© 201 2 M acmillan Publishers Limited. All rights reserved. Frequency comb generation at terahertz frequencies by coherent phonon excitation in silicon Muneaki Hase 1,2 * , Masayuki Katsuragawa 3 , Anca Monia Constantinescu 1 and Hrvoje Petek 1 * High-order nonlinear light–matter interactions in gases enable the generation of X-ray and attosecond light pulses, metrology and spectroscopy 1 . Optical nonlinearities in solid-state materials are particularly interesting for combining optical and electronic functions for high-bandwidth information processing 2 . Third-order nonlinear optical processes in silicon have been used to process optical signals with bandwidths greater than 1 GHz (ref. 2). However, fundamental physical processes for a silicon-based optical modulator in the terahertz bandwidth range have not yet been explored. Here, we demon- strate ultrafast phononic modulation of the optical index of silicon by irradiation with intense few-cycle femtosecond pulses. The anisotropic reflectivity modulation by the resonant Raman susceptibility at the fundamental frequency of the longitudinal optical phonon of silicon (15.6 THz) generates a frequency comb up to seventh order. All-optical >100 THz frequency comb generation is realized by harnessing the coherent atomic motion of the silicon crystalline lattice at its highest mechanical frequency. The coherent modulation of electronic and vibrational nonlinea- rities in atoms and molecular gases by intense few-cycle pulses has been used to generate high-harmonic optical pulses in the soft X-ray and attosecond regime, as well as Raman frequency combs that span multiple octaves from the terahertz to petahertz (infrared to vacuum ultraviolet) frequency regions 1,3–6 . In principle, similar high-order nonlinear processes can be excited efficiently in solids and liquids because of their high nonlinear polarizability densities. In practice, however, optical absorption, phase matching and competition from other nonlinear processes (for example, white light generation) limit applications of solid-state materials to low-order electronic processes, such as second-harmonic generation and optical rectifi- cation 7 . The nonlinear optical responses of solid surfaces, however, have been extended to high-harmonic generation 8 and multi-photon photoemission 9 . One might therefore anticipate that coherent vibrational Raman processes 10–12 in a highly nonlinear regime could also modulate light at multiple phonon frequencies in the 1–100 THz range. The impulsive excitation of a crystalline lattice into a quantum optical coherent state 13 corresponding to oscillation at its highest mechanical frequency, the zone-centre longitudinal optical (LO) phonon (coherent phonon), has been used to study nonlinear light–matter interactions, as well as electron–phonon and phonon– phonon coupling 14 . Coherent LO phonon spectroscopy and dynamics have been studied for insulators (diamond 15 ), semicon- ductors (GaAs 16,17 and silicon 18,19 ), semimetals (bismuth 20 , antimony 20 and graphite 21 ), metals 22 , ferroelectrics 23 and organic crystals 24 . In many materials, including silicon, the sudden gener- ation of dense electron–hole plasma both drives the coherent lattice vibration and induces lattice softening, potentially causing a structural phase transition 25 . The novel environment of such non-equilibrium plasmas could also promote highly nonlinear light–matter interactions that are absent in transparent materials. Here, we explore the coherent phonon-induced refractive index modulation of a Si(001) surface upon excitation at ≏397 nm (3.12 eV) in near-resonance 10 with the direct bandgap of silicon (≏3.4 eV) (Fig. 1) 19 . By means of anisotropic electron–hole pair generation and coherent Raman scattering, laser pulses with an energy of ,1 nJ and duration of ≏10 fs exert a sudden electrostric- tive force on the silicon lattice, launching coherent LO phonon oscillations at a frequency of 15.6 THz. With more than one order of magnitude larger amplitude 19 than for non-resonant impulsive stimulated Raman excitation 18,24 , the LO phonon oscillation strongly modulates the direct bandgap of silicon through the optical deformation potential 26,27 . The concomitant oscillatory change in the optical constants modulates the reflected probe light at the fundamental LO phonon frequency, generating a broad comb of frequencies at exact integer multiples of the fundamental frequency and extending to beyond 100 THz. On the basis of an analytical model, we show that the simultaneous amplitude and phase modulation of the reflected light by the coherent lattice polarization at 15.6 THz generates the frequency comb. Figure 2 shows the transient electro-optic reflectivity of the Si(001) sample upon excitation by a pump pulse with a fluence of 1 mJ cm 22 , which is measured by scanning the pump–probe delay and recording the intensity difference between the orthogonal polarization components of the reflected light. Following an initial aperiodic electronic response, which follows the onset of the electro- strictive force 19 , the signal oscillates with a period of ≏64 fs of the zone-centre coherent LO phonon and decays within several picoseconds 18,19 . The coherent LO phonon response in Fig. 2 can be approximately simulated as a damped oscillator with amplitude A, frequency v LO and chirp h, delay-dependent relaxation time (t LO þ vt) and initial phase f, DR EO (t ) R 0 =A exp −t / t LO + nt ( )   cos v LO + ht ( ) t + f   (1) The LO phonon frequency, relaxation time and phase depend on the photoexcited carrier density, which varies with both pump fluence and delay. The frequency shift and relaxation time change are manifestations of the complex self-energy for LO phonon 1 Department of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara Street, Pittsburgh, Pennsylvania 15260, USA, 2 Institute of Applied Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan, 3 Department of Engineering Science, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan. *e-mail: mhase@bk.tsukuba.ac.jp; petek@pitt.edu LETTERS PUBLISHED ONLINE: 4 MARCH 2012 | DOI: 10.1038/NPHOTON.2012.35 NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics 243