Abstract

This paper investigates a specific kind of solitary pulse, the modulated zero-area (MZA) solitary pulse, when propagating within MgO photonic bandgap medium doped with silver nanoparticles (NPs). It will be shown that two coupled MZA pulses do propagate unattenuated within this medium but for a certain combination of the dipole moments and the density of the NPs. More important, and in contrast to the other kinds of solitary pulses, one of the two MZA pulses exhibits a slowing in its group velocity in comparison to the other one, depending on the amplitudes of the components of the dipole moments of the NPs that are in resonance with the two MZA pulses. With this particular feature, the system has the potential of working as an all-optical switch.

© 2013 Optical Society of America

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2012 (2)

Z.-G. Zang and Y.-J. Zhang, “Low-switching power (<45 mW) optical bistability based on optical nonlinearity of ytterbium-doped fiber with a fiber Bragg grating pair,” J. Mod. Opt. 59, 161–165 (2012).
[CrossRef]

Z. Zang and Y. Zhang, “Analysis of optical switching in a Yb3+-doped fiber Bragg grating by using self-phase modulation and cross-phase modulation,” Appl. Opt. 51, 3424–3430 (2012).
[CrossRef]

2011 (1)

Z.-G. Zang and W.-X. Yang, “Theoretical and experimental investigation of all-optical switching based on cascaded LPFGs separated by an erbium-doped fiber,” J. Appl. Phys. 109, 103106 (2011).
[CrossRef]

2010 (1)

J. Tejada, R. D. Zysler, E. Molins, and E. M. Chudnovsky, “Evidence for quantization of mechanical rotation of magnetic nanoparticles,” Phys. Rev. Lett. 104, 027202 (2010).
[CrossRef]

2009 (1)

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

2008 (1)

2007 (1)

M. R. Singh, “The effect of the dipole–dipole interaction in electromagnetically induced transparency in polaritonic bandgap materials,” J. Mod. Opt. 54, 1739–1757 (2007).
[CrossRef]

2005 (1)

Y. M. Park, Y. J. Park, J. D. Song, and J. I. Lee, “Estimation of built-in dipole moment in InAs quantum dots,” Solid State Commun. 134, 391–395 (2005).
[CrossRef]

2004 (2)

M. R. Singh, “Anomalous electromagnetically induced transparency in photonic-bandgap materials,” Phys. Rev. A 70, 033813 (2004).
[CrossRef]

A. Muller, Q. Q. Wang, P. Bianucci, C. K. Shih, and Q. K. Xue, “Determination of anisotropic dipole moments in self-assembled quantum dots using Rabi oscillation,” Appl. Phys. Lett. 84, 981–983 (2004).
[CrossRef]

2003 (1)

2002 (2)

A. A. Sukhorukov and Y. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
[CrossRef]

S. Chi, B. Luo, and H.-Y. Tseng, “Ultrashort Bragg soliton in a fiber Bragg grating,” Opt. Commun. 206, 115–121 (2002).
[CrossRef]

2001 (1)

E. del Barco, J. Asenjo, X. X. Zhang, R. Pieczynski, A. Julia, J. Tejada, and R. F. Ziolo, “Free rotation of magnetic nanoparticles in a solid matrix,” Chem. Mater. 13, 1487–1490 (2001).
[CrossRef]

2000 (1)

A. B. Aceves, “Optical gap solitons: past, present, and future; theory and experiments,” Chaos 10, 584–590 (2000).
[CrossRef]

1999 (1)

K. Busch and S. John, “Liquid-crystal photonic-bandgap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

1998 (2)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

S. John and N. Akozbek, “Observation of nonlinear self-trapping in triangular photonic lattices,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

1995 (1)

1994 (1)

F. T. Hioe and R. Grobe, “Matched optical solitary waves for three- and five-level systems,” Phys. Rev. Lett. 73, 2559–2562 (1994).
[CrossRef]

1993 (1)

S. John and N. Akozbek, “Nonlinear optical solitary waves in a photonic bandgap,” Phys. Rev. Lett. 71, 1168–1171 (1993).
[CrossRef]

1990 (1)

S. John and J. Wang, “Quantum electrodynamics near a photonic bandgap: photons bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef]

1987 (3)

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

S. John, “Strong localization of photons in certain dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

D. L. Mills and S. E. Trullinger, “Possibilities for the observation of gap solitons in waveguide geometries,” Phys. Rev. B 36, 947–952 (1987).
[CrossRef]

1986 (1)

P. St. J. Russell, “Interference of integrated Floquet–Bloch waves,” Phys. Rev. A 33, 3232–3242 (1986).
[CrossRef]

1981 (1)

M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567–2583 (1981).
[CrossRef]

1969 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Aceves, A. B.

A. B. Aceves, “Optical gap solitons: past, present, and future; theory and experiments,” Chaos 10, 584–590 (2000).
[CrossRef]

Akozbek, N.

S. John and N. Akozbek, “Observation of nonlinear self-trapping in triangular photonic lattices,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

S. John and N. Akozbek, “Nonlinear optical solitary waves in a photonic bandgap,” Phys. Rev. Lett. 71, 1168–1171 (1993).
[CrossRef]

Asenjo, J.

E. del Barco, J. Asenjo, X. X. Zhang, R. Pieczynski, A. Julia, J. Tejada, and R. F. Ziolo, “Free rotation of magnetic nanoparticles in a solid matrix,” Chem. Mater. 13, 1487–1490 (2001).
[CrossRef]

Bianucci, P.

A. Muller, Q. Q. Wang, P. Bianucci, C. K. Shih, and Q. K. Xue, “Determination of anisotropic dipole moments in self-assembled quantum dots using Rabi oscillation,” Appl. Phys. Lett. 84, 981–983 (2004).
[CrossRef]

Broderick, N. G. R.

Broeng, J.

Busch, K.

K. Busch and S. John, “Liquid-crystal photonic-bandgap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

Cao, W.

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Mirrorless lasing in liquid crystalline materials,” Materials Research Society Symposium Proceedings, 776 (2003); http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8084128 .
[CrossRef]

Changmuang, P.

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

Chi, S.

S. Chi, B. Luo, and H.-Y. Tseng, “Ultrashort Bragg soliton in a fiber Bragg grating,” Opt. Commun. 206, 115–121 (2002).
[CrossRef]

Chudnovsky, E. M.

J. Tejada, R. D. Zysler, E. Molins, and E. M. Chudnovsky, “Evidence for quantization of mechanical rotation of magnetic nanoparticles,” Phys. Rev. Lett. 104, 027202 (2010).
[CrossRef]

del Barco, E.

E. del Barco, J. Asenjo, X. X. Zhang, R. Pieczynski, A. Julia, J. Tejada, and R. F. Ziolo, “Free rotation of magnetic nanoparticles in a solid matrix,” Chem. Mater. 13, 1487–1490 (2001).
[CrossRef]

Eberly, J. H.

M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567–2583 (1981).
[CrossRef]

Erdogan, T.

Ghannam, T.

Grobe, R.

F. T. Hioe and R. Grobe, “Matched optical solitary waves for three- and five-level systems,” Phys. Rev. Lett. 73, 2559–2562 (1994).
[CrossRef]

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Hermann, D. S.

Hioe, F. T.

F. T. Hioe and R. Grobe, “Matched optical solitary waves for three- and five-level systems,” Phys. Rev. Lett. 73, 2559–2562 (1994).
[CrossRef]

John, S.

K. Busch and S. John, “Liquid-crystal photonic-bandgap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

S. John and N. Akozbek, “Observation of nonlinear self-trapping in triangular photonic lattices,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

S. John and N. Akozbek, “Nonlinear optical solitary waves in a photonic bandgap,” Phys. Rev. Lett. 71, 1168–1171 (1993).
[CrossRef]

S. John and J. Wang, “Quantum electrodynamics near a photonic bandgap: photons bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef]

S. John, “Strong localization of photons in certain dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Julia, A.

E. del Barco, J. Asenjo, X. X. Zhang, R. Pieczynski, A. Julia, J. Tejada, and R. F. Ziolo, “Free rotation of magnetic nanoparticles in a solid matrix,” Chem. Mater. 13, 1487–1490 (2001).
[CrossRef]

Kanjanachuchai, S.

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kivshar, Y. S.

A. A. Sukhorukov and Y. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
[CrossRef]

Konopnicki, M. J.

M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567–2583 (1981).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kuwata-Gonokami, M.

Laouthaiwattana, K.

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

Lee, J. I.

Y. M. Park, Y. J. Park, J. D. Song, and J. I. Lee, “Estimation of built-in dipole moment in InAs quantum dots,” Solid State Commun. 134, 391–395 (2005).
[CrossRef]

Lemaire, P. J.

Luo, B.

S. Chi, B. Luo, and H.-Y. Tseng, “Ultrashort Bragg soliton in a fiber Bragg grating,” Opt. Commun. 206, 115–121 (2002).
[CrossRef]

Martijn de Sterke, C.

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Mills, D. L.

D. L. Mills and S. E. Trullinger, “Possibilities for the observation of gap solitons in waveguide geometries,” Phys. Rev. B 36, 947–952 (1987).
[CrossRef]

Mizrahi, V.

Mohideen, U.

Molins, E.

J. Tejada, R. D. Zysler, E. Molins, and E. M. Chudnovsky, “Evidence for quantization of mechanical rotation of magnetic nanoparticles,” Phys. Rev. Lett. 104, 027202 (2010).
[CrossRef]

Muller, A.

A. Muller, Q. Q. Wang, P. Bianucci, C. K. Shih, and Q. K. Xue, “Determination of anisotropic dipole moments in self-assembled quantum dots using Rabi oscillation,” Appl. Phys. Lett. 84, 981–983 (2004).
[CrossRef]

Muñoz, A.

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Mirrorless lasing in liquid crystalline materials,” Materials Research Society Symposium Proceedings, 776 (2003); http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8084128 .
[CrossRef]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Palffy-Muhoray, P.

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Mirrorless lasing in liquid crystalline materials,” Materials Research Society Symposium Proceedings, 776 (2003); http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8084128 .
[CrossRef]

Panyakeow, S.

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

Park, Y. J.

Y. M. Park, Y. J. Park, J. D. Song, and J. I. Lee, “Estimation of built-in dipole moment in InAs quantum dots,” Solid State Commun. 134, 391–395 (2005).
[CrossRef]

Park, Y. M.

Y. M. Park, Y. J. Park, J. D. Song, and J. I. Lee, “Estimation of built-in dipole moment in InAs quantum dots,” Solid State Commun. 134, 391–395 (2005).
[CrossRef]

Pieczynski, R.

E. del Barco, J. Asenjo, X. X. Zhang, R. Pieczynski, A. Julia, J. Tejada, and R. F. Ziolo, “Free rotation of magnetic nanoparticles in a solid matrix,” Chem. Mater. 13, 1487–1490 (2001).
[CrossRef]

Ratanathamaphan, S.

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

Rosenthal, A.

Russell, P. St. J.

P. St. J. Russell, “Interference of integrated Floquet–Bloch waves,” Phys. Rev. A 33, 3232–3242 (1986).
[CrossRef]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Shih, C. K.

A. Muller, Q. Q. Wang, P. Bianucci, C. K. Shih, and Q. K. Xue, “Determination of anisotropic dipole moments in self-assembled quantum dots using Rabi oscillation,” Appl. Phys. Lett. 84, 981–983 (2004).
[CrossRef]

Singh, M. R.

M. R. Singh, “The effect of the dipole–dipole interaction in electromagnetically induced transparency in polaritonic bandgap materials,” J. Mod. Opt. 54, 1739–1757 (2007).
[CrossRef]

M. R. Singh, “Anomalous electromagnetically induced transparency in photonic-bandgap materials,” Phys. Rev. A 70, 033813 (2004).
[CrossRef]

Sipe, J. E.

Slusher, R. E.

Song, J. D.

Y. M. Park, Y. J. Park, J. D. Song, and J. I. Lee, “Estimation of built-in dipole moment in InAs quantum dots,” Solid State Commun. 134, 391–395 (2005).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov and Y. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
[CrossRef]

Suraprapapich, S.

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

Taheri, B.

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Mirrorless lasing in liquid crystalline materials,” Materials Research Society Symposium Proceedings, 776 (2003); http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8084128 .
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Tangmattajittakul, O.

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

Tejada, J.

J. Tejada, R. D. Zysler, E. Molins, and E. M. Chudnovsky, “Evidence for quantization of mechanical rotation of magnetic nanoparticles,” Phys. Rev. Lett. 104, 027202 (2010).
[CrossRef]

E. del Barco, J. Asenjo, X. X. Zhang, R. Pieczynski, A. Julia, J. Tejada, and R. F. Ziolo, “Free rotation of magnetic nanoparticles in a solid matrix,” Chem. Mater. 13, 1487–1490 (2001).
[CrossRef]

Thainoi, S.

K. Laouthaiwattana, O. Tangmattajittakul, S. Suraprapapich, S. Thainoi, P. Changmuang, S. Kanjanachuchai, S. Ratanathamaphan, and S. Panyakeow, “Optimization of stacking high-density quantum dot molecules for photovoltaic effect,” Sol. Energy Mater. Sol. Cells 93, 746–749 (2009).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Trullinger, S. E.

D. L. Mills and S. E. Trullinger, “Possibilities for the observation of gap solitons in waveguide geometries,” Phys. Rev. B 36, 947–952 (1987).
[CrossRef]

Tseng, H.-Y.

S. Chi, B. Luo, and H.-Y. Tseng, “Ultrashort Bragg soliton in a fiber Bragg grating,” Opt. Commun. 206, 115–121 (2002).
[CrossRef]

Wang, J.

S. John and J. Wang, “Quantum electrodynamics near a photonic bandgap: photons bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef]

Wang, Q. Q.

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A. Muller, Q. Q. Wang, P. Bianucci, C. K. Shih, and Q. K. Xue, “Determination of anisotropic dipole moments in self-assembled quantum dots using Rabi oscillation,” Appl. Phys. Lett. 84, 981–983 (2004).
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Zang, Z.-G.

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Appl. Opt. (1)

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Chem. Mater. (1)

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[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Three-level lambda configuration of the NPs along with the probe (Ea) and pump (Eb) pulses. The two pulses are in resonance with the energy levels of the NPs, hence, ωa,b=ω12,23.

Fig. 2.
Fig. 2.

Dimensionless pulse area gradient (dΘ/ds) versus distance (103m); the slowing of the a pulse (dashed) in comparison to the b pulse (solid) for pa=2×pb, β=1. (a) The two pulses coinciding at time zero. (b) After 0.875 ns, a difference of 0.1825 ns between the centers of the two pulses.

Fig. 3.
Fig. 3.

Slowing of the b pulse in comparison to the a pulse for pa=0.5×pb, β=16. (a) The two pulses coinciding at time zero (the a pulse is the larger lighter one). (b) After 0.875 ns, a difference of 0.1375 ns between the centers of the two pulses. Vertical and horizontal units are same as in Fig. 2.

Fig. 4.
Fig. 4.

For pa=pb, β=4, no relative slowing is taking place between the two pulses. Vertical and horizontal units are same as in Fig. 2.

Equations (21)

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Z(εij±)=(εvεij±)/[(εcεij±)+κ2].
εij±=12(εvωij)±[(εvωij)2+4ωij(εvεc)]1/2.
i×[dσij]=[[0(Eapa)*Z120EapaZ210EbpbZ230(Ebpb)*Z320],σij].
E⃗a,b=e^a,bAa,b(r,t)e(ikziωa,bt)+c.c.,
P⃗a,b=e^a,bLa,b(r,t)e(ikziωa,bt)+c.c.
σ˙11=i2[Ωa*σ21Z12Ωaσ12Z21],σ˙22=i2[Ωaσ12Z21Ωa*σ21Z12+Ωbσ32Z23Ωb*σ23Z32],σ˙33=i2[Ωb*σ23Z32Ωbσ32Z23],σ˙12=i2[Ωa*(σ22σ11)Z12Ωb*σ13Z32],σ˙32=i2[Ωb*(σ22σ33)Z32Ωa*σ31Z12],σ˙13=i2[Ωa*σ23Z12Ωbσ12Z23],
Ωa,b=2pa,be^Aa,b.
u+=σ23+σ32,u=i(σ23σ32),u3=σ22σ33,v+=σ21+σ12,v=i(σ21σ12),v3=σ22σ11,t+=σ13+σ31,t=i(σ13σ31).
u˙+=Z32su312Z12qt++12Z12pt,u˙=Z32ru312Z12pt+12Z12qt,v˙+=Z12qv312Z32st+12Z32rt,v˙=Z12pv312Z32rt++12Z32st,t˙+=Z12qu++12Z12pu+12Z32sv++12Z32rv,t˙=Z12pu++12Z12qu+12Z32rv++12Z32sv,v˙3=Z12qv+Z12pv12Z32su+12Z32ru,u˙3=12Z12qv+12Z12pvZ32su+Z32ru.
Aa(z,t)z+1vgAa(z,t)t=iωaLa4ε0vg,
Ab(z,t)z+1vgAb(z,t)t=iωbLb4ε0vg,
η(t=0)=(0,0,0,0,0,1,0,0)t.
u+(t)=4S|Θa2|Ψ3[sin(Ψ)sin2(Ψ4)],u(t)=4R|Θa2|Ψ3[sin(Ψ)sin2(Ψ4)],etc.
Ψ=Z122(P2+Q2)+Z322(R2+S2)=Θa2+Θb2.
σ21=12(ν+iν)=12(QiP)Z12Ψ3[Θa2sin(Ψ)+2Θb2sin(Ψ2)].
Aa(z,t)z+1vgAa(z,t)t=iωaLa4ε0vg=npaωa4ε0vgXa,
Ab(z,t)z+1vgAb(z,t)t=iωbLb4ε0vg=npbωb4ε0vgXb,
t(Θaz+1vgΘat)=n|pa|2Z12ωa2ε0vgXa,t(Θbz+1vgΘbt)=n|pb|2Z32ωb2ε0vgXb.
1q2vg2Θaz∂́τ=n|pa|2Z12ωa2ε0vgXaand1q2vg2Θbz∂́τ=n|pb|2Z32ωb2ε0vgXb.
d2Θads2=Xa,
d2Θbds2=βXb,

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