Abstract

We investigate the light emission properties of electrical dipole emitters inside 2-dimensional (2D) and 3-dimensional (3D) silicon slot waveguides and evaluate the spontaneous emission enhancement (Fp) and waveguide coupling ratio (β). Under realistic conditions, we find that greater than 10-fold enhancement in Fp can be achieved, together with a β as large as 0.95. In contrast to the case of high Q optical resonators, such performance enhancements are obtained over a broad wavelength region, which can cover the entire emission spectrum of popular optical dopants such as Er. The enhanced luminescence efficiency and the strong coupling into a limited set of well-defined waveguide modes enables a new class of power-efficient, CMOS-compatible, waveguide-based light sources.

© 2009 Optical Society of America

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2009

2008

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nat. Photonics 2, 433-437 (2008).

C. Creatore and L. C. Andreani, "Quantum theory of spontaneous emission in multilayer dielectric structures," Phys. Rev. A 78, 063825 (2008).

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. L. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).

2006

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

B. Jalali and S. Fathpour, "Silicon photonics," J. Lightwave Technol. 24, 4600-4615 (2006).

2005

2004

2000

Y. Xu, R. K. Lee, and A. Yariv, "Quantum analysis and the classical analysis of spontaneous emission in a microcavity," Phys. Rev. A 61, 033807 (2000).

1998

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (1998).

W. L. Barnes, "Fluorescence near interfaces: the role of photonic mode density," J. Mod. Opt. 45, 661-699 (1998).

1997

A. Polman, "Erbium implanted thin film photonic materials," J. Appl. Phys. 82, 1-39 (1997).

1995

E. Snoeks, A. Lagendijk, and A. Polman, "Measuring and modifying the spontaneous emission rate of erbium near an interface," Phys. Rev. Lett. 74, 2459-2462 (1995).
[PubMed]

1994

S. Coffa, G. Franzo, F. Priolo, A. Polman, and R. Serna, "Temperature dependence and quenching processes of the intra-4f luminescence of Er in crystalline Si," Phys. Rev. B 49, 16313-16320 (1994).

1993

A. M. Vredenberg, N. E. J. Hunt, E. F. Schubert, D. C. Jacobson, J. M. Poate, and G. J. Zydzik, "Controlled Atomic spontaneous emission from Er3+ in a transparent Si/SiO2 microcavity," Phys. Rev. Lett. 71, 517-520 (1993).
[PubMed]

1984

G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195 (1984).

1978

R. Chance, A. Prock, and R. Silby, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978).

Almeida, V. R.

Andreani, L. C.

C. Creatore and L. C. Andreani, "Quantum theory of spontaneous emission in multilayer dielectric structures," Phys. Rev. A 78, 063825 (2008).

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

Barnes, W. L.

W. L. Barnes, "Fluorescence near interfaces: the role of photonic mode density," J. Mod. Opt. 45, 661-699 (1998).

Barrios, C. A.

Beals, M.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nat. Photonics 2, 433-437 (2008).

Belotti, M.

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

Bernardis, S.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nat. Photonics 2, 433-437 (2008).

Brongersma, M. L.

A. C. Hryciw, Y. C. Jun, and M. L. Brongersma, "Plasmon-enhanced emission from optically-doped MOS light sources," Opt. Express 17, 185-192 (2009).
[PubMed]

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. L. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).

Canino, A.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

Chance, R.

R. Chance, A. Prock, and R. Silby, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978).

Chen, Y.

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

Cheng, J.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nat. Photonics 2, 433-437 (2008).

Coffa, S.

S. Coffa, G. Franzo, F. Priolo, A. Polman, and R. Serna, "Temperature dependence and quenching processes of the intra-4f luminescence of Er in crystalline Si," Phys. Rev. B 49, 16313-16320 (1994).

Creatore, C.

C. Creatore and L. C. Andreani, "Quantum theory of spontaneous emission in multilayer dielectric structures," Phys. Rev. A 78, 063825 (2008).

Fathpour, S.

Ford, G. W.

G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195 (1984).

Franzo, G.

S. Coffa, G. Franzo, F. Priolo, A. Polman, and R. Serna, "Temperature dependence and quenching processes of the intra-4f luminescence of Er in crystalline Si," Phys. Rev. B 49, 16313-16320 (1994).

Galli, M.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

Gerace, D.

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

Hryciw, A. C.

Hunt, N. E. J.

A. M. Vredenberg, N. E. J. Hunt, E. F. Schubert, D. C. Jacobson, J. M. Poate, and G. J. Zydzik, "Controlled Atomic spontaneous emission from Er3+ in a transparent Si/SiO2 microcavity," Phys. Rev. Lett. 71, 517-520 (1993).
[PubMed]

Irrera, A.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

Jacobson, D. C.

A. M. Vredenberg, N. E. J. Hunt, E. F. Schubert, D. C. Jacobson, J. M. Poate, and G. J. Zydzik, "Controlled Atomic spontaneous emission from Er3+ in a transparent Si/SiO2 microcavity," Phys. Rev. Lett. 71, 517-520 (1993).
[PubMed]

Jalali, B.

Jun, Y. C.

A. C. Hryciw, Y. C. Jun, and M. L. Brongersma, "Plasmon-enhanced emission from optically-doped MOS light sources," Opt. Express 17, 185-192 (2009).
[PubMed]

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. L. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).

Kekatpure, R. D.

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. L. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).

Kimerling, L. C.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nat. Photonics 2, 433-437 (2008).

Lagendijk, A.

E. Snoeks, A. Lagendijk, and A. Polman, "Measuring and modifying the spontaneous emission rate of erbium near an interface," Phys. Rev. Lett. 74, 2459-2462 (1995).
[PubMed]

Lee, R. K.

Y. Xu, R. K. Lee, and A. Yariv, "Quantum analysis and the classical analysis of spontaneous emission in a microcavity," Phys. Rev. A 61, 033807 (2000).

Lipson, M.

Liscidini, M.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

Liu, J.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nat. Photonics 2, 433-437 (2008).

Lo Savio, R.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

Long, C.

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (1998).

Manolatou, C.

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (1998).

Michel, J.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nat. Photonics 2, 433-437 (2008).

Miritello, M.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

Panepucci, R. R.

Patrini, M.

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

Poate, J. M.

A. M. Vredenberg, N. E. J. Hunt, E. F. Schubert, D. C. Jacobson, J. M. Poate, and G. J. Zydzik, "Controlled Atomic spontaneous emission from Er3+ in a transparent Si/SiO2 microcavity," Phys. Rev. Lett. 71, 517-520 (1993).
[PubMed]

Politi, A.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

Polman, A.

A. Polman, "Erbium implanted thin film photonic materials," J. Appl. Phys. 82, 1-39 (1997).

E. Snoeks, A. Lagendijk, and A. Polman, "Measuring and modifying the spontaneous emission rate of erbium near an interface," Phys. Rev. Lett. 74, 2459-2462 (1995).
[PubMed]

S. Coffa, G. Franzo, F. Priolo, A. Polman, and R. Serna, "Temperature dependence and quenching processes of the intra-4f luminescence of Er in crystalline Si," Phys. Rev. B 49, 16313-16320 (1994).

Pomerene, A.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nat. Photonics 2, 433-437 (2008).

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[PubMed]

Priolo, F.

M. Galli, A. Politi, M. Belotti, D. Gerace, M. Liscidini, M. Patrini, L. C. Andreani, M. Miritello, A. Irrera, F. Priolo, and Y. Chen, "Strong enhancement of Er3+ emission at room temperature in silicon-on-insulator photonic crystal waveguides," Appl. Phys. Lett 88, 251114 (2006).

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, and F. Priolo, "Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides," Appl. Phys. Lett 89, 241114 (2006).

S. Coffa, G. Franzo, F. Priolo, A. Polman, and R. Serna, "Temperature dependence and quenching processes of the intra-4f luminescence of Er in crystalline Si," Phys. Rev. B 49, 16313-16320 (1994).

Prock, A.

R. Chance, A. Prock, and R. Silby, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978).

Robinson, J. T.

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (1998).

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[PubMed]

Schubert, E. F.

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A. M. Vredenberg, N. E. J. Hunt, E. F. Schubert, D. C. Jacobson, J. M. Poate, and G. J. Zydzik, "Controlled Atomic spontaneous emission from Er3+ in a transparent Si/SiO2 microcavity," Phys. Rev. Lett. 71, 517-520 (1993).
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A. M. Vredenberg, N. E. J. Hunt, E. F. Schubert, D. C. Jacobson, J. M. Poate, and G. J. Zydzik, "Controlled Atomic spontaneous emission from Er3+ in a transparent Si/SiO2 microcavity," Phys. Rev. Lett. 71, 517-520 (1993).
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Appl. Phys. Lett

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Figures (9)

Fig. 1.
Fig. 1.

(a) Spontaneous emission enhancement factor Fp as a function of slot width W. (b) Dissipated power density of a dipole as a function of normalized in-plane wavevector (for W = 20 nm). (c) β as a function of slot width W. Normal and parallel dipoles correspond to red and blue lines, while the isotropic average is shown in a dotted black line. In (a) ~ (c), WT = 410nm and the red and blue lines correspond to normal and parallel dipoles. (d) Isotroically averaged Fp as a function of total slab width (WT) and slot width (W). In all cases, we consider a dipole in the middle of the slot and λ0 = 1540 nm.

Fig. 2.
Fig. 2.

(a) Dissipated power density of a dipole as a function of normalized in-plane wavevector. The dipole is 5 nm off from the center. (b) Position dependent Fp inside a slot into individual waveguide and radiation modes. Isotropically averaged Fp is plotted. (c), (d) Isotropically averaged, total Fp and β inside a slot. W = 20 nm for (a) ~ (c), and W = 50 nm for (d). In all cases, WT = 410 nm and λ0 = 1540 nm.

Fig. 3.
Fig. 3.

(a) Spontaneous emission enhancement factor Fp, and (b) waveguide coupling ratio β as a function of the freespace wavelength λ0 for normal (red line) and parallel (blue line) dipoles. We consider a dipole in the middle of the slot and W = 20 nm. Isotropically averaged values are plotted in black lines. In (a), the result from 3D FDTD simulations are also plotted in green dotted lines.

Fig. 4.
Fig. 4.

(a) Isotropically averaged spontaneous emission enhancement factor Fp for different, initial quantum efficiencies. (b) Quantum efficiency (QE) as a function of slot width for two different, initial quantum efficiencies. We consider a dipole in the middle of the slot and λ0 = 1540nm.

Fig. 5.
Fig. 5.

(a) Spontaneous emission enhancement Fp for a 3D slot structure (W = 20 nm, H = 300 nm, WT = 410 nm). Red, blue, and green lines correspond to the enhancements from y, x, and z dipoles. (b) Isotropically averaged Fp for slot widths W = 20, 30, 50 nm.

Fig. 6.
Fig. 6.

Eigenmode profiles of quasi-TE and quasi-TM modes in a 3D silicon slot waveguide. Energy density U=ε0ε∣E∣2 (upper) and Electric field magnitude ∣E∣ (lower). Arrows in the lower figures show E-field directions.

Fig. 7.
Fig. 7.

(a) Waveguide mode excitation enhancement Fwg as a function of freespace wavelength. The black solid squares and circles are based on the approach using the Fermi Golden rule (Eq. 15), while the dotted lines from 3D FDTD flux measurements. (b) Waveguide coupling ratio β obtained from the cacluations based on the Fermi Golden Rule. The inset is the isotropically averaged waveguide coupling ratios for three different slot widths. W = 20 nm.

Fig. 8.
Fig. 8.

(a) Waveguide mode excitation enhancement Fwg as a function of slot width. (b) Normalized effective mode area and mode group velocity as a function of slot width. W = 20 nm.

Fig. 9.
Fig. 9.

Fwg map inside a W = 20 nm slot for (a) y-dipole (which couples to quasi-TE mode) and (b) x-dipole (which couples to quasi-TM mode). λ0 = 1540 nm.

Equations (18)

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γ γ 3 = 3 2 Im 0 du u 3 3 [ 1 + r 345 e 2 3 s ̂ ] [ 1 + r 321 e 2 3 d ̂ ] 1 r 321 r 345 e 2 3 W ̂
γ γ 3 = 3 4 Im 0 du u 3 { [ 1 + r 345 e 2 3 s ̂ ] [ 1 + r 321 e 2 3 d ̂ ] 1 r 321 r 345 e 2 3 W ̂ + ( 1 u 2 ) [ 1 r 345 e 2 3 s ̂ ] [ 1 r 321 e 2 3 d ̂ ] 1 r 321 r 345 e 2 3 W ̂ }
r ijk ⊥,∥ = r ij ⊥,∥ + r jk ⊥,∥ e 2 2 L ̂ 2 1 + r ij ⊥,∥ r jk ⊥,∥ e 2 2 L ̂ 2
r ij = ( i j ) / ( i + j )
r ij = ( i ε j j ε i ) / ( i ε j + j ε i )
F p ⊥,∥ = ( 1 η 0 ) + η 0 γ ⊥,∥ γ 3
F p iso = γ iso γ 3 = 1 3 γ γ 3 + 2 3 γ γ 3
β = γ guided / γ total
β iso = γ ⊥,guided + 2 γ ∥,guide γ ⊥,total + 2 γ ∥,total
η = η 0 γ iso / γ 3 ( 1 η 0 ) + η 0 γ iso / γ 3
F p = γ / γ 0 = P / P 0
F p iso = γ iso γ 0 = γ x + γ y + γ z 3 γ 0
γ wg ( r e , ω ) = 2 π g ( r e , ω ) 2 D ( ω )
g ( r e , ω ) 2 = d 0 · α E ( r e ) / ħ 2 = ω d 0 2 / { 2 ħε ε 0 V ˜ eff }
α 2 = ħ ω / { ε 0 d ( εω ) E 2 + μ 0 H 2 } d r ħ ω / 2 ε 0 ε E 2 d r
A ˜ eff ( r ˜ e ) = ε 0 ε ( r ) E ( r ) 2 d r ε 0 ε ( r e ) E ( r e ) 2 = A eff · max [ ε 0 ε ( r ) E ( r ) 2 ] ε 0 ε ( r e ) E ( r e ) 2
D ( ω ) = / [ π υ g ( ω ) ]
F wg = γ wg γ 0 = 3 4 π · c / n υ g · ( λ 0 / n ) 2 A ˜ eff

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