Abstract

In this work, we report the modeling and the experimental demonstration of intermodal spontaneous as well as stimulated four-wave mixing (FWM) in silicon waveguides. In intermodal FWM, the phase-matching condition is achieved by exploiting the different dispersion profiles of the optical modes in a multimode waveguide. Since both the energy and the wave vectors have to be conserved in the FWM process, this leads to a wide tunability of the generated photon wavelength, allowing us to achieve a large spectral conversion. We measured several waveguides that differ by their widths and demonstrate large signal generation spanning from the pump wavelength (1550 nm) down to 1202 nm. A suited setup evidences that the different waves propagated indeed on different order modes, which supports the modeling. Despite observing a reduced efficiency with respect to intramodal FWM due to the decreased modal overlap, we were able to show a maximum spectral distance between the signal and idler of 979.6 nm with a 1550 nm pump. Our measurements suggest the intermodal FWM is a viable means for large wavelength conversion and heralded photon sources.

© 2018 Chinese Laser Press

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References

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  38. The conversion efficiency is calculated as the ratio between the on-chip idler peak power and the on-chip signal power, both evaluated at the end of the waveguide. The input on-chip signal power was about 47 µW (= −13.3 dB) at 1640 nm on the second-order mode. At the end of the waveguide, considering 4.6  dB·cm−1 of propagation losses and 1.5 cm waveguide length, the signal power on the second-order mode is −20.2  dBm. The off-chip generated average idler power is about −74.2  dBm, as shown in Fig. 8(a). Considering the coupling losses for the first-order mode, 5.3 dB, the on-chip average idler power is −68.9  dBm. Considering that the pump laser has 10 MHz repetition rate and 40 ps pulse width, the on-chip idler peak power, at the end of the waveguide, is −34.9  dBm. Therefore, the conversion between the signal power, −20.2  dBm, and the idler peak power, −34.9  dBm, is −14.7  dB.

2017 (9)

Y. Qiu, X. Li, M. Luo, D. Chen, J. Wang, J. Xu, Q. Yang, and S. Yu, “Mode-selective wavelength conversion of OFDM-QPSK signals in a multimode silicon waveguide,” Opt. Express 25, 4493–4499 (2017).
[Crossref]

M. Borghi, C. Castellan, S. Signorini, A. Trenti, and L. Pavesi, “Nonlinear silicon photonics,” J. Opt. 19, 093002 (2017).
[Crossref]

K. Guo, L. Lin, J. Christensen, E. Christensen, X. Shi, Y. Ding, K. Rottwitt, and H. Ou, “Broadband wavelength conversion in a silicon vertical-dual-slot waveguide,” Opt. Express 25, 32964–32971 (2017).
[Crossref]

J. Yuan, Z. Kang, F. Li, X. Zhang, X. Sang, G. Zhou, Q. Wu, B. Yan, K. Wang, C. Yu, H. Tam, and P. Wai, “Demonstration of intermodal four-wave mixing by femtosecond pulses centered at 1550 nm in an air-silica photonic crystal fiber,” J. Lightwave Technol. 35, 2385–2390 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, J. Fatome, A. Tonello, M. Fabert, V. Couderc, S. Wabnitz, and G. Millot, “Intermodal modulational instability in graded-index multimode optical fibers,” Opt. Lett. 42, 3419–3422 (2017).
[Crossref]

E. A. Kittlaus, N. T. Otterstrom, and P. T. Rakich, “On-chip inter-modal Brillouin scattering,” Nat. Commun. 8, 15819 (2017).
[Crossref]

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref]

S. K. Liao, H. L. Yong, C. Liu, G. L. Shentu, D. D. Li, J. Lin, H. Dai, S. Q. Zhao, B. Li, J. Y. Guan, W. Chen, Y. H. Gong, Y. Li, Z. H. Lin, G. S. Pan, J. S. Pelc, M. M. Fejer, W. Z. Zhang, W. Y. Liu, J. Yin, J. G. Ren, X. B. Wang, Q. Zhang, C. Z. Peng, and J. W. Pan, “Long-distance free-space quantum key distribution in daylight towards inter-satellite communication,” Nat. Photonics 11, 509–513 (2017).
[Crossref]

S. Signorini, M. Borghi, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Oblique beams interference for mode selection in multimode silicon waveguides,” J. Appl. Phys. 122, 113106 (2017).
[Crossref]

2016 (1)

2015 (2)

2014 (5)

2013 (1)

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

2012 (2)

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[Crossref]

2009 (2)

2007 (1)

2006 (3)

1982 (1)

1981 (1)

C. Lin and M. A. Bsch, “Large-Stokes-shift stimulated four-photon mixing in optical fibers,” Appl. Phys. Lett. 38, 479–481 (1981).
[Crossref]

1974 (1)

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Agrawal, G. P.

Ashkin, A.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Baets, R.

Bagheri, S.

S. Bagheri and W. M. Green, “Silicon-on-insulator mode-selective add-drop unit for on-chip mode-division multiplexing,” in Proceedings of IEEE Conference on Group IV Photonics (IEEE, 2009).

Begleris, I.

S. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24, 30338–30349 (2016).
[Crossref]

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” in Proceedings of 42nd European Conference on Optical Communication (2016).

Bendahmane, A.

Berdagu, S.

Bergmen, K.

L. W. Luo, N. Ophir, C. P. Chen, L. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref]

Bernard, M.

S. Signorini, M. Borghi, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Oblique beams interference for mode selection in multimode silicon waveguides,” J. Appl. Phys. 122, 113106 (2017).
[Crossref]

S. Signorini, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Broad wavelength generation and conversion with multi modal four wave mixing in silicon waveguides,” in Proceedings of IEEE Conference on Group IV Photonics (IEEE, 2017), pp. 59–60.

Bjorkholm, J. E.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Boppart, S. A.

H. Tu, Z. Jiang, D. L. Marks, and S. A. Boppart, “Intermodal four-wave mixing from femtosecond pulse-pumped photonic crystal fiber,” Appl. Phys. Lett. 94, 101109 (2009).
[Crossref]

Borghi, M.

M. Borghi, C. Castellan, S. Signorini, A. Trenti, and L. Pavesi, “Nonlinear silicon photonics,” J. Opt. 19, 093002 (2017).
[Crossref]

S. Signorini, M. Borghi, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Oblique beams interference for mode selection in multimode silicon waveguides,” J. Appl. Phys. 122, 113106 (2017).
[Crossref]

Bowers, J. E.

D. Dai and J. E. Bowers, “Silicon-based on-chip multiplexing technologies and devices for peta-bit optical interconnects,” Nanophotonics 3, 283–311 (2014).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Bsch, M. A.

C. Lin and M. A. Bsch, “Large-Stokes-shift stimulated four-photon mixing in optical fibers,” Appl. Phys. Lett. 38, 479–481 (1981).
[Crossref]

Castellan, C.

M. Borghi, C. Castellan, S. Signorini, A. Trenti, and L. Pavesi, “Nonlinear silicon photonics,” J. Opt. 19, 093002 (2017).
[Crossref]

Chang, W. S.

W. S. Chang, Fundamentals of Guided-Wave Optoelectronic Devices (Cambridge University, 2009).

Charan, K.

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Chen, C. P.

L. W. Luo, N. Ophir, C. P. Chen, L. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref]

Chen, D.

Chen, L.

M. Ma and L. Chen, “On-chip silicon mode-selective broadband wavelength conversion based on cross-phase modulation,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper STh3E.3.

Chen, W.

S. K. Liao, H. L. Yong, C. Liu, G. L. Shentu, D. D. Li, J. Lin, H. Dai, S. Q. Zhao, B. Li, J. Y. Guan, W. Chen, Y. H. Gong, Y. Li, Z. H. Lin, G. S. Pan, J. S. Pelc, M. M. Fejer, W. Z. Zhang, W. Y. Liu, J. Yin, J. G. Ren, X. B. Wang, Q. Zhang, C. Z. Peng, and J. W. Pan, “Long-distance free-space quantum key distribution in daylight towards inter-satellite communication,” Nat. Photonics 11, 509–513 (2017).
[Crossref]

Chen, X.

Cheng, J.

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Chraplyvy, A. R.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

Christensen, E.

Christensen, J.

Couderc, V.

Crowder, J. G.

J. G. Crowder, S. D. Smith, A. Vass, and J. Keddie, “Infrared methods for gas detection,” in Mid-Infrared Semiconductor Optoelectronics (Springer, 2006), pp. 595–613.

Dadap, J. I.

Dai, D.

D. Dai and J. E. Bowers, “Silicon-based on-chip multiplexing technologies and devices for peta-bit optical interconnects,” Nanophotonics 3, 283–311 (2014).
[Crossref]

Dai, H.

S. K. Liao, H. L. Yong, C. Liu, G. L. Shentu, D. D. Li, J. Lin, H. Dai, S. Q. Zhao, B. Li, J. Y. Guan, W. Chen, Y. H. Gong, Y. Li, Z. H. Lin, G. S. Pan, J. S. Pelc, M. M. Fejer, W. Z. Zhang, W. Y. Liu, J. Yin, J. G. Ren, X. B. Wang, Q. Zhang, C. Z. Peng, and J. W. Pan, “Long-distance free-space quantum key distribution in daylight towards inter-satellite communication,” Nat. Photonics 11, 509–513 (2017).
[Crossref]

Dam, J. S.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref]

Desgroseilliers, M.

Ding, Y.

Dulkeith, E.

Dupiol, R.

Essiambre, R. J.

Y. Xiao, R. J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, and G. P. Agrawal, “Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers,” Opt. Express 22, 32039–32059 (2014).
[Crossref]

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

Fabert, M.

Facq, P.

Fatome, J.

Fauchet, P. M.

Fejer, M. M.

S. K. Liao, H. L. Yong, C. Liu, G. L. Shentu, D. D. Li, J. Lin, H. Dai, S. Q. Zhao, B. Li, J. Y. Guan, W. Chen, Y. H. Gong, Y. Li, Z. H. Lin, G. S. Pan, J. S. Pelc, M. M. Fejer, W. Z. Zhang, W. Y. Liu, J. Yin, J. G. Ren, X. B. Wang, Q. Zhang, C. Z. Peng, and J. W. Pan, “Long-distance free-space quantum key distribution in daylight towards inter-satellite communication,” Nat. Photonics 11, 509–513 (2017).
[Crossref]

Fontana, G.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref]

Foster, M. A.

Friis, S.

Friis, S. M. M.

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” in Proceedings of 42nd European Conference on Optical Communication (2016).

Gabrielli, L.

L. W. Luo, N. Ophir, C. P. Chen, L. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref]

Gaeta, A. L.

Gao, S.

Ghulinyan, M.

S. Signorini, M. Borghi, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Oblique beams interference for mode selection in multimode silicon waveguides,” J. Appl. Phys. 122, 113106 (2017).
[Crossref]

S. Signorini, M. Mancinelli, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “Broad wavelength generation and conversion with multi modal four wave mixing in silicon waveguides,” in Proceedings of IEEE Conference on Group IV Photonics (IEEE, 2017), pp. 59–60.

Gnauck, A. H.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
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The conversion efficiency is calculated as the ratio between the on-chip idler peak power and the on-chip signal power, both evaluated at the end of the waveguide. The input on-chip signal power was about 47 µW (= −13.3 dB) at 1640 nm on the second-order mode. At the end of the waveguide, considering 4.6  dB·cm−1 of propagation losses and 1.5 cm waveguide length, the signal power on the second-order mode is −20.2  dBm. The off-chip generated average idler power is about −74.2  dBm, as shown in Fig. 8(a). Considering the coupling losses for the first-order mode, 5.3 dB, the on-chip average idler power is −68.9  dBm. Considering that the pump laser has 10 MHz repetition rate and 40 ps pulse width, the on-chip idler peak power, at the end of the waveguide, is −34.9  dBm. Therefore, the conversion between the signal power, −20.2  dBm, and the idler peak power, −34.9  dBm, is −14.7  dB.

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Figures (11)

Fig. 1.
Fig. 1. (a) Cross section of the waveguides used in this work. BOX refers to the buried oxide. The core is made of crystalline Si, and the cladding by deposited SiO2. (b) Computed intensity profiles in the core region of the first three TE modes supported by a 3.5-μm-wide waveguide. (c) Effective index as a function of the wavelength for the first TE mode for the different waveguide widths reported in the legend. (d) Effective index as a function of the wavelength for the second TE mode for the different waveguide widths reported in the legend.
Fig. 2.
Fig. 2. Spectral dependence of the idler generation efficiency for the (1,1,1,1) intramodal FWM and for the (1,2,2,1) intermodal FWM. All waves have TE polarization. A silicon waveguide with a cross section 3.5  μm×243  nm was assumed in the calculation. Each efficiency is normalized to its maximum.
Fig. 3.
Fig. 3. Numerical simulation of Δk according to Eq. (4) for a 2-μm-wide waveguide with (1,2,2,1) intermodal combination and TE polarization. In grayscale, the Δk parameter is a function of the pump wavelength and the signal wavelength. The red line highlights the perfect phase-matching spectral position.
Fig. 4.
Fig. 4. Computed power coupled into the different modes of a waveguide by a tapered lensed fiber as a function of the fiber position with respect to the center of the waveguide. It is assumed a 3.5-μm-wide waveguide and 1 W at the input fiber with a wavelength of 1550 nm. When the fiber position is 0 μm, the fiber is in the middle of the waveguide. These values have been normalized with respect to the measured coupling losses for the first TE waveguide mode.
Fig. 5.
Fig. 5. Setup for the sFWM. The pump is initially filtered with two 1550 nm bandpass filters. Then, the pump and the signal, after a polarization controller stage, are mixed by a free-space beam splitter and coupled in the same tapered lensed fiber through a collimator. The input fiber injects the light in the desired waveguide on the SOI chip. The light is collected from the waveguide by another tapered lensed fiber. The position of both fibers is controlled by means of two XYZ nanopositioning stages via piezocontrollers. The collected light is analyzed with an OSA or a monochromator, depending on the power of the generated signal. In the inset, the waveguide modes involved in the intermodal FWM are sketched by showing the mode profiles at the input and at the output of the waveguide; as an example, the case of the (1,2,2,1) modal combination is considered. For the SFWM, the setup is exactly the same, except for the lack of the input signal.
Fig. 6.
Fig. 6. (a) Spectra of the SFWM in a 3.5-μm-wide waveguide, with the (1,2,2,1) combination and TE polarization. The pump is at 1550 nm. The different lines refer to the different positions of the output fiber: the blue line refers to the spectrum measured with the output fiber in the middle of the waveguide, and the red line refers to the spectrum measured with the output fiber at 1.1 μm with respect to the center of the waveguide. The two vertical arrows indicate a spurious signal due to Raman scattering occurring in the input fiber. (b) Intensity profile at the output facet of the waveguide, for the peak at 1469 nm in (a). (c) Intensity profile at the output facet of the waveguide, for the peak at 1640 nm in (a).
Fig. 7.
Fig. 7. Average on-chip photon generation rate for the SFWM process as a function of the on-chip average pump power. The blue circles are the experimental measurements, while the red line is the quadratic fit of the data belonging to the 0–0.5 mW range. The inset shows the low pump power region.
Fig. 8.
Fig. 8. (a) Spectrum of the sFWM with the (1,2,2,1) TE intermodal combination in a 3.5-μm-wide waveguide. The stimulating CW signal at 1640 nm is converted into the pulsed idler at 1469 nm. The smaller peaks are spurious signals due to the OSA. (b) Spectrum of the stimulated idler generation efficiency with the intermodal FWM combination (1,2,2,1) TE in a 3.5-μm-wide waveguide. The simulation was performed with a 3.66-μm-wide waveguide. The blue circles are the measured data, while the orange line is the simulation. This measurement was performed by synchronously scanning the signal wavelength and the monochromator wavelength in order to read the idler power corresponding to the input signal. The simulated spectrum was shifted by 3.3  dBm in order to match the experimental data.
Fig. 9.
Fig. 9. Phase-matched wavelengths as a function of the waveguide width for the (1,2,2,1) combination, TE polarization, and 1550 nm pump. The experimental idler and signal are reported in blue and red, respectively, while the corresponding simulated values are reported by the light blue and orange points, where the lines are a guide for the eye. The phase-matched signal wavelengths have been deduced by using Eq. (1) and the measured idler wavelengths. For some widths (those with the error bars), we performed repeated measurements (ten measurements) on nominally identical waveguides. The theory and the experiment are in agreement.
Fig. 10.
Fig. 10. Spectrum of the SFWM for the (1,2,2,1) combination, TE polarization, and a 2-μm-wide waveguide. The pump is at 1550 nm with an on-chip peak power of about 3.9 W (3.3 W on the first-order mode, 0.6 W on the second one). The idler is generated at 1202 nm. The peak at 1434 nm is the anti-Stokes peak of the Si Raman scattering. The two vertical arrows indicate the spurious signals due to Raman scattering occurring in the input fiber.
Fig. 11.
Fig. 11. Spectra of intermodal SFWM in a 3.8-μm-wide waveguide. Both TE and TM measurements are reported. The position of the output fiber is reported in the legend. The peaks refer to the indicated combinations. The pump is at 1550 nm.

Tables (2)

Tables Icon

Table 1. Calculated Mode Field Overlap for a 3.5  μm×243  nm Si Waveguidea

Tables Icon

Table 2. Intermodal Combinations Measured in Fig. 11a

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

ωP+ωP=ωS+ωI,
kP1+kP2=kS+kI,
η|fjqlm|2sinc(ΔkL2π)2,
Δk=ΔkL+ΔkNL,
ΔkL=ωPcneffj(ωP)+ωPcneffq(ωP)ωScneffl(ωS)ωIcneffm(ωI),
fjqlm=A0Ej(r,ωj)Eq(r,ωq)El*(r,ωl)Em*(r,ωm)Ai=jqlm[Anwg(r,ωi)2|Ei(r,ωi)|2A]12,
ξx,m=dxdyψxEx,m*dxdyψx*Ex,mdxdy|ψx|2dxdy|Ex,m|2,
Pp,mwg=(1R)ξp,mPp,
du1dz=α12u1F12u1+i[γ1|u1|2+2(γ12|u2|2+γ1s|us|2+γ1i|ui|2)]u1+2iγ1is2P2PiPsP1uiusu2*eiΔkLz,
du2dz=α22u2F22u2+i[γ2|u2|2+2(γ21|u1|2+γ2s|us|2+γ2i|ui|2)]u2+2iγ2is1P1PiPsP2uiusu1*eiΔkLz,
duidz=αi2uiFi2ui+i[γi|ui|2+2(γi1|u1|2+γi2|u2|2+γis|us|2)]ui+2iγi12sP1P2PsPiu1u2us*eiΔkLz,
dusdz=αs2usFs2us+i[γs|us|2+2(γs1|u1|2+γs2|u2|2+γsi|ui|2)]us+2iγs12iP1P2PiPsu1u2ui*eiΔkLz,
Fν=σν(1+iμν)N,
σν=1Ncκνn(ων)vg,νδαFC,μν=1σνN2ωνκνn(ων)vg,νδnFC,
κν=n(ων)2A0|Eν(r,ων)|2dAAnwg(r,ων)2|Eν(r,ων)|2dA.
δαFC=14.5×1018N,
δnFC=8.8×1022N8.5×1018N0.8.
γadcd=3ωang,ang,bng,cng,d4ϵ0A0c2Γabcd,
Γabcd=A0A0Ea*(r,ωa)χ(3)Eb(r,ωb)Ec*(r,ωc)Ed(r,ωd)dAν=abcd[Anwg(r,ων)2|Eν(r,ων)|2dA]1/2,

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