Abstract

Silicon-based plasmonic waveguides can be used to simultaneously transmit electrical signals and guide optical energy with deep subwavelength localization, thus providing us with a well needed connecting link between contemporary nanoelectronics and silicon photonics. In this paper, we examine the possibility of employing the large third-order nonlinearity of silicon to create active and passive photonic devices with silicon-based plasmonic waveguides. We unambiguously demonstrate that the relatively weak dependance of the Kerr effect, two-photon absorption (TPA), and stimulated Raman scattering on optical intensity, prevents them from being useful in μm-long plasmonic waveguides. On the other hand, the TPA-initiated free-carrier effects of absorption and dispersion are much more vigorous, and have strong potential for a variety of practical applications. Our work aims to guide research efforts towards the most promising nonlinear optical phenomena in the thriving new field of silicon-based plasmonics.

© 2011 Optical Society of America

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References

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2010

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-based plasmonics for on-chip photonics,” IEEE J. Sel. Top. Quantum Electron. 16, 295–306 (2010).
[CrossRef]

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

J. Y. Lee, L. Yin, G. P. Agrawal, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express 18, 11514–11523 (2010).
[CrossRef] [PubMed]

J. N. Caspers, N. Rotenberg, and H. M. van Driel, “Ultrafast silicon-based active plasmonics at telecom wavelengths,” Opt. Express 18, 19761–19769 (2010).
[CrossRef] [PubMed]

2009

L. Tang, S. Latif, and D. A. B. Miller, “Plasmonic device in silicon CMOS,” Electron. Lett. 45, 706 (2009).
[CrossRef]

2008

2007

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

M. L. Brongersma, R. Zia, and J. A. Schuller, “Plasmonics—the missing link between nanoelectronics and microphotonics,” Appl. Phys., A Mater. Sci. Process. 89, 221–223 (2007).
[CrossRef]

2006

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 ?m femtosecond pulses,” Opt. Express 14, 8336–8346 (2006).
[CrossRef] [PubMed]

2005

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

2001

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
[CrossRef]

Abedin, K. S.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Agrawal, G. P.

Alivisatos, A. P.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-based plasmonics for on-chip photonics,” IEEE J. Sel. Top. Quantum Electron. 16, 295–306 (2010).
[CrossRef]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-based plasmonics for on-chip photonics,” IEEE J. Sel. Top. Quantum Electron. 16, 295–306 (2010).
[CrossRef]

Baets, R.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Bogaerts, W.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Boyraz, O.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

M. L. Brongersma, R. Zia, and J. A. Schuller, “Plasmonics—the missing link between nanoelectronics and microphotonics,” Appl. Phys., A Mater. Sci. Process. 89, 221–223 (2007).
[CrossRef]

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Caspers, J. N.

Chulkov, E. V.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Cohen, O.

Dekker, R.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
[CrossRef]

Dimitropoulos, D.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-based plasmonics for on-chip photonics,” IEEE J. Sel. Top. Quantum Electron. 16, 295–306 (2010).
[CrossRef]

Driessen, A.

Dumon, P.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Echenique, P. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Pannicia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Fauchet, P. M.

Först, M.

Hak, D.

Jalali, B.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

Jones, R.

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Latif, S.

L. Tang, S. Latif, and D. A. B. Miller, “Plasmonic device in silicon CMOS,” Electron. Lett. 45, 706 (2009).
[CrossRef]

Lee, J. Y.

Liang, T. K.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Liu, A.

Miller, D. A. B.

L. Tang, S. Latif, and D. A. B. Miller, “Plasmonic device in silicon CMOS,” Electron. Lett. 45, 706 (2009).
[CrossRef]

Miyazaki, T.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Moormann, C.

Niehusmann, J.

Nunes, L. R.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Ozbay, E.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Paniccia, M.

Pannicia, M.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Pannicia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Pitarke, J. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Raghunathan, V.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

Rong, H.

Rotenberg, N.

Schröter, U.

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
[CrossRef]

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

M. L. Brongersma, R. Zia, and J. A. Schuller, “Plasmonics—the missing link between nanoelectronics and microphotonics,” Appl. Phys., A Mater. Sci. Process. 89, 221–223 (2007).
[CrossRef]

Sheldon, M. T.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-based plasmonics for on-chip photonics,” IEEE J. Sel. Top. Quantum Electron. 16, 295–306 (2010).
[CrossRef]

Silkin, V. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Soref, R. A.

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-based plasmonics for on-chip photonics,” IEEE J. Sel. Top. Quantum Electron. 16, 295–306 (2010).
[CrossRef]

Tang, L.

L. Tang, S. Latif, and D. A. B. Miller, “Plasmonic device in silicon CMOS,” Electron. Lett. 45, 706 (2009).
[CrossRef]

Thourhout, D. V.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Tsang, H.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Tsuchiya, M.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

van Driel, H. M.

Wahlbrink, T.

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Yin, L.

Zia, R.

M. L. Brongersma, R. Zia, and J. A. Schuller, “Plasmonics—the missing link between nanoelectronics and microphotonics,” Appl. Phys., A Mater. Sci. Process. 89, 221–223 (2007).
[CrossRef]

Appl. Phys., A Mater. Sci. Process.

M. L. Brongersma, R. Zia, and J. A. Schuller, “Plasmonics—the missing link between nanoelectronics and microphotonics,” Appl. Phys., A Mater. Sci. Process. 89, 221–223 (2007).
[CrossRef]

Electron. Lett.

L. Tang, S. Latif, and D. A. B. Miller, “Plasmonic device in silicon CMOS,” Electron. Lett. 45, 706 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-based plasmonics for on-chip photonics,” IEEE J. Sel. Top. Quantum Electron. 16, 295–306 (2010).
[CrossRef]

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

Nat. Mater.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Nature

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Pannicia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Opt. Commun.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
[CrossRef]

Opt. Express

Phys. Rev. B

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
[CrossRef]

Rep. Prog. Phys.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[CrossRef]

Science

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Other

G. T. Reed, and A. P. Knights, Silicon Photonics: An Introduction (Wiley, Hoboken, 2004).
[CrossRef]

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, “Silicon Raman amplifiers, lasers and their applications,” in Active and Passive Optical Components for WDM Communications V, A. K. Dutta, Y. Ohishi, N. K. Dutta, and J. Moerk, Eds., Proc. SPIE 6014, 21–26 (2005).

S. A. Maier, Plasmonics: Fundamentals and Applications, (Springer, 2007).

S. M. Sederberg, V. Van, and A. Y. Elezzabi, “Silicon-based plasmonic waveguides interfaced to silicon photonic platform,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper JThE4.

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Figures (4)

Fig. 1
Fig. 1

Geometry of a metal–silicon–metal plasmonic waveguide and electromagnetic field components of the TM SPP mode. Two metallic regions of permittivity ɛ2 are separated by a silicon layer of thickness d and permittivity ɛ1.

Fig. 2
Fig. 2

(a) Parameters Γ, η, ℬ, and �� for the fundamental mode as functions of silicon layer thickness d in MSM (Ag/Si/Ag) and SOI waveguides; circles correspond to �� = d. (b) Plasmonic attenuation factor ϖ and SPP propagation length LSPP for the same Ag/Si/Ag waveguide.

Fig. 3
Fig. 3

(a) Group velocity vg = (Reβ1)−1 and dispersion parameter Re(β2) as a function of the silicon-layer thickness in an MSM plasmonic waveguide; dashed curve shows the group velocity in an SOI waveguide. (b) Ratios R1 = |βeff|/(ζr��) and R2 = γeff/(|ζi|��) as functions of waveguide thickness d for τeff = 5 ns, 50 ps, and 5 ps. Other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

(a) Intensity and (b) nonlinear phase shift as functions of d for a quasi-CW SPP excited inside a plasmonic waveguide (Ag/Si/Ag) when free-carrier effects dominate over TPA and the Kerr effect; dashed curve shows propagation length of SPPs. Simulation parameters are the same as in Fig. 2.

Equations (31)

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E ( x , z , t ) 2 𝒩 A ( z , t ) E 0 ( x , ω ) exp [ i ( β z ω t ) ] + c . c . ,
H ( x , z , t ) 2 𝒩 A ( z , t ) H 0 ( x , ω ) exp [ i ( β z ω t ) ] + c . c . ,
𝒩 = 1 2 + e z [ E 0 ( x , ω ) × H 0 * ( x , ω ) ] d x + c . c . ,
P ( z , t ) = | A ( z , t ) | 2 exp ( z / L SPP ) ,
𝒟 = 1 Q ( + | E 0 | 2 d x ) 2 , Q = + | E 0 | 4 d x ,
E x ( x , ω ) = { cos ( k 1 x ) , | x | < d / 2 , cos ( k 1 d / 2 ) ( ɛ 1 / ɛ 2 ) exp [ i k 2 ( | x | d / 2 ) ] , | x | > d / 2 ,
E z ( x , ω ) = k 1 i β × { sin ( k 1 x ) , | x | d / 2 , sin ( k 1 d / 2 ) sign ( x ) exp [ i k 2 ( | x | d / 2 ) ] , | x | d / 2 ,
tanh [ i k 1 ( ω ) d 2 ] = ɛ 1 ( ω ) ɛ 2 ( ω ) k 2 ( ω ) k 1 ( ω ) ,
β ( ω , d ) 1 d { ɛ 2 ( k d ) 2 + [ ɛ 1 / ɛ 2 ( ɛ 1 / ɛ 2 ) 2 + ( ɛ 1 ɛ 2 ) ( k d ) 2 ] 2 } 1 / 2 .
i ( β + β * ) | E x | 2 + E x d E z * d x E x * d E z d x = i ω μ 0 ( E 0 × H 0 * + E 0 * + H 0 ) e z ,
A z + n = 1 i n 1 β b n ! n A t n = i γ 1 + ir 1 + i ϖ | A | 2 A 1 + i ϖ ( σ α 2 + i k σ n ) N A ,
N t = N τ eff + β ^ TPA 2 h ¯ ω 𝒟 d | A | 4 ,
γ = k n 2 η 𝒟 ( n 0 k Γ Re β ) 2 , r = β TPA 2 k n 2 , = ɛ 0 n 0 c 𝒩 d / 2 + d / 2 | E 0 | 2 d x , Γ = Re β μ 0 ω 𝒩 + | E 0 | 2 d x ,
η = 1 Q κ λ μ v d / 2 + d / 2 κ λ μ v E κ * E λ * E μ * E ν d x , β ^ TPA = β TPA η 1 + ϖ 2 ( n 0 k Γ Re β ) 2 .
ϖ = Im β μ 0 ω 𝒩 + | E x | 2 d x ,
1 A A z = ( β eff 2 i γ eff ) | A | 2 ( ζ r 2 + i ζ i ) 0 | A ( z , τ τ eff q ) | 4 exp ( q ) d q ,
β eff = 2 γ r ϖ 1 + ϖ 2 , γ eff = γ 1 + ϖ r 1 + ϖ 2 , ζ r = β ^ TPA τ eff 2 h ¯ ω 𝒟 d σ α + 2 k σ n ϖ 1 + ϖ 2 , ζ i = 2 k σ n σ α ϖ σ α + 2 k σ n ϖ ζ r 2 ,
τ eff 1 = τ c 1 + 2 μ V / d 2 ,
I K ( z , τ ) = I 0 ( τ ) 1 + β eff I 0 ( τ ) 𝒟 z , φ K ( z , τ ) = γ eff β eff ln | 1 + β eff I 0 ( τ ) 𝒟 z | ;
I FC ( z , τ ) = I 0 ( τ ) 1 + 2 ζ r I 0 2 ( τ ) 𝒟 2 z , φ FC ( z , τ ) = ζ i 2 ζ r ln | 1 + 2 ζ r I 0 2 ( τ ) 𝒟 2 z | .
max [ I K ( z , τ ) ] min ( | β eff | ζ r 𝒟 , γ eff | ζ i | 𝒟 ) ,
I FC ( z , τ ) max ( | β eff | ζ r 𝒟 , γ eff | ζ i | 𝒟 ) .
i ( β + β * ) | E ˜ T | 2 + E ˜ T T E ˜ z * E ˜ T * T E ˜ z = i ω μ 0 ( E ˜ × H ˜ * + E ˜ × H ˜ ) e z ,
E ˜ ( r ) = E 0 ( x , y ) exp ( i β z ) , H ˜ ( r ) = H 0 ( x , y ) exp ( i β z ) .
i ω μ 0 ( E ˜ × H ˜ * + E ˜ * × H ˜ ) e z = e z [ E ˜ * × [ × E ˜ ] ] c . c .
e z [ E ˜ * × [ × E ˜ ] ] = e z [ ( E ˜ * E ˜ ) ( E ˜ * ) E ˜ ] = E ˜ * E ˜ z ( E ˜ * ) E z = i β | E ˜ | 2 ( E ˜ T * T ) E z i β E ˜ z 2 = i β | E ˜ T | 2 ( E ˜ T * T ) E z .
1 d / 2 + d / 2 | E 0 | 2 d x = Π + ( 1 ) | k 1 β | 2 Π ( 1 ) , Π ± ( ν ) = 1 2 [ sin ( ν k 1 d ) k 1 ± sinh ( ν k 1 d ) k 1 ] , + | E 0 | 2 d x = 1 + 1 k 2 { | ɛ 1 ɛ 2 cos ( k 1 d 2 ) | 2 + | k 1 β sin ( k 1 d 2 ) | 2 } , + | E x | 2 d x = Π + ( 1 ) + 1 k 2 | ɛ 1 ɛ 2 cos ( k 1 d 2 ) | 2 ,
Q = Λ + + | k 1 β | 4 Λ 1 4 | k 1 β | 2 Π ( 2 ) + 1 2 k 2 { | ɛ 1 ɛ 2 cos ( k 1 d 2 ) | 2 + | k 1 β sin ( k 1 d 2 ) | 2 } 2 , Λ ± = d 4 + Π + ( 2 ) 8 ± 1 2 Re [ sin ( k 1 d ) k 1 ] ,
𝒩 = ɛ 0 ω { Re ( ɛ 1 β ) Π + ( 1 ) + Re ( ɛ 2 β ) 1 k 2 | ɛ 1 ɛ 2 cos ( k 1 d 2 ) | 2 } .
x x x x = 1 , z z z z = 1 + ρ 2 , x x z z = z z x x = x z z x = z x x z = x z x z = z x z x = ρ 3 ,
η Q = Λ + + 1 + ρ 2 | k 1 β | 4 Λ ρ 6 | k 1 β | 2 Π ( 2 ) 2 3 ρ { Re ( k 1 2 β 2 ) [ d 4 Π + ( 2 ) 8 ] + 1 2 Im ( k 1 2 β 2 ) Im [ sin ( k 1 d ) k 1 ] }

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