Abstract

The nonlinear optics of Si photonic wires is discussed. The distinctive features of these waveguides are that they have extremely large third-order susceptibility χ(3) and dispersive properties. The strong dispersion and large third-order nonlinearity in Si photonic wires cause the linear and nonlinear optical physics in these guides to be intimately linked. By carefully choosing the waveguide dimensions, both linear and nonlinear optical properties of Si wires can be engineered. We review the fundamental optical physics and emerging applications for these Si wires. In many cases, the relatively low threshold powers for nonlinear optical effects in these wires make them potential candidates for functional on-chip nonlinear optical devices of just a few millimeters in length; conversely, the absence of nonlinear optical impairment is important for the use of Si wires in on-chip interconnects. In addition, the characteristic length scales of linear and nonlinear optical effects in Si wires are markedly different from those in commonly used optical guiding systems, such as optical fibers or photonic crystal fibers, and therefore guiding structures based on Si wires represent ideal optical media for investigating new and intriguing physical phenomena.

© 2009 Optical Society of America

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2008 (9)

J. I. Dadap, N. C. Panoiu, X. Chen, I. Hsieh, X. Liu, C. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008).
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M. A. Foster, A. C. Turner, M. Lipson, A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16, 1300–1320 (2008).
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X. Liu, W. M. J. Green, X. Chen, I-W. Hsieh, J. I. Dadap, Y. A. Vlasov, R. M. Osgood, “Conformal dielectric overlayers for engineering dispersion and effective nonlinearity of silicon nanophotonic wires,” Opt. Lett. 33, 2889–2891 (2008).
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J. M. Dudley, G. Genty, B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16, 3644–3651 (2008).
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K. K. Tsia, S. Fathpour, B. Jalali, “Electrical control of parametric processes in silicon waveguides,” Opt. Express 16, 9838–9843 (2008).
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M. Krause, H. Renner, S. Fathpour, B. Jalali, E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[CrossRef]

L. Ding, C. Benton, A. V. Gorbach, L. Ding, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. Gnan, M. Sorrel, R. M. De La Rue, “Solitons and spectral broadening in long silicon-on-insulator photonic wires,” Opt. Express 16, 3310–3319 (2008).
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R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2008).
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B. G. Lee, X. G. Chen, A. Biberman, X. P. Liu, I. W. Hsieh, C. Y. Chou, J. I. Dadap, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks” IEEE Photon. Technol. Lett. 20, 398–400 (2008).
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2007 (14)

A. Demircan, U. Bandelow, “Analysis of the interplay between soliton fission and modulation instability in supercontinuum generation,” Appl. Phys. B 86, 31–39 (2007).
[CrossRef]

E.-K. Tien, N. S. Yuksek, F. Qian, Ö. Boyraz, “Pulse compression and modelocking by using TPA in silicon waveguides,” Opt. Express 15, 6500–6506 (2007).
[CrossRef] [PubMed]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express 15, 7682–7688 (2007).
[CrossRef] [PubMed]

L. Yin, G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32, 391–393 (2007).
[CrossRef] [PubMed]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, W. Freude, “Nonlinear silicon-on-insulator waveguides for all optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef] [PubMed]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
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A. D. Bristow, N. Rotenberg, H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
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Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
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M. A. Foster, A. C. Turner, R. Salem, M. Lipson, A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express 15, 12949–12958 (2007).
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I.-W. Hsieh, X. Chen, X. Liu, J. I. Dadap, N. C. Panoiu, C.-Y. Chou, F. Xia, W. M. Green, Y. A. Vlasov, R. M. Osgood, “Supercontinuum generation in silicon photonic wires,” Opt. Express 15, 15242–15249 (2007).
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F. Xia, L. Sekaric, Y. A. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
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Q. Lin, O. J. Painter, G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
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R. Dekker, N. Usechak, M. Först, A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D 40, R249–R271 (2007).
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I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, Y. A. Vlasov, “Cross-phase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 1135–1146 (2007).
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2006 (32)

C. Manolatou, M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24, 1433–1439 (2006).
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T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171–174 (2006).
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R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses,” Opt. Express 14, 8336–8346 (2006).
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E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, R. M. Osgood, “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006).
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I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express 14, 12380–12387 (2006).
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Y.-H. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, O. Cohen, “Demonstration of wavelength conversion at 40 Gb∕s data rate in silicon waveguides,” Opt. Express 14, 11721–11726 (2006).
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M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, A. L. Gaeta, “Broadband optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
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K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, S. Itabashi, “All-optical efficient wavelength conversion using silicon photonic wire waveguide,” IEEE Photon. Technol. Lett. 18, 1046–1048 (2006).
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Q. Lin, J. Zhang, P. M. Fauchet, G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006).
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A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, J. E. Bowers, “Electrically pumped hybrid AlGaInAs–silicon evanescent laser,” Opt. Express 14, 9203–9210 (2006).
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J. F. McMillan, X. Yang, N. C. Panoiu, R. M. Osgood, C. W. Wong, “Enhanced stimulated Raman scattering in slow-light photonic crystal waveguides,” Opt. Lett. 31, 1235–1237 (2006).
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H. Rong, Y.-H. Kuo, S. Xu, A. Liu, R. Jones, M. Paniccia, O. Cohen, O. Raday, “Monolithic integrated Raman silicon laser,” Opt. Express 14, 6705–6712 (2006).
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B. Jalali, S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24, 4600–4615 (2006).
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R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
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P. Dumon, G. Priem, L. R. Nunes, W. Bogaerts, D. van Thourhout, P. Bienstman, T. K. Liang, M. Tsuchiya, P. Jaenen, S. Beckx, J. Wouters, R. Baets, “Linear and nonlinear nanophotonic devices based on silicon-on-insulator wire waveguides,” Jpn. J. Appl. Phys. Part 1 45, 6589–6602 (2006).
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R. M. Roth, N.-C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, M. B. Raschke, “Resonant-plasmon field enhancement from asymmetrically illuminated conical metallic-probe tips,” Opt. Express 14, 2921–2931 (2006).
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W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027–1030 (2006).
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W. Fan, S. Zhang, K. J. Malloy, R. J. Brueck, N.-C. Panoiu, R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express 14, 9570–9575 (2006).
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C. G. Poulton, C. Koos, M. Fujii, A. Pfrang, T. Schimmel, J. Leuthold, W. Freude, “Radiation modes and roughness loss in high index-contrast waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1306–1321 (2006).
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J. Lou, L. Tong, Z. Ye, “Dispersion shifts in optical nanowires with thin dielectric coatings,” Opt. Express 14, 6993–6998 (2006).
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J. M. Dudley, G. Genty, S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
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X. Chen, N. C. Panoiu, R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
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E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14, 3853–3863 (2006).
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A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
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L. Yin, Q. Lin, G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295–1297 (2006).
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X. Chen, N. Panoiu, I. Hsieh, J. I. Dadap, R. M. Osgood, “Third-order dispersion and ultrafast pulse propagation in silicon wire waveguides,” IEEE Photon. Technol. Lett. 18, 2617–2619 (2006).
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H. Garcia, R. Kalyanaraman, “Phonon-assisted two-photon absorption in the presence of a dc-field: the nonlinear Franz–Keldysh effect in indirect gap semiconductors,” J. Phys. B 39, 2737–2746 (2006).
[CrossRef]

N. C. Panoiu, X. Chen, R. M. Osgood, “Modulation instability in silicon photonic nanowires,” Opt. Lett. 31, 3609–3611 (2006).
[CrossRef] [PubMed]

Y. Liu, H. K. Tsang, “Nonlinear absorption and Raman gain in helium-ion-implanted silicon waveguides,” Opt. Lett. 31, 1714–1716 (2006).
[CrossRef] [PubMed]

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2005 (25)

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Figures (24)

Fig. 1
Fig. 1

Typical experimental setup. In nonlinear optical experiments, input tapered fiber is often replaced by a microscope objective to mitigate SPM in the fiber because of the high intensity of the ultrashort pulses used.

Fig. 2
Fig. 2

Plots of computed effective index of refraction, and first-order dispersion through TOD as a function of wavelength for four different Si-wire dimensions. Blue, 220 n m × 350 n m ; green, 220 n m × 360 n m ; red, 220 n m × 450 n m ; light blue, 330 n m × 450 n m . Inset, expanded view of the second-order dispersion indicating the ZGVD line. Also shown is the waveguide geometry.

Fig. 3
Fig. 3

ZGVD map for SPW channel waveguides. The ZGVD wavelengths are expressed in units of micrometers.

Fig. 4
Fig. 4

Left, plot of GVD, D, versus different Si 3 N 4 overlayer thicknesses, for the fundamental TE-like mode of a Si nanophotonic wire with dimensions h = 220 nm , w = 450 nm , for different Si 3 N 4 overlayer thicknesses. Inset, cross-section geometry. Right, ZGVD wavelength contours for lowest quasi-TE mode versus Si 3 N 4 layer thickness, for several wire widths. Waveguide height is 220 nm . From [97].

Fig. 5
Fig. 5

Relevant nonlinear optical processes in Si photonics: (a) spontaneous Raman (Stokes and anti-Stokes) emission, (b) stimulated Raman emission, (c) CARS, (d) Kerr effect and TPA, and (e) FWM.

Fig. 6
Fig. 6

Measured value of β TPA and n 2 versus wavelength as described in [110]. The solid curve in the left panel is a best fit based on the theory of Garcia and Kalyanaraman [112]. Right, n 2 (squares) versus wavelength; in this panel the dashed curve is a guide to the eye and the solid curve is based on a Kramers–Krönig transformation of the solid curve for β TPA shown in the left panel. Most data shown here were taken by Bristow et al. [110].

Fig. 7
Fig. 7

(a) Wavelength dependence of the susceptibility χ 1111 (3) . (b) Real and imaginary parts of the nonlinear coefficient γ versus wavelength. Inset, interpolated values of experimentally measured bulk parameters n 2 and β TPA . In (b), the thin and thick curves correspond to the case in which only the waveguide dispersion is considered and the case when both the material and the waveguide dispersion are included, respectively.

Fig. 8
Fig. 8

Propagation of a pulse with T 0 = 100 fs , peak power P 0 = 6.5 W , and pulse wavelength λ = 1500 nm in a SPW with dimensions h × w = 220 n m × 360 n m , for which the ZGVD wavelength λ 0 = 1550 nm , β 2 = 4 ps 2 m , β 3 = 0.0915 ps 3 m , and γ = 446.5 W 1 m 1 . (a) Influences of the TPA and free carriers are neglected; (b) influence of TPA is ignored while free carriers effects are included; (c) both TPA and free carriers are fully accounted for.

Fig. 9
Fig. 9

Spectrogram of a pulse with T 0 = 100 fs , peak power P 0 = 6.5 W , and pulse wavelength λ = 1500 nm after it propagated a distance z = 10 mm in a SPW with dimensions h × w = 220 n m × 360 n m . The ZGVD wavelength λ 0 = 1550 nm , β 2 = 4 ps 2 m , β 3 = 0.0915 ps 3 m , and γ = 446.5 W 1 m 1 . (a) Influences of the TPA and free carriers are neglected; (b) influence of TPA is ignored while the effects of the free carriers are included; (c) both TPA and free carriers are accounted for.

Fig. 10
Fig. 10

Real part of the parameter τ s , calculated in two cases: dashed curve, the frequency dependence of χ ( 3 ) is neglected; solid curve, the frequency dependence of χ ( 3 ) is fully accounted for. Inset, imaginary part of τ s versus λ.

Fig. 11
Fig. 11

Temporal and spectral profiles of a pulse that propagates a distance z = 10 mm in a SPW with dimensions h × w = 300 n m × 450 n m .

Fig. 12
Fig. 12

Pulse spectrograms calculated for the three cases shown in Fig. 9. (a) τ 0 = 0 and τ w m = 0 ( τ r = 0 and τ i = 0 ); (b) τ 0 = 1.03 fs and τ w m = 0 ( τ r = 0 and τ i = 0 ); (c) τ 0 = 1.03 fs , τ r = 2.99 fs , and τ i = 0.51 fs . In all cases the propagation distance is z = 10 mm .

Fig. 13
Fig. 13

On–off gain versus input pump power. The maximum gain is 0.7 dB (17%) with a pump power of 29 mW . A linear fit with a slope of 0.029 dB mW corresponds to an SRS coefficient g R 29 cm GW . From [43].

Fig. 14
Fig. 14

Dependence of output power on coupled input power for (a) 1.8 ps (from [72]) and (b) 200 fs pulses (from [73]). Experiment, squares; simulations, curves.

Fig. 15
Fig. 15

Self-phase modulation observed experimentally using picosecond and femtosecond pulses. (a) 1.8 ps pulses (figure from [72]) (b) 200 fs pulses (data from [73]).

Fig. 16
Fig. 16

Comparison of simulation with experimental measurements of pulses propagating in a SPW waveguide with γ P 0 = 56.3 cm 1 . Left, measured spectra (brown). Right, simulation using sech input pulse (red). Blue curves on both panels correspond to γ P 0 = 1.1 cm 1 (with sech input pulse for simulation). Dashed line, OSA noise floor. The numbers illustrate features of the output spectrum common to experiment and simulation. From [73].

Fig. 17
Fig. 17

Change in output spectra due to SPM for different excitation levels γ P 0 = 1.1 , 11.3, 33.8, 45.0 cm 1 (bottom to top) spectra for both (a) theory and (b) experiment. Note also the evolution of soliton radiation (dashed line) at 1590 nm .

Fig. 18
Fig. 18

Demonstration of XPM in SPWs. Dependence of probe spectrum on pump power and pump–probe delay.

Fig. 19
Fig. 19

Experimental (red) and numerical simulation (blue) of the shift in center frequency due to XPM with the time delay of the pump and probe pulse. The center wavelength of the probe is 1590 nm .

Fig. 20
Fig. 20

(a) Converted wavelength and phase mismatch versus pump–wavelength separation. (b) Conversion efficiency versus phase mismatch for Δ λ = 0.148 nm at two possible propagation losses of α = 3.5 dB cm (solid curve) and 0.1 dB cm (dashed-dotted curve). From [60].

Fig. 21
Fig. 21

Calculated MI gain spectra for the two case of (a) normal and (b) anomalous dispersion described in the text.

Fig. 22
Fig. 22

(a) Continuum generation in a 220 n m × 520 n m Si wire showing the pump-power dependence of the output spectra. At P 0 1 W the spectral broadening is 350 nm . (b) Spectral width as a function of coupled peak power. From [80].

Fig. 23
Fig. 23

Simulation of pulse compression as a result of XPM in a h × w = 220 n m × 360 nm Si wire. Signal (left) and pump (right) field envelopes versus time and propagation distance. The temporal width is 200 fs for both the pump and the signal pulses. Here γ p s P p 100 cm 1 for the pump, with a center frequency of 1625 nm . For the signal P s P p , and its center frequency is 1451 nm . Insets, initial and final pulse envelopes. From [37].

Fig. 24
Fig. 24

Data taken for SRS-induced delays. Measured delay versus the measured Raman gain parameter. From Okawachi et al. [173].

Tables (2)

Tables Icon

Table 1 Comparison of Characteristic Lengths for Ultrashort ( 200 fs ) and Long ( 10 ps ) Pulses and γ Parameter in a SPW (dimensions h × w = 220 × 450 nm 2 ) and a Single-Mode Optical Fiber for λ = 1550 nm

Tables Icon

Table 2 Use of Four Wavelength Configurations Based on FWM Wavelength-Conversion Signal-Regeneration Schemes in a SPW for Three Wavelengths λ 1 , λ 2 , λ 3 Where λ 1 < λ 2 < λ 3 a

Equations (64)

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n ( λ ) = ε + A λ 2 + B λ 1 2 λ 2 λ 1 2 .
P = D ε 0 E = ε 0 ( χ ( 1 ) E + χ ( 2 ) : E E + χ ( 3 ) E E E + ) ,
P L = ε 0 χ ( 1 ) E ,
P NL = ε 0 χ ( 2 ) : E E + ε 0 χ ( 3 ) E E E + .
E ( r , t ) = 1 2 E ( r , ω ) e i ω t + c.c . ,
P NL ( r , ω ) = 3 4 ε 0 χ ( 3 ) ( ω ; ω , ω , ω ) E ( r , ω ) E * ( r , ω ) E ( r , ω ) ,
ω c n 2 + i 2 β TPA = 3 ω 4 ε 0 c 2 n 2 χ eff ( 3 ) ,
χ eff ( 3 ) = χ 1122 ( 3 ) [ ( a ̂ * b ̂ ) ( c ̂ d ̂ ) + ( a ̂ * c ̂ ) ( b ̂ d ̂ ) + ( a ̂ * d ̂ ) ( b ̂ c ̂ ) ] + ( χ 1111 ( 3 ) 3 χ 1122 ( 3 ) ) i = 1 3 a ̂ i * b ̂ i c ̂ i d ̂ i .
χ i j k l R ( Ω ) = π N v 3 σ ω σ ( α i j , σ α k l , σ + α i k , σ α l j , σ ) ω σ 2 Ω 2 + 2 i Ω Δ ω ,
R 1 = ( 0 0 0 0 0 1 0 1 0 ) , R 2 = ( 0 0 1 0 0 0 1 0 0 ) , R 3 = ( 0 1 0 1 0 0 0 0 0 ) .
χ i j k l R ( Ω ) = π N v 6 σ α i j , σ α k l , σ + α i k , σ α l j , σ ω R Ω + i Δ ω .
P R = 3 2 ε 0 [ χ R ( ω R ) E ( r , ω s ) E * ( r , ω s ) E ( r , ω p ) δ ( ω ω p ) + χ R ( ω R ) E ( r , ω p ) E * ( r , ω p ) E ( r , ω s ) δ ( ω ω s ) ] .
Γ = A 0 A 0 e * ( r ; ω ) χ ( 3 ) ( r ; ω , ω , ω , ω ) e ( r ; ω ) e * ( r ; ω ) e ( r ; ω ) d A ( A n 2 ( r ) e ( r , ω ) 2 d A ) 2 ,
δ n FC = e 2 2 ε 0 n ω 2 ( N m c e * + N 0.8 m c h * ) ,
α FC = e 3 N ε 0 c n ω 2 ( 1 μ e m c e * + 1 μ h m c h * 2 ) ,
P z TPA = 3 ω β 1 2 P 0 2 Γ 2 ε 0 A 0 u 4 ,
N t = N t c + 3 β 1 2 P 0 2 Γ 4 ε 0 A 0 2 u 4 ,
z A F c e ̂ z d A = A F c d A ,
F c = E 1 * × H 2 + E 2 × H 1 * .
z A ( E 0 * × H + E × H 0 * ) e ̂ z d A = i ω A δ P E 0 d A ,
δ ε L = i ε 0 c n α in ω + 2 ε 0 n δ n FC + i ε 0 c n α FC ω ,
E 0 = 1 2 Z 0 P 0 A 0 e ( r , ω 0 ) e i ( β 0 z ω 0 t ) ,
H 0 = 1 2 P 0 Z 0 A 0 h ( r , ω 0 ) e i ( β 0 z ω 0 t )
1 4 A 0 A ( e × h * + e * × h ) e ̂ z d A = 1.
E = 1 2 Z 0 P 0 A 0 u ( z , ω ) e ( r , ω ) e i ( β z ω t ) ,
H = 1 2 P 0 Z 0 A 0 u ( z , ω ) h ( r , ω ) e i ( β z ω t ) ,
i ( u z + 1 v g u t ) β 2 2 2 u t 2 i β 3 6 3 u t 3 = i c κ 2 n v g ( α in + α FC ) u ω κ n v g δ n FC u 3 ω P 0 Γ 4 ε 0 A 0 v g 2 u 2 u ,
κ = n 2 A 0 e ( r ) 2 d A A n 2 ( r ) e ( r ) 2 d A .
W t = Z 0 P 0 2 A 0 A ε 0 n 2 ( r ) e ( r ) 2 d A .
P NL ( r , ω ) = 3 4 ε 0 { [ χ ( 3 ) ( ω s ; ω s , ω s , ω s ) E ( r , ω s ) E * ( r , ω s ) E ( r , ω s ) + 2 χ ( 3 ) ( ω s ; ω s , ω p , ω p ) E ( r , ω s ) E * ( r , ω p ) E ( r , ω p ) ] δ ( ω ω s ) + [ χ ( 3 ) ( ω p ; ω p , ω p , ω p ) E ( r , ω p ) E * ( r , ω p ) E ( r , ω p ) + 2 χ ( 3 ) ( ω p ; ω p , ω s , ω s ) E ( r , ω p ) E * ( r , ω s ) E ( r , ω s ) ] δ ( ω ω p ) } ,
i ( u p z + 1 v g , p u p t ) β 2 , p 2 2 u p t 2 i β 3 , p 6 3 u p t 3 = i c κ p 2 n v g , p ( α in + α FC p ) u p
ω p κ p n v g , p δ n FC p u p 3 ω p 4 ε 0 A 0 v g , p ( P p Γ p v g , p u p 2 + 2 P s Γ s p v g , s u s 2 ) u p ,
i ( u s z + 1 v g , s u s t ) β 2 , s 2 2 u s t 2 i β 3 , s 6 3