## Design and investigation of surface addressable photonic crystal cavity confined band edge modes for quantum photonic devices |

Optics Express, Vol. 19, Issue 6, pp. 5014-5025 (2011)

http://dx.doi.org/10.1364/OE.19.005014

Acrobat PDF (1457 KB)

### Abstract

We propose to use a localized Γ-point slow Bloch mode in a 2D-Photonic Crystal (PC) membrane to realize an efficient surface emitting source. This device can be used as a quantum photonic device, e.g. a single photon source. The physical mechanisms to increase the *Q*/*V* factor and to improve the directivity of the PC microcavity rely on a fine tuning of the geometry in the three directions of space. The PC lateral mirrors are first engineered in order to optimize photons confinement. Then, the effect of a Bragg mirror below the 2DPC membrane is investigated in terms of out-of-plane leakages and far field emission pattern. This photonic heterostructure allows for a strong lateral confinement of photons, with a modal volume of a few (*λ*/n)^{3} and a Purcell factor up to 80, as calculated by two different numerical methods. We finally discuss the efficiency of the single photon source for different collection set-up.

© 2011 OSA

## 1. Introduction

1. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim I, “Two-dimensional photonic band-Gap defect mode laser,” Science **284**(5421), 1819–1821 (1999). [CrossRef] [PubMed]

2. H. Y. Ryu, H. G. Park, and Y. H. Lee, “Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. **8**(4), 891–908 (2002). [CrossRef]

3. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. **95**(1), 013904 (2005). [CrossRef] [PubMed]

*F*) which is proportional to the ratio between the quality factor (

_{p}*Q*) and the modal volume (

*V*) of the microcavity resonant mode. Assuming a dipole source with a perfect spatial and spectral matching with the cavity mode, its spontaneous emission rate is increased by the factor

*F*. This effect impacts strongly the efficiency of a SPS which is proportional to the

_{p}*β*factor defined by

*β*represents the proportion of emitted photons into the optical mode of interest.

*Q-*factor [4

4. T. Asano, B. S. Song, and S. Noda, “Analysis of the experimental Q factors (~ 1 million) of photonic crystal nanocavities,” Opt. Express **14**(5), 1996–2002 (2006). [CrossRef] [PubMed]

6. F. Bordas, M. J. Steel, C. Seassal, and A. Rahmani, “Confinement of band-edge modes in a photonic crystal slab,” Opt. Express **15**(17), 10890–10902 (2007). [CrossRef] [PubMed]

7. K. Nozaki and T. Baba, “Laser characteristics with ultimate-small modal volume in photonic crystal slab point-shift nanolasers,” Appl. Phys. Lett. **88**(21), 211101 (2006). [CrossRef]

8. M. Toishi, D. Englund, A. Faraon, and J. Vucković, “High-brightness single photon source from a quantum dot in a directional-emission nanocavity,” Opt. Express **17**(17), 14618–14626 (2009). [CrossRef] [PubMed]

9. J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics **4**(3), 174–177 (2010). [CrossRef]

*Q*PC microcavity [3

3. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. **95**(1), 013904 (2005). [CrossRef] [PubMed]

10. J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. **39**(6), 526 (2003). [CrossRef]

## 2. Design of the basic PC structure

*n*= 3.49 at the target wavelength. The first step is to select a slow light mode located in the center of the first Brillouin zone (Γ-point Bloch modes). For that purpose we use the Plane Wave Expansion method (PWE) developed by the MIT [11

11. S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “MIT Photonic Bands,” http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands.

10. J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. **39**(6), 526 (2003). [CrossRef]

*β*-factor, we must ensure that the QD emitters couple to only one resonant mode. Accordingly, the Γ-point Bloch mode must be non-degenerate and sufficiently spectrally isolated from the other slow Bloch modes. We see in Fig. 1 that two modes satisfy this requirement: the monopolar and hexapolar modes. The spatial distribution intensity of the monopolar mode is mainly located in the material as compared to the intensity of the hexapolar mode: 82% of the intensity is in the GaAs material. This promotes the interaction of the modes with the emitters. On the other hand, the maximum intensity of the hexapolar mode is close to holes sidewalls in the semiconductor zone of the PC: this mode will therefore couple preferentially to QDs close to the holes surface, which promotes unwanted non-radiative recombination.

*A*PC lattice parameter

*A*= 420nm is determined so as to operate around

*λ*~915nm. This corresponds to a distance between the closest holes of 240nm.

*ff*= 37% (defined for the honeycomb structure as

*ff*as high as 50% should be chosen, as it can be shown by calculating band diagrams for different

*ff*; however, such a

*ff*leads to a separation between closest holes of only 20nm. To keep reasonable technological constraints, we limit

*ff*to 37%, which corresponds to a 50 nm separation.

## 3. Laterally confined Γ-point Bloch mode

12. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B **63**(12), 125107 (2001). [CrossRef]

6. F. Bordas, M. J. Steel, C. Seassal, and A. Rahmani, “Confinement of band-edge modes in a photonic crystal slab,” Opt. Express **15**(17), 10890–10902 (2007). [CrossRef] [PubMed]

13. L. Ferrier, P. Rojo-Romeo, E. Drouard, X. Letatre, and P. Viktorovitch, “Slow Bloch mode confinement in 2D photonic crystals for surface operating devices,” Opt. Express **16**(5), 3136–3145 (2008). [CrossRef] [PubMed]

*Q*/

*V*ratio. As the band curvature of the monopolar mode is negative around the Γ-point, the light confinement may be obtained by decreasing the radii of air holes in barrier region: the energy of the barrier band decreases and the photons should stand preferably in the core area, where the holes radii are larger. Such a photonic heterostructure based on the honeycomb structure described above was simulated using the 3D-Finite Difference Time Domain method (FDTD) [14

14. J. Mouette, and L. Carrel., “Tessa Project,” http://alioth.debian.org/projects/tessa/

*ff*

_{core}= 37%). We then simulate five structures, with different lower air hole radii in the cladding region (

*ff*

_{clad}= 36, 35, 33, 31 and 29%). For each structure we determine numerically the

*Q*-factor and the modal volume (

*V*), in order to calculate

_{m}*F*. To determine the spectrum of the PC cavity, a dipole source is located inside the middle-plane membrane; a narrow temporal Gaussian pulse is emitted in order to probe the PC structure over a wide spectral range. Figure 3 shows a typical spectrum of the photonic heterostructure, obtained from a Fourier transform of the field at a specific location in the structure. The resonance peaks are associated to different Γ-point Bloch modes (namely monopolar and hexapolar). For the monopolar mode, harmonics are observed which are due to the lateral confinement.

_{p}*V*is calculated by integrating the electric energy density over the FDTD simulation workspace:

_{m}*F*is calculated using the well known formula:

_{p}15. Y. Xu, J. S. Vučković, R. K. Lee, O. J. Painter, A. Scherer, and A. Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B **16**(3), 465 (1999). [CrossRef]

*F*, the dipole must be located at an electric field antinode, and collinearly oriented with respect to the electric field polarization), and we calculate the total flux of Poynting vector through the total area of the workspace. A similar simulation is performed in the case of a dipole in a uniform GaAs material. In this work, both methods have been used and the discrepancy between the obtained Purcell factors is found to be less than 5%.

_{p}*Q*-factor, the mode volume

*V*and the resulting Purcell factor are calculated as a function of the barrier filling factor (see Fig. 4(a) and 4(b)). For small

_{m}*ff*

_{bar}, the

*Q*factor increases when the filling factor difference decreases (see Fig. 4(b)), indicating reduced scattering losses at the microcavity boundaries [6

6. F. Bordas, M. J. Steel, C. Seassal, and A. Rahmani, “Confinement of band-edge modes in a photonic crystal slab,” Opt. Express **15**(17), 10890–10902 (2007). [CrossRef] [PubMed]

*ff*

_{bar}tends to

*ff*

_{core}, lateral losses through the barrier become dominant and the

*Q*-factor is decreased. The lateral confinement process is well illustrated by the evolution of

*V*with

_{m}*ff*

_{bar.}: a large filling factor difference leads to a large photonics “band offset” and then to a better spatial confinement. Finally, a maximum Purcell factor

*F*~24 is reached for

_{p}*ff*

_{bar}= 31%, with

*Q*~1000 and

*Q*-factor, indicating that optical tunneling through the barrier is negligible. This indicates that the main losses are out of plane losses through scattering at the interfaces between the core area of the PC and cladding areas, and within the core area of the PC. It is worthwhile to notice that

*V*is significantly smaller than the physical volume of the cavity (

_{m}## 4. Control of the vertical and lateral losses of the Confined Bloch Mode

*Q*-factor is mainly controlled by the coupling between the CBM and the radiative continuum. This coupling can be controlled by an accurate design of the electromagnetic environment of the PC membrane in the vertical direction. As it has been shown in previous works [13

13. L. Ferrier, P. Rojo-Romeo, E. Drouard, X. Letatre, and P. Viktorovitch, “Slow Bloch mode confinement in 2D photonic crystals for surface operating devices,” Opt. Express **16**(5), 3136–3145 (2008). [CrossRef] [PubMed]

16. B. Ben Bakir, C. Seassal, X. Letartre, P. Viktorovitch, M. Zussy, L. Di Cioccio, and J. M. Fedeli, “Surface-emitting microlaser combining two-dimensional photonic crystal membrane and vertical Bragg mirror,” Appl. Phys. Lett. **88**(8), 081113 (2006). [CrossRef]

*Q*-factor.

*F*~24), is positioned above a Bragg mirror composed by three pairs of

_{p}_{2}layers, with a reflectivity around 98% at 915nm (Fig. 5 ). The first advantage of this configuration is to direct photons only upwards. Moreover, as it can be shown in Fig. 6(a) , the

*Q*-factor can be increased to about 2100 (instead of ~1000 without Bragg mirror) for an air gap

*a*between the PC membrane and the mirror of about 830nm. In this configuration, we obtain

_{g}*F*~44.

_{p}*V*can be explained by considering that the field is spread across the PC membrane and the air gap located below. To be more precise, it can be shown [17

_{m}17. P. Viktorovitch, B. Ben Bakir, S. Boutami, J. L. Leclercq, X. Letartre, P. Rojo-Romeo, C. Seassal, M. Zussy, L. Di Cioccio, and J. M. Fedeli, “3D harnessing of light with 2.5D photonic crystals,” Laser Photon. Rev. **4**(3), 401–413 (2010). [CrossRef]

*η*Eq. (1) is given by the ratio between the times spent by photons respectively in the air gap and in the PC membrane, providing the relation:where

_{gap}*η*~2%. Considering a PC membrane (resp. an air gap) thickness of 135nm (resp. 830nm), the increase of the effective mode thickness can be estimated to about 10%, in agreement with the value obtained by direct numerical calculation of the modal volume.

_{gap}16. B. Ben Bakir, C. Seassal, X. Letartre, P. Viktorovitch, M. Zussy, L. Di Cioccio, and J. M. Fedeli, “Surface-emitting microlaser combining two-dimensional photonic crystal membrane and vertical Bragg mirror,” Appl. Phys. Lett. **88**(8), 081113 (2006). [CrossRef]

*Q*-factor for a gap size

*Q*-factor dependence on the air-gap thickness, for GaAs membranes with three different thicknesses: 130, 150 and 180nm.

*Q*~2500 is found for a 150nm thick membrane, resulting in an increased Purcell factor

*F*= 52. Moreover, a quasi symmetric evolution of the

_{p}*Q*-factor around this maximum is observed. Assuming that this thickness is optically “

*λ*/2”, we deduce an average refractive index of 3.05 for the PC membrane. A similar value is obtained by weighting the refractive indices of GaAs and air by the spatial profile of the electromagnetic energy.

*Q*-factor, indicating a strong inhibition of radiative coupling. We can now guess that optical losses may be, to a certain extent, determined by the lateral tunneling through the barrier. To check this hypothesis, we increase the number of rows of graphite unit cell in the barrier region from 5 to 11 (for more than 11 rows the Q remains constant). We achieve a maximum quality factor of

*Q*~2950 for the 130nm membrane’s thickness; this increase indicates that the lateral losses were not completely inhibited. For a 150nm membrane’s thickness (optically “

*λ*/2”), the quality factor finally reaches

*Q*~3800 leading to

*F*= 80.

_{p}## 5. Far-field pattern of the surface emitting Confined Bloch Mode

18. D. Englund, I. Fushman, and J. Vucković, “General recipe for designing photonic crystal cavities,” Opt. Express **13**(16), 5961–5975 (2005). [CrossRef] [PubMed]

19. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express **12**(8), 1562–1568 (2004). [CrossRef] [PubMed]

20. L. Ferrier, O. El Daif, X. Letartre, P. Rojo Romeo, C. Seassal, R. Mazurczyk, and P. Viktorovitch, “Surface emitting microlaser based on 2D photonic crystal rod lattices,” Opt. Express **17**(12), 9780–9788 (2009). [CrossRef] [PubMed]

*Q*-factor (3

*λ*/4 equivalent optical thickness), the losses around the gamma point are inhibited and the relative weight of the losses away from the vertical direction is then increased, leading to a less directional emission pattern. On the contrary, for a distance corresponding to a minimum

*Q*factor (

*λ*/2 equivalent optical thickness), the FFD is very similar to those of the CBM without Bragg mirror. More specifically, the function

*I(θ)*, where

*I(θ)*d

*θ*is the emitted power between the cones of angle

*θ*and

*θ + dθ*, is plotted on Fig. 9 for 3 configurations: the 150nm thick PC membrane (a) without Bragg mirror (

*F*26), (b) with a Bragg mirror located for constructive interference (

_{p}~*F*~10), (c) with a Bragg mirror located for destructive interference (

_{p}*F*~80). It can be seen that, with this approach, a trade-off has to be found between the Purcell factor and the directivity of the device.

_{p}*η*) can be defined as the product of the

*β*factor, which corresponds to the fraction of the dipole emission that couples to the mode of interest, by the collection efficiency (

*γ*) of an optical device (e.g. an objective lens) with a numerical aperture

*NA*. Then

*η*can be expressed as Eq. (2):For the three configurations, the efficiency of the device (

*η*) is plotted as a function of the

*NA*of the collection (Fig. 9(b)). We thus see that the optimum configuration will depend heavily on the collection mechanism. For a single-mode optical fiber with a

*NA*~0.2, only the solution with Bragg mirror in constructive interference is suitable, achieving a moderate efficiency of about 35%. We expect a better collection with a more sophisticated optical system such as a lensed fiber or a spherical lens with, e.g., a numerical aperture of ~0.95. In this case, a very high efficiency (

*η*~0.99) can be reached with the

*F*-optimized structure. This value can be compared with those obtained for different devices proposed in the literature:

_{p}- - for a PhC microcavity, TRAN et al. [21] get, if considering backward and upward collections, comparable (resp. three time lower) efficiencies for large (resp. small) NA .
21. N. Tran, S. Combrié, and A. De Rossi, “Directive emission from high-

*Q*photonic crystal cavities through band folding,” Phys. Rev. B**79**(4), 041101 (2009). [CrossRef] - - “patch” nano-antennas [22] demonstrate, for
22. R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett.

**104**(2), 026802 (2010). [CrossRef] [PubMed]*NA*= 0.95, an efficiency of 0.7, instead of 0.99 in our study (case of destructive interferences). - - vertical nanowires [9] exhibit, for
9. J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics

**4**(3), 174–177 (2010). [CrossRef]*NA*= 0.75, a theoretical efficiency of 0.95, instead of 0.90 in this paper (both for destructive or constructive interferences)

## 6. Conclusion

## Appendix

23. X. Letartre, J. Mouette, J. L. Leclercq, P. R. Romeo, C. Seassal, and P. Viktorovitch, “Switching devices with spatial and spectral resolution combining photonic crystal and MOEMS structures,” J. Lightwave Technol. **21**(7), 1691–1699 (2003). [CrossRef]

*ω*and its coupling to vertical plane wave

_{0}*1/τ*(

*Q*=

*ω*). Then, the complex resonant frequency of this mode is:

_{0}τ*r*and

_{u}*r*are the reflectivity seen from the PCM at its interface:

_{d}*n*is the effective optical index of the PCM.

_{PCM}*r*is the reflectivity of the Bragg mirror.

_{bragg}*φ*(resp.

_{PCM}*φ*) are the dephasing of a vertical plane wave through the PCM (resp. the spacer). They depends on the optical frequency and equation (Eq. (A1)) has to be solved numerically.

_{S}*ℓ*,

_{PCM}*ℓ*) leads to the following results:

_{S}## Acknowledgement

## References and links

1. | O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim I, “Two-dimensional photonic band-Gap defect mode laser,” Science |

2. | H. Y. Ryu, H. G. Park, and Y. H. Lee, “Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. |

3. | D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. |

4. | T. Asano, B. S. Song, and S. Noda, “Analysis of the experimental Q factors (~ 1 million) of photonic crystal nanocavities,” Opt. Express |

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

6. | F. Bordas, M. J. Steel, C. Seassal, and A. Rahmani, “Confinement of band-edge modes in a photonic crystal slab,” Opt. Express |

7. | K. Nozaki and T. Baba, “Laser characteristics with ultimate-small modal volume in photonic crystal slab point-shift nanolasers,” Appl. Phys. Lett. |

8. | M. Toishi, D. Englund, A. Faraon, and J. Vucković, “High-brightness single photon source from a quantum dot in a directional-emission nanocavity,” Opt. Express |

9. | J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics |

10. | J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. |

11. | S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “MIT Photonic Bands,” http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands. |

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

13. | L. Ferrier, P. Rojo-Romeo, E. Drouard, X. Letatre, and P. Viktorovitch, “Slow Bloch mode confinement in 2D photonic crystals for surface operating devices,” Opt. Express |

14. | J. Mouette, and L. Carrel., “Tessa Project,” http://alioth.debian.org/projects/tessa/ |

15. | Y. Xu, J. S. Vučković, R. K. Lee, O. J. Painter, A. Scherer, and A. Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B |

16. | B. Ben Bakir, C. Seassal, X. Letartre, P. Viktorovitch, M. Zussy, L. Di Cioccio, and J. M. Fedeli, “Surface-emitting microlaser combining two-dimensional photonic crystal membrane and vertical Bragg mirror,” Appl. Phys. Lett. |

17. | P. Viktorovitch, B. Ben Bakir, S. Boutami, J. L. Leclercq, X. Letartre, P. Rojo-Romeo, C. Seassal, M. Zussy, L. Di Cioccio, and J. M. Fedeli, “3D harnessing of light with 2.5D photonic crystals,” Laser Photon. Rev. |

18. | D. Englund, I. Fushman, and J. Vucković, “General recipe for designing photonic crystal cavities,” Opt. Express |

19. | D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express |

20. | L. Ferrier, O. El Daif, X. Letartre, P. Rojo Romeo, C. Seassal, R. Mazurczyk, and P. Viktorovitch, “Surface emitting microlaser based on 2D photonic crystal rod lattices,” Opt. Express |

21. | N. Tran, S. Combrié, and A. De Rossi, “Directive emission from high- |

22. | R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. |

23. | X. Letartre, J. Mouette, J. L. Leclercq, P. R. Romeo, C. Seassal, and P. Viktorovitch, “Switching devices with spatial and spectral resolution combining photonic crystal and MOEMS structures,” J. Lightwave Technol. |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(270.5580) Quantum optics : Quantum electrodynamics

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: September 30, 2010

Revised Manuscript: January 5, 2011

Manuscript Accepted: February 20, 2011

Published: March 2, 2011

**Citation**

P. Nedel, X. Letartre, C. Seassal, Alexia Auffèves, L. Ferrier, E. Drouard, A. Rahmani, and P. Viktorovitch, "Design and investigation of surface addressable photonic crystal cavity confined band edge modes for quantum photonic devices," Opt. Express **19**, 5014-5025 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5014

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### References

- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999). [CrossRef] [PubMed]
- H. Y. Ryu, H. G. Park, and Y. H. Lee, “Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. 8(4), 891–908 (2002). [CrossRef]
- D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005). [CrossRef] [PubMed]
- T. Asano, B. S. Song, and S. Noda, “Analysis of the experimental Q factors (~ 1 million) of photonic crystal nanocavities,” Opt. Express 14(5), 1996–2002 (2006). [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]
- F. Bordas, M. J. Steel, C. Seassal, and A. Rahmani, “Confinement of band-edge modes in a photonic crystal slab,” Opt. Express 15(17), 10890–10902 (2007). [CrossRef] [PubMed]
- K. Nozaki and T. Baba, “Laser characteristics with ultimate-small modal volume in photonic crystal slab point-shift nanolasers,” Appl. Phys. Lett. 88(21), 211101 (2006). [CrossRef]
- M. Toishi, D. Englund, A. Faraon, and J. Vucković, “High-brightness single photon source from a quantum dot in a directional-emission nanocavity,” Opt. Express 17(17), 14618–14626 (2009). [CrossRef] [PubMed]
- J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4(3), 174–177 (2010). [CrossRef]
- J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526 (2003). [CrossRef]
- S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “MIT Photonic Bands,” http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands .
- T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001). [CrossRef]
- L. Ferrier, P. Rojo-Romeo, E. Drouard, X. Letatre, and P. Viktorovitch, “Slow Bloch mode confinement in 2D photonic crystals for surface operating devices,” Opt. Express 16(5), 3136–3145 (2008). [CrossRef] [PubMed]
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