Abstract

We present a method for performing time domain simulations of a microphotonic system containing a four level gain medium based on the finite element method. This method includes an approximation that involves expanding the pump and probe electromagnetic fields around their respective carrier frequencies, providing a dramatic speedup of the time evolution. Finally, we present a two dimensional example of this model, simulating a cylindrical spaser array consisting of a four level gain medium inside of a metal shell.

© 2012 OSA

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References

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  1. K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
    [CrossRef]
  2. A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
    [CrossRef]
  3. A. Fang, “Reducing the losses of optical metamaterials,” Ph.D. thesis, Iowa State University (2010).
  4. A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12, 024013 (2010).
    [CrossRef]
  5. A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102 (2010).
    [CrossRef]
  6. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
    [CrossRef] [PubMed]
  7. A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express 19, 12688–12699 (2011).
    [CrossRef] [PubMed]
  8. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A 369, 3525–3550 (2011).
    [CrossRef]
  9. J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
    [CrossRef] [PubMed]
  10. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007).
    [CrossRef] [PubMed]
  11. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16, 6692–6716 (2008).
    [CrossRef] [PubMed]
  12. A. E. Siegman, Lasers (University Science Books, 1986). See chapters 2, 3, and 6.
  13. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
    [CrossRef] [PubMed]
  14. W. B. J. Zimmerman, Process Modelling and Simulation with Finite Element Methods (World Scientific, 2004).
  15. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).
  16. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
    [CrossRef] [PubMed]
  17. M. I. Stockman, “Spasers explained,” Nat. Photon. 2, 327–329 (2008).
    [CrossRef]
  18. E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. 51, 2641–2651 (2003).
    [CrossRef]
  19. C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
    [CrossRef]

2011 (3)

A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express 19, 12688–12699 (2011).
[CrossRef] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A 369, 3525–3550 (2011).
[CrossRef]

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef] [PubMed]

2010 (4)

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12, 024013 (2010).
[CrossRef]

A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102 (2010).
[CrossRef]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[CrossRef] [PubMed]

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

2009 (1)

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

2008 (3)

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[CrossRef]

J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16, 6692–6716 (2008).
[CrossRef] [PubMed]

M. I. Stockman, “Spasers explained,” Nat. Photon. 2, 327–329 (2008).
[CrossRef]

2007 (1)

2003 (2)

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. 51, 2641–2651 (2003).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

2000 (1)

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Böhringer, K.

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[CrossRef]

Fang, A.

A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express 19, 12688–12699 (2011).
[CrossRef] [PubMed]

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12, 024013 (2010).
[CrossRef]

A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102 (2010).
[CrossRef]

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

A. Fang, “Reducing the losses of optical metamaterials,” Ph.D. thesis, Iowa State University (2010).

Fietz, C.

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

Gordon, J. A.

Hamm, J. M.

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A 369, 3525–3550 (2011).
[CrossRef]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[CrossRef] [PubMed]

Hess, O.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A 369, 3525–3550 (2011).
[CrossRef]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[CrossRef] [PubMed]

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[CrossRef]

Hess, T.

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef] [PubMed]

Holloway, C. L.

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. 51, 2641–2651 (2003).
[CrossRef]

Huang, Z.

Jiang, X.

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

Koschny, T.

A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express 19, 12688–12699 (2011).
[CrossRef] [PubMed]

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12, 024013 (2010).
[CrossRef]

A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102 (2010).
[CrossRef]

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

Kuester, E. F.

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. 51, 2641–2651 (2003).
[CrossRef]

Mohamed, M. A.

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. 51, 2641–2651 (2003).
[CrossRef]

Piket-May, M.

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. 51, 2641–2651 (2003).
[CrossRef]

Pusch, A.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A 369, 3525–3550 (2011).
[CrossRef]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[CrossRef] [PubMed]

Shvets, G.

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986). See chapters 2, 3, and 6.

Soukoulis, C. M.

A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express 19, 12688–12699 (2011).
[CrossRef] [PubMed]

A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102 (2010).
[CrossRef]

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12, 024013 (2010).
[CrossRef]

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, “Spasers explained,” Nat. Photon. 2, 327–329 (2008).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Tsakmakidis, K. L.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A 369, 3525–3550 (2011).
[CrossRef]

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[CrossRef] [PubMed]

Wegener, M.

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

Wuestner, S.

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A 369, 3525–3550 (2011).
[CrossRef]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[CrossRef] [PubMed]

Zimmerman, W. B. J.

W. B. J. Zimmerman, Process Modelling and Simulation with Finite Element Methods (World Scientific, 2004).

Ziolkowski, R. W.

IEEE Trans. Antennas Propag. (1)

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. 51, 2641–2651 (2003).
[CrossRef]

J. Opt. (1)

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12, 024013 (2010).
[CrossRef]

Nat. Photon. (1)

M. I. Stockman, “Spasers explained,” Nat. Photon. 2, 327–329 (2008).
[CrossRef]

Opt. Express (3)

Phil. Trans. R. Soc. London Ser. A (1)

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. London Ser. A 369, 3525–3550 (2011).
[CrossRef]

Phys. Rev. B (3)

A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102 (2010).
[CrossRef]

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[CrossRef] [PubMed]

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and T. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[CrossRef]

Other (4)

W. B. J. Zimmerman, Process Modelling and Simulation with Finite Element Methods (World Scientific, 2004).

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

A. E. Siegman, Lasers (University Science Books, 1986). See chapters 2, 3, and 6.

A. Fang, “Reducing the losses of optical metamaterials,” Ph.D. thesis, Iowa State University (2010).

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Figures (5)

Fig. 1
Fig. 1

Simple model of a four level gain medium. The lasing and pump transitions are assumed to be electric dipole transistions with frequencies of ωa and ωb respectively. The decay processes between the i-th and j-th energy levels are described by the decay rates 1/τij.

Fig. 2
Fig. 2

(a) Diagram of the simulation domain for the one dimensional cylindrical spaser array with a core gain medium (blue) and outer Ag shell (gray). A periodic boundary condition is imposed on the top and bottom boundaries, and a matched boundary condition (Sec. 2.3) is imposed on the left and right boundaries. Real and imaginary parts of the electric surface polarizability α y y e e (b) and magnetic surface polarizability α z z m m (c) are plotted, clearly indicating separate electric and magnetic resonnaces. Inset are the field profiles for the two resonances and their corresponding wavelengths and Q factors. Color indicates magnetic field Hz, and arrows indicate the electric polarization P = DE.

Fig. 3
Fig. 3

(a) Electric surface polarizability α y y e e and (b) magnetic sufrace polarizability α z z m m for the cylindrical array with gain medium relative permittivity of ɛ G = 9 σ N int / ( ω 2 ω b 2 i Γ b ω ). (c) Total, absorption as well as absorption in Ag and absorption in the gain ωb − Γbω medium. It is clear that the presence of the electronic transition in the gain medium strongly modifies the spectrum of the cylindrical array.

Fig. 4
Fig. 4

(a) Lasing intensity defined as the power emitted outward from the array in either direction and (b) population inversion measured as ∫Ω d2x (Ny2 − Ny1), where Ω is the domain of the simulation, and as a function of time normalized to lasing periods. The time domain simulation begins at t = 0 with a steady state solution where the pump has been on for a very long time (tτ21) and the system has population inversion without lasing due to the lack of spontaneous emmision. (c) Plot of steady state lasing intensity vs. pump intensity. A linear fit indicates a pump threshold intensity of 7.15W/mm2 and a slope of 0.145.

Fig. 5
Fig. 5

Real (solid lines) and imaginary (dashed lines) parts of the electric surface polarizability α y y e e for the lasing electric resonance shown in Fig. 2 for various pump intensities (shown in legend). In Fig. 5(a) we see that at higher pump intensities the linewidth of the resonance narrows. In Fig. 5(b) we see that at even higher pump intensities the imaginary part of the surface polarizability flips (indicating gain) and the linewidth of the resonance begins to broaden.

Equations (23)

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× ( 1 μ 0 × A ) + ɛ r ɛ 0 2 A t 2 = P t ,
E = A t , B = × A
P t + γ P = ɛ 0 ω p 2 A
2 P a i t 2 + Γ a P a i t + ω a 2 P a i = σ a ( N 2 i N 1 i ) E i , 2 P b i t 2 + Γ b P b i t + ω b 2 P b i = σ b ( N 3 i N 0 i ) E i .
N 3 i t = 1 h ¯ ω b E i P b i t ( 1 τ 30 + 1 τ 32 ) N 3 i , N 2 i t = N 3 i τ 32 + 1 h ¯ ω a E i P a i t N 2 i τ 21 , N 1 i t = N 2 i τ 21 1 h ¯ ω a E i P a i t N 1 i τ 10 , N 0 i t = N 3 i τ 30 + N 1 i τ 10 1 h ¯ ω b E i P b i t ,
A ( t , x ) = A 1 ( t , x ) e i ω 1 t + A 2 ( t , x ) e i ω 2 t + c . c . 2 ,
× ( 1 μ 0 × A 1 ) + ɛ r ɛ 0 ( ω 1 2 A 1 + 2 i ω 1 A 1 t + 2 A 1 t 2 ) = P 1 t , × ( 1 μ 0 × A 2 ) + ɛ r ɛ 0 ( ω 2 2 A 2 + 2 i ω 2 A 2 t + 2 A 2 t 2 ) = P 2 t .
P ( t , x ) = P 1 ( t , x ) e i ω 1 t + P 2 ( t , x ) e i ω 2 t + c . c 2 .
i ω 1 P 1 d + P 1 d t + γ P 1 d = ɛ 0 ω p 2 A 1 , i ω 2 P 2 d + P 2 d t + γ P 2 d = ɛ 0 ω p 2 A 2 ,
ω 2 2 P 1 i g + 2 i ω 1 P 1 i g t + 2 P 1 i g t 2 + Γ a ( i ω 1 P 1 i g + P 1 i g t ) + ω a 2 P 1 i g = σ a ( N 2 i N 1 i ) E 1 i , ω 2 2 P 2 i g + 2 i ω 2 P 2 i g t + 2 P 2 i g t 2 + Γ b ( i ω 2 P 2 i g + P 2 i g t ) + ω b 2 P 2 i g = σ b ( N 3 i N 0 i ) E 2 i .
N 3 i t = 1 h ¯ ω b E 2 i P 2 i t ( 1 τ 30 + 1 τ 32 ) N 3 i , N 2 i t = N 3 i τ 32 + 1 h ¯ ω a E 1 i P 1 i t N 2 i τ 21 , N 1 i t = N 2 i τ 21 1 h ¯ ω a E 1 i P 1 i t N 1 i τ 10 , N 0 i t = N 3 i τ 30 + N 1 i τ 10 1 h ¯ ω b E 2 i P 2 i t .
E 1 i P 1 i t = 1 2 Re [ ( i ω 1 A 1 i A 1 i t ) * ( i ω 1 P 1 i + P 1 i t ) ] , E 2 i P 2 i t = 1 2 Re [ ( i ω 1 A 2 i A 2 i t ) * ( i ω 1 P 2 i + P 2 i t ) ] ,
F A 1 ( A ˜ 1 , A 1 ) = ( × A ˜ 1 ) 1 μ 0 ( × A 1 ) + ɛ r ɛ 0 A ˜ 1 ( ω 1 2 A 1 + 2 i ω 1 A 1 t + 2 A 1 t 2 ) A ˜ 1 P 1 t , F A 2 ( A ˜ 2 , A 2 ) = ( × A ˜ 2 ) 1 μ 0 ( × A 2 ) + ɛ r ɛ 0 A ˜ 2 ( ω 2 2 A 2 + 2 i ω 2 A 2 t + 2 A 2 t 2 ) A ˜ 2 P 2 t .
0 = Ω d 3 x F A 1 = Ω d 3 x A ˜ 1 [ × ( 1 μ 0 × A 1 ) + ɛ r ɛ 0 ( ω 1 2 A 1 + 2 i ω 1 A 1 t + 2 A 1 t 2 ) P 1 t ] Ω d A A ˜ 1 [ n ^ × ( 1 μ 0 × A 1 ) ] .
A 1 ( t , x ) = a ( t n ^ x c ) + b ( t + n ^ x c ) .
n ^ × ( 1 μ 0 × A 1 ) = n ^ × ( n ^ μ 0 c × a t + n ^ μ 0 c × b t ) = 1 z 0 ( n ^ × n ^ × ( E 1 out E 1 inc ) ) = 1 z 0 n ^ × n ^ × ( A 1 t + 2 E 1 inc ) ,
B A 1 ( A ˜ 1 , A 1 ) = Ω d A 1 z 0 A ˜ 1 [ n ^ × n ^ × ( A 1 t + 2 E 1 inc ) ] .
F P D 1 ( P ˜ 1 d , P 1 d ) = P ˜ 1 d [ i ω 1 P 1 d + P 1 d t + γ P 1 d + ɛ 0 ω p 2 A 1 ] , F P D 2 ( P ˜ 2 d , P 2 d ) = P ˜ 2 d [ i ω 2 P 2 d + P 2 d t + γ P 2 d + ɛ 0 ω p 2 A 2 ] ,
F P G 1 ( P ˜ 1 g , P 1 g ) = P ˜ 1 g [ ω 1 2 P 1 i g + 2 i ω 1 P 1 i g t + 2 P 1 i g t 2 + Γ a ( i ω 1 P 1 i g + P 1 i g t ) + ω a 2 P 1 i g + σ a ( N 2 i N 1 i ) E 1 i ] , F P G 2 ( P ˜ 2 g , P 2 g ) = P ˜ 2 g [ ω 2 2 P 2 i g + 2 i ω 2 P 2 i g t + 2 P 2 i g t 2 + Γ b ( i ω 2 P 2 i g + P 2 i g t ) + ω b 2 P 2 i g + σ b ( N 3 i N 0 i ) E 2 i ] ,
F N 3 i ( N ˜ 3 i , N 3 i ) = N ˜ 3 i [ N 3 i t 1 h ¯ ω b E 2 i P 2 i t + ( 1 τ 30 + 1 τ 32 ) N 3 i ] , F N 2 i ( N ˜ 2 i , N 2 i ) = N ˜ 2 i [ N 2 i t N 3 i τ 32 1 h ¯ ω a E 1 i P 1 i t + N 2 i τ 21 ] , F N 1 i ( N ˜ 1 i , N 1 i ) = N ˜ 1 i [ N 1 i t N 2 i τ 21 + 1 h ¯ ω a E 1 i P 1 i t + N 1 i τ 10 ] ,
α ^ = ( α y y e e α y z e m α z y m e α z z m m ) = 2 i ω / c ( 1 + S 12 + S 21 det ( S ) ) × ( [ 1 + det ( S ) ( S 11 + S 22 ) ] ɛ 0 [ ( S 12 S 21 ) ( S 11 S 22 ) ] / c [ ( S 12 S 21 ) + ( S 11 S 22 ) ] / c [ 1 + det ( S ) + ( S 11 + S 22 ) ] μ 0 ) .
A 2 = A pu e ^ y 1 2 [ 1 + erf ( t 5 τ pu 2 τ pu ) ] ,
α y y e e = α inf α 0 ω 2 ω α 2 i γ α ω .

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