Abstract

The paper addresses the leaky stop bands associated with resonant photonic crystal slabs and periodic waveguides. We apply a semianalytical model pertinent to the second band to compute the dispersion curves describing the leaky stop band and verify its correctness by rigorous band computations. This approximate model provides clear insights into the physical properties of the leaky stop band in terms of explicit analytical expressions found. In particular, it enables comparison of the structure of the bands computed in complex propagation constant, implying spatially decaying leaky modes, with the bands computed in complex frequency, implying temporally decaying modes. It is shown that coexisting Bragg-coupling and energy-leakage mechanisms perturb the bands in complex propagation constant whereas these mechanisms are decoupled in complex frequency. As a result, the bands in complex frequency are well defined exhibiting a clear gap. These conclusions are verified by numerical diffraction computations for both weak and strong grating modulations where the resonance peaks induced by external illumination are shown to closely track the band profile computed in complex frequency. Thus, in general, phase matching to a resonant leaky mode occurs via real propagation constant that is found by dispersion computations employing complex frequency.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2004 (5)

2002 (1)

B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002).
[CrossRef]

2001 (1)

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
[CrossRef]

2000 (2)

D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, "Optical fiber endface biosensor based on resonances in dielectric waveguide gratings," Proceedings of the SPIE 3911, 86-94 (2000).
[CrossRef]

D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, and 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]

1997 (2)

D. Rosenblatt, A. Sharon, and A. A. Friesem, "Resonant grating waveguide structures," IEEE J. Quantum Electron. 33, 2038-2059 (1997).
[CrossRef]

T. Tamir and S. Zhang, "Resonant scattering by multilayered dielectric gratings," J. Opt. Soc. Am. A 14, 1607-1616 (1997).
[CrossRef]

1996 (1)

T. Tamir and S. Zhang, "Modal transmission-line theory of multilayered grating structures," J. of Lightwave Technol. 14, 914-927 (1996).
[CrossRef]

1995 (1)

1985 (2)

R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation loss," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
[CrossRef]

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

1979 (1)

P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979).
[CrossRef]

1977 (1)

T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235-254 (1977).
[CrossRef]

1976 (1)

1975 (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguide," IEEE Trans. Microwave Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

Andreani, L. C.

D. Gerace and L. C. Andreani, "Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs," Phys. Rev. E 69, 056603 (2004).
[CrossRef]

Bendickson, J. M.

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguide," IEEE Trans. Microwave Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

Brundrett, D. L.

Challener, W.

Cowan, A. R.

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
[CrossRef]

Cunningham, B. T.

B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002).
[CrossRef]

Ding, Y.

Elachi, C.

Friesem, A. A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, "Resonant grating waveguide structures," IEEE J. Quantum Electron. 33, 2038-2059 (1997).
[CrossRef]

Gaylord, T. K.

Gerace, D.

D. Gerace and L. C. Andreani, "Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs," Phys. Rev. E 69, 056603 (2004).
[CrossRef]

Glytsis, E. N.

Grann, E. B.

Henry, C. H.

R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation loss," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
[CrossRef]

Hugh, B.

B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002).
[CrossRef]

Jaggard, D. L.

Kazarinov, R. F.

R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation loss," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
[CrossRef]

Li, P.

B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002).
[CrossRef]

Lin, B.

B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002).
[CrossRef]

Liu, H.

D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, "Optical fiber endface biosensor based on resonances in dielectric waveguide gratings," Proceedings of the SPIE 3911, 86-94 (2000).
[CrossRef]

Magnusson, R.

Maldonado, T. A.

Moharam, M. G.

Neviere, M.

P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979).
[CrossRef]

Pacradouni, V.

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
[CrossRef]

Paddon, P.

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
[CrossRef]

Peng, C.

Peng, S. T.

T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235-254 (1977).
[CrossRef]

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguide," IEEE Trans. Microwave Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

Pepper, J.

B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002).
[CrossRef]

Pomerantz, M.

Pommet, D. A.

Priambodo, P. S.

Purvinis, G.

Qiu, J.

B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002).
[CrossRef]

Rosenblatt, D.

D. Rosenblatt, A. Sharon, and A. A. Friesem, "Resonant grating waveguide structures," IEEE J. Quantum Electron. 33, 2038-2059 (1997).
[CrossRef]

Sharon, A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, "Resonant grating waveguide structures," IEEE J. Quantum Electron. 33, 2038-2059 (1997).
[CrossRef]

Tamir, T.

T. Tamir and S. Zhang, "Resonant scattering by multilayered dielectric gratings," J. Opt. Soc. Am. A 14, 1607-1616 (1997).
[CrossRef]

T. Tamir and S. Zhang, "Modal transmission-line theory of multilayered grating structures," J. of Lightwave Technol. 14, 914-927 (1996).
[CrossRef]

T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235-254 (1977).
[CrossRef]

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguide," IEEE Trans. Microwave Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

Tibuleac, S.

D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, "Optical fiber endface biosensor based on resonances in dielectric waveguide gratings," Proceedings of the SPIE 3911, 86-94 (2000).
[CrossRef]

Vincent, P.

P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979).
[CrossRef]

Wawro, D.

D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, "Optical fiber endface biosensor based on resonances in dielectric waveguide gratings," Proceedings of the SPIE 3911, 86-94 (2000).
[CrossRef]

Young, J. F.

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
[CrossRef]

Zhang, S.

T. Tamir and S. Zhang, "Resonant scattering by multilayered dielectric gratings," J. Opt. Soc. Am. A 14, 1607-1616 (1997).
[CrossRef]

T. Tamir and S. Zhang, "Modal transmission-line theory of multilayered grating structures," J. of Lightwave Technol. 14, 914-927 (1996).
[CrossRef]

Zhou, M.

Appl. Phys. (2)

P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979).
[CrossRef]

T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235-254 (1977).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation loss," IEEE J. Quantum Electron. QE-21, 144-150 (1985).
[CrossRef]

D. Rosenblatt, A. Sharon, and A. A. Friesem, "Resonant grating waveguide structures," IEEE J. Quantum Electron. 33, 2038-2059 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguide," IEEE Trans. Microwave Theory Tech. MTT-23, 123-133 (1975).
[CrossRef]

J. of Lightwave Technol. (1)

T. Tamir and S. Zhang, "Modal transmission-line theory of multilayered grating structures," J. of Lightwave Technol. 14, 914-927 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. E (1)

D. Gerace and L. C. Andreani, "Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs," Phys. Rev. E 69, 056603 (2004).
[CrossRef]

Proc. IEEE (1)

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

Proceedings of the SPIE (1)

D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, "Optical fiber endface biosensor based on resonances in dielectric waveguide gratings," Proceedings of the SPIE 3911, 86-94 (2000).
[CrossRef]

Sens. Actuators B (1)

B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002).
[CrossRef]

Other (3)

A. Yariv, Optical Electronics in Modern Communications, 5th edition (Oxford University Press, New York, 1997).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995).

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).

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Figures (15)

Fig. 1.
Fig. 1.

The periodic waveguide under study. Thickness d=0.5μm, period Λ=1μm, fill factor F= Λh /Λ=0.4, cover region index nc =1.0, substrate index ns =1.5, navg =2, and n avg = F n h 2 + ( 1 F ) n l 2 . θ is the angle of incidence. The incident wave is in the TE polarization state such that the electric vector is normal to the plane of incidence.

Fig. 2.
Fig. 2.

Dispersion diagram for the TE0 mode in a periodic waveguide. The propagation constant of the leaky mode is separated into real part and imaginary part with β=βR +I . (a) Real part of the propagation constant. (b) Imaginary part of the propagation constant; βI is presented on a logarithmic scale. (c) Schematics showing the directions of modes and excited waves in and outside of the stop bands. The plot in (a) is reduced to the first Brillouin zone and only half of that zone is presented. The structure is that in Fig. 1 with nh ,=2.05, Δε=0.34, and nf= navg .

Fig. 3.
Fig. 3.

Stop band details. (a) The first stop band. (b) The second stop band. (c) The third stop band. The dashed curves show the corresponding homogeneous waveguide dispersion valid for Δε approaching zero.

Fig. 4.
Fig. 4.

Coupling processes at the second stop band.

Fig. 5.
Fig. 5.

Details of a nonleaky stop band. (a) Dispersion represented with real frequency and complex propagation constant. (b) Dispersion represented with complex frequency and real propagation constant. The complex frequency is defined by k0 =k0,R +jk0,I . Only half of the first Brillouin zone is plotted. The plots are calculated for the structure in Fig. 1 (nh =2.05) while forcing the coupling coefficient h1 to zero.

Fig. 6.
Fig. 6.

Details of a leaky stop band at small modulation (nh =2.05 and Δε=0.34). The dispersion curves are formulated in real frequency and complex propagation constant.

Fig. 7.
Fig. 7.

Details of a leaky stop band at small modulation (nh =2.05 and Δε=0.34). The dispersion curves are formulated in complex frequency and real propagation constant.

Fig. 8.
Fig. 8.

Rigorously computed leaky stop band structure in real and complex frequency confirming the predictions of the KH model in Figs. 6 and 7 and in Table I.

Fig. 9.
Fig. 9.

Details of a leaky stop band with strong modulation (nh ,=2.6 and Δε=4.6) computed with the KH coupled-mode formalism and with RCWA. (a) Dispersion represented with real frequency and complex propagation constant. (b) Dispersion represented with complex frequency and real propagation constant. (c) Comparison of the real parts of β using RCWA.

Fig. 10.
Fig. 10.

Interaction between a periodic waveguide and an incident plane wave. (a) Excitation of a leaky mode in a higher-order diffraction regime. (b) Excitation of a leaky mode in the zero-order diffraction regime.

Fig. 11.
Fig. 11.

Spectra of a single-layer waveguide grating under normal incidence θ=0°. The GMRs are marked by small circles. Here, η denotes diffraction efficiency, R reflectance, and T transmittance. The structure is that in Fig. 1 with nh =2.05 and Δε=0.34 with corresponding dispersion diagram shown in Fig. 2.

Fig. 12.
Fig. 12.

Dispersion and diffraction picture for an obliquely incident plane wave. The grating has nh =2.05 and Δε=034. (a) Frequency dependence of the real part of the propagation constant and the dispersion curve of the incident wave at θ=28.6°. (b) GMR reflectance of the 0th order labeled with TEm,ν (c) Efficiency of higher diffraction orders.

Fig. 13.
Fig. 13.

GMRs around the second stop band. (a) Diffraction diagram of the GMR reflectance as function of the normalized wave vector kx /K and frequency k0 /K. (b) Comparison of the β curve (complex frequency) and the locations of GMRs. (c) Comparison of the βR curve (real frequency) and the locations of GMRs.

Fig. 14.
Fig. 14.

Comparison of the dispersion curves and the locations of GMRs under strong modulation. The dots represent the resonance locations estimated from diffraction computations for incident angles from 0° to 5°. The structure is the same as that in Fig. 9 with nh =2.6 and Δε=4.6.

Fig. 15.
Fig. 15.

Diffraction efficiency and the maximum amplitude of the first orders at normal incidence. The structure has nh =2.6 and Δε=4.6.

Tables (1)

Tables Icon

Table I. Main features of leaky stop bands in real and complex frequency

Equations (9)

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k 0 n c m K = R e ( β ) or k 0 n s m K = R e ( β )
Δ β 2 = [ ( Δ k · h 3 ) 2 h 2 2 ] + 2 j h 1 [ ( Δ k · h 3 ) Re ( h 2 ) ]
( Δ k · h 3 + j h 1 ) 2 = Δ β 2 + ( h 2 2 + 2 j h 1 Re ( h 2 ) h 1 2 )
TE mode { h 1 = j k 0 4 γ 1 2 2 k d 0 d 0 ϕ ( x ) ϕ * ( x ) G ( x , x ) d x d x h 2 = k 0 2 γ 2 2 k d 0 ϕ ( x ) ϕ * ( x ) d x h 3 = k 0 K γ 0 ( x ) ϕ ( x ) ϕ * ( x ) d x
Δ β = ± ( Δ k · h 3 ) 2 h 2 2
Δ k = ± ( Δ β ) 2 + h 2 2 h 3
k 0 n c sin θ m K = ± k 0 n c or k 0 n c sin θ m K = ± k 0 n s
k 0 n c sin θ m K = Re ( β ν )
k 0 n c sin θ m K = β ν

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