## Creation of large band gap with anisotropic annular photonic crystal slab structure

Optics Express, Vol. 18, Issue 5, pp. 5221-5228 (2010)

http://dx.doi.org/10.1364/OE.18.005221

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### Abstract

A two-dimensional anisotropic annular photonic crystal slab structure composed of circular air holes and dielectric rods with finite thickness in a triangular lattice is presented to achieve an absolute photonic band gap. Positive uniaxial crystal Tellurium is introduced to the structure with the extraordinary axis parallel to the extension direction of rods. The role of each geometric parameter is investigated by employing the conjugate-gradient method. A large mid-gap ratio is realized by the parameter optimization. A flat band called as anomalous group velocity within two large gaps is discovered and can be widely applied in many fields. A hybrid structure with GaAs slab and Te rods is designed to achieve a large gap and demonstrates that the annular structure can improve the gap effectively.

© 2010 OSA

## 1. Introduction

1. S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B **43**(16), 12772–12789 (1991). [CrossRef]

3. S. Y. Zhu, H. Chen, and H. Huang, “Quantum Interference Effects in Spontaneous Emission from an Atom Embedded in a Photonic Band Gap Structure,” Phys. Rev. Lett. **79**(2), 205–208 (1997). [CrossRef]

4. T. F. Krauss, R. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature **383**(6602), 699–702 (1996). [CrossRef]

6. K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattices as revealed by transmittance measurement,” Phys. Rev. B **53**(3), 1010–1013 (1996). [CrossRef]

7. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. **74**(1), 7–9 (1999). [CrossRef]

8. Y. Chen, Z. Li, Z. Zhang, D. Psaltis, and A. Scherer, “Nanoimprinted circular grating distributed feedback dye laser,” Appl. Phys. Lett. **91**(5), 051109 (2007). [CrossRef]

9. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science **302**(5649), 1374–1377 (2003). [CrossRef] [PubMed]

10. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**(6961), 944–947 (2003). [CrossRef] [PubMed]

11. M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science **308**(5726), 1296–1298 (2005). [CrossRef] [PubMed]

*)*structure composed of dielectric rods and air holes with each dielectric rod centered in air hole was proposed and analyzed [13

13. H. Kurt and D. S. Citrin, “Annular photonic crystals,” Opt. Express **13**(25), 10316–10326 (2005). [CrossRef] [PubMed]

14. H. Kurt, R. Hao, Y. Chen, J. Feng, J. Blair, D. P. Gaillot, C. Summers, D. S. Citrin, and Z. Zhou, “Design of annular photonic crystal slabs,” Opt. Lett. **33**(14), 1614–1616 (2008). [CrossRef] [PubMed]

15. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large Absolute Band Gap in 2D Anisotropic Photonic Crystals,” Phys. Rev. Lett. **81**(12), 2574–2577 (1998). [CrossRef]

17. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B **48**(11), 8434–8437 (1993). [CrossRef]

18. Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B **58**(7), 3721–3729 (1998). [CrossRef]

15. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large Absolute Band Gap in 2D Anisotropic Photonic Crystals,” Phys. Rev. Lett. **81**(12), 2574–2577 (1998). [CrossRef]

14. H. Kurt, R. Hao, Y. Chen, J. Feng, J. Blair, D. P. Gaillot, C. Summers, D. S. Citrin, and Z. Zhou, “Design of annular photonic crystal slabs,” Opt. Lett. **33**(14), 1614–1616 (2008). [CrossRef] [PubMed]

19. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B **17**(3), 387–400 (2000). [CrossRef]

27. M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. **3**(4), 211–219 (2004). [CrossRef] [PubMed]

*n*and

_{o}*n*, respectively. The ordinary axis of uniaxial crystal is chosen horizontal (perpendicular to the extension direction of rods) while the extraordinary axis is chosen vertical for the sake of simplicity. The uniaxial material Tellurium with

_{e}*n*= 4.8 and

_{o}*n*= 6.2 in the wavelength regime between 3.5 and 35μm is used as the material of slab and rods here. The semiconductor material Tellurium which has large anisotropy can be effectively realized in photonic crystal through the current advanced semiconductor technology. In slabs the modes are not purely TE or TM modes, but they can still be classified as vertically even or odd modes with respect to the horizontal symmetry plane bisecting the slab. However, since the modes are purely TE- or TM- polarization in the mirror plane, we can roughly regard even and odd modes as TE-like and TM-like modes respectively [16].

_{e}29. J. B. Feng, Y. Chen, J. Blair, H. Kurt, R. Hao, D. S. Citrin, C. J. Summers, and Z. Zhou, “Fabrication of annular photonic crystals by atomic layer deposition and sacrificial etching,” J. Vac. Sci. Technol. B **27**(2), 568–572 (2009). [CrossRef]

## 2. Structure

*r*

_{1}and dielectric rods with radius

*r*

_{2}are arranged in triangular lattice. The symmetrical structure with same refractive index in the solid substrate and cover is used here. The refractive index of solid substrate is equal to 1.45. To reduce fabrication difficulties, the dielectric-rod radius

*r*

_{2}should be sufficiently small. Furthermore, the air-holes radius

*r*

_{1}should be large enough to provide enough space for the inner dielectric pillars and produce a wide band gap for TE-modes [16]. If the slab is too thin, the modes are weakly guided. The bands lie just below the light line, so that the modes decay slowly into the substrate or cover region. The band gap is absent for the small frequency difference between the fundamental band and light line. The band in the fundamental guided modes approaches that of infinite 2D system when the slab becomes very thick. Higher-order modes are pulled down and the gap closes. Considering the restrictions mentioned above, the relevant design parameters are initially taken as:

*r*

_{1}= 0.47

*a*, 0.48

*a*and

*h =*0.6

*a*, 0.8

*a*respectively, where

*a*is the lattice constant. The optimization process of

*r*

_{2}is performed below.

## 3. Results and discussion

*r*

_{1}and

*h*are picked at reasonable values and the dielectric-rod radius is scanned from 0 to 0.32

*a*to see mid-gap ratio variation with

*r*

_{2}. The mid-gap ratio is defined as the ratio of the width of gap and the mid-frequency of gap. The results are summarized in Fig. 2 . The gaps between high bands, which are useless, are ignored here.

*r*

_{2}equals zero, an ordinary PCS structure is attained. As

*r*

_{2}scales up, the band frequency decreases for the effective refractive index scaling up and the PBG width reduces for the permittivity contrast scaling down. Therefore, the mid-gap ratio of 3-4 bands gap reduces with the addition of dielectric inside the air hole. As

*r*

_{2}increases further, the bands become flat. The 3-4 bands gap closes while a gap between 4 and 5 bands opens. This gap can also exhibit a large mid-gap ratio.

*r*

_{2}equal to 0 and

*r*

_{1}fixed at 0.47

*a*and 0.48

*a*. The results are shown in Fig. 3 .

*a*, the

*h*corresponding to the maximal mid-gap ratio is 0.7

*a*, but as air-hole radius is equal to 0.47

*a*the value is 0.8

*a*. This can be understood intuitively by considering an extreme case. The TE-like mode primarily refers to the horizontal dielectric constant, so the band structure of TE-like mode slightly varies with the change of slab thickness. When the air-hole radius scales up, the band frequency increases owing to the scale-down of the horizontal effective refractive index. But for TM-like mode, the electric field vector is vertical and the band structure is significantly affected by the increment of

*h*. Bands are pulled down into the light line and populate the guided mode. The band gap opens and overlaps with the TE-like mode to form the PBG. Since the large air-hole radius results in the high band frequency, the frequency of TM-like band gap matching the TE-like band gap is high. Therefore, the height of slab to sustain the peak of PBG is lower as air-hole radius is larger.

*h*. Because of the large vertical permittivity, the frequency of TM-like band is lower than that of TE-like band. Moreover, the variation of band frequency of TM-like mode is faster than that of TE-like mode. When the

*h*increases further after the mid-gap ratio attaining a peak, the band frequency decreases and the bands populate the guided mode. The guided bands of TM-like mode become dense and locate in the gap of TE-like mode. Thus, the absolute PBG decreases. Because of the

*h*sustaining the peak of mid-gap ratio scale-down as

*r*

_{1}increases, the absolute band gap of large air-hole radius is affected more critically and rapidly than that of small air-hole radius owing to the large band frequency. Therefore, the maximum band gap scales down as air-hole radius scales up.

*a*and

*r*

_{1}is taken on 0.48

*a*, a maximal mid-gap ratio equal to 16.5% is achieved, which is a considerable improvement relative to the previous results [14

14. H. Kurt, R. Hao, Y. Chen, J. Feng, J. Blair, D. P. Gaillot, C. Summers, D. S. Citrin, and Z. Zhou, “Design of annular photonic crystal slabs,” Opt. Lett. **33**(14), 1614–1616 (2008). [CrossRef] [PubMed]

*a*)~0.3988(2πc/

*a*). Moreover, a gap with mid-gap ratio equal to 16.4% is achieved when

*r*

_{1},

*r*

_{2}and

*h*are taken as 0.47

*a*, 0 and 0.8

*a*respectively. If the photonic band of slab structure is calculated using isotropic dielectric with the same effective refractive index 5.3 [18

18. Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B **58**(7), 3721–3729 (1998). [CrossRef]

*a*to 0.8

*a*, the largest mid-gap ratio is 9% when air-hole radius equal 0.47

*a*while it is 12% as air-hole radius is taken as 0.48

*a*. Thus, a conclusion can be drawn that the anisotropy in dielectric can improve the mid-gap ratio significantly. Although the annular structure does not improve the gap obviously, the annular structure has the potential to be applied in photonic crystal microcavity. The increase in vertical quality factor

*Q*

_{⊥}needs a large fill factor to reduce frequency for a defect mode in the slab to decrease the size of light cone, which determines the degree of vertical radiation loss [31

31. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express **11**(6), 579–593 (2003). [CrossRef] [PubMed]

32. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express **10**(15), 670–684 (2002). [PubMed]

*Q*

_{∥}and the total quality factor

*Q*=

*Q*

_{⊥}

*Q*

_{∥}/(

*Q*

_{⊥}+

*Q*

_{∥}) decrease. However, large

*Q*

_{⊥}and

*Q*

_{∥}can be achieved simultaneously in the annular structure. Furthermore, the annular structures can design the donor and acceptor mode microcavity easily [33

33. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. **67**(24), 3380–3383 (1991). [CrossRef] [PubMed]

*a*, 0.48

*a*and 0 respectively, the band structure is obtained and plotted in Fig. 4(a) . Notice that the 4th band is nearly flat within two large PBGs. The values of mid-gap ratio are 13.9% between 3rd-4th bands (0.3593(2πc/

*a*)~0.4129(2πc/

*a*)) and 5.43% between 4th-5th bands (0.4180(2πc/

*a*)~0.4414(2πc/

*a*)) respectively. A detailed diagram of this band can be seen in Fig. 4(b) and the group velocity which is obtained by the slope of band (∂ω∕∂k) is shown in Fig. 4(c). The maximum group velocity is about 0.037

*c*. Because of the tiny frequency span of this band and the large band gap at 3-4bands and 4-5bands, the structure can be used as filter for the EM wave cannot radiate in the forbidden band region and propagates selectively in APC slab. The spontaneous emission and stimulated emission are inhibited in the band gap while they are enhanced in the guided band region owing to the small group velocity which increases the interaction of light with matter. Thus a PC laser can be achieved and manufactured [28].

30. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B **60**(8), 5751–5758 (1999). [CrossRef]

## 4. Hybrid Structure and Result Analysis

*a*, 0.48

*a*and the slab thickness

*h*is 0.6

*a*, 0.8

*a*, respectively. The dielectric rod radius is scanned from 0 to 0.32

*a.*The calculated results are summarized in Fig. 5 .

*r*

_{2}is zero, a normal PCS structure is obtained. There is not gap in the pure slab structure. The maximum mid-gap ratio is 12.1% The frequency region of this gap is 0.2621(2πc/

*a*)~0.2957(2πc/

*a*). Although the result is lower than the former owing to the small effective refractive index of slab, we can find that the anisotropic APC slab structure can improve the gap dramatically.

*r*

_{2}sustaining the maximum mid-gap ratio scales up as the

*r*

_{1}increases. For TE-like mode, the band frequency scales down and the mid-gap ratio decreases when the

*r*

_{2}increases. The gap of low frequency bands closes while the gap between high bands opens and becomes large. During this process, the frequency of the upper band of the gap decreases slower than that of band below. Since the permittivity of rods in horizontal direction is smaller than that in vertical direction, the band frequency of TE-like mode is higher than that of TM-like. Simultaneously, the band frequency of large

*r*

_{1}is higher than that of low

*r*

_{1}. Therefore, more dielectric rod is needed to be added in the air hole to overlap the two gaps to achieve the largest gap when

*r*

_{1}is larger.

## 5. Conclusion

*r*

_{1},

*r*

_{2}and

*h*, has been presented. The slab thickness sustaining the maximum mid-gap ratio scales down as the air-hole radius increases. Furthermore, when the slab thickness increases further, the mid-gap ratio scales down as the air-hole radius scales up in the slab structure. Such a concept is also applicative to other lattice types and configurations. Optimizing geometry parameters yields a maximum PBG, whose mid-gap ratio equals 16.5% between 3 and 4 bands. It is a considerable improvement comparing with the results reported. More specifically, a flat band, which has widespread applications, is found. The structure is easy to be realized experimentally with the current advanced semiconductor technology. Additionally, a structure with GaAs slab and Te rods is calculated and also exhibits a large gap. The annular structure improves the mid-gap ratio dramatically in this case. This structure will be of highly desire for the mode-insensitivity nature of PBG.

## Acknowledgment

## References and links

1. | S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B |

2. | S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A |

3. | S. Y. Zhu, H. Chen, and H. Huang, “Quantum Interference Effects in Spontaneous Emission from an Atom Embedded in a Photonic Band Gap Structure,” Phys. Rev. Lett. |

4. | T. F. Krauss, R. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature |

5. | U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. |

6. | K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattices as revealed by transmittance measurement,” Phys. Rev. B |

7. | M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. |

8. | Y. Chen, Z. Li, Z. Zhang, D. Psaltis, and A. Scherer, “Nanoimprinted circular grating distributed feedback dye laser,” Appl. Phys. Lett. |

9. | R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science |

10. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

11. | M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science |

12. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995), pp.66–93. |

13. | H. Kurt and D. S. Citrin, “Annular photonic crystals,” Opt. Express |

14. | H. Kurt, R. Hao, Y. Chen, J. Feng, J. Blair, D. P. Gaillot, C. Summers, D. S. Citrin, and Z. Zhou, “Design of annular photonic crystal slabs,” Opt. Lett. |

15. | Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large Absolute Band Gap in 2D Anisotropic Photonic Crystals,” Phys. Rev. Lett. |

16. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995), pp.135–155. |

17. | R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B |

18. | Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B |

19. | Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B |

20. | M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B |

21. | J. F. McMillan, X. Yang, N. C. Panoiu, R. M. Osgood, and C. W. Wong, “Enhanced stimulated Raman scattering in slow-light photonic crystal waveguides,” Opt. Lett. |

22. | Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. |

23. | C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express |

24. | K. Inoue, H. Oda, N. Ikeda, and K. Asakawa, “Enhanced third-order nonlinear effects in slow-light photonic-crystal slab waveguides of line-defect,” Opt. Express |

25. | T. Baba, “Slow light in photonic crystals,” Nat. Photonics |

26. | B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O'Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics |

27. | M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. |

28. | K. Sakoda, Optical properties of photonic crystals (Springer, Berlin, Allemagne, 2001). |

29. | J. B. Feng, Y. Chen, J. Blair, H. Kurt, R. Hao, D. S. Citrin, C. J. Summers, and Z. Zhou, “Fabrication of annular photonic crystals by atomic layer deposition and sacrificial etching,” J. Vac. Sci. Technol. B |

30. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

31. | K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express |

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

33. | E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(160.5293) Materials : Photonic bandgap materials

(230.5298) Optical devices : Photonic crystals

(130.5440) Integrated optics : Polarization-selective devices

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: December 2, 2009

Revised Manuscript: January 13, 2010

Manuscript Accepted: January 30, 2010

Published: February 26, 2010

**Citation**

Peng Shi, Kun Huang, Xue-liang Kang, and Yong-ping Li, "Creation of large band gap with anisotropic annular photonic crystal slab structure," Opt. Express **18**, 5221-5228 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-5221

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### References

- S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43(16), 12772–12789 (1991). [CrossRef]
- S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50(2), 1764–1769 (1994). [CrossRef] [PubMed]
- S. Y. Zhu, H. Chen, and H. Huang, “Quantum Interference Effects in Spontaneous Emission from an Atom Embedded in a Photonic Band Gap Structure,” Phys. Rev. Lett. 79(2), 205–208 (1997). [CrossRef]
- T. F. Krauss, R. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature 383(6602), 699–702 (1996). [CrossRef]
- U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68(6), 747–749 (1996). [CrossRef]
- K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattices as revealed by transmittance measurement,” Phys. Rev. B 53(3), 1010–1013 (1996). [CrossRef]
- M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74(1), 7–9 (1999). [CrossRef]
- Y. Chen, Z. Li, Z. Zhang, D. Psaltis, and A. Scherer, “Nanoimprinted circular grating distributed feedback dye laser,” Appl. Phys. Lett. 91(5), 051109 (2007). [CrossRef]
- R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003). [CrossRef] [PubMed]
- Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]
- M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308(5726), 1296–1298 (2005). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995), pp.66–93.
- H. Kurt and D. S. Citrin, “Annular photonic crystals,” Opt. Express 13(25), 10316–10326 (2005). [CrossRef] [PubMed]
- H. Kurt, R. Hao, Y. Chen, J. Feng, J. Blair, D. P. Gaillot, C. Summers, D. S. Citrin, and Z. Zhou, “Design of annular photonic crystal slabs,” Opt. Lett. 33(14), 1614–1616 (2008). [CrossRef] [PubMed]
- Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large Absolute Band Gap in 2D Anisotropic Photonic Crystals,” Phys. Rev. Lett. 81(12), 2574–2577 (1998). [CrossRef]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995), pp.135–155.
- R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48(11), 8434–8437 (1993). [CrossRef]
- Z. Y. Li, J. Wang, and B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Phys. Rev. B 58(7), 3721–3729 (1998). [CrossRef]
- Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000). [CrossRef]
- M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19(9), 2052–2059 (2002). [CrossRef]
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