Abstract

We present a design and analysis study of guided-mode resonances in photonic crystal slabs. Three-dimensional finite-difference time-domain (FDTD) simulations are used in parallel with a simplified model of guided-mode resonances to produce a representation of their evolution with structural parameters. From the analysis of the effective medium behavior of the system, we propose a simplified method able to predict the first guided-mode resonances at normal incidence with a good accuracy (1%) for holes with radius-to-period ratio smaller than 0.3 for the transverse magnetic polarization created internally. A substantial gain of time is, therefore, provided compared to FDTD (from the hours level to the seconds level). We also focus on two other important parameters, the quality factor and asymmetry of peaks, and present a way to design symmetric peaks with low sidebands.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90, 031911 (2007).
    [CrossRef]
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    [CrossRef]
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2009

2007

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

2006

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

2005

C. Kappel, A. Selle, M. A. Bader, and G. Marowsky, “Double grating waveguide structures: 350-fold enhancement of two-photon fluorescence applying ultrashort pulses,” Sens. Actuators B 107, 135–139 (2005).
[CrossRef]

2004

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 μm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004).
[CrossRef]

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84, 4905–4907 (2004).
[CrossRef]

2003

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82, 1999–2001 (2003).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

2000

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

1997

1996

1994

1992

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

1968

S. R. Coriell and J. L. Jackson, “Bounds on transport coefficients of two-phase materials,” J. Appl. Phys. 39, 4733–4736 (1968).
[CrossRef]

1965

1902

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. Phys. Soc. London 18, 269–275 (1902).
[CrossRef]

Astratov, V. N.

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

Bader, M. A.

C. Kappel, A. Selle, M. A. Bader, and G. Marowsky, “Double grating waveguide structures: 350-fold enhancement of two-photon fluorescence applying ultrashort pulses,” Sens. Actuators B 107, 135–139 (2005).
[CrossRef]

Bakir, B. Ben

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 790–852.

Boutami, S.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Chang-Hasnain, C. J.

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 μm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004).
[CrossRef]

Chen, L.

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 μm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004).
[CrossRef]

Chow, E.

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

Coriell, S. R.

S. R. Coriell and J. L. Jackson, “Bounds on transport coefficients of two-phase materials,” J. Appl. Phys. 39, 4733–4736 (1968).
[CrossRef]

Crozier, K. B.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Culshaw, I.

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

Cunningham, B. T.

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

De La Rue, R. M.

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

Fan, S.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84, 4905–4907 (2004).
[CrossRef]

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82, 1999–2001 (2003).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

Ganesh, N.

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

Garrigues, M.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Ghatak, A. K.

M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, 1977), pp. 5–29.

Hane, K.

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

Hessel, A.

Huang, M. C. Y.

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 μm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004).
[CrossRef]

Hutley, M.

P. Lalanne and M. Hutley, “Artificial media optical properties—subwavelength scale,” in Encyclopedia of Optical Engineering, R.Driggers, ed. (Marcel Dekker, 2003), Vol.  1, pp. 62–71.

Jackson, J. L.

S. R. Coriell and J. L. Jackson, “Bounds on transport coefficients of two-phase materials,” J. Appl. Phys. 39, 4733–4736 (1968).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

S. G. Johnson, “Meep,” http://ab-initio.mit.edu/wiki/index.php/Meep.

S. G. Johnson, “Meep Tutorial/Band diagram, resonant modes, and transmission in a holey waveguide,” http://ab-initio.mit.edu/wiki/index.php/Meep_Tutorial/Band_diagram%2C_resonant_modes%2C_and_transmission_in_a_holey_waveguide.

Kanamori, Y.

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

Kappel, C.

C. Kappel, A. Selle, M. A. Bader, and G. Marowsky, “Double grating waveguide structures: 350-fold enhancement of two-photon fluorescence applying ultrashort pulses,” Sens. Actuators B 107, 135–139 (2005).
[CrossRef]

Kilic, O.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Kim, S.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Kitani, T.

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

Krauss, T. F.

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

Lalanne, P.

P. Lalanne and M. Hutley, “Artificial media optical properties—subwavelength scale,” in Encyclopedia of Optical Engineering, R.Driggers, ed. (Marcel Dekker, 2003), Vol.  1, pp. 62–71.

Leclercq, J.-L.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Letartre, X.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Lousse, V.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Magnusson, R.

Malyarchuk, V.

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

Marowsky, G.

C. Kappel, A. Selle, M. A. Bader, and G. Marowsky, “Double grating waveguide structures: 350-fold enhancement of two-photon fluorescence applying ultrashort pulses,” Sens. Actuators B 107, 135–139 (2005).
[CrossRef]

Mateus, C. F. R.

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 μm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004).
[CrossRef]

Mathias, P. C.

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

Morris, G. M.

Nevière, M.

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 123–157.
[CrossRef]

Oliner, A. A.

Peng, S.

Peter, Y.-A.

Pottier, P.

Regreny, P.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Rojo-Romeo, P.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991), pp. 238–271.
[CrossRef]

Seassal, C.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Selle, A.

C. Kappel, A. Selle, M. A. Bader, and G. Marowsky, “Double grating waveguide structures: 350-fold enhancement of two-photon fluorescence applying ultrashort pulses,” Sens. Actuators B 107, 135–139 (2005).
[CrossRef]

Shi, L.

Skolnick, M. S.

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

Skorobogatiy, M.

Smith, A. D.

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

Soares, J. A. N. T.

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

Sodha, M. S.

M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, 1977), pp. 5–29.

Solgaard, O.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82, 1999–2001 (2003).
[CrossRef]

Stevenson, R. M.

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

Suh, W.

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84, 4905–4907 (2004).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82, 1999–2001 (2003).
[CrossRef]

Suzuki, Y.

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 μm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991), pp. 238–271.
[CrossRef]

Tibuleac, S.

Viktorovitch, P.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, C. Seassal, P. Rojo-Romeo, P. Regreny, M. Garrigues, and P. Viktorovitch, “Photonic crystal-based MOEMS devices,” IEEE J. Sel. Top. Quantum Electron. 13, 244–252 (2007).
[CrossRef]

Wang, S. S.

Whittaker, D. M.

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 790–852.

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. Phys. Soc. London 18, 269–275 (1902).
[CrossRef]

Yanik, M. F.

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82, 1999–2001 (2003).
[CrossRef]

Zhang, W.

N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. A. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nat. Nanotechnol. 2, 515–520 (2007).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

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

V. N. Astratov, R. M. Stevenson, I. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. 77, 178–180 (2000).
[CrossRef]

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84, 4905–4907 (2004).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Positions of guided-mode resonances in normalized frequency a / λ versus the normalized thickness of the PhC membrane (a)  t / λ or (b)  t / a . The color dotted lines are for the limit case of infinitely small holes. The color solid lines correspond to 3D FDTD simulations for several hole normalized radii r / a . Red denotes TE polarization and blue is TM for the light inside the membrane, as defined in the main text.

Fig. 2
Fig. 2

Unit cell of the PhC membrane used in the FDTD calculations, with period a, radius r, and indices of refraction n 1 and n 2 . A plane wave is launched from the top (bold arrow). Reflected and transmitted power is monitored, respectively, at the top and bottom surfaces.

Fig. 3
Fig. 3

Reflection R (color scale) of the PhC membrane versus normalized frequency a / λ (vertical axis) and normalized thickness t / a (horizontal axis) for varying normalized radius r / a . For r / a = 0.3 are exemplified the first guided-mode resonances (arrow A) corresponding to the waveguide mode m = 0 and the additional ones (arrow B) corresponding to the mode m = 1 .

Fig. 4
Fig. 4

Quality factor (superimposed number and color scale) of the PhC membrane reflection peaks versus a / λ and (a)  t / λ or (b)  t / a , for several normalized radii r / a (see the correspondence with Fig. 1), for TE internal polarization. (c) and (d) are the same, respectively, for TM internal polarization. The theoretical limits as in Fig. 1 are also represented (dotted curves).

Fig. 5
Fig. 5

Asymmetry (superimposed number and color scale) of the PhC membrane reflection peaks versus a / λ and (a)  t / λ or (b)  t / a , for several normalized radii r / a (see the correspondence with Fig. 1), for TE internal polarization. (c) and (d) are the same, respectively, for TM internal polarization. The theoretical limits as in Fig. 1 are also represented (dotted curves). The black and white solid curves correspond (from left to right) to the characteristic thicknesses t / λ = 1 / ( 4 n eff- v ) , 1 / ( 2 n eff- v ) , 3 / ( 4 n eff- v ) , and 1 / n eff- v (described in the main text).

Fig. 6
Fig. 6

Reflection spectrum of a PhC membrane ( r / a = 0.2 , t = 134 nm , a = 462 nm ), without (green curve) and with (blue curve) an antireflection (AR) layer above and below it designed for λ = 550 nm . Inset: schematic of the PhC membrane with antireflection layers.

Fig. 7
Fig. 7

Reflection spectrum of a PhC membrane ( r / a = 0.2 , t / a = 0.8 ) (blue curve, with peaks) and reflection spectrum of a Fabry–Perot membrane of index n m = 1.95 (red curve, slowly varying), which has been matched to the former.

Fig. 8
Fig. 8

Effective index n eff- v of the PhC membrane versus area filling factor (f) of the holes and their normalized radius r / a . Dots represent the value extracted from FDTD simulations, and colored curves correspond to different theoretical models.

Fig. 9
Fig. 9

Positions of guided-mode resonances in normalized frequency a / λ versus the normalized thickness of the PhC membrane (a)  t / λ or (b)  t / a for varying hole normalized radius r / a for TM internal polarization, obtained using the “horizontal” effective index in the model of infinitely small holes.

Equations (19)

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a λ 1 = 1 n eff .
f ( x ) = 2 π t λ n 2 2 x 2 2 arctan ( η 21 x 2 n 1 2 n 2 2 x 2 ) m π ,
1 n 2 < a λ < 1 n 1 .
t λ < 1 2 n 2 2 n 1 2 ,
a λ 2 = 2 n eff .
a λ > 1 n 1 .
Q = ν r Δ ν ,
A = ν m ν mean Δ ν / 2 ,
t 3 / a = ( t 3 / λ ) / ( a / λ ) = [ 1 / ( 4 n 3 ) ] / ( a / λ ) ,
R FP = 1 ( 1 R i ) 2 1 + R i 2 2 R i cos ( 2 π λ 2 n m t ) ,
R i = ( n 1 n m n 1 + n m ) 2 .
n eff- v 1 = f n 1 + ( 1 f ) n 2 .
n eff- v 2 = ( f n 1 2 + ( 1 f ) n 2 2 ) 1 2 ,
n eff v 3 = ( f n 1 2 + ( 1 f ) n 2 2 ) 1 2 .
ε p = ε 2 · ( 1 2 r a + 2 1 L 1 · [ π 4 arctan [ ( 1 G 1 + G ) 1 2 ] ( 1 G 2 ) 1 2 ] ) ,
ε s = ε 2 · ( 1 2 r a + π 2 ( L 1 ) ln [ H + 1 + ( H 2 1 ) 1 2 H + 1 ( H 2 1 ) 1 2 ] ( L 1 ) ( H 2 1 ) 1 2 ) 1 ,
n eff- h = ( f n a 2 + ( 1 f ) n b 2 ) 1 2 ,
n eff- h = ( f n 1 2 + ( 1 f ) n eff 2 ) 1 2 .
a λ 1 = 1 n eff- h ,

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