Abstract

We investigate the electromagnetically induced transparency (EIT) and nonlinear pulse propagation in a Λ-type three-level atomic gas filled in a slot waveguide, in which electric field is strongly confined inside the slot of the waveguide due to the discontinuity of dielectric constant. We find that EIT effect can be greatly enhanced due to the reduction of optical-field mode volume contributed by waveguide geometry. Comparing with the atomic gases in free space, the EIT transparency window in the slot waveguide system can be much wider and deeper, and the Kerr nonlinearity of probe laser field can be much stronger. We also prove that using slot waveguide ultraslow optical solitons can be produced efficiently with extremely low generation power.

© 2013 OSA

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2012 (3)

2011 (1)

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

2010 (5)

2009 (7)

M. P. Hiscocks, C. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express17, 7295–7303 (2009).
[CrossRef] [PubMed]

P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express17, 9282–9287 (2009).
[CrossRef] [PubMed]

F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A.79, 013818 (2009).
[CrossRef]

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009).
[CrossRef]

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon.3, 706–714 (2009).
[CrossRef]

2008 (1)

2007 (2)

2006 (3)

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006).
[CrossRef]

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. 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).
[CrossRef]

2005 (4)

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E.72, 016617 (2005).
[CrossRef]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

F. Benabid, P. Light, F. Couny, and P. Russell, “Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF,” Opt. Express13, 5694–5703 (2005).
[CrossRef] [PubMed]

2004 (3)

2003 (2)

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
[CrossRef]

2001 (1)

1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature.397, 594–598 (1999).
[CrossRef]

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Almeida, V. R.

Andreani, L. C.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. 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).
[CrossRef]

Aras, M. S.

Artoni, M.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

Bajcsy, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Balic, V.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Bao, X.-H.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Barrios, C. A.

Beausoleil, R. G.

Bednar, C. J.

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature.397, 594–598 (1999).
[CrossRef]

Benabid, F.

Bhagwat, A. R.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
[CrossRef]

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

Bulu, I.

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009).
[CrossRef]

Canino, A.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. 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).
[CrossRef]

Cataliotti, F.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

Chen, S.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Chen, Z.-B.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Cluzel, B.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Couny, F.

Dai, H. N.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

Dai, H.-N.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

de Fornel, F.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Deng, L.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E.72, 016617 (2005).
[CrossRef]

Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett.93, 143904 (2004).
[CrossRef] [PubMed]

Deng, Y.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Deng, Y.-J.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature.397, 594–598 (1999).
[CrossRef]

Engeness, T.

Fink, Y.

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

Foubert, K.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Freude, W.

Fry, E. S.

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Gaeta, A. L.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
[CrossRef]

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

Galli, M.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. 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).
[CrossRef]

Gao, J.

Gerace, D.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. 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).
[CrossRef]

Ghosh, S.

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

Gibson, B. C.

Goh, S.

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

Greentree, A. D.

Guo, R.

Hadji, E.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Hafezi, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
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M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Left: Schematic of slot waveguide structure. The slot width (index nS) is 2a, the width of the high-index silicon slabs (index nH) is 2b − 2a. The index of the cladding material is nC. Right: Level diagram of the three-level atomic system. Ground state |1〉 couples to the exited |2〉 and |3〉 with the control field Ωc and the probe field Ωp. Δ2 and Δ3 are the detunings of control and probe fields, respectively. Γ3132) is the spontaneous emission decay rate from |3〉 to |1〉 (|2〉). Γ12 and Γ21 are incoherent population exchange rates. Below the level diagram is the coordinate system chosen for theoretical calculations. The slot region is |z| ≤ a, and the slabs are in the region a < |z| < b.

Fig. 2
Fig. 2

(a): Im(K) as a function of frequency ω for different slot width 2a. The red solid, black dashed, and blue dashed-dotted lines are for the slot width 2a = 50, 30 and 10 nm, respectively. (b): Im(K0) as a function of |Ωc| for different slot width 2a. The red solid, black dashed, and blue dashed-dotted lines are for the slot width 2a = 50, 30, 10 nm, respectively.

Fig. 3
Fig. 3

(a): Im(K) as a function of ω. (b): Im(K0) as a function of |Ωc| with Γ21 = 0 (red solid line), Γ21 = 0.5Γ12 (black dashed line) and Γ21 = Γ12 (blue dashed-dotted line), respectively.

Fig. 4
Fig. 4

Third-order susceptibility Re ( χ p p ( 3 ) ) as a function of detunning Δ3 for different slot width 2a. The red solid, black dashed and blue dashed-dotted lines are for 2a = 50,25 and 5 nm, respectively.

Fig. 5
Fig. 5

(a) The three-dimensional plot of the wave shape |Ωp/U0|2 as a function of z/LD and t/τ0. The solution is numerically obtained from Eq. (14) with full complex coefficients taken into account. The values of parameters are given in the text. (b): The interaction between two identical bright solitons.

Equations (50)

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E TM ( r , t ) = k h ¯ ω 1 ( k ) 2 ε 0 V 1 u 1 , k ( z ) a 1 ( k ) e i [ k y ω 1 ( k ) t ] + c . c . ,
u 1 , k ( z ) = c 2 N 1 ω 1 ( k ) n 2 ( z ) [ k H 1 , k ( z ) e z + i d H 1 , k ( z ) d z e y ] .
E TM ( r , t ) = l = p , c 1 ( W 1 V 1 ) 1 2 u 1 , l ( z ) exp { i [ k ( ω l ) y ω l t ] } + c . c ..
^ = h ¯ j 3 Δ j | j j | h ¯ [ ζ p * ( z ) Ω p * | 1 3 | + ζ c * ( z ) Ω c * | 2 3 | + h . c . ] ,
i t σ 11 + i Γ 21 σ 11 i Γ 12 σ 22 i Γ 13 σ 33 + ζ p * ( z ) Ω p * σ 31 ζ p ( z ) Ω p σ 31 * = 0 ,
i t σ 22 i Γ 21 σ 11 + i Γ 12 σ 22 i Γ 23 σ 33 + ζ c * ( z ) Ω c * σ 32 ζ c ( z ) Ω c σ 32 * = 0 ,
i t σ 33 + i ( Γ 13 + Γ 23 ) σ 33 ζ p * ( z ) Ω p * σ 31 + ζ p ( z ) Ω p σ 31 * ζ c * ( z ) Ω c * σ 32 + ζ c ( z ) Ω c σ 32 * = 0 ,
( i t + d 21 ) σ 21 ζ p ( z ) Ω p σ 32 * + ζ c * ( z ) Ω c * σ 31 = 0 ,
( i t + d 31 ) σ 31 ζ p ( z ) Ω p ( σ 33 σ 11 ) + ζ c ( z ) Ω c σ 21 = 0 ,
( i t + d 32 ) σ 32 ζ c ( z ) Ω c ( σ 33 σ 22 ) + ζ p ( z ) Ω p σ 21 * = 0 ,
P = 𝒩 a d v f ( v ) [ p 13 σ 31 e i ( k p y ω p t ) + p 23 σ 32 e i ( k c y ω c t ) + c . c . ] ,
i ( y + 1 c n 2 ( z ) n eff t ) Ω p + c 2 ω p n eff 2 Ω p x 2 + κ 13 d v f v σ 31 ( v , z ) = 0 ,
σ 11 ( 0 ) = Γ 12 ( Γ 13 + Γ 23 ) X 1 + ( Γ 12 + Γ 13 ) | ζ ( z ) Ω c | 2 X 2 ,
σ 22 ( 0 ) = Γ 21 ( Γ 13 + Γ 23 ) X 1 + Γ 21 | ζ ( z ) Ω c | 2 X 2 ,
σ 33 ( 0 ) = Γ 21 | ζ ( z ) Ω c | 2 X 2 ,
σ 32 ( 0 ) = ζ ( z ) Ω c d 32 Γ 21 ( Γ 13 + Γ 23 ) X 1 X 2 ,
K ( ω ) = 1 d z | ζ ( z ) | 2 d z | ζ ( z ) | 2 { ω c n 2 ( z ) n eff + κ 13 d v f ( v ) ζ ( z ) Ω c σ 32 * ( 0 ) + ( ω + d 21 ) ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) | ζ ( z ) Ω c | 2 ( ω + d 21 ) ( ω + d 31 ) } .
K ( ω ) = d z | ζ ( z ) | 2 ( ω c n 2 ( z ) n eff + 𝒦 1 ( z ) + 𝒦 2 ( z ) ) / d z | ζ ( z ) | 2 ,
𝒦 1 ( z ) = π κ 13 { | ζ ( z ) Ω c | 2 Γ 21 Γ 3 ( i Δ ω D i γ 32 ) + ( ω + i γ 21 ) [ Γ 12 Γ 3 ( Δ ω D 2 + γ 32 2 ) + 2 γ 32 ( Γ 12 Γ 21 + Γ 13 ) | ζ ( z ) Ω c | 2 ] } / { ( Δ ω D 2 + η 2 ) ( Γ 12 + Γ 21 ) Γ 3 [ | ζ ( z ) Ω c | 2 ( ω + i γ 21 ) ( ω + i Δ ω D + i γ 31 ) ] } ,
𝒦 2 ( z ) = π κ 13 Δ ω D { | ζ ( z ) Ω c | 2 Γ 21 Γ 3 ( i η i γ 32 ) + ( ω + i γ 21 ) [ Γ 12 Γ 3 ( η 2 + γ 32 2 ) + 2 γ 32 ( Γ 12 Γ 21 + Γ 13 ) | ζ ( z ) Ω c | 2 ] } / { η ( Δ ω D 2 η 2 ) ( Γ 12 + Γ 21 ) Γ 3 [ | ζ ( z ) Ω c | 2 ( ω + i γ 21 ) ( ω + i η + i γ 31 ) ] } .
Im ( K 0 ) = 1 d z | ζ ( z ) | 2 d z | ζ ( z ) | 2 d v f ( v ) κ 13 γ 21 | ζ ( z ) Ω c | 2 + γ 21 γ 31 × ( 1 Γ 21 Γ 3 X 1 + 3 Γ 21 | ζ ( z ) Ω c | 2 ( Γ 21 + Γ 12 ) Γ 3 X 1 + ( 2 Γ 21 + Γ 12 + Γ 13 ) | ζ ( z ) Ω c | 2 ) .
χ p = d v f ( v ) 𝒩 a | p 13 | 2 ε 0 h ¯ σ 31 ( v , z ) Ω p χ p ( 1 ) + χ p p ( 3 ) | p | 2 ,
χ p ( 1 ) = 𝒩 a | p 13 | 2 ε 0 h ¯ d z | ζ ( z ) | 2 d v f ( v ) × d 21 d 32 * ( σ 33 ( 0 ) σ 11 ( 0 ) ) | ζ ( z ) Ω c | 2 ( σ 33 ( 0 ) σ 22 ( 0 ) ) d 32 * ( d 21 d 31 | ζ ( z ) Ω c | 2 ) / d z | ζ ( z ) | 2 ,
χ p p ( 3 ) = 𝒩 a | p 13 | 4 ε 0 h ¯ 3 1 d z | ζ ( z ) | 2 d z | ζ ( z ) | 2 d v f ( v ) × | ζ ( z ) | 2 [ i d 32 ζ ( z ) Ω c Z 1 Z 4 ( 2 d 21 | d 32 | 2 d 32 | ζ ( z ) Ω c | 2 ) Z 2 ( d 21 | d 32 | 2 2 d 32 | ζ ( z ) Ω c | 2 ) Z 3 ] / [ i Z 1 | d 32 | 2 ( | ζ ( z ) Ω c | 2 d 21 d 31 ) ] ,
Ω p ( 1 ) = Fe i θ ,
σ 31 ( 1 ) = ( ω + d 21 ) ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) + ζ ( z ) Ω c σ 32 * ( 0 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 ) ( ω + d 31 ) ζ ( z ) Fe i θ ,
σ 21 ( 1 ) = ( ω + d 31 ) σ 32 * ( 0 ) + ζ c * ( z ) Ω c * ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 ) ( ω + d 31 ) ζ ( z ) Fe i θ ,
i F y 2 + c 2 ω p n eff 2 F x 1 2 K 2 2 2 F t 1 2 W | F | 2 Fe 2 α ¯ y 2 = 0 ,
W = κ 13 d z | ζ ( z ) | 4 d v f ( v ) ζ ( z ) Ω c a 32 * ( 2 ) + ( ω + d 21 ) ( 2 a 11 ( 2 ) + a 22 ( 2 ) ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 ) ( ω + d 31 ) / d z | ζ ( z ) | 2 ,
i ( y + α ) U + c 2 ω p n eff 2 U x 2 K 2 2 2 U τ 2 W | U | 2 U = 0 ,
i u s + 2 u σ 2 + 2 u | u | 2 = i d 0 u + d 1 2 u ξ 2 ,
u = 2 β sech [ 2 β ( σ σ 0 + 4 δ s ) ] exp [ 2 i δ σ 4 i ( δ 2 β 2 ) s i ϕ 0 ] ,
Ω p = 1 τ 0 K ˜ 2 W ˜ sech [ 1 τ 0 ( t y v ˜ g ) ] exp [ i K ˜ 0 y + i y 2 L D ] ,
H m , k | | TM ( r , t ) = ( k ^ | | × e z ) H m , k | | ( z ) e i ( k | | r ω m t ) + c . c . ,
E m , k | | TM ( r , t ) = i ω m ( k | | ) ε 0 ε ( z ) [ i k | | H m , k | | ( z ) e z + d H m , k | | ( z ) d z k ^ | | ] e i ( k | | r ω m t ) + c . c . ,
d 2 d z 2 H m , k | | ( z ) + [ ( ω c ) 2 n 2 ( z ) k | | 2 ] H m , k | | ( z ) = 0 ,
H m , k | | TM ( z ) = { cosh ( γ S m z ) , | z | < a C m cos [ κ H m ( | z | a ) ] + D m sin [ κ H m ( | z | a ) ] , a < | z | < b E m exp [ γ S m ( | z | b ) ] , | z | > b
tan [ κ H m ( b a ) arctan ( γ S m n H 2 κ H m n S 2 ) ] = γ S n n H 2 κ H m n S 2 tanh ( γ S m a ) .
E TM ( r , t ) = k x , k y m = 1 h ¯ ω m 2 ε 0 V m u m , k | | ( z ) a ^ m ( k ) e i ( k | | r ω m t ) + h . c . ,
H TM ( r , t ) = k x , k y m = 1 h ¯ ω m 2 ε 0 V m ( k ^ | | × e z ) ε 0 c N m H m , k | | ( z ) a ^ m ( k ) e i ( k | | r ω m t ) + h . c . ,
V m = S { 1 2 γ S m [ sinh ( 2 γ S m a ) + 2 E m 2 ] + a 2 + ( C m 2 + D m 2 ) ( b a ) + 1 2 κ H m ( C m 2 D m 2 ) sin [ 2 κ H m ( b a ) ] } / N m ,
N m = ω m 2 / ( c P ) 2 ,
H m , k | | TM ( z ) = { sin ϕ m exp ( z + b 2 b ψ m cos ϕ m ) , z < b cos ( ψ m z b sin ϕ m ) , b < z < b sin ϕ m exp ( b z 2 b ψ m cos ϕ m ) , z > b
E TM ( r , t ) = k x , k y m = 1 h ¯ ω m 2 ε 0 W m u m , k | | ( z ) a ^ n ( k ) e i ( k | | r ω m t ) + h . c . ,
H TM ( r , t ) = k x , k y m = 1 h ¯ ω m 2 ε 0 W m ( k ^ | | × e z ) ε 0 c M m H m , k | | ( z ) a ^ m ( k ) e i ( k | | r ω m t ) + h . c . ,
W m = S { b [ 2 sin 2 ϕ m + ψ m cos ϕ m ] / ( ψ m cos ϕ m ) + b sin ( 2 ψ m sin ϕ m ) / ( 2 ψ m sin ϕ m ) } / M m ,
M m = ω m 2 / ( c G ) 2 ,
a 11 ( 2 ) = i { [ i ( Γ 12 + Γ 23 ) 2 | ζ 2 ( z ) Ω c | 2 ( 1 d 32 1 d 32 * ) ] [ ( ω + d 21 * ) ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) + ζ * ( z ) Ω c * σ 32 ( 0 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 * ) ( ω + d 31 * ) c . c . ] + i ( Γ 12 Γ 13 ) [ ζ * ( z ) Ω c * d 32 ( ω + d 31 * ) σ 32 ( 0 ) + ζ ( z ) Ω c ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 * ) ( ω + d 31 * ) + c . c . ] } / [ i ( Γ 12 + Γ 21 ) ( Γ 13 + Γ 23 ) ( 2 Γ 21 + Γ 12 + Γ 13 ) | ζ 2 ( z ) Ω c | 2 ( 1 d 32 1 d 32 * ) ] ,
a 22 ( 2 ) = i Γ 12 Γ 13 [ ( ω + d 21 * ) ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) + ζ * ( z ) Ω c * σ 32 ( 0 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 * ) ( ω + d 31 * ) c . c . i ( Γ 21 + Γ 13 ) a 11 ( 2 ) ] ,
a 32 ( 2 ) = 1 d 32 [ ( ω + d 31 * ) σ 32 ( 0 ) + ζ ( z ) Ω c ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 * ) ( ω + d 31 * ) ζ ( z ) Ω c ( a 11 ( 2 ) + 2 a 22 ( 2 ) ) ] .

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