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

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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).

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]

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]

2010

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).

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]

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]

2009

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]

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]

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]

2008

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]

2007

2006

2004

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).

2002

2001

1999

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]

1996

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]

1993

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]

1992

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]

1966

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]

Ahmed, I.

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).

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]

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).

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]

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]

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).

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. 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]

Asano, T.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics1(8), 449–458 (2007).
[CrossRef]

Cao, H.

Chang, H. C.

Chen, Z.

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]

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]

Chua, E. K.

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]

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]

Dziekonski, A.

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]

Fang, W.

Fujita, M.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics1(8), 449–458 (2007).
[CrossRef]

Goh, R. S. M.

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).

Ho, S. T.

Hsu, S. M.

Huang, Y.

Hung, T. G. G.

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).

Khoo, E. H.

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).

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]

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]

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]

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).

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]

Kogami, Y.

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]

Krakiwsky, S. E.

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).

Kurniawan, O.

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]

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).

Lee, K. H.

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).

Levi, A. F. J.

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]

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]

Li, E. P.

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).

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]

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).

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]

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]

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).

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. 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]

Liu, Z. H.

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]

Logan, R. A.

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]

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]

Ma, Y.

Matsumura, K.

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]

McCall, S. L.

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]

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]

Mohideen, U.

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]

Mrozowski, M.

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]

Noda, S.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics1(8), 449–458 (2007).
[CrossRef]

Okoniewski, M. M.

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).

Pearton, S. J.

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]

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]

Rezac, J.

Rosenberger, A.

Sadeghi, P.

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]

See, K. Y.

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]

Shams, R.

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]

Slusher, R. E.

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]

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]

Solomon, G. S.

Sypek, R.

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]

Tomabechi, Y.

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]

Turner, L. E.

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).

Xu, J. Y.

Yamilov, A.

Yee, K. S.

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]

Zhang, J.

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]

Zheng, F.

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]

Appl. Phys. Lett.

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]

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]

IEEE Int. Microw. Sym. Digest

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).

IEEE J. Quantum Electron.

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]

IEEE Microw. Guided Wave Lett.

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]

IEEE Photon. J.

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).

IEEE Trans. Antenn. Propag.

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]

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Figures (10)

Fig. 1
Fig. 1

Flowchart of the GPU implementation for the DTEQM-FDTD algorithm

Fig. 2
Fig. 2

Layout of the elliptical microcavity in three dimensional (3D) schematic. The semi-major and semi-minor lengths are 400 and 250 nm respectively. The thickness is kept at 200 nm and refractive index is 3.5 for all simulation results.

Fig. 3
Fig. 3

Ez field distribution at different planes. (a)-(c) Ez field simulated by using CPU and (d)-(f) Ez field simulated by using GPU. (a) and (d) at the middle of x-y plane. (b) and (e) at the middle of x-z plane. (c) and (f) at the middle of y-z plane.

Fig. 4
Fig. 4

The resonance spectrum of the elliptical microcavity with semi-major and semi-minor length of 400 and 200 nm respectively. The resonance wavelength is 752 nm. The resonance spectrum calculated by GPU and CPU matches extremely well.

Fig. 5
Fig. 5

(a) Resonance spectra of the elliptical microcavity at selected length ratios. The semi-minor length is varied from 150 to 250 nm. It is observed that the blue shift occurs at Lr = 2.5 and 1.818. The lines and symbols for respective length ratios calculated by CPU and GPU matches very well. (b) Graph of the resonance peak wavelength at different semi-minor length ratios. The resonance peaks calculated by CPU and GPU agree very well.

Fig. 6
Fig. 6

Ez field distribution at the x-y plane of the elliptical microcavity at Lr of 1.86, 1.818, 1.739 respectively. (a) and (c) Ez field shows the whispering gallery mode as the dominate mode. (b) Ez field shows the cavity mode as the dominate mode. Whispering gallery mode is not shown in the field distribution.

Fig. 7
Fig. 7

Ez field distribution of the elliptical microcavity at Lr of 2.5 and 2.353 respectively. (a) Ez shows that the cavity mode is the dominate mode. (b) Whispering gallery mode is excited at Lr = 2.353.

Fig. 8
Fig. 8

(a) Spectra of the elliptical microcavity at various Lr. The resonance wavelength shift occur at Lr = 1.9 and 2.1. (b) Plot of resonance wavelength peaks at different Lr. The plot shows a “stair-step” like characteristic. Sharp drop or rise in the resonance wavelength occurs at certain Lr values.

Fig. 9
Fig. 9

Ez field distribution of the elliptical microcavity. (a) Lr = 2.05. (b) Lr = 2.1. (c) Lr = 2.15. It is shown that the field distribution for Lr = 2.1 does not have whispering gallery mode but a normal cavity mode.

Fig. 10
Fig. 10

(a) Simulation time comparison between CPU and GPU. (b) It is shown that the speed up is approximately 300 time for the minor length between 180 to 250 nm.

Equations (12)

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×H= D t +P
×E= B t
E x | u,v+ 1 2 ,w+ 1 2 n+1 = E x | u,v+ 1 2 ,w+ 1 2 n 1 ε i=1 M ( P i,x | u,v+ 1 2 ,w+ 1 2 n+1 P i,x | u,v+ 1 2 ,w+ 1 2 n ) + Δt εΔy ( H z | u,v+1,w+ 1 2 n+ 1 2 H z | u,v,w+ 1 2 n+ 1 2 ) Δt εΔz ( H y | u,v+ 1 2 ,w+1 n+ 1 2 H y | u,v+ 1 2 ,w n+ 1 2 )
E y | u 1 2 ,v+1,w+ 1 2 n+1 = E y | u 1 2 ,v+1,w+ 1 2 n 1 ε i=1 M ( P i,y | u 1 2 ,v+1,w+ 1 2 n+1 P i,y | u 1 2 ,v+1,w+ 1 2 n ) + Δt εΔz ( H x | u 1 2 ,v+1,w+1 n+ 1 2 H x | u 1 2 ,v+1,w n+ 1 2 ) Δt εΔx ( H z | u,v+1,w+ 1 2 n+ 1 2 H z | u1,v+1,w+ 1 2 n+ 1 2 )
E z | u 1 2 ,v+ 1 2 ,w+1 n+1 = E z | u 1 2 ,v+ 1 2 ,w+1 n 1 ε i=1 M ( P i,z | u 1 2 ,v+ 1 2 ,w+1 n+1 P i,z | u 1 2 ,v+ 1 2 ,w+1 n ) + Δt εΔx ( H y | u,v+ 1 2 ,w+1 n+ 1 2 H y | u1,v+ 1 2 ,w+1 n+ 1 2 ) Δt εΔy ( H x | u 1 2 ,v+1,w+1 n+ 1 2 H x | u 1 2 ,v,w+1 n+ 1 2 )
P i,x | u,v+ 1 2 ,w+ 1 2 n+1 = 42Δ t 2 ( ω ai 2 +4 ω ai 2 2 | μ ix | 2 A x 2 | u,v+ 1 2 ,w+ 1 2 n ) 2+Δt γ i P i,x | u,v+ 1 2 ,w+ 1 2 n + Δt γ i 2 Δt γ i +2 P i,x | u,v+ 1 2 ,w+ 1 2 n1 4Δ t 2 ω ai ( Δt γ i +2 ) | μ ix | 2 ( N Ci | u,v+ 1 2 ,w+ 1 2 n N Vi | u,v+ 1 2 ,w+ 1 2 n ) E x | u,v+ 1 2 ,w+ 1 2 n
P i,y | u 1 2 ,v+1,w+ 1 2 n+1 = 42Δ t 2 ( ω ai 2 +4 ω ai 2 2 | μ iy | 2 A y 2 | u 1 2 ,v+1,w+ 1 2 n ) 2+Δt γ i P i,y | u 1 2 ,v+1,w+ 1 2 n + Δt γ i 2 Δt γ i +2 P i,y | u 1 2 ,v+1,w+ 1 2 n1 4Δ t 2 ω ai ( Δt γ i +2 ) | μ iy | 2 ( N Ci | u 1 2 ,v+1,w+ 1 2 n N Vi | u 1 2 ,v+1,w+ 1 2 n ) E y | u 1 2 ,v+1,w+ 1 2 n
P i,z | u 1 2 ,v+ 1 2 ,w+1 n+1 = 42Δ t 2 ( ω ai 2 +4 ω ai 2 2 | μ iz | 2 A z 2 | u 1 2 ,v+ 1 2 ,w+1 n ) 2+Δt γ i P i,z | u 1 2 ,v+ 1 2 ,w+1 n + Δt γ i 2 Δt γ i +2 P i,z | u 1 2 ,v+ 1 2 ,w+1 n1 4Δ t 2 ω ai ( Δt γ i +2 ) | μ iz | 2 ( N Ci | u 1 2 ,v+ 1 2 ,w+1 n N Vi | u 1 2 ,v+ 1 2 ,w+1 n ) E z | u 1 2 ,v+ 1 2 ,w+1 n
Δ N C/V( i,i1 ) ( r,t )= N C/V( i ) ( r,t )[ 1 N C/V( i1 ) ( r,t ) N C/V( i1 ) 0 ( r ) ] τ C/V( i,i1 ) N C/V( i1 ) ( r,t )[ 1 N C/V( i ) ( r,t ) N C/V( i ) 0 ( r ) ] τ C/V( i1,i )
Δ N i ( r,t )= ΔE( r,t )ΔP( r,t ) 2 ω ai + N Ci ( r,t )[ 1 N Vi ( r,t ) N Vi 0 ( r ) ] τ i + W pump [ 1 N C/V( i ) ( r,t ) N C/V( i ) 0 ( r ) ]Δt
i=1 M N Ci 0 = i=1 M N Vi 0 = i=1 M N Ci n+1 + i=1 M N Vi n+1
τ C/V( i1,i ) τ C/V( i,i1 ) = N C/V( i1 ) 0 ( r ) N C/V( i ) 0 ( r ) exp( E C/V( i ) E C/V( i1 ) k B T )

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