Pure Appl. Opt. 4, 588–594 (2002).
[CrossRef]

1999 (1)

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

1998 (4)

1995 (2)

1994 (1)

1993 (1)

1992 (2)

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

M. Lenoir, M. Vullierme-Ledard, C. Hazard, “Variational formulations for determination of resonant states in scattering problems,” SIAM J. Math. Anal. 23, 579–608 (1992).
[CrossRef]

1965 (1)

Bao, G.

G. Bao, K. Huang, “Optimal design of guided-mode grating resonance filters,” IEEE Photonics Technol. Lett. 16, 141–144 (2004).
[CrossRef]

G. Bao, K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 16, 755–760 (2002).
[CrossRef]

G. Bao, D. Dobson, J. A. Cox, “Mathematical studies of rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
[CrossRef]

Bobitski, Y. V.

V. M. Fitio, Y. V. Bobitski, “Resonance effects in a dielectric grating; total absorption of electromagnetic waves by a dielectric grating on metal system,” J. Opt. A Pure Appl. Opt. 6, 943–951 (2004).
[CrossRef]

Brundrett, D. L.

Cambril, E.

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

Cox, J. A.

G. Bao, D. Dobson, J. A. Cox, “Mathematical studies of rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
[CrossRef]

J. A. Cox, “Overview and applications of diffractive optics technology,” in Mathematical Modeling in Optical Sciences, SIAM Frontiers in Applied Mathematics, G. Bao, L. Cowsar, and W. Masters, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).
[CrossRef]

Dobson, D.

Elschner, J.

J. Elschner, G. Schmidt, “Diffraction in periodic structures and optimal design of binary gratings. I. Direct problems and gradient formulas,” Math. Methods Appl. Sci. 21, 1297–1342 (1998).
[CrossRef]

Fehrembach, A. L.

A. L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” J. Opt. A Pure Appl. Opt. 4, 588–594 (2002).
[CrossRef]

Fitio, V. M.

V. M. Fitio, Y. V. Bobitski, “Resonance effects in a dielectric grating; total absorption of electromagnetic waves by a dielectric grating on metal system,” J. Opt. A Pure Appl. Opt. 6, 943–951 (2004).
[CrossRef]

Gaylord, T. K.

Giovannini, H.

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

Glytsis, E. N.

Hazard, C.

M. Lenoir, M. Vullierme-Ledard, C. Hazard, “Variational formulations for determination of resonant states in scattering problems,” SIAM J. Math. Anal. 23, 579–608 (1992).
[CrossRef]

Hessel, A.

Huang, K.

G. Bao, K. Huang, “Optimal design of guided-mode grating resonance filters,” IEEE Photonics Technol. Lett. 16, 141–144 (2004).
[CrossRef]

K. Huang, “Optimal design of diffractive optics,” Ph.D. thesis (Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1024, 2002).

Lemarchand, F.

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

Lenoir, M.

M. Lenoir, M. Vullierme-Ledard, C. Hazard, “Variational formulations for determination of resonant states in scattering problems,” SIAM J. Math. Anal. 23, 579–608 (1992).
[CrossRef]

Liu, Z. S.

Magnusson, R.

Maldonado, T. A.

P. S. Priambodo, T. A. Maldonado, R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83, 3248–3250 (2003).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).

Maystre, D.

A. L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” J. Opt. A Pure Appl. Opt. 4, 588–594 (2002).
[CrossRef]

Morris, M.

Nevière, M.

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Heidelberg, Germany, 1980).
[CrossRef]

Oliner, A. A.

Priambodo, P. S.

P. S. Priambodo, T. A. Maldonado, R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83, 3248–3250 (2003).
[CrossRef]

Ramdani, K.

G. Bao, K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 16, 755–760 (2002).
[CrossRef]

Schmidt, G.

J. Elschner, G. Schmidt, “Diffraction in periodic structures and optimal design of binary gratings. I. Direct problems and gradient formulas,” Math. Methods Appl. Sci. 21, 1297–1342 (1998).
[CrossRef]

Sentenac, A.

A. L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” J. Opt. A Pure Appl. Opt. 4, 588–594 (2002).
[CrossRef]

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

Shin, D.

Thurman, S. T.

Tibuleac, S.

Vullierme-Ledard, M.

M. Lenoir, M. Vullierme-Ledard, C. Hazard, “Variational formulations for determination of resonant states in scattering problems,” SIAM J. Math. Anal. 23, 579–608 (1992).
[CrossRef]

Wang, S.

Young, P. P.

Appl. Math. Lett. (1)

G. Bao, K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 16, 755–760 (2002).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

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

P. S. Priambodo, T. A. Maldonado, R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83, 3248–3250 (2003).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

G. Bao, K. Huang, “Optimal design of guided-mode grating resonance filters,” IEEE Photonics Technol. Lett. 16, 141–144 (2004).
[CrossRef]

J. Opt. A Pure Appl. Opt. (2)

V. M. Fitio, Y. V. Bobitski, “Resonance effects in a dielectric grating; total absorption of electromagnetic waves by a dielectric grating on metal system,” J. Opt. A Pure Appl. Opt. 6, 943–951 (2004).
[CrossRef]

A. L. Fehrembach, D. Maystre, A. Sentenac, “Filtering of unpolarized light by gratings,” J. Opt. A Pure Appl. Opt. 4, 588–594 (2002).
[CrossRef]

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

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

J. Opt. Soc. Am. A (1)

Math. Methods Appl. Sci. (1)

J. Elschner, G. Schmidt, “Diffraction in periodic structures and optimal design of binary gratings. I. Direct problems and gradient formulas,” Math. Methods Appl. Sci. 21, 1297–1342 (1998).
[CrossRef]

Opt. Lett. (4)

SIAM J. Math. Anal. (1)

M. Lenoir, M. Vullierme-Ledard, C. Hazard, “Variational formulations for determination of resonant states in scattering problems,” SIAM J. Math. Anal. 23, 579–608 (1992).
[CrossRef]

Other (7)

K. Huang, “Optimal design of diffractive optics,” Ph.D. thesis (Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1024, 2002).

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Heidelberg, Germany, 1980).
[CrossRef]

J. A. Cox, “Overview and applications of diffractive optics technology,” in Mathematical Modeling in Optical Sciences, SIAM Frontiers in Applied Mathematics, G. Bao, L. Cowsar, and W. Masters, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).
[CrossRef]

R. Petit ed. Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, (Springer-Verlag, Heidelberg, Germany, 1980).
[CrossRef]

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).

G. Bao, L. Cowsar, and W. Masters, eds. Mathematical Modeling in Optical Sciences, SIAM Frontiers in Applied Mathematics (Society of Industrial and Applied Mathematics, Philadelphia, Pa., 2001).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Waveguide grating.

Fig. 2
Fig. 2

Design parameters.

Fig. 3
Fig. 3

Reflectance curve with initial data (TE).

Fig. 4
Fig. 4

Reflectance curve with initial data (TM).

Fig. 5
Fig. 5

Reflectance curve with new fill factor (TE).

Fig. 6
Fig. 6

Reflectance curve with new period (TE).

Fig. 7
Fig. 7

Reflectance curve with new period (TM).

Equations (29)

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Δ α u α + ω 2 μ ϵ u α = 0 ( TE ) ,
α ( 1 μ ϵ α u α ) + ω 2 u α = 0 ( TM ) ,
a = + i ( α , 0 ) = ( x 1 + i α , x 2 ) ,
Δ α = α 2 = ( x 1 + i α ) 2 + ( x 2 ) 2 .
u α n = T u α on Γ ,
u α n = T + u α 2 β exp ( i β b ) exp ( i α ) on Γ + ,
T ± ( v ) = p Z i β p ± v ̂ exp [ i ( 2 p π Λ ) x 1 ] ,
( TE )
Δ α u α + ω 2 u ϵ u α = 0 in Ω ,
n u α = T + u α 2 i β 1 exp ( i β 1 b ) on Γ + ,
n u α = T u α on Γ .
( TM )
α ( 1 μ ϵ α u α ) + ω 2 u α = 0 in Ω ,
n u α = T + u α 2 i β 1 exp ( i β 1 b ) on Γ + ,
n u α = T u α on Γ .
Ω ( α u α α v ¯ ω 2 μ ϵ u α v ¯ ) Γ + T + u α v ¯ Γ T u α v ¯
= Γ + 2 i β exp ( i β b ) v ¯ .
Ω ( 1 μ ϵ α u α α v ¯ ω 2 u α v ¯ ) 1 μ ϵ + Γ + T + u α v ¯ 1 μ ϵ Γ T u α v ¯ = 1 μ ϵ + Γ + 2 i β exp ( i β b ) v ¯ .
n eff 2 = f n H 2 + ( 1 f ) n L 2
1 n eff 2 = f n H 2 + 1 f n L 2
Ω u α v ¯ + α ( ν ) 2 u α v ¯ 2 i α ( ν ) 1 u α v ¯ Γ + T + ( ν ) u α v ¯ Γ T ( ν ) u α v ¯ = ν Ω ϵ μ u α v ¯ .
A ( ν ) x = ν B x ,
tan κ d = κ ( κ + δ ) κ 2 γ δ ,
κ = ( ϵ eff k 2 ξ 2 ) 1 2 , γ = ( ξ 2 ϵ + k 2 ) 1 2 , δ = ( ξ 2 ϵ k 2 ) 1 2
ξ = k ( n 0 sin θ ± i λ Λ ) .
R ( f j + 1 , d j , Λ ) = λ * .
e 0 ( f j + 1 , d j + 1 , Λ ) = min d e 0 ( f j + 1 , d , Λ ) .
R ( f ) = 610 .
R ( Λ ) = 610 .

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