Abstract

We developed a general numerical method to calculate the spontaneous emission lifetime in an arbitrary microcavity, using a finite-difference time-domain algorithm. For structures with rotational symmetry we also developed a more efficient but less general algorithm. To simulate an open radiation problem, we use absorbing boundaries to truncate the computational domain. The accuracy of this method is limited only by numerical error and finite reflection at the absorbing boundaries. We compare our result with cases that can be solved analytically and find excellent agreement. Finally, we apply the method to calculate the spontaneous emission lifetime in a slab waveguide and in a dielectric microdisk, respectively.

© 1999 Optical Society of America

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References

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  1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  2. T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Topics Quantum Electron. 3, 808–830 (1997).
    [CrossRef]
  3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  4. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).
  5. E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics, P. R. Berman, ed. (Academic, New York, 1994).
  6. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  7. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  8. Y. Chen, R. Mittra, and P. Harms, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
    [CrossRef]
  9. C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
    [CrossRef]
  10. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  11. R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
    [CrossRef] [PubMed]
  12. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
    [CrossRef]
  13. J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
    [CrossRef]
  14. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
    [CrossRef]
  15. M. Fusco, “FDTD algorithm in curvilinear coordinates,” IEEE Trans. Antennas Propag. 38, 76–89 (1990).
    [CrossRef]
  16. M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302–3307 (1994).
    [CrossRef]
  17. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

1997

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Topics Quantum Electron. 3, 808–830 (1997).
[CrossRef]

1996

Y. Chen, R. Mittra, and P. Harms, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
[CrossRef]

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[CrossRef]

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

1994

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302–3307 (1994).
[CrossRef]

1991

R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

1990

M. Fusco, “FDTD algorithm in curvilinear coordinates,” IEEE Trans. Antennas Propag. 38, 76–89 (1990).
[CrossRef]

1987

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1981

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

1966

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

1946

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Baba, T.

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Topics Quantum Electron. 3, 808–830 (1997).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Chen, Y.

Y. Chen, R. Mittra, and P. Harms, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
[CrossRef]

Chin, M. K.

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302–3307 (1994).
[CrossRef]

Chu, D. Y.

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302–3307 (1994).
[CrossRef]

Fusco, M.

M. Fusco, “FDTD algorithm in curvilinear coordinates,” IEEE Trans. Antennas Propag. 38, 76–89 (1990).
[CrossRef]

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Glauber, R. J.

R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Harms, P.

Y. Chen, R. Mittra, and P. Harms, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
[CrossRef]

Henry, C. H.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[CrossRef]

Ho, S. T.

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302–3307 (1994).
[CrossRef]

Kazarinov, R. F.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[CrossRef]

Lewenstein, M. L.

R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Mittra, R.

Y. Chen, R. Mittra, and P. Harms, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
[CrossRef]

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yee, K. S.

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

IEEE J. Sel. Topics Quantum Electron.

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Topics Quantum Electron. 3, 808–830 (1997).
[CrossRef]

IEEE Trans. Antennas Propag.

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

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

M. Fusco, “FDTD algorithm in curvilinear coordinates,” IEEE Trans. Antennas Propag. 38, 76–89 (1990).
[CrossRef]

IEEE Trans. Electromagn. Compat.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

Y. Chen, R. Mittra, and P. Harms, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
[CrossRef]

J. Appl. Phys.

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302–3307 (1994).
[CrossRef]

J. Comput. Phys.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Phys. Rev.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. A

R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Rev. Mod. Phys.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).

E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics, P. R. Berman, ed. (Academic, New York, 1994).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

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Figures (7)

Fig. 1
Fig. 1

Position of field components in the FDTD lattice in the 3D algorithm. The components of the dipole moment are placed in the same position as the electric-field components and are not shown.

Fig. 2
Fig. 2

Position of field components in the FDTD lattice in the cylindrical algorithm. The components of the dipole moment are placed in the same position as the electric-field components and are not shown.

Fig. 3
Fig. 3

Spontaneous emission rate of a radial dipole between two metal plates. The dipole is polarized parallel to the metal plates and placed at the center of the two metal plates. The spontaneous emission rate is normalized to the corresponding free-space value.

Fig. 4
Fig. 4

Spontaneous emission rate of an axial dipole between two metal plates. The dipole is polarized normal to the metal plates and placed at the center of the two metal plates. The spontaneous emission rate is normalized to the corresponding free-space value.

Fig. 5
Fig. 5

Spontaneous emission rate of axial and radial dipoles in a semiconductor waveguide. The dipole is placed at the center of the semiconductor waveguide. The spontaneous emission rate is normalized to the corresponding free-space value. The arrows indicate the cutoff thickness of the TE modes.

Fig. 6
Fig. 6

Spontaneous emission rate of a radial dipole in a semiconductor microdisk (star), spontaneous emission rate of a radial dipole in a slab waveguide (circle). The dipole polarizes in the x direction. The spontaneous emission rate is normalized to the corresponding free-space value.

Fig. 7
Fig. 7

Spontaneous emission rate of an axial dipole in a semiconductor microdisk (star), spontaneous emission rate of an axial dipole in a slab waveguide (circle). We connected the circles with a solid curve to make them more obvious. The dipole polarizes in the z direction. The spontaneous emission rate is normalized to the corresponding free-space value.

Equations (78)

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·[(x)E(x, t)]=0,
×[×E(x, t)]+(x)μ0 2E(x, t)t2
+μ0 2P(x, t)t2+μ0 J(x, t)t=0,
P(x, t)=d(t)dˆδ(x-x0),
J(x, t)=γ(x)E(x, t).
E(x, t)=nαn(t)Fn(x).
×[×Fn(x)]=(x)μ0ωn2Fn(x),
d3x(x)Fn*(x)·Fm(x)=δm,n.
μ0(x)n{α¨n(t)+γα˙n(t)}Fn(x)+nαn(t)×[×Fn(x)]
=-μ0d¨(t)dˆδ(x-x0),
α¨n(t)+γα˙n(t)+ωn2αn(t)=-d¨(t)[dˆ·Fn*(x0)].
d(t)=μ exp(-iω0t).
αn(t)=d(t) ω02ωn2-ω02-iω0γ[dˆ·Fn*(x0)].
Pclassical=-12Red3x P*(x, t)t·E(x, t).
Pclassical=14ω02μ2n|dˆ·Fn(x0)|2 γ/2(ωn-ω0)2+γ2/4.
limγ0 γ/2(ωn-ω0)2+γ2/4=πδ(ωn-ω0).
Pclassical=14πμ2ω02n|dˆ·Fn(x0)|2δ(ωn-ω0).
Pquantum=ωegτspon=πωeg2μeg2n|dˆ·Fn(x0)|2δ(ωn-ωeg),
τsponbulkτsponcavity=PclassicalcavityPclassicalbulk,
μ0 Ht=-×E,
 Et=×H-Pt.
(i, j, k)=(iΔx, jΔy, kΔz)
Fn(i, j, k)=F(iΔx, jΔy, kΔz, nΔt),
Ezn+1(i, j, k+12)=Ezn(i, j, k+12)+Δt(i, j, k+12)×Hyn+1/2(i+12, j, k+12)-Hyn+1/2(i-12, j, k+12)Δx-Hxn+1/2(i, j+12, k+12)-Hxn+1/2(i, j-12, k+12)Δy-(P˙z)n+1/2(i, j, k+12),
Hzn+1/2(i+12, j+12, k)
=Hzn-1/2(i+12, j+12, k)+Δtμ0Exn(i+12, j+1, k)-Exn(i+12, j, k)Δy-Eyn(i+1, j+12, k)-Eyn(i, j+12, k)Δx.
×H=iω˜E,
×E=-iωμ0μ˜H,
˜=μ˜=sysz/sx000sxsz/sy000sxsy/sz,
sx=1+σxjω0,
sy=1+σyjω0,
sz=1+σzjω0.
σz(z)=σmax|z-z0|mdm,
E(r, z, ϕ)=exp(imϕ)[Er(r, z)eˆr+iEϕ(r, z)eˆϕ+Ez(r, z)eˆz],
P(r, z, ϕ)=exp(imϕ)[Pr(r, z)eˆr+iPϕ(r, z)eˆϕ+Pz(r, z)eˆz],
H(r, z, ϕ)=exp(imϕ)[iHr(r, z)eˆr+Hϕ(r, z)eˆϕ+iHz(r, z)eˆz],
 Ert=-Hϕz-mrHz-Prt,
 Eϕt=Hrz-Hzr-Pϕt,
 Ezt=mrHr+1r(rHϕ)r-Pzt,
-μ0 Hrt=-Eϕz+mrEz,
-μ0 Hϕt=Erz-Ezr,
-μ0 Hzt=-mrEr+1r(rEϕ)r.
Ezn+1(i, j+12)=Ezn(i, j+12)+Δt(i, j+12)×mHrn+1/2(i, j+12)ri+ri+1/2Hϕn+1/2(i+12, j+12)-ri-1/2Hϕn+1/2(i-12, j+12)riΔr-(P˙z)n+1/2(i, j+12),
Hzn+1/2(i+12, j)=Hzn-1/2(i+12, j)+Δtμ0mErn(i+12, j)ri+1/2-ri+1Eϕn(i+1, j)-riEϕn(i, j)ri+1/2Δr,
Hzn+1/2(0, j)=Hzn-1/2(0, j)-4Δtμ0ΔrEϕn(1, j).
ψ(t, r, z, ϕ)2πx expiωt-x+νπ2+π4×exp(±ikzz)exp(imϕ),
keωψt+ψr+12rψ=0.
1c22t2+1c2rt+12crt-122z2ψ=0.
Eϕn+1(i0, j)
=-11+cΔt4ri0-1/2+cΔtΔrEϕn+1(i0-1, j)×1+cΔt4ri0-1/2-cΔtΔr+Eϕn-1(i0, j)×1-cΔt4ri0-1/2-cΔtΔr+Eϕn-1(i0-1, j)×1-cΔt4ri0-1/2+cΔtΔr+-2+c2Δt2Δz2{Eϕn(i0, j)+Eϕn(i0-1, j)}-c2Δt22Δz2{Eϕn(i0, j+1)+Eϕn(i0, j-1)+Eϕn(i0-1, j+1)+Eϕn(i0-1, j-1)}.
1c2zt-1c22t2+122x2+2y2ψ=0.
transv2=2x2+2y2=1rrr r+1r22ϕ2.
1c2zt-1c22t2+121rr+2r2-m2r2ψ=0.
Eϕn+1(i, 0)
=cΔt-ΔzcΔt+Δz[Eϕn+1(i, 1)+Eϕn-1(i, 0)]-Eϕn-1(i, 1)+Δz(2Δr2-c2Δt2)Δr2(Δz+cΔt)[Eϕn(i, 0)+Eϕn(i, 1)]-m22ri2c2Δt2ΔzcΔt+Δz[Eϕn(i, 0)+Eϕn(i, 1)]+c2Δt2ΔzcΔt+Δz12Δr1Δr+12ri×[Eϕn(i+1, 0)+Eϕn(i+1, 1)]+c2Δt2ΔzcΔt+Δz12Δr1Δr-12ri×[Eϕn(i-1, 0)+Eϕn(i-1, 1)].
Dcutoffλ=n22(n22-n12)1/2m,
Dcutoffλ=0.52m,m=0, 1, 2,.
·D=ρ,
×E=-Bt,
·B=0,
×H=Dt+J
B=×A,
E=-Φ-At.
·[(x)F]=0.
·[(x)Φ]=-ρ,
×[×A]+μ0 2At2=μ0Jtran=μ0J- tΦ.
d3xJ·E¯=d3xJ·Elong¯+d3xJ·Etran¯=d3xJ·[-Φ]¯+d3xJ·-At¯,
d3xJ·Φ¯=d3xΦ[-·J]¯=d3xΦ ρt¯.
d3xJ·Φ¯=12d3x t[Φρ]¯.
d3xJ·Elong¯=-limT 12Td3xΦ(x, t)ρ(x, t)t=T-d3xΦ(x, t)ρ(x, t)t=0.
Pclassical=d3xJ·E¯=d3xJ·Etran¯,
×E=-Bt,
×H=Dt+Pt.
·[×F(x)]=0,
t·B=0,
t·D=-t·P.
·D=-·P,
·B=0.

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