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 OSA

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2010 (13)

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. 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. 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]

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,” Nature Mater. 9, 193–204 (2010).
[Crossref]

M. Makarova, Y. Gong, S.-L. Cheng, Y. Nishi, S. Yerci, R. Li, L. D. Negro, and J. Vuc̆ović, “Photonic crystal and plasmonic silicon-based light sources,” IEEE J. Sel. Top. Quantum Electron. 16, 132–140 (2010).
[Crossref]

A. Hryciw, Y. C. Jun, and M. L. Brongersma, “Plasmonics: Electrifying plasmonics on silicon,” Nature Mater. 9, 3–4 (2010).
[Crossref]

R. J. Walters, R. V. A. van Loon, I. Brunets, J. Schmitz, and A. Polman, “A silicon-based electrical source of surface plasmon polaritons,” Nature Mater. 9, 21–25 (2010).
[Crossref]

A. V. Krasavin and A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18, 11791–11799 (2010).
[Crossref] [PubMed]

B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B 27, 956–965 (2010).
[Crossref]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18, 21427–21448 (2010).
[Crossref] [PubMed]

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2010).
[Crossref] [PubMed]

2009 (6)

2008 (3)

J. Basak, L. Liao, A. Liu, D. Rubin, Y. Chetrit, H. Nguyen, D. Samara-Rubio, R. Cohen, N. Izhaky, and M. Paniccia, “Developments in gigascale silicon optical modulators using free carrier dispersion mechanisms,” Adv. Opt. Technol. 2008, 678948(1–10) (2008).

M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express 16, 1385–1392 (2008).
[Crossref] [PubMed]

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007(1–9) (2008).
[Crossref]

2007 (5)

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[Crossref] [PubMed]

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 89, 221–223 (2007).
[Crossref]

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115(1–3) (2007).
[Crossref]

Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15, 924–929 (2007).
[Crossref] [PubMed]

2006 (9)

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

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[Crossref]

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 μm femtosecond pulses,” Opt. Express 14, 8336–8346 (2006).
[Crossref] [PubMed]

Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31, 341–343 (2006).
[Crossref] [PubMed]

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(1–9) (2006).
[Crossref]

2005 (2)

2004 (2)

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12, 4072–4079 (2004).
[Crossref] [PubMed]

M. W. Geis, S. J. Spector, R. C. Williamson, and T. M. Lyszczarz, “Submicrosecond, submilliwatt, silicon-on-insulator thermooptic switch,” IEEE Photon. Technol. Lett. 16, 2514–2516 (2004).
[Crossref]

2003 (2)

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
[Crossref]

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

2001 (1)

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

Abdollahi, S.

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]

Adegoke, J. A.

Afshar, S.

Agrawal, G. P.

B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B 27, 956–965 (2010).
[Crossref]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18, 21427–21448 (2010).
[Crossref] [PubMed]

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010).
[Crossref] [PubMed]

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]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2010).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach,” J. Lightwave Technol. 27, 3241–3248 (2009).
[Crossref]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[Crossref] [PubMed]

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]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(1–9) (2006).
[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]

Bahoura, M.

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,” Nature 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]

Basak, J.

J. Basak, L. Liao, A. Liu, D. Rubin, Y. Chetrit, H. Nguyen, D. Samara-Rubio, R. Cohen, N. Izhaky, and M. Paniccia, “Developments in gigascale silicon optical modulators using free carrier dispersion mechanisms,” Adv. Opt. Technol. 2008, 678948(1–10) (2008).

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.

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115(1–3) (2007).
[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]

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).

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,” Nature Mater. 9, 193–204 (2010).
[Crossref]

A. Hryciw, Y. C. Jun, and M. L. Brongersma, “Plasmonics: Electrifying plasmonics on silicon,” Nature Mater. 9, 3–4 (2010).
[Crossref]

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

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U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
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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,” Nature 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 89, 221–223 (2007).
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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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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).
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J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(1–9) (2006).
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Yuksek, N. S.

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Adv. Opt. Technol. (1)

J. Basak, L. Liao, A. Liu, D. Rubin, Y. Chetrit, H. Nguyen, D. Samara-Rubio, R. Cohen, N. Izhaky, and M. Paniccia, “Developments in gigascale silicon optical modulators using free carrier dispersion mechanisms,” Adv. Opt. Technol. 2008, 678948(1–10) (2008).

Appl. Opt. (1)

Appl. Phys. A (1)

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

Appl. Phys. Lett. (2)

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115(1–3) (2007).
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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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