CW-pumped single-pass frequency comb generation by resonant optomechanical nonlinearity in dual-nanoweb fiber

Recent experiments in the field of strong optomechanical interactions have focused on either structures that are simultaneously optically and mechanically resonant, or photonic crystal fibers pumped by a laser intensity modulated at a mechanical resonant frequency of the glass core. Here, we report continuous-wave (CW) pumped self-oscillations of a fiber nanostructure that is only mechanically resonant. Since the mechanism has close similarities to stimulated Raman scattering by molecules, it has been named stimulated Raman-like scattering. The structure consists of two submicrometer thick glass membranes (nanowebs), spaced by a few hundred nanometers and supported inside a 12-cm-long capillary fiber. It is driven into oscillation by a CW pump laser at powers as low as a few milliwatts. As the pump power is increased above threshold, a comb of Stokes and anti-Stokes lines is generated, spaced by the oscillator frequency of ∼6 MHz. An unprecedentedly high Raman-like gain of ∼4 × 106 m−1 W−1 is inferred after analysis of the experimental data. Resonant frequencies as high as a few hundred megahertz are possible through the use of thicker and less-wide webs, suggesting that the structure can find application in passive mode-locking of fiber lasers, optical frequency metrology, and spectroscopy. © 2014 Optical Society of America


INTRODUCTION
In recent years, a major thrust in optomechanics has been the observation of light-matter interactions at the single quantum level [1,2].Remarkable microstructures have been developed in which both light and acoustic vibrations are tightly confined within a small volume for relatively long periods of time, and thus forced to interact strongly.An extensive review of the recent progress in this field can be found in [3].
Another goal of optomechanics has been the design of micro/nanostructures that display very high optomechanical nonlinearities, through either electrostrictive changes in refractive index [4,5] or radiation-pressure-driven changes in morphology [6,7].For example, in small-core silica-air photonic crystal fibers, it has been shown that acoustic core resonances at few-gigahertz frequencies can be excited electrostrictively by pumping with dual-frequency laser light [8].These core resonances then act back on the light, resulting in the generation of an optical frequency comb.This process, which was named stimulated Raman-like scattering (SRLS), has been used to passively mode lock a fiber ring laser at gigahertz frequencies [9].
Another example is the dual-nanoweb fiber (the subject of this work), a structure that displays a giant optomechanical nonlinearity as a result of the high mechanical compliance of two very thin, wide, and closely spaced glass "nanowebs" mounted inside a fiber capillary [see Fig. 1(a)] [10].When light of only a few milliwatts is launched into the nanowebs so that the phase across them is constant (i.e., the even mode is excited), optical gradient forces cause the webs to be pulled together, increasing the effective index of the mode.If the odd mode is instead excited, the webs are pushed apart, but the modal index still rises [11].
In a previous study, the frequency response of this nonlinearity was measured at different gas pressures [12].When driven by a laser beam intensity-modulated at the frequency of the fundamental flexural resonance of the nanowebs, effective optomechanical nonlinearities m −1 W −1 some 60,000 times higher than the Kerr-related nonlinearity were measured.
Something more intriguing is observed, however, when the gas pressure is reduced to the μbar range (thus eliminating viscous damping and squeezed-film effects) and the structure is pumped with CW light.Above a sharp threshold of a few milliwatts, the output signal begins to oscillate in intensity and sidebands appear in the optical frequency spectrum [13].Unlike in previous experiments with CW light where optical cavities with high Q factors (for example, in highly nonlinear fibers [14] or whispering gallery mode resonators [15]) were used to generate frequency combs via the electronic Kerr effect, the underlying mechanism in our case is SRLS.As we will show, this effect is initiated by scattering of light at thermally excited phonons.This gives rise to weak uncorrelated Stokes (S) and anti-Stokes (AS) signals.For certain combinations of the randomly fluctuating phases of these signals, the beat note with the pump light drives the acoustic resonance more strongly, further enhancing scattering into the sidebands.An optical frequency comb spaced by the acoustic resonant frequency (∼6 MHz) is created.Symmetry between S and AS scattering causes suppression of pump-to-Stokes Raman gain in gases in the special case when both S and AS are phase-matched to the same coherence wave [16].In our case, however, because of the much stronger thermal vibrations at 6 MHz (kT-driven molecular excitations are vanishingly weak at the multiterahertz frequencies typical of gases) enhanced by the high mechanical Q factor, a substantial population of stochastic thermal phonons is available to stimulate Stokes photon creation, or to cause frequency up-shifting to the anti-Stokes.As a result, no gain suppression is seen.
Here, we report in detail on this new phenomenon, which can be viewed as the first example of noise-seeded, optomechanical SRLS; the dual-nanoweb structure behaves like a sort of "artificial Raman-active molecule."

STRUCTURE, SETUP, AND EXPERIMENTAL RESULTS
The ∼22 μm wide waveguide region of the dual-nanoweb fiber consists of two optically coupled nanowebs with slightly convex thickness profiles.The thicknesses of the upper and lower nanowebs are ∼460 nm and ∼480 nm at the center, and the gap between them is ∼550 nm wide [Fig.1(a)].In the experiment, a 12 cm long sample was used, mounted in a gas cell with windows at each end and evacuated to a pressure of ∼1 μbar.This strongly enhanced the Q factor of the acoustic vibrations and the strength of the resonant optomechanical nonlinearity [12].As explained above, the system began oscillating when a few milliwatts of CW laser light at 1550 nm was launched into the fiber [13].The resulting RF spectrum was measured with high resolution using the heterodyne setup depicted in Fig. 1(b).
The laser system comprised a narrow-linewidth single-mode fiber laser (3 dB linewidth ∼3 kHz) and an erbium-doped fiber amplifier (EDFA).Using a combination of λ∕2 plate and polarizer before and after the polarizing beam splitter, the power and polarization state in both the sample and the local oscillator (LO) paths could be controlled.TE-polarized light was launched into the sample and the transmitted signal, containing the pump and the optomechanically created sidebands, was coupled into a single mode fiber and mixed with the LO signal at a fiber coupler.In the LO path, an acousto-optical modulator was used to upshift the optical carrier frequency by 200 MHz.The beat note between the signals transmitted through the sample and LO path was then detected using a fast photodiode and visualized using a radio-frequency spectrum analyzer.The RF power P RF n of the beat note between the nth comb component with power P n and the LO with power P LO is proportional to the product of both optical powers, i.e., P RF n ∝ P n P LO [17].In Fig. 2, a series of four RF spectra are shown, measured at different launched pump powers P IN and constant P LO , the RF power therefore being proportional to the optical power of the corresponding comb component.Note that the laser system exhibits sidebands at 210 kHz relative frequency, which are, however, suppressed by more than 53 dB relative to the main laser line and cannot seed any optomechanical sidebands due to the frequency mismatch.The spectrum at 4.8 mW exhibits, on either side of the pump peak (at zero relative frequency and ∼70 dB above the background noise), small S and AS peaks (∼10 dB above the noise level) at 6.022 MHz, corresponding to the fundamental flexural resonance of the structure.As the input power is increased to 5.3 mW, six sidebands spaced by 6.022 MHz can already be distinguished.At 7.8 mW, the number of sidebands increases to ten.When the launched power is raised above ∼8 mW, a fine structure of the comb lines with a frequency spacing of 39 kHz appears, and at 10.2 mW, four to six secondary comb lines can be observed around each of the main peaks.When the air pressure inside the fiber is increased to a few mbar, viscous damping of the flexural vibrations causes the threshold power for the onset of comb generation to rise considerably, until above ∼100 mbar it is no longer possible to generate a frequency comb at the power levels available in the experiment.
To further characterize the system, we measured the SRLS gain spectrum using the copolarized dual-frequency (pump+S) excitation technique [8].This involved inserting an electrooptic modulator driven by a function generator before the EDFA, its DC bias adjusted so that two equal amplitude sidebands are synthesized and the carrier wave strongly suppressed.Figure 3 shows the energy transfer from the pump to the S wave, after propagation through the dual-web fiber, as a function of the frequency spacing between them.The total launched power level was kept at a low ∼2 mW so as to minimize conversion to higher-order sidebands.Interestingly, the spectrum reveals clusters of closely spaced sharp peaks, similar to those observed previously [12].These peaks we attribute to flexural resonances localized at structural nonuniformities along the fiber sample.Two distinct resonances appear at 6.022 and 6.061 MHz, spaced 39 kHz and, as we will show later, the interaction between these resonances leads to the generation of the fine structure of the comb lines.

MECHANISM AND THEORY
In this section, we set up a theoretical model for the noiseinitiated SRLS and determine its gain characteristics.We restrict the analysis to the lowest-order TE-polarized optical mode [single-lobed in the y and x directions, Fig. 1(a)] and the fundamental acoustic flexural mode (single-lobed in the x direction) [10].The flat dispersion curve of the flexural mode and the small frequency shift (cut-off frequency ∼6 MHz) ensure that the same phonon can cause phase-matched coupling between successive S and AS components (Fig. 4).This means that the model must take into account a large number of comb components, spaced by ∼6 MHz.
The nonlinear wave equation for the electric field may be written as [8] ∂ where the nonlinear polarization P NL describes the coupling between optical and acoustic fields.The refractive index of the optical mode is n m , ε 0 is the dielectric permittivity, and c is the speed of light in vacuo.The optomechanical nonlinearity originates from the interdependence of optical gradient forces and deflection-dependent changes in n m , and strongly dominates over the contributions of photoelasticity and electrostriction, which we therefore neglect.P NL can therefore be expressed to first order as [18] P NL ε 0 ∂ε ∂δ δE 2ε 0 n m ∂n m ∂δ δE: (2) The web deflection δ is governed by the flexural wave equation for a thin plate [19,20]: where D Y h 3 ∕121 − ν 2 is the flexural rigidity of a nanoweb with thickness h; Y is the Young's modulus, and ν is the Poisson's ratio.The parameter τ is related to the damping rate (see below).The driving term is given by the optical gradient pressure p opt , which can be calculated using the Maxwell stress tensor [11].To simplify the analysis, we neglect magnetic field components (they contribute only ∼6% to the interweb force) and approximate p opt acting on one web by evaluating the difference between the squared electric fields at its upper and lower surfaces [18].Further, in Eq. ( 3), we have introduced a Langevin noise source f describing the thermal excitation of acoustic waves, based on δ-correlated Gaussian noise [21].
Assuming that the fiber has no structural nonuniformities along its length, the electric field, containing an infinite number of copolarized equidistant-in-frequency components, can be written in the form Ex; y; z; t sxf y ffiffiffiffiffiffiffiffiffiffi ffi P 0 Z 0 2n m s X n a n z; te iβ n z−ω n t c:c:; where Z 0 is the impedance of vacuum, P 0 is the launched optical power, and sxf y represents the transverse field distribution of the optical mode, normalized so that R w∕2 −w∕2 jsxj 2 dx 1 and R ∞ −∞ jf yj 2 dy 1.For the structures studied here, sxf y can be taken to be independent of power for launched powers in the milliwatts range [10].Within this normalization, sx and f y have units m −1∕2 .The dimensionless functions a n z; t represent the slowly varying field amplitudes of the comb lines, with frequency ω n ω 0 nΩ and axial propagation constant β n β 0 nq, where ω 0 is the angular frequency, β 0 is the wavevector of the pump mode, and n 1; 2; … is the sideband order, negative values corresponding to S frequencies.
For a flexural wave with axial propagation constant q and frequency Ω close to the cut-off frequency, which is given approximately by Ω co ≈ π∕w is the transverse profile of the flexural resonance (with units m −1∕2 ) normalized so that R w∕2 −w∕2 jδ 0 xj 2 dx 1, and bz; t is the dimensionless slowly varying envelope of the flexural mode.The transverse profile has been taken for simplicity to be δ 0 x 2∕w 1∕2 cosπx∕w, which approximates to its shape at the cut-off frequency.Substituting this expression into Eq.( 3) in the absence of any driving terms and assuming exponential decay of sinusoidal oscillation in time yields a decay rate Γ τΩ 2 co .Note that, although q is non-zero in the experiment, it is very small, taking the value q β p − β S ≈ 2πf ac n m ∕c 0.15 m −1 for acoustic frequency f ac 6 MHz and modal index n m 1.2.This yields an axial acoustic wavelength of ∼42 m, i.e., much longer than the fiber sample.Thus, in Eq. ( 3), we can assume that the z derivative of the deflection is negligibly small compared to its x derivative.Now we apply the slowly varying envelope approximation to Eqs. ( 1) and ( 3), considering only those components of the optical driving term that oscillate with the same frequency and wavevector as the acoustic wave.Since the group velocity of the guided flexural wave at Ω 0 ≈ Ω co is nearly zero (Fig. 4), we can neglect phonon propagation, i.e., set ∂b∕∂z ≈ 0, and obtain the following set of coupled equations: where v g ∂ω∕∂β is the group velocity of the light (which is effectively identical for every frequency component) and ξ L is the rate of seeding by Langevin noise, which is related to the forcing term in Eq. (3) via f −iΩξ L ζ expiqz −Ωtc:c:., where ζ and the other parameters in Eq. ( 6) are defined in Appendix A. By eliminating bz from Eqs. ( 6), the evolution of the fields can be rewritten for exact phase matching (Ω Ω 0 ) and in the steady-state (∂∕∂t 0 for time-averaged values of all quantities as follows: Since the frequency spacing between different optical comb lines and the spectral width of the comb in the experiment are much smaller than the carrier frequency of the pump wave, we have approximated ω n by ω 0 for all n.The uncorrelated Langevin noise terms ξ n−1 L and ξ n1 L (causing coupling to a n from the lower-and higher-frequency sidebands) have the same statistics and are calculated anew for each realization of the code.
To quantify the gain factor g 0 , we calculate the overlap between the optical mode and the flexural mode, using the numerical technique described in [11] and considering the geometry of the experimental structure.This results in n m 1.24, ∂n m ∕∂δ −64 × 10 3 m −1 at the center of the optical mode, h p −6.8 × 10 −6 m, and Q om 661 m −1∕2 .Further, we take the area density σ 10 −3 kg∕m 2 , the experimental values of the mechanical resonance frequency Ω 0 2π × 6.022 MHz and linewidth Γ ≈ 1 kHz, and calculate the gain coefficient at λ 1.55 μm to be g 0 ≈ 10 6 m −1 W −1 .This exceeds the SRLS gain in small-core PCF [8] by six orders of magnitude.

DISCUSSION
The data points in Fig. 5 represent the measured powers in each optical sideband (normalized to the total power) at several different values of launched power.A pronounced threshold is observed at ∼4.9 mW, which is not observed in numerical solutions of Eq. ( 7), which show that the sidebands begin to grow immediately even at infinitesimal power levels.
To explain the existence of the oscillation threshold in the experiment, we have found it necessary to consider acoustic mode competition in the system, by analogy with cavity mode competition in lasers.As mentioned before, structural nonuniformities are known to exist along the fiber.These will cause the appearance of many acoustic resonances, each with a slightly different frequency, localized at different positions along the fiber (see Fig. 3).Since all these resonances interact with the same guided optical mode and have similar gain characteristics, they destructively interfere, suppressing the SRLS process as long as the input power stays below a critical value.Full lines show the theoretical expectations for the pump (black), the first-order (red), second-order (blue), third-order (green), fourth-order (magenta), and fifth-order (cyan) S and AS using the fit parameters g 0 4 × 10 6 m −1 W −1 and an effective comb generation length L eff 6 cm.A distinct threshold for the onset of frequency comb generation occurs at a launched power of ∼4.9 mW, marked by the dashed vertical line.Below threshold, the power in each comb line decreases exponentially with its order.The inset shows the experimentally relevant data range above a noise floor of ∼10 -7 relative to the pump power.
In our model we assume that, due to the Lorentzian lineshape of the gain profile, there is a power-dependent line narrowing that gradually reduces the overlap between competing acoustic modes.Above the critical power, the overlap is weak enough and the mode competition eliminated, so that a subset of the localized acoustic modes switches from random motion with zero mean phonon amplitude to coherent vibration with finite mean phonon amplitude.Thus, the effective nonlinear interaction length L eff increases and consecutive optical sidebands are generated and amplified via SRLS.
The theoretical plots in Fig. 5 (solid lines) were obtained by numerically solving a model for acoustic mode competition and calculating the average from a set of 1000 different realizations seeded by stochastic acoustic noise.Good quantitative agreement with the experimental data is obtained using g 0 4 × 10 6 m −1 W −1 and L eff 6 cm as the effective nonlinear length.Given the uncertainties in the values of acoustic linewidth and L eff , the agreement between the theoretical and experimental values of gain is good.Due to the resolution limit of the measurement, the estimated acoustic linewidth (1 kHz) is likely to be too high, leading to an underestimate of the experimental gain.Also, uncertainty in L eff leads to uncertainty in the value of g 0 , since only the product g 0 L eff can be measured experimentally.
Finally, we address the appearance of secondary frequency combs around each primary sideband at pump powers above ∼8 mW [Fig.2(d)].These arise from the presence of a second localized resonance at a slightly different frequency, caused by nonuniformities in the nanoweb structure (we have recently observed this using a side-scattering technique, to be reported elsewhere).This second resonance has a higher threshold power and acts on all the frequency components of the primary comb, so that each of the primary sidebands (spaced by f 1 6.022 MHz) pumps a secondary comb with a slightly different frequency spacing (f 2 6.061 MHz), resulting in the generation of a multiplicity of finely spaced frequencies f f 0 nf 1 mf 2 .As the spacing of the secondary comb is 39 kHz larger than that of the primary comb, the fine structure of the comb lines is offset by nf 2 − f 1 n × 39 kHz from the center of the nth primary sideband.The remarkably high gain factor of SRLS results in full energy transfer from the primary fifth S and AS signals to the secondary comb lines at a pump power of 10.2 mW.An extended theory (to be published in detail elsewhere), including acoustic mode competition and dual-comb generation, confirms this picture.

CONCLUSIONS
A mechanically highly compliant nanostructure, consisting of two very thin glass membrane waveguides, experiences strong optical gradient forces when light is launched into it.The resulting mechanical deformation results in a large increase in effective modal phase index, i.e., a giant optomechanical nonlinearity.As a result, when the structure is pumped with CW laser light at the few-milliwatts level, it behaves like an artificial Raman-active molecule, causing the generation of a frequency comb with spacing equal to the acoustic resonant frequency.This is the first time that single-pass stimulated Raman-like scattering, seeded from noise, has been observed in a nanomechanical resonator.By using thicker and less-wide webs, resonant frequencies as high as a few hundred megahertz seem possible (albeit with lower gain), suggesting that the structure may be useful in optical frequency metrology, spectroscopy, and passive mode locking of fiber lasers.

Fig. 1 .
Fig. 1.(a) Scanning electron micrograph of the fiber core region: the width w of the dual-nanoweb waveguide is ∼22 μm, the upper and lower web thicknesses in the center are h u ∼ 460 nm and h l ∼ 480 nm, and the gap thickness is h g ∼ 550 nm.The sample length L is 12 cm.(b) Schematic of the heterodyne detection setup with an evacuated dual-nanoweb fiber sample.FL, fiber laser; EDFA, erbium-doped fiber amplifier; L, lens; P, polarizer; PBS, polarizing beam splitter; AOM, acousto-optical modulator; SMF, single mode fiber; PC, polarization controller; FC, fiber coupler; PD, photodiode; RF-SA, radio-frequency spectrum analyzer; PM, power meter.

Fig. 2 .
Fig. 2. RF spectra of the transmitted optical signal measured at different CW input powers.The frequency was scanned over 500 kHz around each of the comb sidebands at multiples of 6.022 MHz.(a) Initial amplification of first-order S and AS components at 4.8 mW input power.(b) At 5.3 mW, three sidebands appear on each side of the pump peak.(c) S and AS components up to fifth order are detected at 7.8 mW input power.(d) At 10.2 mW, a fine structure of lines with a spacing of 39 kHz appears around each of the main comb components.The tick spacing on the frequency axis of each pane is 39 kHz.

Fig. 3 .
Fig. 3. Energy transfer from pump to Stokes as a function of the frequency spacing of the dual-frequency light (50% pump and 50% Stokes) at 2 mW launched total power.

2 Fig. 4 .
Fig.4.Schematic of the dispersion diagram for SRLS by guided flexural waves with a cut-off angular frequency Ω co .A phonon with frequency Ω 0 ≈ Ω co and propagation constant q automatically provides phase matching between successive optical S and AS components.

Fig. 5 .
Fig.5.Evolution of the power in the comb lines (normalized to the total output power) as a function of the launched pump power.Full and open circles represent experimental data for the S and AS components, respectively.Full lines show the theoretical expectations for the pump (black), the first-order (red), second-order (blue), third-order (green), fourth-order (magenta), and fifth-order (cyan) S and AS using the fit parameters g 0 4 × 10 6 m −1 W −1 and an effective comb generation length L eff 6 cm.A distinct threshold for the onset of frequency comb generation occurs at a launched power of ∼4.9 mW, marked by the dashed vertical line.Below threshold, the power in each comb line decreases exponentially with its order.The inset shows the experimentally relevant data range above a noise floor of ∼10 -7 relative to the pump power.