## Electromagnetic field structure and normal mode coupling in photonic crystal nanocavities

Optics Express, Vol. 13, Issue 13, pp. 4980-4985 (2005)

http://dx.doi.org/10.1364/OPEX.13.004980

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### Abstract

The electromagnetic field of a high-quality photonic crystal nanocavity is computed using the finite difference time domain method. It is shown that a separatrix occurs in the local energy flux discriminating between predominantly near and far field components. Placing a two-level atom into the cavity leads to characteristic field modifications and normal-mode splitting in the transmission spectra.

© 2005 Optical Society of America

## 1. Introduction

2. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H.M. Gibbs, G. Rupper, C. Ell, O.B. Shchekin, and D.G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature **432**, 200–203 (2004) [CrossRef] [PubMed]

## 2. The structure

3. T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, “High quality two-dimensional photonic crystal slab cavities,” Appl. Phys. Lett. **79**, 4289–4291 (2001). [CrossRef]

*h*=0.348

*µm*) photonic crystal slab of semiconductor material, index

*n*=3.4, patterned with a triangular lattice (lattice constant

*a*=0.464

*µm*) of air columns of radius

*r*=0.275

*a*. Light is confined in the vertical (

*z*) direction by total internal reflection at the interface between the slab and the air clad and in the in-plane direction by the photonic-bandgap effect. The slab is thin, less than the light wavelength, which ensures TE (dominant

*E*

_{x},

*E*

_{y},

*H*

_{z}components) single mode confinement in the vertical direction. The structure does not support a bandgap for TM modes.

*r*=0.275

*a*to

*r*=0.2

*a*at the center of the lattice. This defect forms a nanocavity centered at the defect and is surrounded by six layers of air columns arranged in a hexagonal pattern, see, e.g., Fig. 4. The nanocavity supports a doubly-degenerate pair of donor defect eigenstates that can be separated into

*X*and

*Y*-dipole modes according to the orientation of the dominant electric field component at the center of the defect [3

3. T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, “High quality two-dimensional photonic crystal slab cavities,” Appl. Phys. Lett. **79**, 4289–4291 (2001). [CrossRef]

*Q*of the cavity to ≈ 1500 an additional

*fractional edge dislocation*defect of size 46.5

*nm*is introduced through the central defect air column along the x-direction of the photon crystal lattice [4

4. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E **65**, 016608-1–11 (2001). [CrossRef]

*φ*

_{i}are excited, the system can be described by the Hamiltonian:

*H*=∑

_{i}

*E*

_{i}

*a*

_{i}-∑

_{i,i′}

*h̄*Ω

_{i,i′}(

*t*)

*a*

_{i}

^{′}, the level energies

*E*

_{i}and the Rabi frequency Ω

_{i,i′}(

*t*)=

_{i,i′}·

*t*)/

*h̄*with the optical dipole moment

*µ*

_{i,i}

^{′}have been introduced. Assuming that the extension of the atom is negligible compared to the wavelength of the electromagnetic field, the macroscopic polarization which enters the Maxwell equations can be written as:

_{m}(

*t*,

*δ*(

_{atom})∑

_{i,i′}

_{i,i}〈

*a*

_{i}

_{′}〉.

*a*

_{i}

^{′}〉 can be derived using the Heisenberg equation. Considering only linear excitation of the lowest transition (between levels 1 and 2) the dynamics of the relevant microscopic polarization

*p*=〈

*a*

_{2}〉 is determined by [6]:

*iℏ*

*ṗ*=(

*E*

_{2}-

*E*

_{1})

*p*+Ω

_{1,2}(

*t*)

## 3. The simulation

*E*

_{2}-

*E*

_{1})/

*h*=190

*THz*in agreement with the dominant cavity mode frequency. In order to have strong light-matter coupling effects and to reduce the computational effort, we assume the relatively large dipole moment of

*µ*

_{12}=12.5

*enm*, however from scaling considerations we expect the results presented here to be transferable to similar systems in the strong coupling regime. The atomic dipole moment vector is chosen in the plane of the photonic crystal, parallel to the x-direction.

*nm*is used which translates to 20 points per wavelength in the semiconductor slab. To improve simulation accuracy and offset errors due to the staircasing approximation of the air holes, which possess stronger curvature relative to the wavelength, an effective dielectric constant is computed for each cell on the air-semiconductor boundaries using a volume average of the dielectric constants at a resolution of Δ

*x*/10 [8]. However, the discretized representation should only be considered as an approximation to the geometrically defined structure.

## 4. Results

*X*-dipole mode, the point polarization source is driven by a spectrally narrow pulse centered at the mode frequency and

*x*-polarized. In order to focus on the eigen-modes of the cavity, all measurements are performed after the driving source is turned off and the rapidly decaying components have radiated away. In Fig. 1 we show a plot of the energy density of the electromagnetic field in the x-y and the x-z plane averaged over one period (left) and instantenous with carrier oscillations (right). Within the slab a hexagonal radiation pattern with strongly localized outgoing beams can be seen. Around the cavity a region with predominantly evanescent field components is visible. The far field emissions above the structure do not follow the hexagonal scheme and are much weaker compared to the in-plane losses.

*k*

_{x}|

^{2}+|

*k*

_{y}|

^{2}≤

*ω*

_{0}/

*c*) can propagate into the upper far field, the remaining components only contribute to the near field and form the evanescent (inhomogeneous) field [9]. Figure 3 shows the energy density of the propagating, evanescent and total field along the vertical direction (z) above the center of the structure. Near the structure the evanescent part energetically dominates. However, since it falls of exponentially (note the logarithmic scale) the initially weaker but slower decaying propagating component is already of the same magnitude at a distance of about 1

*µm*≈2/3λ above the structure and clearly dominates the far field.

*µm*above the plane of the photonic crystal, where the presence of the atom increases the energy density up to a factor of five.

*µm*below the structure (transmission direction) for excitation with a plane wave pulse. The spectrum shows, that in comparison to the single mode for the empty cavity, the coupled system exhibits two modes with are spectrally separated by 873 GHz. As expected, the new linewidth is the average of the original cavity mode and atom linewidths.

2. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H.M. Gibbs, G. Rupper, C. Ell, O.B. Shchekin, and D.G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature **432**, 200–203 (2004) [CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgments

## References and links

1. | J.D. Joannopoulos, R.D. Meade, and J.N. Winn, |

2. | T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H.M. Gibbs, G. Rupper, C. Ell, O.B. Shchekin, and D.G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature |

3. | T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, “High quality two-dimensional photonic crystal slab cavities,” Appl. Phys. Lett. |

4. | J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E |

5. | H. Haug and S.W. Koch, |

6. | L. Allen and J.H. Eberly, |

7. | A. Taflove and S.C. Hagness, |

8. | N. Kaneda, B. Houshmand, and T. Itoh, “FDTD Analysis of Dielectric Resonators with Curved Surfaces,” IEEE Trans. on Microwave Theory and Techniques |

9. | L. Mandel and E. Wolf, |

10. | P.R. Berman (Editor), |

11. | Full versions of animations are available at: http://acms.arizona.edu/oe/ |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(020.0020) Atomic and molecular physics : Atomic and molecular physics

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 5, 2005

Revised Manuscript: June 6, 2005

Published: June 27, 2005

**Citation**

C. Dineen, J. Förstner, A. Zakharian, J. Moloney, and S. Koch, "Electromagnetic field structure and normal mode coupling in photonic crystal nanocavities," Opt. Express **13**, 4980-4985 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-13-4980

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### References

- J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995)
- T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H.M. Gibbs, G. Rupper, C. Ell, O.B. Shchekin, and D.G. Deppe, �??Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,�?? Nature 432, 200-203 (2004) [CrossRef] [PubMed]
- T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, �??High quality two-dimensional photonic crystal slab cavities,�?? Appl. Phys. Lett. 79, 4289 - 4291 (2001). [CrossRef]
- J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, �??Design of photonic crystal microcavities for cavity QED,�?? Phys. Rev. E 65, 016608-1-11 (2001). [CrossRef]
- H. Haug and S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors 4th ed., (World Scientific, Singapore, 2004).
- L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms. (Dover, 1975)
- A. Taflove and S.C. Hagness, Computational Electrodynamics: the FDTD method 2nd ed. (Artech House, Boston, London, 2000)
- N. Kaneda, B. Houshmand, and T. Itoh, �??FDTD Analysis of Dielectric Resonators with Curved Surfaces,�?? IEEE Trans. On Microwave Theory and Techniques 45, 1645-1649 (1997).
- L. Mandel, and E. Wolf, Optical coherence and quantum optics (Cambridge Univ. Press, 1995)
- P.R. Berman (Editor), Cavity Quantum Electrodynamics (Academic Press, San Diego, 1994)
- Full versions of animations are available at: <a href="http://acms.arizona.edu/oe/">http://acms.arizona.edu/oe/</a>

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