## Efficient analysis of mode profiles in elliptical microcavity using dynamic-thermal electron-quantum medium FDTD method |

Optics Express, Vol. 21, Issue 5, pp. 5910-5923 (2013)

http://dx.doi.org/10.1364/OE.21.005910

Acrobat PDF (1344 KB)

### Abstract

The dynamic-thermal electron-quantum medium finite-difference time-domain (DTEQM-FDTD) method is used for efficient analysis of mode profile in elliptical microcavity. The resonance peak of the elliptical microcavity is studied by varying the length ratio. It is observed that at some length ratios, cavity mode is excited instead of whispering gallery mode. This depicts that mode profiles are length ratio dependent. Through the implementation of the DTEQM-FDTD on graphic processing unit (GPU), the simulation time is reduced by 300 times as compared to the CPU. This leads to an efficient optimization approach to design microcavity lasers for wide range of applications in photonic integrated circuits.

© 2013 OSA

## 1. Introduction

1. J. Rezac and A. Rosenberger, “Locking a microsphere whispering-gallery mode to a laser,” Opt. Express **8**(11), 605–610 (2001). [CrossRef] [PubMed]

4. R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk laser,” Appl. Phys. Lett. **63**(10), 1310–1312 (1993). [CrossRef]

6. S. M. Hsu and H. C. Chang, “Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy,” Opt. Express **15**(24), 15797–15811 (2007). [CrossRef] [PubMed]

9. I. Ahmed, E. H. Khoo, and E. P. Li, “Development of the CPML for three-dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antenn. Propag. **58**(3), 832–837 (2010). [CrossRef]

11. I. Ahmed, E. K. Chua, E. P. Li, and Z. Chen, “Development of the three dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antenn. Propag. **56**(11), 3596–3600 (2008). [CrossRef]

12. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. **14**(3), 302–307 (1966). [CrossRef]

## 2. Theoretical model: DTEQM-FDTD

17. Y. Huang and S. T. Ho, “Computational model of solid-state, molecular, or atomic media for FDTD simulation based on a multi-level multi-electron system governed by Pauli exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,” Opt. Express **14**(8), 3569–3587 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-8-3569. [CrossRef] [PubMed]

18. E. H. Khoo, S. T. Ho, I. Ahmed, E. P. Li, and Y. Huang, “Light energy extraction from the minor surface arc of an electrically pumped elliptical microcavity laser,” IEEE J. Quantum Electron. **46**(1), 128–136 (2010). [CrossRef]

20. E. H. Khoo, I. Ahmed, and E. P. Li, “Enhancement of light energy extraction from elliptical microcavity using external magnetic field for switching applications,” Appl. Phys. Lett. **95**(12), 121104 (2009). [CrossRef]

21. I. Ahmed, E. H. Khoo, O. Kurniawan, and E. P. Li, “Modeling and simulation of plasmonic with FDTD method by using solid state and Lorentz -Drude dispersion model,” J. Opt. Soc. Am. B **28**(3), 352–359 (2011). [CrossRef]

*P*are written as For three dimensional (3D) case, the electric field equations are given as

*P*(where

_{i,r}*r*represent the

*x*,

*y*and

*z*components) that is different from the conventional electric field equations for the FDTD method. The parameter,

*i*represent the number of electron energy levels and

*M*is the maximum number of energy levels considered in the simulation. The energy level separation is represented by Δλ and is defined according to the number of energy level defined in the DTEQM-FDTD algorithm. For the magnetic field, the equations are identical to the equations used in conventional FDTD because the material is non-magnetic and is not affected by any external factors. However, if the magnetic properties of the material are affected by external magnetic field, then the magnetization density would need to be considered in the FDTD algorithm to model the effect of magnetic torque [24

24. R. Shams and P. Sadeghi, “On optimization of finite-difference time-domain (FDTD) computation on heterogeneous and GPU clusters,” J. Parallel Distrib. Comput. **71**(4), 584–593 (2011). [CrossRef]

*x*,

*y*, and

*z*directions are defined in [17

17. Y. Huang and S. T. Ho, “Computational model of solid-state, molecular, or atomic media for FDTD simulation based on a multi-level multi-electron system governed by Pauli exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,” Opt. Express **14**(8), 3569–3587 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-8-3569. [CrossRef] [PubMed]

18. E. H. Khoo, S. T. Ho, I. Ahmed, E. P. Li, and Y. Huang, “Light energy extraction from the minor surface arc of an electrically pumped elliptical microcavity laser,” IEEE J. Quantum Electron. **46**(1), 128–136 (2010). [CrossRef]

*μ*is the atomic dipole moment of electrons involved in the interband transition between the

_{ir}*i*th level in the conduction and valence bands for

*x*,

*y*and

*z*directions. The expression for the atomic dipole moment

*μ*, is given by

_{ir}*is the interband transition time between the*

_{i}*i*th energy level. The interband frequency is denoted by ω

_{αi}. The parameter

*A*is the vector potential, where r =

_{r}*x*,

*y*or

*z*direction and

*γ*is atomic dipole dephrasing time at

_{i}*i*th energy level after excitation. The polarization density,

*P*is dependent on the atomic properties of the medium. It relates the quantized electromagnetic field and atomic dynamics.

_{ir}*i*th energy level in the conduction band and valence band respectively is defined by

*N*and

_{Ci}*N*. The rate equation [17

_{Vi}17. Y. Huang and S. T. Ho, “Computational model of solid-state, molecular, or atomic media for FDTD simulation based on a multi-level multi-electron system governed by Pauli exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,” Opt. Express **14**(8), 3569–3587 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-8-3569. [CrossRef] [PubMed]

*represent the intraband transition time within the energy levels in conduction and valence band respectively. For the interband, the electron density that is being transferred from the valence to conduction band is given bywhere W*

_{C/V}_{pump}is the electron pumping rate to the active semiconductor media. The carrier density at the initial energy level is given by

## 3. GPU implementation for DTEQM-FDTD

24. R. Shams and P. Sadeghi, “On optimization of finite-difference time-domain (FDTD) computation on heterogeneous and GPU clusters,” J. Parallel Distrib. Comput. **71**(4), 584–593 (2011). [CrossRef]

## 4. Design, simulation and results of DTEQM-FDTD implemented on GPU

### 4.1 Elliptical microcavity parameters and setup

_{air}= 1. An optical pulse is injected into the elliptical microcavity to generate resonance for lasing [17

**14**(8), 3569–3587 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-8-3569. [CrossRef] [PubMed]

18. E. H. Khoo, S. T. Ho, I. Ahmed, E. P. Li, and Y. Huang, “Light energy extraction from the minor surface arc of an electrically pumped elliptical microcavity laser,” IEEE J. Quantum Electron. **46**(1), 128–136 (2010). [CrossRef]

20. E. H. Khoo, I. Ahmed, and E. P. Li, “Enhancement of light energy extraction from elliptical microcavity using external magnetic field for switching applications,” Appl. Phys. Lett. **95**(12), 121104 (2009). [CrossRef]

*i*-states in the conduction and valence band is then defined. There are infinite numbers of energy level

*i*-states in the conduction and valence band due to the large number of atoms in the semiconductor structure. For more practical modeling, the infinite energy levels are grouped together due to short dipole dephrasing time of the electron as compared to the decay lifetime of the photon. In this paper, 20 energy levels are defined in the conduction and valence band for sufficient modeling accuracy. The energy level separation is set to Δλ = 20 nm. The energy discrete levels (in electron volt units) are given by the Eq., Ei = 1.65 + 0.129*

*i*(eV), where

*i*is the

*i*th energy level and

*i*= 1, 2, 3…..20, for 20 energy levels in the conduction and valence band. The value of 1.65 eV for E

_{i}comes from the designated resonance wavelength at 752 nm. The interband transition wavelength between the conduction and valence bands is 850 nm. The effective mass of the electrons and holes in the valence and conduction band are given as 0.047 m

_{e}and 0.36 m

_{e}, where m

_{e}is the free electron mass. This effective mass is obtained by considering the internal atomics binding and repulsion forces.

_{c(i)}to E

_{c(i-1)}) in the conduction and valence band is given as 1 picosecond (ps) and 100 femtosecond (fs) respectively. The upward intraband transition time can be obtained from the Fermi-Dirac thermalization steady state equations and is given as [25

25. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics **1**(8), 449–458 (2007). [CrossRef]

26. W. Fang, J. Y. Xu, A. Yamilov, H. Cao, Y. Ma, S. T. Ho, and G. S. Solomon, “Large enhancement of spontaneous emission rates of InAs quantum dots in GaAs microdisks,” Opt. Lett. **27**(11), 948–950 (2002). [CrossRef] [PubMed]

^{9}/s. For electron dynamic model in this paper, the spontanenous noise equations are not considered. Hence, a short optical pulse is manually added into the computational algorithm to excite the whispering gallery mode in the elliptical microcavity to instigate lasing. The excitation pulse is a 18 fs plane wave.

### 4.2 Field distribution and resonance spectrum for elliptical microcavity

*z*-direction,

*E*is shown in Fig. 3(a) . The

_{z}*E*field distribution is observed at the middle of the three dimensional elliptical microcavity by slicing along the

_{z}*x*-

*y*plane at thickness of 100 nm from the top. Figures 3(b) and 3(c) shows the

*E*field distribution by slicing along the

_{z}*x-z*and

*y-z*direction at the major and minor length path. The whispering gallery mode [27

27. Y. Kogami, Y. Tomabechi, and K. Matsumura, “Resonance characteristic of whispering-gallery mode in an elliptic disk resonator,” IEEE Trans. Microw. Theory Tech. **44**(3), 473–475 (1996). [CrossRef]

*E*field distributions and the resonance spectrum match very well with the results that are obtained from CPU simulation. This supports the facts that GPU implementation is correct and accurate.

_{z}*E*field distribution in Figs. 3(a)-3(c) and the spectrum in Fig. 4 are obtained at fixed semi-major and semi-minor lengths of 400 and 200 nm respectively. In the next section, the semi-major and semi-minor lengths are varied separately to investigate the effect on the resonance wavelength and electric field distribution in the elliptical microcavity.

_{z}### 4.3 Case study 1: Varying the semi-minor length and its effects on wavelength shift and the electric field distribution

*L*, is defined as the ratio of the semi-major length to semi-minor length and is used to plot the results obtained from the simulation. The length ratio,

_{r}*L*is varied between 1.6 and 2.8 (i.e. the minor length is varied between 142 and 250). In Fig. 5(a) , the spectra for selected length ratio, L

_{r}_{r}is plotted on CPU and GPU. The plot in Fig. 5(b) shows the resonance peak wavelength for both CPU and GPU simulation. It is observed that the respective spectra and the resonance peak wavelength obtained from CPU and GPU agreed very well. In Fig. 5(a), the spectra for different L

_{r}1.739, 1.818 and 1.86 is shown for both CPU and GPU. The line and scatter symbol represent CPU and GPU implementation respectively and both results match very well.

_{r}of the elliptical microcavity. The decrease in length ratio results in the increase of size of the elliptical microcavity, which increases optical path length of the whispering gallery mode. Hence, the resonance wavelength increases because longer wavelength can be supported in the larger microcavity. It also satisfies the condition of the appearance of whispering gallery mode. In general, the resonance peak is red shifted with decrease in L

_{r}(or increase in semi-minor length). However, there are two distinguish points in Figs. 5(a) and 5(b), at which the wavelength is blue shifted significantly. At the length ratio of 1.818 or minor length of 220 nm, the resonance wavelength drops to 568 nm. Similarly, at length ratio of 2.5, the resonance wavelength drops to 591 nm. In order to investigate this abnormal drop in resonance wavelength, we plot the only resonance peak wavelength at selected length ratios as shown in Fig. 5(b). It is shown clearly that the resonance peak wavelength steadily increases with decrease in the length ratio except for L

_{r}values at 1.818 and 2.5.

_{r}= 1.86, 1.818 and 1.739. The electric field (

*E*) distribution in the x-y plane is plotted for the length ratio of 1.818 as shown in Fig. 6(b) , whereas for the neighboring length ratios 1.86 and 1.739 is also plotted in Figs. 6(a) and 6(c) respectively to demonstrate the difference in resonance wavelengths. Figures 6(a) and 6(c) show the whispering gallery mode in the elliptical microcavity, where field is mainly found on the outer part of the micorcavity and there is no field presence at the centre. However, the E

_{z}_{z}field distribution in Fig. 6(b) shows presence of more than one mode in the elliptical microcavity. It is observed that the field distribution consists of mixture of fundamental whispering gallery and cavity modes. The fields are present on the circumference as well as on the internal area of the elliptical microcavity. The whispering gallery mode in the elliptical microcavity is much weaker whereas the cavity mode is stronger. The cavity mode is the peak resonance mode as shown in Fig. 5(a). It has lower resonance wavelength compared to the whispering gallery mode. This is the reason why lower resonance wavelength is observed. If there is only whispering gallery mode, then the field distribution would be similar to that in Figs. 6(a) or 6(c). Since the cavity mode dominates in the elliptical microcavityat 568 nm wavelength, therefore, a change in the resonance wavelength is detected. This explains the blue shift at length ratio, L

_{r}= 1.818. The second blue-shift in the wavelength peak occurs when the length ratio is 2.5. The

*E*field distribution at length ratio of 2.5 and 2.353 is shown in Figs. 7(a) and 7(b) respectively. It is observed that in Fig. 7(a), the

_{z}*E*field distribution consists of mixture of cavity and whispering gallery modes. Again the cavity mode is a dominate mode and it causes a blue shift in the resonance peak.

_{z}### 4.4 Case study 2: Varying the semi-major length and its effects on wavelength shift and the electric field distribution

_{r}from 1.75 to 2.5 (between 350 to 500 nm), while the semi-minor length is fixed at 200 nm. Figure 8(a) shows the wavelength spectra for different length ratios by varying the semi-major length. Figure 8(b) shows the different resonance peaks for different length ratios. It is observed that the peak resonance wavelength varies with the length ratio in a “stair-step” manner. For example, the peak resonance wavelength increases from length ratio of 1.75 to 1.85. Then it drops to wavelength of 632 nm at L

_{r}= 1.9. This trend continues until the length ratio of 2.2. After L

_{r}= 2.2, the resonance wavelength trend increases up to length ratio of 2.5.

_{r}= 2.05, 2.1 and 2.15 are selected to investigate the drop in resonance peak wavelength. The

*E*field distribution pattern at selected L

_{z}_{r}is shown in Figs. 9(a) -9(c) respectively. The

*E*field distribution in Fig. 9(a) is for L

_{z}_{r}= 2.05 and the whispering gallery mode is present in the elliptical microcavity. However, Fig. 9(b) shows that the cavity mode is excited along the

*x*axis and oscillating in the elliptical microcavity. This cavity mode is the dominate mode, hence this explain the blue shift in resonance wavelength. As the length ratio increases to 2.15, the whispering gallery mode is excited again as shown in Fig. 9(c).

## 5. Simulation speed improvement with GPU

## 6. Conclusions

_{r}of elliptical microcavity were varied and some changes in the reasonance wavelength were observed. These include shifting of resonance peak wavelength at some particular length ratios, increasing the complexity of the mode profiles with different modes existing together, and mixing of the different modal field distribution in the elliptical microcavity. At few selected L

_{r}values, the cavity mode was excited and becomes dominate mode. As a result, sharp blue shift in the resonance wavelength was observed. At other values of L

_{r}, only whispering gallery mode was observed. GPU based DTEQM-FDTD approach drastically reduced the simulation time by approximately 300 times as compared to CPU. This achievement also results in more efficient and comprehensive study and understanding of the mode properties in the elliptical microcavity at different length ratios. This approach is believed to be useful for the simulation of large complex optical system with different complex materials. It also encourages the inclusion of more complex physical theories to the DTEQM-FDTD method for higher accuracy with minimal demand of computational resource. It is believed that the achievement in this paper is useful for simulation of nanophotonics devices such as surface plasmonic waveguides, optical antennae, nanolaser and exciton-plasmon interaction systems by changing the material conditions and formulations.

## Acknowledgment

## References and links

1. | J. Rezac and A. Rosenberger, “Locking a microsphere whispering-gallery mode to a laser,” Opt. Express |

2. | M. Ohtsu, |

3. | S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk laser,” Appl. Phys. Lett. |

4. | R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk laser,” Appl. Phys. Lett. |

5. | J. D. Jackson, |

6. | S. M. Hsu and H. C. Chang, “Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy,” Opt. Express |

7. | Z. H. Liu, E. K. Chua, and K. Y. See, “Accurate and efficient evaluation of method of moments matrix based on a generalized analytical approach,” PIERS |

8. | G. Strang and G. Fix, |

9. | I. Ahmed, E. H. Khoo, and E. P. Li, “Development of the CPML for three-dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antenn. Propag. |

10. | F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guided Wave Lett. |

11. | I. Ahmed, E. K. Chua, E. P. Li, and Z. Chen, “Development of the three dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antenn. Propag. |

12. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. |

13. | S. E. Krakiwsky, L. E. Turner, and M. M. Okoniewski, “Acceleration of finite different time domain (FDTD) using graphics processor units (GPU),” IEEE Int. Microw. Sym. Digest |

14. | S. Adam, J. Payne, and R. Boppana, “Finite different time domain (FDTD) simulations using graphics processor,” Proceedings of the Department of Defense High Performance Computing Modernization Program Users Group Conference, 334–338 (2007). |

15. | R. Sypek, A. Dziekonski, and M. Mrozowski, “How to render FDTD computations more effective using a graphics accelerator,” IEEE Trans. Magn. |

16. | K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude model on GPU for plasmonics applications,” PIERS |

17. | Y. Huang and S. T. Ho, “Computational model of solid-state, molecular, or atomic media for FDTD simulation based on a multi-level multi-electron system governed by Pauli exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,” Opt. Express |

18. | E. H. Khoo, S. T. Ho, I. Ahmed, E. P. Li, and Y. Huang, “Light energy extraction from the minor surface arc of an electrically pumped elliptical microcavity laser,” IEEE J. Quantum Electron. |

19. | R. K. Chang and A. J. Campillo, |

20. | E. H. Khoo, I. Ahmed, and E. P. Li, “Enhancement of light energy extraction from elliptical microcavity using external magnetic field for switching applications,” Appl. Phys. Lett. |

21. | I. Ahmed, E. H. Khoo, O. Kurniawan, and E. P. Li, “Modeling and simulation of plasmonic with FDTD method by using solid state and Lorentz -Drude dispersion model,” J. Opt. Soc. Am. B |

22. | E. H. Khoo, I. Ahmed, and E. P. Li, “Investigation of light energy extraction efficiency using surface plasmonics in electrically pumped semiconductor microcavity,” Proc. SPIE |

23. | O. Kurniawan, I. Ahmed, and E. P. Li, “Generation of surface plasmon polariton using plasmonic resonant cavity based on microdisk laser,” IEEE Photon. J. |

24. | R. Shams and P. Sadeghi, “On optimization of finite-difference time-domain (FDTD) computation on heterogeneous and GPU clusters,” J. Parallel Distrib. Comput. |

25. | S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics |

26. | W. Fang, J. Y. Xu, A. Yamilov, H. Cao, Y. Ma, S. T. Ho, and G. S. Solomon, “Large enhancement of spontaneous emission rates of InAs quantum dots in GaAs microdisks,” Opt. Lett. |

27. | Y. Kogami, Y. Tomabechi, and K. Matsumura, “Resonance characteristic of whispering-gallery mode in an elliptic disk resonator,” IEEE Trans. Microw. Theory Tech. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(200.4960) Optics in computing : Parallel processing

(140.3948) Lasers and laser optics : Microcavity devices

(250.5960) Optoelectronics : Semiconductor lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: August 31, 2012

Revised Manuscript: October 25, 2012

Manuscript Accepted: October 25, 2012

Published: March 4, 2013

**Citation**

E. H. Khoo, I. Ahmed, R. S. M. Goh, K. H. Lee, T. G. G. Hung, and E. P. Li, "Efficient analysis of mode profiles in elliptical microcavity using dynamic-thermal electron-quantum medium FDTD method," Opt. Express **21**, 5910-5923 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5910

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### References

- J. Rezac and A. Rosenberger, “Locking a microsphere whispering-gallery mode to a laser,” Opt. Express8(11), 605–610 (2001). [CrossRef] [PubMed]
- M. Ohtsu, Principles of Nanophotonics (CRC Press/Taylor & Francis NW, 2008).
- S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk laser,” Appl. Phys. Lett.60(3), 289–291 (1992). [CrossRef]
- R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk laser,” Appl. Phys. Lett.63(10), 1310–1312 (1993). [CrossRef]
- J. D. Jackson, Classical Electrodynamic (Wiley Press, 1999).
- S. M. Hsu and H. C. Chang, “Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy,” Opt. Express15(24), 15797–15811 (2007). [CrossRef] [PubMed]
- Z. H. Liu, E. K. Chua, and K. Y. See, “Accurate and efficient evaluation of method of moments matrix based on a generalized analytical approach,” PIERS94, 367–382 (2009). [CrossRef]
- G. Strang and G. Fix, An Analysis of The Finite Element Method (Prentice Hall Press, 1973).
- I. Ahmed, E. H. Khoo, and E. P. Li, “Development of the CPML for three-dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antenn. Propag.58(3), 832–837 (2010). [CrossRef]
- F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guided Wave Lett.9(11), 441–443 (1999). [CrossRef]
- I. Ahmed, E. K. Chua, E. P. Li, and Z. Chen, “Development of the three dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antenn. Propag.56(11), 3596–3600 (2008). [CrossRef]
- K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag.14(3), 302–307 (1966). [CrossRef]
- S. E. Krakiwsky, L. E. Turner, and M. M. Okoniewski, “Acceleration of finite different time domain (FDTD) using graphics processor units (GPU),” IEEE Int. Microw. Sym. Digest2, 1033–1036 (2004).
- S. Adam, J. Payne, and R. Boppana, “Finite different time domain (FDTD) simulations using graphics processor,” Proceedings of the Department of Defense High Performance Computing Modernization Program Users Group Conference, 334–338 (2007).
- R. Sypek, A. Dziekonski, and M. Mrozowski, “How to render FDTD computations more effective using a graphics accelerator,” IEEE Trans. Magn.45(3), 1324–1327 (2009). [CrossRef]
- K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude model on GPU for plasmonics applications,” PIERS116, 441–456 (2011).
- Y. Huang and S. T. Ho, “Computational model of solid-state, molecular, or atomic media for FDTD simulation based on a multi-level multi-electron system governed by Pauli exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,” Opt. Express14(8), 3569–3587 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-8-3569 . [CrossRef] [PubMed]
- E. H. Khoo, S. T. Ho, I. Ahmed, E. P. Li, and Y. Huang, “Light energy extraction from the minor surface arc of an electrically pumped elliptical microcavity laser,” IEEE J. Quantum Electron.46(1), 128–136 (2010). [CrossRef]
- R. K. Chang and A. J. Campillo, Optical processes in Microcavities, Advanced series in Applied Physics (World Scientific, Singapore 1996).
- E. H. Khoo, I. Ahmed, and E. P. Li, “Enhancement of light energy extraction from elliptical microcavity using external magnetic field for switching applications,” Appl. Phys. Lett.95(12), 121104 (2009). [CrossRef]
- I. Ahmed, E. H. Khoo, O. Kurniawan, and E. P. Li, “Modeling and simulation of plasmonic with FDTD method by using solid state and Lorentz -Drude dispersion model,” J. Opt. Soc. Am. B28(3), 352–359 (2011). [CrossRef]
- E. H. Khoo, I. Ahmed, and E. P. Li, “Investigation of light energy extraction efficiency using surface plasmonics in electrically pumped semiconductor microcavity,” Proc. SPIE7764, 7764B (2010).
- O. Kurniawan, I. Ahmed, and E. P. Li, “Generation of surface plasmon polariton using plasmonic resonant cavity based on microdisk laser,” IEEE Photon. J.3, 344–352 (2011).
- R. Shams and P. Sadeghi, “On optimization of finite-difference time-domain (FDTD) computation on heterogeneous and GPU clusters,” J. Parallel Distrib. Comput.71(4), 584–593 (2011). [CrossRef]
- S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics1(8), 449–458 (2007). [CrossRef]
- W. Fang, J. Y. Xu, A. Yamilov, H. Cao, Y. Ma, S. T. Ho, and G. S. Solomon, “Large enhancement of spontaneous emission rates of InAs quantum dots in GaAs microdisks,” Opt. Lett.27(11), 948–950 (2002). [CrossRef] [PubMed]
- Y. Kogami, Y. Tomabechi, and K. Matsumura, “Resonance characteristic of whispering-gallery mode in an elliptic disk resonator,” IEEE Trans. Microw. Theory Tech.44(3), 473–475 (1996). [CrossRef]

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