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1/56 Nonlinear effects in Silicon Waveguides Govind P. Agrawal Institute of Optics University of Rochester Rochester, NY 14627 JJ II J I c 2007 G. P. Agrawal Back Close Outline • Why use silicon for photonics? 2/56 • Silicon-on-Insulator (SOI) Waveguides • Two-Photon Absorption and Free-Carrier Effects • Self-Phase Modulation and Soliton Formation • Higher-Order Solitons and Supercontinuum Generation • Cross-Phase Modulation and Optical Switching • Four-Wave Mixing and Parametric Amplification • Raman Amplification and Silicon Raman Lasers • Concluding Remarks JJ II J I Back Close Silicon Photonics • Silicon dominates microelectronics industry totally. 3/56 • Silicon photonics is a new research area trying to capitalize on the huge investment by the microelectronics industry. • It has the potential for providing a monolithically integrated optoelectronic platform on a silicon chip. JJ II J I Credit: Intel and IBM Websites Back Close Silicon-on-Insulator Technology 4/56 • Core waveguide layer is made of silicon (n1 = 3.45). • A silica layer under the core layer is used for lower cladding. • Silica layer formed by implanting oxygen, followed with annealing. • A rib or ridge structure used for two-dimensional confinement. • Air on top confines mode tightly (large index difference). JJ II J I Back Close SOI Waveguide Geometry [0 0 1] 5/56 [1 1 0] • SEM image of an SOI device; Izhaky et al., JSTQE, (2006). • Typically W ∼ 1 µm, H ∼ 1 µm, h ∼ H/2. • Dispersion normal at telecom wavelengths (λ0 ∼ 2.2 µm). JJ II J I Back Close Waveguide Dispersion • Mode index n̄(ω) = n1(ω) − δ nW (ω). 6/56 • Material dispersion results from n1(ω) (index of silicon). • Waveguide dispersion results from δ nW (ω). • Total dispersion D = DM + DW can be controlled. JJ II J I Back Close Major Nonlinear Effects • Self-Phase Modulation (SPM) 7/56 • Cross-Phase Modulation (XPM) • Four-Wave Mixing (FWM) • Stimulated Brillouin Scattering (SBS) • Stimulated Raman Scattering (SRS) Origin of Nonlinear Effects in Silicon • Third-order nonlinear susceptibility χ (3). • Real part leads to SPM, XPM, and FWM. • Imaginary part leads to two-photon absorption (TPA). JJ II J I Back Close Third-order Nonlinear Response 3ε0 (3) e e∗ e = χ i jkl (−ωi ; ω j , −ωk , ωl )E j (ω j )Ek (ωk )El (ωl )dω j dωk 2 16π • Two distinct contributions from electronic and Raman responses: (3) Pei (ωi) ZZ 8/56 (3) χi jkl = χiejkl + χiRjkl . • Electronic response for silicon is governed by e e e χiejkl = χ1122 δi j δkl + χ1212 δik δ jl + χ1221 δil δ jk + χdeδi jkl . • Raman response for silicon is governed by eR(ωl − ωk )(δik δ jl + δil δ jk − 2δi jkl ) + a similar term. χiRjkl = χRH • A relatively narrow Lorentzian Raman-gain spectrum: 2 Ω Ω Γ R R R eR(Ω) = H = 15.6 THz, = 105 GHz . π Ω2R − Ω2 − 2iΓRΩ 2π JJ II J I Back Close Third-order Nonlinear Susceptibility • The tensorial nature of χ (3) makes theory quite complicated. 9/56 • It can be simplified considerably when a single optical beam excites the fundamental TE or TM mode of the Si waveguide. e • Only the component χ1111 (−ω; ω, −ω, ω) is relevant in this case. • Its real and imaginary part the Kerr coefficient n2 and the TPA coefficient βT as i 3ω ω e n2(ω) + βTPA(ω) = (−ω; ω, −ω, ω). χ1111 2 2 c 2 4ε0c n0 • A recent review provides more details: Q. Lin, O. J. Painter, G. P. Agrawal, Opt. Express 15, 16604 (2007). JJ II J I Back Close Measured Nonlinear Parameters 4 0.7 10/56 0.6 3 0.5 0.4 2 0.3 0.2 1 0.1 0.0 0 1200 1400 1600 1800 2000 2200 2400 1200 1400 1600 1800 2000 2200 2400 Lin et al, Appl. Phys. Lett. 91, 021111 (2007) • We measured n2 and βTPA for silicon in the spectral range of 1.2 to 2.4 µm using the z scan technique. • n2 larger for silicon by >100 compared with silica fibers. • As expected, βTPA vanishes for λ > 2.2 µm JJ II J I Back Close Nonlinear Parameters • Refractive index depends on intensity as (Kerr effect): 11/56 n(ω, I) = n̄(ω) + n2(1 + ir)I(t). • Material parameter n2 = 3 × 10−18 m2/W is larger for silicon by a factor of 100 compared with silica fibers. • Dimensionless parameter r = βTPA/(2k0n2) is related to two-photon absorption (TPA) occurring when hν exceeds Eg/2. • TPA parameter: βTPA = 5×10−12 m/W at wavelengths near 1550 nm. • Dimensionless parameter r ≈ 0.1 for silicon near 1550 nm. • TPA is a major limiting factor for SOI waveguides because it creates free carriers (in addition to nonlinear losses). JJ II J I Back Close Free-Carrier Generation • TPA creates free carriers inside a silicon waveguide according to 12/56 ∂ Nc βTPA 2 Nc = I (z,t) − . ∂t 2hν0 τc • Carrier lifetime is relatively large for silicon (τc > 10 ns). • It limits the device response time if carriers cannot be removed quickly enough. • Free carriers also introduce loss and change the refractive index. • Pulse propagation inside silicon waveguides is governed by ∂ A iβ2 ∂ 2A σ αl 2 + = ik n (1 + ir)|A| A − (1 + iµ)N A − A. 0 2 c ∂z 2 ∂t 2 2 2 JJ II J I Back Close Free-Carrier Absorption (FCA) • Loss induced by FCA: α f = σ Nc with σ = 1.45 × 10−21 m2. 13/56 • Free carriers also change the refractive index by ∆n = −(µ/2k0)σ Nc (free-carrier dispersion). • This change is opposite to the index change n2I resulting from the Kerr effect. • Parameter µ is known as the “linewidth enhancement factor” in the context of semiconductor lasers. • Its value for silicon is close to 7.5 in the spectral region near 1550 nm. • Absorption and index changes resulting from free carriers affect the performance of silicon waveguides. • Quick removal of carriers helps (e.g., by applying a dc electric field). JJ II J I Back Close Removal of Free Carriers 14/56 Jones et al, Opt. Exp. 13, 519 (2005) • A reversed-biased p-n junction is used for this purpose. • Electric field across the waveguide removes electrons and holes. • Drift time becomes shorter for larger applied voltages. • Effective carrier lifetime can be shortened from >20 to <1 ns. JJ II J I Back Close Self-Phase Modulation (SPM) • Refractive index depends on intensity as n0 = n̄ + n2I(t). 15/56 • Propagation constant also becomes intensity-dependent: β 0 = β + k0n2(P/Aeff) ≡ β + γP. • Nonlinear parameter γ = k0n2/Aeff can be larger by a factor of 10,000 compared with silica fibers. • Nonlinear Phase shift: φNL = Z 0 L (β − β ) dz = 0 Z L γP(z) dz = γPinLeff. 0 Here, P(z) = Pine−αz and Leff = (1 − e−αL )/α. • Optical field modifies its own phase (SPM). • Phase shift varies with time for pulses (chirping). JJ II J I Back Close Chirping and Spectral Broadening 1 2 (a) (b) 16/56 0.8 Chirp, δωT0 Phase, φNL 1 0.6 0.4 0 −1 0.2 −2 0 −2 −1 0 1 Time, T/T0 2 −2 −1 0 1 Time, T/T0 2 • In the case of optical pulses, φNL(t) = γP(t)Leff. • Chirp is related to the phase derivative dφNL/dt. • Phase and chirp profiles for super-Gaussian pulses are shown using P(t) = P0 exp[−(t/T )2m] with m = 1 and m = 3. • SPM creates new frequencies and leads to spectral broadening. JJ II J I Back Close Self-Phase Modulation and TPA • Preceding analysis neglected two-photon absorption. 17/56 • Its impact on SPM can be studied by solving: ∂A αl = iγ(1 + ir)|A|2A − A. ∂z 2 • This equation ignores dispersive and free-carrier effects. √ • Using A = P exp(iφNL), we obtain the following analytic solution: P(L,t) = P(0,t) exp(−αl L) , 1 + 2rγP(0,t)Leff 1 ln[1 + 2rγP(0,t)Leff]. 2r • TPA converts linear dependence of φNL on power to a logarithmic one. φNL(L,t) = JJ II J I Back Close Impact of Two-Photon Absorption • In the absence of TPA, φ0 = φmax = γP0Leff. • Inset shows the reduction using r = 0.1. φmax= 3π φmax= 2π 2 φmax= π 8 φ0 / π 2.5 0 φ0 = ln(1 + 2rφmax)/(2r). 10 18/56 nonlinear phase shift φ / π • TPA reduces the maximum phase shift: 3 6 4 2 0 0 1.5 2 4 φmax/ π 6 8 10 1 0.5 0 0.2 0.4 0.6 βTPA/(k0n2) • TPA-induced reduction becomes severe at high powers. • When φmax = 100, φ0 is limited to a value of 15. 0.8 1 JJ II J I Back Close Impact of Free-Carrier Generation Spectral density J/(cm2⋅nm) φmax= 1.5π, I0= 1.2 GW/cm2 0.015 TPA=0 0.01 max 19/56 0.025 TPA only 0.02 Full 2 0.015 0.01 0.005 0.005 0 1.549 1.5495 1.55 1.5505 1.551 λ (µm) 0 φmax= 15.5π, I0= 12.5 GW/cm2 4 0.015 TPA 0.01 only TPA +FCA 0.005 Full 1.548 φ 1.55 λ (µm) 1.552 = 15.5π, I = 12.5 GW/cm2 max 3 Phase shift φ / π Spectral density J/(cm2⋅nm) = 7.5π, I0= 6.0 GW/cm2 φ 0 TPA only 2 1 0 −1 −2 TPA +FCA P(t) = P0e−t /T T0 = 10 ps L = 2 cm τc = 1 ns αl = 1 dB/cm 2 Yin and Agrawal, Opt. Lett. (2007) Full −3 0 1.548 1.549 1.55 1.551 λ (µm) 1.552 −5 −4 −3 −2 −1 0 1 2 3 4 5 t / T0 Free carriers produce a nonlinear phase shift in the opposite direction. JJ II J I Back Close Experimental Results 20/56 Boyraz et al., Opt. Exp. 12, 829 (2004). • First observation of SPM-induced spectral broadening in 2004. • 4-ps pulses launched inside a 2-cm-long SOI waveguide. • The 3-peak output spectrum broadened by a factor of 2 when peak intensity was 2.2 GW/cm2 (P0 ≈ 100 W). JJ II J I Back Close mitted intensity but not in spectral shape and position. The spectral full width at half m (FWHM) is identical to pulses measured without sample. Hence, the applied f the two low-power spectra belong to the regime where the waveguide is responding linear. However, by increasing the peak power to P = 1.55 W (I ≈ 2.6 GW/cm2), the hape of the pulses broadens dramatically; now exhibiting several spectral side wings. dening continues for higher powers which can be seen in the spectrum with P = 6.85 .6 GW/cm2). Experimental Results -70 Transmission (dBm) 100 -75 -80 -85 6.85 W 1.55 W 100 200 300 400 8 mW 1.8 mW 200 300 400 53.2 49.9 43.6 37.4 31.2 24.9 18.7 12.5 6.23 0 500 600 -90 • Larger broadening by 2006. 21/56 • 1.8-ps pulses launched inside a 4-mm-long waveguide. • Width 470 nm height 226 nm. -95 -100 -105 1480 1500 1520 Wavelength (nm) • Spectral asymmetry is due to free-carrier effects. Fig. 1. Transmission spectra of a 4 mm long SOI waveguide as a function of coupled peak power. Linear optical response is observed for low powers (black and red) whereas SPMinduced spectral broadening occurs for high powers (green and blue). Inset: Time- and spectrally resolved FROG trace of input pulses at 1500 nm. Dulkeith et al., Opt. Exp. 14, 5524 (2006). • Inset shows the FROG trace. measured spectral pulse distortions can be explained by self-phase-modulation, known to cause an intensity dependent phase shift of the pulse carrier frequency. s own duration, the pulse experiences an intensity- and thus time-dependent index. New frequency components are generated and the initial pulse spectrum JJ II J I Back Close re broadened and asymmetric. The significant difference is that SPM-induced ng becomes more efficient for longer wavelengths. The spectral distance between the velength SPM oscillation and the originally injected laser wavelength increases from m when the OPA is tuned from 1400 to 1650 nm. As will be discussed in the next his enhancement for longer wavelengths is probably due to an increase of group nd nonlinear refractive index n2. Wavelength Dependence of SPM Transmission (dB) -75 • SPM-broadened spectra at three different wavelengths. -80 -85 -90 22/56 • 1.8-ps pulses tunable from 1400 to 1650 nm. -95 -100 -105 1400 1450 1500 1550 1600 1650 Wavelength (nm) • Larger broadening at longer wavelengths. • Consistent with a larger n2 near 1550 nm Fig. 4. SPM-induced spectral broadening of optical pulses for the same coupled peak power (6.85 W) injected into the SOI waveguide at 1400, 1500 and 1600 nm. The spectral broadening increases with the wavelength. Dulkeith et al., Opt. Exp. 14, 5524 (2006). Experimental Results agree with theoretical predictions. arison with theory and discussion l description ze the experimental data we have used a recently developed theoretical model to he pulse dynamics in Si photonic wires. It accounts for GVD, parametric and nonc nonlinear effects such as SPM, XPM, TPA, or Raman interaction; free carrier- JJ II J I Back Close Formation of Optical Solitons • SPM-induced phase shift φNL > 1 is easily realized. 23/56 • φNL = 1 occurs at z = LNL = 1/(γP0). • Nonlinear length LNL ∼1 cm at moderate peak powers <10 W. • Dispersion length (LD = T02/|β2) can also be made ∼1 cm for pulses 100-fs wide or less. • Pulses propagate as fundamental solitons (N = 1) when LD γP0T02 N = = = 1. LNL |β2 2 • Formation of optical solitons possible if waveguide is designed to provide anomalous dispersion at the operating wavelength. • Numerical simulations confirm this expectation. JJ II J I Back Close Formation of Optical Solitons Normalized P ower P uls e S hape 1 24/56 0.8 P ath averaged 0.6 0.4 TPA Input 0.2 0 500 400 300 200 100 0 100 time (fs ) P uls e S pectrum 200 300 400 500 S pectral Dens ity 1 0.8 0.6 0.4 P ath averaged TPA Input α 0.2 0 1.51 1.52 1.53 1.54 1.55 1.56 Wavelength (µm) 1.57 1.58 1.59 Yin, Lin, and Agrawal, Opt. Lett. 31, 1295 (2006) 130-fs pulses launched inside a 5-mm-long waveguide (N = 1). JJ II J I Back Close Path-Averaged Solitons • Perfect solitons do not for inside silicon waveguides. 25/56 • Linear and nonlinear (TPA) losses broaden the pulse as N = 1 is not maintained along the whole device. • Concept of path-averaged solitons can be used to reduce pulse broadening. • Input peak power increased such that N = 1 on average. • Required peak power is estimated using 1 P̄0 = L Z 0 L P0(z) dz = 1 . γLD • Solion formation has been observed in a recent experiment. JJ II J I Back Close Observation of Optical Solitons 26/56 • In a 2007 experiment, Zhang et al. [Opt. Exp. 15, 7682 (2007)] launched 110-fs Gaussian pulses inside a 5-mm-long waveguide. • Gaussian spectrum broadened at 1250 nm because of β2 > 0. • It narrowed and acquired a “sech” shape at 1484 nm where β2 < 0. JJ II J I Back Close 0 0 −5 −5 −10 −10 Normalized Spectrum (dB) Normalized Intensity (dB) Numerical Simulations −15 −20 −25 −15 −20 −25 −30 −30 −35 −35 −40 −8 −6 −4 −2 0 Time τ/T0 2 4 6 8 27/56 −40 1420 1440 1460 1480 1500 1520 1540 Wavelength (nm) • Numerical results under experimental conditions. • Input: blue curves; Output: red curves; low power case (green). • Experimental data in agreement with numerical predictions. JJ II J I Back Close Supercontinuum Generation • Ultrashort pulses are affected by a multitude of nonlinear effects, such as SPM, XPM, FWM, and SRS, together with dispersion. 28/56 • All of these nonlinear processes are capable of generating new frequencies outside the input pulse spectrum. • For sufficiently intense pulses, the pulse spectrum can become so broad that it extends over a frequency range exceeding 100 THz. • Such extreme spectral broadening is referred to as supercontinuum generation. • This phenomenon was first observed in solid and gases more than 35 years ago (late 1960s.) • Since 2000, microstructure fibers have been used for supercontinuum generation. JJ II J I Back Close SC Generation in a microstructured fiber 29/56 Ranka et al., Opt. Lett. 25, 25 (2000) • Output spectrum generated in a 75-cm section of microstructured fiber using 100-fs pules with 0.8 pJ energy. • Even for such a short fiber, supercontinuum extends from 400 to 1600 nm. • Supercontinuum is also relatively flat over the entire bandwidth. JJ II J I Back Close SC Generation through Soliton Fission • Input pulses correspond to a higher-order soliton of large order; typically N = (γP0T02/|β2|)1/2 exceeds 5. 30/56 • Higher-order effects lead to their fission into much narrower fundamental solitons: Tk = T0/(2N + 1 − 2k). • Each of these solitons is affected by intrapulse Raman scattering. • Spectrum of each soliton is shifted toward longer and longer wavelengths with propagation inside the fiber. • At the same time, each soliton emits nonsolitonic radiation at a different wavelength on the blue side. • XPM and FWM generate additional bandwidth and lead to a broad supercontinuum. JJ II J I Back Close SC Generation in Silicon Waveguides • SOI waveguides also support higher-order solitons when N = (γP0T02/|β2|)1/2 exceeds 1. 31/56 • Higher-order dispersion leads to their fission into much shorter fundamental solitons: Tk = T0/(2N + 1 − 2k). • Spectrum of each soliton shifts toward longer wavelengths by a smaller amount because of a narrower Raman-gain spectrum of silicon. • Similar to the case of optical fibers, each soliton is expected to emit dispersive waves with frequencies on the blue side. • Numerical simulations confirm the potential of SOI waveguides for SC generation. • Spectral broadening over 350 nm is predicted for femtoseond pulses. JJ II J I Back Close Normalized Power Spectral and Temporal Evolution 0.1 32/56 0.05 0 −0.2 Solitons 0 0.2 Dispersive wave 0.4 0.6 0.8 Normalized Power (dB) time (ps) 0 −10 −20 −30 −40 1.2 Cherenkov radiation 1.3 1.4 Solitons 1.5 1.6 λ (µm) 1.7 1.8 1.9 Yin, Lin, and Agrawal, Opt. Lett. 32, 391 (2007) 50-fs pulses launched with 25-W peak power into a 1.2-cm waveguide. JJ II J I Back Close Normalized Power (dB) Impact of TPA on SC Generation 0 33/56 L=3 mm TPA=0 −10 −20 −30 −40 1.2 L=3 mm with TPA 1.3 1.4 1.5 1.6 λ (µm) 1.7 1.8 1.9 Yin, Lin, and Agrawal, Opt. Lett. 32, 391 (2007) • TPA reduces SC bandwidth but is not detrimental. • Nearly 400-nm-wide supercontinuum created within a 3-mm-long waveguide. • Required pulse energies are relatively modest (∼1 pJ). JJ II J I Back Close Optical Switching 34/56 • A Mach-Zehnder interferometer can be made using two Si waveguides. Such a device exhibits SPM-induced optical switching. • In each arm, optical field accumulates linear and nonlinear phase shifts. • Transmission through the bar port of theinterferometer: T = sin2(φL + φNL); φNL = (γP0/4)(L1 − L2). • T changes with input power P0 in a nonlinear fashion. JJ II J I Back Close Cross-Phase Modulation • Consider two optical fields propagating simultaneously. 35/56 • Nonlinear refractive index seen by one wave depends on the intensity of the other wave as ∆nNL = n2(|A1|2 + b|A2|2). • Total nonlinear phase shift in a fiber of length L: φNL = (2πL/λ )n2[I1(t) + bI2(t)]. • An optical beam modifies not only its own phase but also of other copropagating beams (XPM). • XPM induces nonlinear coupling among overlapping optical pulses. JJ II J I Back Close XPM-Induced Spectral Changes 36/56 Hsieh et al., Opt. Exp. 15, 1135 (2007) • 200-fs pump and probe pulses (at 1527 and 1590 nm) launched into a 4.7-mm-long SOI waveguide (w = 445 nm, h = 220 nm). • Pump and probe pulses travel at different speeds (walk-off effect). • XPM-induced phase shifts occurs as long as pulses overlap. • Asymmetric XPM-induced spectral broadening depends on pump power (blue curve); Probe spectra without pump (red curve). JJ II J I Back Close XPM-Induced Switching 37/56 • A Mach–Zehnder interferometer is often used for optical switching. • Output switched to a different port using a control signal that shifts the phase through XPM. • If control signal is in the form of a pulse train, a CW signal can be converted into a pulse train. • Turn-on time quite fast but the generation of free carriers widens the switching window (depends on the carrier lifetime). JJ II J I Back Close Experimental Demonstration 38/56 Boyraz et al., Opt. Exp. 12, 4094 (2004) • A Mach–Zehnder interferometer used for optical switching. • Short pump pulses (<1 ps) at 1560 nm pass through the arm containing a 2.5-cm-long SOI waveguide. • CW probe experiences XPM-induced phase shift in that arm. • Temporal slice of the probe overlapping with the pump is optically switched. JJ II J I Back Close XPM-Induced Switching 39/56 Boyraz al., Opt. Exp. 12, 4094 (2004) • Instantaneous switching on the leading edge, as expected, with high on–off contrast. • Long trailing edge results from the free-carrier effects. • Free carriers provide an additional contribution to the probe phase by changing the refractive index. JJ II J I Back Close Four-Wave Mixing (FWM) 40/56 • FWM is a nonlinear process that transfers energy from pumps to signal and idler waves. • FWM requires conservation of ? Energy ω1 + ω2 = ω3 + ω4 ? Momentum β1 + β2 = β3 + β4 • Degenerate FWM: Single pump (ω1 = ω2). JJ II J I Back Close Parametric Amplifiers • FWM can be used to amplify a weak signal. 41/56 • Pump power is transferred to signal through FWM. • The idler (generated as a byproduct) acts as a copy of the signal at a new wavelength (useful for wavelength conversion). • Parametric amplifiers can provide gain at any wavelength using suitable pumps. • They are also useful for all-optical signal processing. • Optical fibers are often used, but the use of SOI waveguides would result in a much more compact device. • Impact of two-photon and free-carrier absorption requires further study. JJ II J I Back Close FWM Theory for Silicon Waveguides • Theory should include TPA and free-carrier effects fully. 42/56 • Polarization and Raman effects should also be included. • Full vector theory by Lin et al., Opt. Exp. 14, 4786 (2006). • Free-carrier absorption limits the gain for a CW pump. • β2 < 0 (red); β2 = 0 (blue); β2 > 0 (green). JJ II J I Back Close FWM with Short Pump Pulses 43/56 Lin et al., Opt. Exp. 14, 4786 (2006) • FCA is reduced significantly for pump pulses much shorter than carrier lifetime τc. • Figure shows the case of 10-ps pump pulses with τc = 1 ns. • Phase-matching condition is satisfied even for signal that is shifted by 70 nm from the pump wavelength. JJ II J I Back Close Singe and Dual-Pump Configurations 44/56 Lin et al., Opt. Exp. 14, 4786 (2006) • Parametric amplifiers with a large bandwidth can be realized by pumping an SOI waveguide with two pumps. • This is possible because of a relatively short device length. • Recent experiments with SOI waveguides are encouraging. JJ II J I Back Close Experimental Results (Single Pump) 45/56 Foster et al., Nature 441, 760 (2006) • 3.5-ps pump pulses at 1550 nm; signal tunable over 80 nm. • Up to 5 dB of gain observed in devices <2 cm long. • 150-nm bandwidth realized in 2007 [Opt. Exp. 15, 12949 (2007)]. JJ II J I Back Close Photon-Pair Generation 46/56 Lin et al., Opt. Lett. 31, 3140 (2006) • Spontaneous FWM in fibers creates entangled photon pairs but suffers from the noise induced by Raman scattering. • The use of SOI waveguides avoids this problem because Raman scattering does not occur when TM mode is excited. JJ II J I Back Close Experimental Results 47/56 Sharping et al., Opt. Exp. 14, 12388 (2006) • 5-ps pump pulses launched into a 9-mm-long SOI waveguide. • Total (red) and accidental (blue) coincidence rates measured. • Their ratio exceeded 10 at low pump powers. JJ II J I Back Close Stimulated Raman Scattering • Scattering of a pump beam from vibrating molecules creates a Stokes beam down-shifted in frequency by a specific amount. 48/56 • Frequency shift is set by a vibrational mode (phonons). • Raman gain spectrum exhibits a dominant peak at 15.6 THz with a 105-GHz bandwidth (≈1 nm wide near 1550 nm). • Peak gain for silicon >1000 larger compared with silica. Claps et al., Opt. Exp. 13, 2459 (2006) JJ II J I Back Close Raman Amplifiers 49/56 Jalali et al., IEEE JSTQE 12, 412 (2006) • CW pumping leads to accumulation of free carriers through TPA. • Free-carrier absorption introduces losses for pump and signal. • No signal gain occurs for τeff > 10 ns. JJ II J I Back Close Pulsed Raman Amplifiers 50/56 Jalali et al., IEEE JSTQE 12, 412 (2006) • Pulsed pumping can provide >20-dB gain if spacing among pulses is much larger than τeff (R pτeff 1). • Free carriers can then decay before the next pulse arrives. • Pump pulses (∼30 ps) at 1540 used to amplify a 1673-nm signal. • 20-dB net gain realized at 37-W peak power of pump pulses. JJ II J I Back Close Raman Lasers 51/56 Boyraz and Jalali, Opt. Exp. 12, 5269 (2004) • Pumped with 30-ps pulses at 1540 nm at 25-MHz repetition rate. • Produced 18 ps pulses at 1675 nm at the same repetition rate. JJ II J I Back Close CW Raman Amplifiers 52/56 Jones et al, Opt. Exp. 13, 519 (2005) • CW pumping can be used if free carriers are removed quickly. • A reversed-biased p-n junction is used for this purpose. • Electric field across the waveguide removes electrons and holes. • Drift time of carriers is shorter for larger applied voltages. JJ II J I Back Close CW Raman Amplifiers 53/56 Jones et al, Opt. Exp. 13, 519 (2005) • A 4.8-cm-long waveguide CW pumped at 1458 nm (signal at 1684 nm). • Output pump and signal powers increase with applied voltage. • Effective carrier lifetime decreases from 16 to 1 ns. JJ II J I Back Close Theory behind Raman Amplifiers and Lasers • Pump and signal powers satisfy the set of two coupled equations: 54/56 ∂ Pp = −(αlp + αfp)Pp − βTppPp2 − 2βTpsPsPp − gRPsPp ∂z ∂ Ps = −(αls + αfs)Ps − βTssPs2 − 2βTspPpPs + gRPpPs. ∂z • Signal loss αfs by free carriers (generated from the pump-induced TPA) limit the performance severely. • For net amplification to occur, the carrier lifetime should satisfy h̄ω pnsā2pp(gR − 2βTsp)2 τ0 < τth ≡ . 2αlsσasn0sβTppā2sp • A Raman laser cannot function if this condition does not hold. JJ II J I Back Close Concluding Remarks • Nonlinear effects in silicon waveguides can be used to make many active and passive components. 55/56 • SPM is useful for soliton formation and supercontinuum generation. • SPM and XPM can also be used for optical switching, wavelength conversion, and all-optical regeneration. • Four-wave mixing converts silicon waveguides into parametric amplifiers. • It can also be used for quantum applications requiring entangled photon pairs. • Stimulated Raman scattering converts silicon waveguides into Raman amplifiers. JJ II J I Back Close Further Reading • L. Pavesi and D. J. Lockwood, Silicon Photonics (Springer, 2004). 56/56 • G. T. Reed, Silicon Photonics: State of the Art (Wiley, 2007). • IEEE J. Sel. Topics Quantum Electron., Special issue on Silicon Photonics, 12 (2006). • B. Jalali and S. Fathpour, “Silicon Photonics,” J. Lightwave Technol. 24, 4600 (2006). • B. Jalali, “Teaching silicon new tricks,” Nature Photonics 1, 193 (2007). • Q. Lin, O. J. Painter, G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides,” Opt. Express 15, 16604 (2007). JJ II J I Back Close