DERIVATION OF TRANSDUCED SIGNALWe begin by modeling the optomechanical system with the Hamiltonian H =hΔâ †â +hω mb †b +hg(b † +b)â †â + ih κ e 2 α in,where Δ = ω o − ω l , with laser frequency ω l , optical mode frequency ω o and mechanical mode frequency ω m . Herê a (â † ) andb (b † ) are respectively the annihilation (creation) operators of photon and phonon resonator quanta, g is the optomechanical coupling rate corresponding physically to the shift in the optical mode frequency due to the zero-point fluctuations (x zpf = h/2mω m , m motional mass) of the phonon mode. Classical Derivation of Observed SpectraBy making the substitutionsâwe can treat the system classically by representing the photon amplitudes as a Fourier decomposition of sidebands. Notice that the infinite summation over each sideband order q, can be relaxed to a few orders in the sideband resolved regime (κ ω m ). The phonon amplitude, β 0 , is the classical mechanical excitation amplitude. For an oscillator undergoing thermal Brownian motion, β 0 , is a stochastic process. We assert the stochastic nature of the variable, at the end of the derivation where the power spectral density is calculated. The equation of motion for the slowly varying component is thenwhere we introduce the cavity (optical) energy loss rate, κ, and the cavity coupling rate, κ e . This can be written as a system of equations M · α = a in whereBy truncating and inverting the coupling matrix M one can determine each one of the sidebands amplitude as α q = (M −1 ) qp a in,p and therefore determine the steady state power leaving the cavity to beSUPPLEMENTARY INFORMATION

THEORY OF OPTOMECHANICAL EIT, EIA AND PARAMETRIC AMPLIFICATIONHere we provide a theoretical treatment of some of the main aspects of EIT [1][2][3][4], EIA [5] and parametric amplification [6][7][8] in optomechanical systems. Modeling the optomechanical system with the Hamiltonianit is possible to linearize the operation of the system, under the influence of a control laser at ω c , about a particular steady-state given by intracavity photon amplitude α 0 and a static phonon shift β 0 . The interaction of the mechanics and pump photons at ω c with secondary "probe" photons at ω s = ω c ± ∆ with two-photon detuning ∆ can then be modeled by making the substitutionŝAssuming that the pump is much larger than the probe, |α 0 | |α ± |, the pump amplitude is left unaffected and the equations for each sideband amplitude α ± are found to beWe have defined ∆ OC = ω o − ω c as the pump detuning from the optical cavity (including the static optomechanical shift, ω o ), and β + = β * − . In these situations it is typical to define G = gα 0 , as the effective optomechanical coupling rate between a sideband and the mechanical subsystem, mediated by the pump. Red-detuned pump: Electromagnetically Induced TransparencyWith the pump detuned from the cavity by a two-photon detuning ∆, the spectral selectivity of the optical cavity causes the sideband populations to be skewed in a drastic fashion. It is then an acceptable approximation to neglect one of these sidebands, depending on whether the pump is on the red or blue side of the cavity. When the pump resides on the red side (∆ OC > 0), the α + is reduced and can be neglected. This is the rotating wave approximation (RWA) and is valid so long as ∆ κ. Then Eqs. (S3-S4) may be solved for the reflection and transmission coefficients r(ω s ) and t(ω s ) of the side-coupled cavity system. We find that These equations are plotted in Figs. S1 and S2.

A laser cavity formed from a single defect in a two-dimensional photonic crystal is demonstrated. The optical microcavity consists of a half wavelength-thick waveguide for vertical confinement and a two-dimensional photonic crystal mirror for lateral localization. A defect in the photonic crystal is introduced to trap photons inside a volume of 2.5 cubic half-wavelengths, approximately 0.03 cubic micrometers. The laser is fabricated in the indium gallium arsenic phosphide material system, and optical gain is provided by strained quantum wells designed for a peak emission wavelength of 1.55 micrometers at room temperature. Pulsed lasing action has been observed at a wavelength of 1.5 micrometers from optically pumped devices with a substrate temperature of 143 kelvin.

I. MEASURED AND SIMULATED PROPERTIES OF THE OPTOMECHANICAL CRYSTAL NANOBEAM RESONATOR Table S1 summarizes the properties of the breathing mechanical modes. Measured values are denoted with a tilde. The necessary RF amplitudes and linewidths are extracted from the spectra of Fig. 3c using a nonlinear least squares fit with linear background and a sum of as many Lorentzian functions as are visible in the spectrum. Simulated values are calculated using methods described below. Fig. 3c of main text) and dividing by the m eff from the model. The superscript, n, in n L OM , indicates coupling of that mechanical mode to the nth optical mode (see Fig. 1b of main text). See §V F for discussion on modeling Q m .

Several kinds of nonlinear optical effects have been observed in recent years using silicon waveguides, and their device applications are attracting considerable attention. In this review, we provide a unified theoretical platform that not only can be used for understanding the underlying physics but should also provide guidance toward new and useful applications. We begin with a description of the third-order nonlinearity of silicon and consider the tensorial nature of both the electronic and Raman contributions. The generation of free carriers through two-photon absorption and their impact on various nonlinear phenomena is included fully within the theory presented here. We derive a general propagation equation in the frequency domain and show how it leads to a generalized nonlinear Schrödinger equation when it is converted to the time domain. We use this equation to study propagation of ultrashort optical pulses in the presence of self-phase modulation and show the possibility of soliton formation and supercontinuum generation. The nonlinear phenomena of cross-phase modulation and stimulated Raman scattering are discussed next with emphasis on the impact of free carriers on Raman amplification and lasing. We also consider the four-wave mixing process for both continuous-wave and pulsed pumping and discuss the conditions under which parametric amplification and wavelength conversion can be realized with net gain in the telecommunication band. 1678-1687 (2006). 4. R. A. Soref, S. J. Emelett, and W. R. Buchwald, "Silicon waveguided components for the long-wave infrared region," J. Opt. A: Pure Appl. Opt. 8, 840-848 (2006). 5. M. Dinu, F. Quochi, and H. Garcia, "Third-order nonlinearities in silicon at telecom wavelengths," Appl. Phys.Lett. 82, 2954-2956 (2003). 6. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Opt. Express 11, 1731-1739 (2003). 7. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption, and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002 3685-3697 (1973). 109. T. R. Hart, R. L. Aggarwal, and B. Lax, "Temperature dependence of Raman scattering in silicon," Phys. Rev. B 1, 638-642 (1970). 110. A. Zwick and R. Carles, "Multiple-order Raman scattering in crystalline and amorphous silicon," Phys. Rev. B 48, 6024-6032 (1993). 111. R. Loudon, "The Raman effect in crystals," Adv. Phys. 50, 813-864 (2001). 112. J. R. Sandercock, "Brillouin-scattering measurements on silicon and germanium," Phys. Rev. Lett. 28, 237-240 (1972). 113. M. Dinu, "Dispersion of phonon-assisted nonresonant third-order nonlinearities," IEEE J. Quantum Electron.39, 1498-1503 (2003).