## Controlling uncoupled resonances in photonic crystals through breaking the mirror symmetry

Optics Express, Vol. 16, Issue 17, pp. 13090-13103 (2008)

http://dx.doi.org/10.1364/OE.16.013090

Acrobat PDF (1180 KB)

### Abstract

We show that modes in a photonic crystal slab that are uncoupled to outside radiation in a symmetric structure can be excited by breaking the mirror symmetry through introducing a protrusion on the side of the photonic crystal holes. We show that coupling to these resonances can be controlled by the strength of this asymmetry, and that it is also possible to choose among modes to couple to, through the shape of the asymmetry introduced. We provide simple theoretical arguments that explain the effect, and present eigenmode simulations and time-domain simulations. We confirm this predicted behavior with measurements on a photonic crystal with a broken mirror symmetry that exhibits an additional sharp resonant feature with a linewidth of 0.5 nm, in agreement with both calculated and simulated predictions.

© 2008 Optical Society of America

## 1. Introduction

1. V. N. Astratov, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B **60**, R16255 (1999). [CrossRef]

2. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B **65**, 235112 (2002). [CrossRef]

3. O. Kilic, S. Kim, W. Suh, Y. Peter, A. Sudbø, M. Yanik, S. Fan, and O. Solgaard, “Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders,” Opt. Lett. **29**, 2782 (2004). [CrossRef] [PubMed]

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

5. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. **68**, 2023 (1992). [CrossRef] [PubMed]

7. M. Galli, M. Agio, L. C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, and P. Bellutti, “Spectroscopy of photonic bands in macroporous silicon photonic crystals,” Phys. Rev. B **65**, 113111 (2002). [CrossRef]

11. V. Pacradouni, W. J. Mandeville, A. R. Crown, P. Paddon, J. F. Young, and S. R. Johnson, “Photonic band structure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B **62**, 4204 (2000). [CrossRef]

12. P. Paddon and J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B **61**, 2090 (2000). [CrossRef]

13. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B **63**, 125107 (2001). [CrossRef]

7. M. Galli, M. Agio, L. C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, and P. Bellutti, “Spectroscopy of photonic bands in macroporous silicon photonic crystals,” Phys. Rev. B **65**, 113111 (2002). [CrossRef]

14. K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. **84**, 4146 (2004). [CrossRef]

*et al*analyze mode symmetries in a surface-emitting photonic-crystal microcavity laser and employ uncoupled non-degenerate modes in the operation of the laser. In principle, these uncoupled modes have infinite vertical

*Q*. Therefore, a way to tune the coupling to these modes can give rise to very sharp resonant features that can be especially useful in sensor applications where the sensitivity is typically dependent upon the width of a spectral line. A method for adjusting the linewidth is also important for filter applications. An interesting filter application is presented in Ref. [15

15. A.-L. Fehrembach and A. Sentenac, “Unpolarized narrow-band filtering with resonant gratings,” Appl. Phys. Lett. **86**, 121105 (2005). [CrossRef]

## 2. Symmetry concepts

13. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B **63**, 125107 (2001). [CrossRef]

16. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B **68**, 035110 (2003). [CrossRef]

*Q*cavities [17

17. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express **10**, 670 (2002). [PubMed]

*x̑*axis, the

*y̑*axis, and the two diagonal axes. In addition to these, the PCS also has three rotational symmetries, and one additional mirror symmetry with respect to the

*z̑*axis. We will denote operators, such as mirror-reflection operators, by

*, and their respective eigenvalues by*σ ^

*σ*.

*E*

^{(1)}and

*E*

^{(2)}(to classify the modes with respect to their symmetries, we will employ the irreducible representation notation in the convention used in Ref. [8

8. K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B **52**, 7982 (1995). [CrossRef]

*A*

_{1},

*A*

_{2},

*B*

_{1}, and

*B*

_{2}. The symmetries of these modes is depicted in Fig. 2 (after Ref. [18

18. T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B **64**, 045117 (2001). [CrossRef]

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

*) as symmetric (*σ ^

_{z}*σ*=+1) or antisymmetric (

_{z}*σ*=-1), which are usually referred to as even and odd modes, respectively.

_{z}σ ^

^{2}=

*I*, where

*I*is the identity operator. Also, the reflection operator has real eigenvalues

*σ*=+1 or -1, since the eigenmodes can have either odd or even symmetries (i.e., can be symmetric or antisymmetric). These two properties show that the reflection operator is unitary, i.e.,

*†*σ ^

*=*σ ^

*I*. By using this identity, it is straightforward to show that non-degenerate modes are uncoupled. As an example, an

*A*

_{1}mode is uncoupled to external radiation because:

*e*〉 and |

_{x}*e*〉 represent plane waves that are

_{y}*x̑*-polarized and

*y̑*-polarized, |

*A*

_{1}〉 represents an

*A*

_{1}-type non-degenerate mode, and

*and*σ ^

_{x}*represent mirror symmetry reflections with respect to the*σ ^

_{y}*x̑*axis and

*y̑*axis, respectively. We can demonstrate the absence of coupling to outside radiation in the same way for the other non-degenerate modes, i.e.,

*A*

_{2},

*B*

_{1}, and

*B*

_{2}.

19. R. Wang, X-H. Wang, B-Y. Gu, and G-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. **90**, 4307 (2001). [CrossRef]

20. R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. **41**, 295 (1994). [CrossRef]

*A*

_{1}-type and a

*B*

_{2}-type non-degenerate mode is perturbed in a keyhole PCS. We will refer to these perturbed modes through a prime sign, such as

*A*′

_{1}or

*B*′

_{2}. Note that in Fig. 3, we depict the

*A*

_{1},

*B*

_{2},

*A*′

_{1}, and

*B*′

_{2}modes with representative electric fields. Slightly different arrow sizes mean that the mode is not symmetric with respect to that axis. We observe that by breaking the PCS symmetry along the

*y̑*axis (

*), we also break the symmetry of the modes along that axis, while the symmetry along the orthogonal*σ ^

_{y}*x̑*axis is not affected. A non-symmetric mode can be separated into a symmetric (

*s*) and an antisymmetric (

*as*) part:

*y̑*axis, we can analyze through our symmetry arguments how these parts couple to outside radiation. These two parts correspond to a mode with a non-degenerate-mode symmetry, and a mode with a degenerate-mode symmetry. This means that the original non-symmetric mode can be considered as a sum of a non-degenerate mode and a degenerate mode, hence it is able to couple to outside radiation due to its degenerate component. More interesting, we see that for the same type of protrusion (along the

*y̑*axis) the degenerate parts of the

*A*′

_{1}and

*B*′

_{2}modes have different polarizations. Therefore, the protrusion perturbs the

*A*

_{1}and

*B*

_{2}modes such that they couple to

*y̑*-polarized and

*x̑*-polarized plane waves, respectively. Similarly, the other modes

*A*′

_{2}and

*B*′

_{1}can be shown to couple to

*x̑*polarization and

*y̑*polarization, respectively. This shows the potential for exploiting these non-degenerate modes to control polarization.

*A*″

_{1}. In this structure, neither the

*x̑*nor the

*y̑*symmetry is broken, so we can write

*|*σ ^

_{x}*A*″

_{1}〉=+|

*A*″

_{1}〉, and

*|*σ ^

_{y}*A*″

_{1}〉=+|

*A*″

_{1}〉. This shows that a perturbed non-degenerate mode in a double-keyhole structure does not have a degenerate part. These symmetry properties make the coupling zero when calculated as in Eq. (1) above by using the unitary reflection operators.

## 3. Results

21. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173 (2001). [CrossRef] [PubMed]

22. W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi A **197**, 620 (2003). [CrossRef]

*Mathematica*to assign symmetries to modes calculated with MPB. The profile of the modes is plotted as the electric field energy density (power in the displacement field) at the center plane, i.e., at the middle of the PCS. The values are normalized to the peak power in the mode.

*E*

^{(1)}and

*E*

^{(2)}modes are not degenerate (they have different eigenfrequencies) only because of a small numerical birefringence due to the finite grid size used in the simulations. This breaks the 90° rotational symmetry and artificially lifts the degeneracy.

*A*′

_{1}and

*B*′

_{2}shown) are perturbed when a protrusion is introduced. The protrusion is a circle of smaller radius placed at the top of the original circular hole. This shape was chosen so that the hole shapes in the simulations were similar to the ones that were fabricated. In the same figure, the symmetric and antisymmetric parts of these modes are shown, verifying our symmetry argument that a perturbed non-degenerate mode will have a degenerate part to it.

23. O. Kilic, M. Digonnet, G. Kino, and O. Solgaard, “External fibre Fabry-Perot acoustic sensor based on a photonic-crystal mirror,” Meas. Sci. Technol. **18**, 3049 (2007). [CrossRef]

^{2}intensity spot diameter of ~100 µm and was launched on the PCS at normal incidence. The light transmitted by the PCS was collected with an optical spectrum analyzer, which provided us with the spectral data shown in Fig. 9. To obtain the spectra, we passed the light first through a linear polarizer, oriented to the

*x̑*axis, so that we only observed a single polarization. We see that in the keyhole structure at 1394 nm there is an additional, much sharper resonant feature, with a linewidth of 0.5 nm, that is missing in the two other structures.

^{th}odd mode with an

*A*

_{2}symmetry (shown in Fig. 4) corresponds to a wavelength of 1393 nm, closely matching the center wavelength of the measured resonance at 1394 nm.

## 4. Discussions

*x̑*polarization. Hence, our symmetry arguments require that we should see five non-degenerate resonances. However, we see only one non-degenerate resonance in the measured spectrum. Before seeking an explanation for the absence of some of the resonances, we should emphasize that throughout the upcoming discussion the terms symmetry and asymmetry will refer to the mirror symmetry of the structure with respect to the

*y̑*axis. When discussing the results shown in Fig. 7, we were comparing different keyhole PCS structures that had different amounts of asymmetry. Specifically, we were suggesting that a larger protrusion size should induce a larger asymmetry in the PCS structure, and hence lead to larger coupling to the non-degenerate modes. The results presented in Fig. 7 were supporting this simple argument. We will now elaborate on this issue to see if we can relate the coupling strength directly to the protrusion size or position.

*ε*(

*x*,

*y*) can give us information about the amount of symmetry, which we will denote as

*S*, in the following way:

*y̑*positions, so that we can seek the maximum overlap, and normalize to obtain a measure of how close the dielectric function is to being symmetric. We assume that the fill factor is large enough so that there is sufficient overlap between the original hole and its mirror image. With this definition, we can say that for

*S*=1 the PCS is symmetric and there will be no coupling to any non-degenerate mode. For

*S*<1 the PCS is not symmetric, and although we know that for this case non-degenerate modes will couple to outside radiation, we cannot quantify the coupling through this parameter. To quantify the effect of asymmetry, we need to investigate the mode shapes for different perturbations.

*S*), the four modes are affected differently. This example also demonstrates that by choosing the proper shape and position of the protrusion, we can select a specific non-degenerate mode to couple to, and also pick out the polarization that it couples to. This gives us good control over the strength of coupling to these modes, and therefore over the transmission and reflection spectra of the PCS.

^{th}odd and 16

^{th}even) in the keyhole PCS structure. The two modes are separated into their symmetric and antisymmetric parts. The normalized power in each part is given below the mode profiles. While the power in the degenerate part does not directly provide the quantitative strength of coupling, it still shows how much the mode is perturbed, so that we can assume that the coupling will be larger for a more strongly perturbed mode. The reason that we chose these two particular modes is because they are perturbed the most (the other modes have at least six times less power in their degenerate parts compared to the second strongest perturbed mode). The mode in Fig. 13 that is perturbed the strongest is actually the 24

^{th}odd mode centered at the wavelength 1393 nm, as we saw in our measurements. We see that the degenerate part of this mode carries more than an order of magnitude more power compared to the other mode (16

^{th}even). This shows that due to the mode profile, the 24

^{th}odd mode is affected utmost by the protrusion that breaks the mirror symmetry. This fact is also evident from the strong asymmetry in the mode (Fig. 13). The coupling to the other modes is one to two orders of magnitude weaker, so that the expected sharp resonances are averaged out probably either due to fabrication disorders in the hole size and lattice constant [3

3. O. Kilic, S. Kim, W. Suh, Y. Peter, A. Sudbø, M. Yanik, S. Fan, and O. Solgaard, “Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders,” Opt. Lett. **29**, 2782 (2004). [CrossRef] [PubMed]

24. H. Takeda and K. Yoshino, “Disappearances of uncoupled modes in two-dimensional photonic crystals due to anisotropies of liquid crystals,” Phys. Rev. E **67**, 056612 (2003). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

1. | V. N. Astratov, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B |

2. | S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B |

3. | O. Kilic, S. Kim, W. Suh, Y. Peter, A. Sudbø, M. Yanik, S. Fan, and O. Solgaard, “Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders,” Opt. Lett. |

4. | K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelength,” Phys. Rev. B |

5. | W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. |

6. | W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of the photon dispersion relation in two-dimensional ordered dielectric arrays,” J. Opt. Soc. Am. B |

7. | M. Galli, M. Agio, L. C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, and P. Bellutti, “Spectroscopy of photonic bands in macroporous silicon photonic crystals,” Phys. Rev. B |

8. | K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B |

9. | K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. III. Group-theoretical treatment,” J. Phys. Soc. Jpn. |

10. | V. Karathanos, “Inactive frequency bands in photonic crystals,” J. Mod. Opt. |

11. | V. Pacradouni, W. J. Mandeville, A. R. Crown, P. Paddon, J. F. Young, and S. R. Johnson, “Photonic band structure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B |

12. | P. Paddon and J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B |

13. | T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B |

14. | K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. |

15. | A.-L. Fehrembach and A. Sentenac, “Unpolarized narrow-band filtering with resonant gratings,” Appl. Phys. Lett. |

16. | O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B |

17. | K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express |

18. | T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B |

19. | R. Wang, X-H. Wang, B-Y. Gu, and G-Z. Yang, “Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals,” J. Appl. Phys. |

20. | R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. |

21. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

22. | W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi A |

23. | O. Kilic, M. Digonnet, G. Kino, and O. Solgaard, “External fibre Fabry-Perot acoustic sensor based on a photonic-crystal mirror,” Meas. Sci. Technol. |

24. | H. Takeda and K. Yoshino, “Disappearances of uncoupled modes in two-dimensional photonic crystals due to anisotropies of liquid crystals,” Phys. Rev. E |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(230.5440) Optical devices : Polarization-selective devices

(230.5750) Optical devices : Resonators

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: May 20, 2008

Revised Manuscript: July 16, 2008

Manuscript Accepted: July 22, 2008

Published: August 12, 2008

**Citation**

Onur Kilic, Michel Digonnet, Gordon Kino, and Olav Solgaard, "Controlling uncoupled resonances in
photonic crystals through breaking
the mirror symmetry," Opt. Express **16**, 13090-13103 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13090

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### References

- V. N. Astratov, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, "Photonic band-structure effects in the reflectivity of periodically patterned waveguides," Phys. Rev. B 60, R16255 (1999). [CrossRef]
- S. Fan and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002). [CrossRef]
- O. Kilic, S. Kim, W. Suh, Y. Peter, A. Sudbø, M. Yanik, S. Fan, and O. Solgaard, "Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders," Opt. Lett. 29, 2782 (2004). [CrossRef] [PubMed]
- K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006). [CrossRef]
- W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Measurement of photonic band structure in a two-dimensional periodic dielectric array," Phys. Rev. Lett. 68, 2023 (1992). [CrossRef] [PubMed]
- W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Measurement of the photon dispersion relation in two-dimensional ordered dielectric arrays," J. Opt. Soc. Am. B 10, 322 (1993). [CrossRef]
- M. Galli, M. Agio, L. C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, and P. Bellutti, "Spectroscopy of photonic bands in macroporous silicon photonic crystals," Phys. Rev. B 65, 113111 (2002). [CrossRef]
- K. Sakoda, "Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices," Phys. Rev. B 52, 7982 (1995). [CrossRef]
- K. Ohtaka and Y. Tanabe, "Photonic bands using vector spherical waves. III. Group-theoretical treatment," J. Phys. Soc. Jpn. 65, 2670 (1996). [CrossRef]
- V. Karathanos, "Inactive frequency bands in photonic crystals," J. Mod. Opt. 45, 1751 (1998). [CrossRef]
- V. Pacradouni, W. J. Mandeville, A. R. Crown, P. Paddon, J. F. Young, and S. R. Johnson, "Photonic band structure of dielectric membranes periodically textured in two dimensions," Phys. Rev. B 62, 4204 (2000). [CrossRef]
- P. Paddon and J. F. Young, "Two-dimensional vector-coupled-mode theory for textured planar waveguides," Phys. Rev. B 61, 2090 (2000). [CrossRef]
- T. Ochiai and K. Sakoda, "Dispersion relation and optical transmittance of a hexagonal photonic crystal slab," Phys. Rev. B 63, 125107 (2001). [CrossRef]
- K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, M. Troccoli, and F. Capasso, "Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser," Appl. Phys. Lett. 84, 4146 (2004). [CrossRef]
- A.-L. Fehrembach and A. Sentenac, "Unpolarized narrow-band filtering with resonant gratings," Appl. Phys. Lett. 86, 121105 (2005). [CrossRef]
- O. Painter and K. Srinivasan, "Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis," Phys. Rev. B 68, 035110 (2003). [CrossRef]
- K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670 (2002). [PubMed]
- T. Ito and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001). [CrossRef]
- R. Wang, X-H. Wang, B-Y. Gu, and G-Z. Yang, "Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals," J. Appl. Phys. 90, 4307 (2001). [CrossRef]
- R. Padjen, J. M. Gerard, and J. Y. Marzin, "Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems," J. Mod. Opt. 41, 295 (1994). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173 (2001). [CrossRef] [PubMed]
- W. Hergert and M. Däne, "Group theoretical investigations of photonic band structures," Phys. Status Solidi A 197, 620 (2003). [CrossRef]
- O. Kilic, M. Digonnet, G. Kino, and O. Solgaard, "External fibre Fabry-Perot acoustic sensor based on a photonic-crystal mirror," Meas. Sci. Technol. 18, 3049 (2007). [CrossRef]
- H. Takeda and K. Yoshino, "Disappearances of uncoupled modes in two-dimensional photonic crystals due to anisotropies of liquid crystals," Phys. Rev. E 67, 056612 (2003). [CrossRef]

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