Abstract

Recent years have seen increased interest in the plasmonic enhancement of nonlinear optical effects, yet there remains an uncertainty as to the limits of this enhancement. We present a simple and physically transparent theory for the plasmonic enhancement of third order nonlinear optical processes and show that while a huge enhancement of the effective nonlinear index can be attained, the most relevant figure of merit, the phase shift per one absorption length, remains very low. This suggests that while nonlinear plasmonic materials are not suitable for applications requiring high efficiency, for example in all-optical switching and wavelength conversion, they can be very useful for applications where overall high efficiency is not critical, such as in sensing.

© 2013 Optical Society of America

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2012

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett.100(1), 011105 (2012).
[CrossRef]

G. Sun and J. B. Khurgin, “Origin of giant difference between fluorescence, resonance and non-resonance Raman scattering enhancement by surface plasmons,” Phys. Rev. A85(6), 063410 (2012).
[CrossRef]

M. B. Dühring, N. Asger Mortensen, and O. Sigmund, “Plasmonic versus dielectric enhancement in thin-film solar cells,” Appl. Phys. Lett.100(21), 211914 (2012).
[CrossRef]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics6(11), 737–748 (2012).
[CrossRef]

B. Sharma, R. R. Frontiera, A. Henry, E. Ringe, and R. P. van Duyne, “SERS: Materials, applications, and the future,” Mater. Today15(1-2), 16–25 (2012).
[CrossRef]

S. Linden, F. B. P. Niesler, J. Förstner, Y. Grynko, T. Meier, and M. Wegener, “Collective effects in second-harmonic generation from split-ring-resonator arrays,” Phys. Rev. Lett.109(1), 015502 (2012).
[CrossRef] [PubMed]

B. Borchers, C. Brée, S. Birkholz, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect in solids,” Opt. Lett.37(9), 1541–1543 (2012).
[CrossRef] [PubMed]

2011

G. Sun and J. B. Khurgin, “Optimization of the nanolens consisting of coupled metal nanoparticles: an analytical approach,” Appl. Phys. Lett.98(15), 153115 (2011).
[CrossRef]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

H. Baida, D. Mongin, D. Christofilos, G. Bachelier, A. Crut, P. Maioli, N. Del Fatti, and F. Vallée, “Ultrafast nonlinear optical response of a single gold nanorod near its surface plasmon resonance,” Phys. Rev. Lett.107(5), 057402 (2011).
[CrossRef] [PubMed]

M. Abb, P. Albella, J. Aizpurua, and O. L. Muskens, “All-optical control of a single plasmonic nanoantenna-ITO hybrid,” Nano Lett.11(6), 2457–2463 (2011).
[CrossRef] [PubMed]

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett.11(6), 2231–2235 (2011).
[CrossRef] [PubMed]

A. V. Krasavin, S. Randhawa, J.-S. Bouillard, J. Renger, R. Quidant, and A. V. Zayats, “Optically-programmable nonlinear photonic component for dielectric-loaded plasmonic circuitry,” Opt. Express19(25), 25222–25229 (2011).
[CrossRef] [PubMed]

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express19(22), 22029–22106 (2011).
[CrossRef] [PubMed]

G. Sun, J. B. Khurgin, and A. Bratkovsky, “Coupled-mode theory of field enhancement in complex metal nanostructures,” Phys. Rev. B84(4), 045415 (2011).
[CrossRef]

G. Sun and J. B. Khurgin, “Theory of optical emission enhancement by coupled metal nanoparticles: An analytical approach,” Appl. Phys. Lett.98(11), 113116 (2011).
[CrossRef]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett.99(21), 211106 (2011).
[CrossRef]

2010

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt.12(10), 104003 (2010).
[CrossRef]

G. Sun and J. B. Khurgin, “Comparative study of field enhancement between isolated and coupled metal nanoparticles: an analytical approach,” Appl. Phys. Lett.97(26), 263110 (2010).
[CrossRef]

I. Karakurt, C. H. Adams, P. Leiderer, J. Boneberg, and R. F. Haglund., “Nonreciprocal switching of VO2 thin films on microstructured surfaces,” Opt. Lett.35(10), 1506–1508 (2010).
[CrossRef] [PubMed]

2009

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(20), 203902 (2009).
[CrossRef] [PubMed]

S. C. Lee, S. Krishna, and S. R. J. Brueck, “Quantum dot infrared photodetector enhanced by surface plasma wave excitation,” Opt. Express17(25), 23160–23168 (2009).
[CrossRef] [PubMed]

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett.94(10), 101103 (2009).
[CrossRef]

J. B. Khurgin, G. Sun, and R. A. Soref, “Practical limits of absorption enhancement near metal nanoparticles,” Appl. Phys. Lett.94(7), 071103 (2009).
[CrossRef]

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and M. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics3(1), 55–58 (2009).
[CrossRef]

2008

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev.108(2), 462–493 (2008).
[CrossRef] [PubMed]

N. Feth, S. Linden, M. W. Klein, M. Decker, F. B. P. Niesler, Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, J. V. Moloney, and M. Wegener, “Second-harmonic generation from complementary split-ring resonators,” Opt. Lett.33(17), 1975–1977 (2008).
[CrossRef] [PubMed]

J. B. Khurgin, G. Sun, and R. A. Soref, “Electroluminescence efficiency enhancement using metal nanoparticles,” Appl. Phys. Lett.93(2), 021120 (2008).
[CrossRef]

E. J. R. Vesseur, R. de Waele, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Surface plasmon polariton modes in a single-crystal Au nanoresonator fabricated using focused-ion-beam milling,” Appl. Phys. Lett.92(8), 083110 (2008).
[CrossRef]

2007

J.-C. Weeber, A. Bouhelier, G. Colas de Francs, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett.7(5), 1352–1359 (2007).
[CrossRef] [PubMed]

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells,” J. Appl. Phys.101(9), 093105 (2007).
[CrossRef]

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express15(21), 14266–14274 (2007).
[CrossRef] [PubMed]

T. Xu, X. Jiao, G. P. Zhang, and S. Blair, “Second-harmonic emission from sub-wavelength apertures: effects of aperture symmetry and lattice arrangement,” Opt. Express15(21), 13894–13906 (2007).
[CrossRef] [PubMed]

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics1(7), 402–406 (2007).
[CrossRef]

2006

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature440(7083), 508–511 (2006).
[CrossRef] [PubMed]

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett.96(9), 097401 (2006).
[CrossRef] [PubMed]

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1961

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V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett.81(19), 4136–4139 (1998).
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N. Bloembergen and Y. R. Shen, “Quantum-theoretical comparison of nonlinear susceptibilities in parametric media, lasers, and Raman Lasers,” Phys. Rev.133(1A), A37–A49 (1964).
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J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127(6), 1918–1939 (1962).
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A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett.11(6), 2231–2235 (2011).
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C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett.46(2), 145–148 (1981).
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H. Baida, D. Mongin, D. Christofilos, G. Bachelier, A. Crut, P. Maioli, N. Del Fatti, and F. Vallée, “Ultrafast nonlinear optical response of a single gold nanorod near its surface plasmon resonance,” Phys. Rev. Lett.107(5), 057402 (2011).
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P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
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F. G. Della Corte, M. Esposito Montefusco, L. Moretti, I. Rendina, and G. Cocorullo, “Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models,” J. Appl. Phys.88(12), 7115–7119 (2000).
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S. I. Bozhevolnyi, J. Beermann, and V. Coello, “Direct observation of localized second-harmonic enhancement in random metal nanostructures,” Phys. Rev. Lett.90(19), 197403 (2003).
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J.-C. Weeber, A. Bouhelier, G. Colas de Francs, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett.7(5), 1352–1359 (2007).
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J. L. Coutaz, M. Nevière, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B Condens. Matter32(4), 2227–2232 (1985).
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H. Baida, D. Mongin, D. Christofilos, G. Bachelier, A. Crut, P. Maioli, N. Del Fatti, and F. Vallée, “Ultrafast nonlinear optical response of a single gold nanorod near its surface plasmon resonance,” Phys. Rev. Lett.107(5), 057402 (2011).
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K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett.78(9), 1667–1670 (1997).
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I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett.88(18), 187402 (2002).
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I. I. Smolyaninov, A. V. Zayats, and C. C. Davis, “Near-field second harmonic generation from a rough metal surface,” Phys. Rev. B56(15), 9290–9293 (1997).
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C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett.46(2), 145–148 (1981).
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E. J. R. Vesseur, R. de Waele, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Surface plasmon polariton modes in a single-crystal Au nanoresonator fabricated using focused-ion-beam milling,” Appl. Phys. Lett.92(8), 083110 (2008).
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Del Fatti, N.

H. Baida, D. Mongin, D. Christofilos, G. Bachelier, A. Crut, P. Maioli, N. Del Fatti, and F. Vallée, “Ultrafast nonlinear optical response of a single gold nanorod near its surface plasmon resonance,” Phys. Rev. Lett.107(5), 057402 (2011).
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F. G. Della Corte, M. Esposito Montefusco, L. Moretti, I. Rendina, and G. Cocorullo, “Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models,” J. Appl. Phys.88(12), 7115–7119 (2000).
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A. J. DeMaria, D. A. Stetson, and H. Heyma, “Mode locking of a Nd3+‐doped glass laser,” Appl. Phys. Lett.8(1), 22–24 (1966).
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Demircan, A.

Dereux, A.

J.-C. Weeber, A. Bouhelier, G. Colas de Francs, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett.7(5), 1352–1359 (2007).
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S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature440(7083), 508–511 (2006).
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A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett.11(6), 2231–2235 (2011).
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H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett.95(25), 257403 (2005).
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J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127(6), 1918–1939 (1962).
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M. B. Dühring, N. Asger Mortensen, and O. Sigmund, “Plasmonic versus dielectric enhancement in thin-film solar cells,” Appl. Phys. Lett.100(21), 211914 (2012).
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S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature440(7083), 508–511 (2006).
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F. G. Della Corte, M. Esposito Montefusco, L. Moretti, I. Rendina, and G. Cocorullo, “Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models,” J. Appl. Phys.88(12), 7115–7119 (2000).
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K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett.78(9), 1667–1670 (1997).
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Fleischmann, M.

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett.26(2), 163–166 (1974).
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S. Wu, X. C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett.61(8), 919–921 (1992).
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L. Hargrove, R. L. Fork, and R. L. Pollack, “Locking of HeNe laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett.5(1), 4–5 (1964).
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E. J. R. Vesseur, R. de Waele, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Surface plasmon polariton modes in a single-crystal Au nanoresonator fabricated using focused-ion-beam milling,” Appl. Phys. Lett.92(8), 083110 (2008).
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D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys.78(9), 5324 (1983).
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D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys.78(9), 5324 (1983).
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Figures (6)

Fig. 1
Fig. 1

(a) Spherical Ag nanoparticle with the electric field distribution. (b) Elliptical nanoparticle with resonance at 1320 nm and its associated electric field distribution. (c) Extinction spectrum of the elliptical nanoparticle.

Fig. 2
Fig. 2

Fields and polarizations in the plasmonically enhanced nonlinear metamaterials: (a) average E ¯ ω and local E ω electric fields, and dipole p ω at the pump frequency ω ; (b) local nonlinear field E ω' , dipole moment p nl ω' and average nonlinear polarization P ¯ nl ω' at the nonlinear output signal frequency ω' .

Fig. 3
Fig. 3

Dispersions of Q-factors for gold and silver nanoparticles.

Fig. 4
Fig. 4

(a) Spherical nanoparticle dimer with the electric field distribution. (b) Elliptical nanoparticle dimer resonant at 1320 nm and associated electric field distribution. (c) Extinction spectrum of the elliptical dimer.

Fig. 5
Fig. 5

Nonlinear phase shift in the chalcogenide waveguide doped with isolated Ag spheroids (a-c) and the dimers (d-f) with different filling factors f at various intensities. The phase shift of undoped waveguides is shown by dashed line.

Fig. 6
Fig. 6

Conversion efficiency in the FWM in the waveguide doped with isolated Ag spheroids (a) and the dimers (d-f) with different filling factors f. The phase shift of undoped waveguides is shown by dashed line.

Equations (40)

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p ω = ε m ε d ε m +2 ε d 4π ε 0 ε d a 3 E ¯ ω
p ω = α ω 0 2 E ¯ ω ω 0 2 ω 2 jωγ α Q L(ω) E ¯ ω
d 2 p d t 2 +γ dp dt = ω 0 2 p+ ω 0 2 α E ¯
E ω (r)={ p 4π ε 0 ε d a 3 r<a 1 4π ε 0 ε d r 3 [ 3( p r ^ ) r ^ p ] r>a
E max,ω 2βQ L(ω) E ¯ ω
ε eff = ε d + N s α ε 0 Q L(ω) = ε d ( 1+3fβ Q L(ω) )
n eff = ε eff 1/2 n d [ 1+ 3fβ 2 Q 2 (1 ω 2 / ω 0 2 ) | L(ω) | 2 + 3fβ 2 jQ | L(ω) | 2 ],
α a = 2π n d λ 3fβQ | L(ω) | 2 ,
p nl (r,t)= P max,nl ω' G(r) e jω't
2 E(r,t) ε r (r) c 2 д 2 д t 2 E(r,t)= 1 ε 0 0 c 2 д 2 д t 2 P nl (r,t).
E(r,t)= l E max,l ω' F l (r) e jω't
E max ω' = P max,nl ω κ ε 0 ε d Q L(ω')
κ= ε d F 1 (r)G(r)dV / ( ε r ' ω ) ω F 1 2 (r)dV .
p nl ω' = 3 2 V ε 0 ε d E max,nl ω' = 3 2 Vκ Q L(ω) P max,nl ω' .
P ¯ nl ω = 3 2 fκ Q L(ω) P max,nl ω
P nl ω 1 ω 2 + ω 3 (r)= ε 0 χ (3) ( ω 3 , ω 2 , ω 1 ) E ω 1 (r) E ω 2 * (r) E ω 3 (r).
E ω k (r)= 2βQ L( ω k ) E ¯ ω k F 1 (r).
P nl ω 1 ω 2 + ω 3 (r)= P max,nl ω 1 ω 2 + ω 3 G(r)
P max,nl ω 1 ω 2 + ω 3 = ε 0 | χ (3) ( ω 3 , ω 2 , ω 1 ) | ( 2βQ ) 3 L( ω 1 ) L * ( ω 2 )L( ω 3 ) E ¯ ω 1 E ¯ ω 2 * E ¯ ω 3 ,
G(r)= χ ( 3 ) /| χ ( 3 ) | F 1 (r) F 1 (r) F 1 (r),
P nl ω 4 = ε 0 χ eff (3) ( ω 3 , ω 2 , ω 1 ) E ¯ ω 1 E ¯ ω 2 * E ¯ ω 3
χ eff (3) 3 2 f κ 3 ( 2β ) 3 Q 4 L 2 (ω) | L(ω) | 2 χ (3)
κ 3 = ε d V eff,1 r>a F 1 (r) χ (3) | χ (3) | F 1 (r) F 1 (r) F 1 (r) d 3 r .
n 2,eff 3 2 f κ 3 ( 2β ) 3 Q 4 L 2 ( ω 2 ) | L( ω 1 ) | 2 n 2 .
ΔΦ(z)= 2π λ n 2,eff 0 z I ¯ ω 1 (z) dz= 2π λ α a n 2,eff I ¯ 0 (1 e α a z )
Δ Φ max = | L( ω 1 ) | 2 3fβQ n 2,eff n d I ¯ 0 κ 3 ( 2β ) 2 Q 3 L 2 ( ω 2 ) n 2 n d I ¯ 0 .
Δ n eff,max = n 2,eff I ¯ 3f κ 3 β Q 2 L 2 ( ω 2 ) Δ n max .
Δ Φ max = 2π λ α a Δ n eff,max = κ 3 Q | L( ω 1 ) | 2 L 2 ( ω 2 ) Δ n max n d .
E ¯ i (z)= 2π λ α a n 2,eff I ¯ 0 E s [ 1 e α a z ] e α a z/2
E ¯ i,max = 2π λ α a 2 3 3/2 n 2,eff I ¯ 0 E s = 2 3 3/2 Δ Φ max E s .
I i / I s ~0.15Δ Φ max 2
d 2 p 1(2) d t 2 +γ d p 1(2) dt = ω 0 2 p 1(2) + ω 0 2 α 1(2) E ¯ ω + ω 0 2 α 1(2) 2 p 2(1) 4π ε 0 ε d r 12 3 .
E max,1(2) ω =2Q L(ω)β+2 β 2 Q ( a 2(1) r 12 ) 3 L 2 ( ω )4 β 2 Q 2 ( a 1 a 2 r 12 2 ) 3 E ¯ ω .
E max,1 ω 2βQ L(ω) E ¯ ω ; E max,2 ω [ 2βQ L(ω) ] 2 E ¯ ω .
P nl,2 ω (r,t)= P max,2 ω G 2 (r) e jωt
p nl,1 ω =2π a 2 3 Q 2 2β ( a 1 r 12 ) 3 κ P max,2 ω L 2 (ω)4 β 2 Q 2 ( a 1 a 2 r 12 2 ) 3 p nl,2 ω =2π a 2 3 Q L(ω)κ P max,2 ω L 2 (ω)4 β 2 Q 2 ( a 1 a 2 r 12 2 ) 3 .
χ eff (3) =24f χ (3) κ 3 β Q 5 [ L(ω) ( a 2 r 12 ) 3 +2βQ ( a 1 a 2 r 12 2 ) 3 ] | L(ω)+2βQ ( a 1 r 12 ) 3 | 2 [ L 2 ( ω )4 β 2 Q 2 ( a 1 a 2 r 12 2 ) 3 ] 2 | L 2 ( ω )4 β 2 Q 2 ( a 1 a 2 r 12 2 ) 3 | 2
χ eff (3) 5f χ (3) κ 3 β 2 Q 6 .
Δ Φ max 1.7 κ 3 β Q 5 n 2 n d I ¯ 0 .
Δ Φ max 0.2 κ 3 Q β 3 Δ n max n d .

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