Abstract

Stimulated Brillouin scattering (SBS) is a third-order nonlinear process that involves the interaction of two light fields and an acoustic wave in a medium. It has been exploited for applications of optical communication, sensing, and signal processing. This effect, originally demonstrated in long optical fibers, has recently been realized in silicon waveguides on a chip-scale integrated platform. However, due to the weak per-unit-length SBS gain, the length of the silicon waveguides is usually several centimeters, which prevents device miniaturization for high-density integration. Here, we engineer a phoxonic crystal waveguide structure to achieve significantly enhanced SBS gain in the entire C band, by taking advantage of its simultaneous confinement of slow propagating optical and acoustic waves. The resulting SBS gain coefficient is greater than 3 × 104 W−1 m−1 in the wavelength range of 1520–1565 nm with the highest value beyond 106 W−1 m−1, which is at least an order of magnitude higher than the existing demonstrations. This giant enhancement of SBS gain enables ultracompact and high-performance SBS-based integrated optoelectronic devices such as Brillouin lasers, amplifiers, and signal processors.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Stimulated Brillouin scattering (SBS) is a nonlinear optical process in a bulk material where traveling light waves are scattered by traveling acoustic waves [1]. This effect is greatly enhanced in nanosized waveguides owing to the mutual optomechanical interaction: the interplay between the pump and the Stokes waves generates dynamic optical forces via optical radiation pressure and electrostrictive effect; the mechanical vibration of the waveguide excited by the optical forces in turn scatters light from the pump into the Stokes wave [2]. Exploiting this physical mechanism, Brillouin photonics has rapidly grown as a new branch of cavity optomechanics, producing various applications for wideband RF and photonic signal processing [3–6]. For example, traveling-wave Brillouin processes have enabled unique schemes for optical pulse compression, pulse and waveform synthesis [4, 7–9], coherent frequency comb generation [4, 7, 10], variable-bandwidth optical amplifiers [9, 11], reconfigurable filters [12], coherent beam combining [13], and nonreciprocal transmission and light storage [14, 15].

More recently, the creation of strong Brillouin nonlinearities has been achieved in nanoscale silicon waveguides on an integrated platform [16–18]. SBS was first achieved in a bare silicon waveguide with amplification of 0.5 dB (12%) by using suspended silicon nanowire structures [16]. Later, Kittlaus et al. reported large Brillouin amplification in a membrane-suspended silicon waveguide, which efficiently suppresses other nonlinear effects such as the two-photon absorption and the subsequent carriers-induced free-carrier absorption [18]. The devices achieved amplification levels greater than 5 dB and demonstrated a record low (5 mW) threshold for net amplification. However, all the realizations are based on waveguides extending several centimeters in length to obtain sufficiently strong interaction between light and sound. Such long waveguide lengths are incompatible with device miniaturization and thus undesirable for on-chip integration. Additionally, compared with silica, the relatively high stiffness of silicon makes guiding of acoustic waves a challenging task, thus motivating fabrication of a delicately suspended waveguide [16] or a complicated membrane waveguide [18] to reduce the phonon dissipation.

Photonic and phononic crystals are artificial periodic structures which exhibit frequency bandgaps for optical (light) and acoustic (sound) waves, respectively. Structures that are designed to have simultaneous photonic and phononic bandgaps are referred to as phoxonic crystals. They possess all the optical and acoustic properties of the individual photonic and phononic crystals. As such, phoxonic crystal waveguides can guide propagating optical and acoustic waves [19] and can even achieve simultaneous slow light and slow sound if properly designed [20]. Phoxonic crystal cavities can confine standing optical and acoustic waves, producing large modal overlap and enabling strong optomechanical interaction [21, 22]. These phoxonic crystal structures have been employed for a wide range of optomechanical applications.

In this paper, we engineer a two-dimensional phoxonic crystal waveguide structure [20, 23–30] which simultaneously confines and guides slow traveling optical and acoustic waves in a nanoscale line defect to achieve strong SBS and to obtain giant enhancement of SBS gain. The defect-guided optical and acoustic waves can both be slowed down due to their relatively flat dispersion, leading to substantially increased interaction time between the light and sound. In addition, the phoxonic crystal structure also reduces phonon dissipation from the defect waveguide due to the complete structural suspension and the absolute phononic bandgap. The proposed structure is also easy to fabricate by using one step of electron-beam lithography and dry etching followed by underlying oxide removal on a standard silicon-on-insulator wafer. Our theoretical and numerical results show that the slow light and the phase-matching condition among the pump light, Stokes light, and acoustic wave can both be achieved in the entire C band, covering the wavelength range of 1520–1565 nm. As a result, the SBS gain is enhanced significantly, with the calculated gain coefficient greater than 3 × 104 W−1 m−1 in the entire C band and the highest value beyond 106 W−1 m−1. This is at least an order of magnitude higher than those in the existing demonstrations, which are 3,055 W−1 m−1, 2,750 W−1 m−1, and 1,020 W−1 m−1 in [16], [17], and [18], respectively. This giant enhancement of SBS gain will lead to significant reduction of device footprint and thus can tremendously increase the integration density of the SBS-based optoelectronic devices such as Brillouin lasers, amplifiers, and signal processors on a chip.

2. Derivation of the SBS gain of a phoxonic crystal waveguide

We use the Bloch wavevector k (q) to characterize the propagating optical (acoustic) mode of a phoxonic crystal waveguide with periodicity along the x direction. In a typical SBS process, the pump (p) and Stokes (s) waves can be described by vector fields

Ep(x,y,z,t)=ap(x,t)ep(x,y,z,t)+c.c.,
Es(x,y,z,t)=as(x,t)es(x,y,z,t)+c.c.,
where the pump and Stokes modal fields with respective wavevector kp and ks are expressed as
ep(x,y,z,t)=e˜p(x,y,z)exp[i(ωptkpx)],
es(x,y,z,t)=e˜s(x,y,z)exp[i(ωstksx)].
Here, ap and as are the envelope functions of the electric fields. p and s are the basis functions of the electric fields which vary periodically in the x direction. ωp and ωs are the angular frequencies of the pump light and the Stokes light. The pump and Stokes light’s energy flux and energy stored within one period of a phoxonic crystal waveguide can be expressed as [31]
Pp=2x^{[ep(x,y,z,t)]*×hp(x,y,z,t)}dydz,
Ps=2x^{[es(x,y,z,t)]*×hs(x,y,z,t)}dydz,
ξp=20adxε(x,y,z)[ep(x,y,z,t)]*ep(x,y,z,t)dydz,
ξs=20adxε(x,y,z)[es(x,y,z,t)]*es(x,y,z,t)dydz,
where a denotes the period of phoxonic crystal waveguide in the x direction. The energy flux is independent of position in the phoxonic crystal waveguide along the traveling direction [31]. The energy flow rates, or group velocities, of the pump and Stokes light can be expressed as vp = aPp/ξp and vp = aPs/ξs, respectively.

Similarly, an optically generated acoustic wave can be expressed as

U(x,y,z,t)=b(x,t)u(x,y,z,t)+c.c.,
where the acoustic modal field with wavevector q = kpks and angular frequency Ω = ωpωs is expressed as
u(x,y,z,t)=u˜(x,y,z)exp[i(Ωtqx)].
Here, b is the envelope function of the acoustic wave, and ũ is the basis function of the acoustic wave which varies periodically in the x direction. The acoustic energy flux and energy stored within one period of a phoxonic crystal waveguide can be expressed as
Pb=2iΩjmlcxjmluj*ulmdydz,
ξb=2Ω20adxρ|u|2dydz,
where c is the stiffness tensor and ρ is the density of the medium. ∂/∂m denotes the spatial derivative with respect to the m direction (m = x, y, z). The acoustic energy flux is also independent of position in the phoxonic crystal waveguide along the traveling direction. The acoustic energy flow rate can be expressed as vb = aPb/ξb.

Next, we adopt the method in [32] and extend it to the case of a longitudinally varying waveguide [33] to derive the light–sound interaction equations. The acoustic field induces perturbations to both electric fields (∆E) and electric displacements (∆D), which should also satisfy the Maxwell equations, so we can write the wave equation as

××(E+ΔE)+μ0(D+ΔD)t=0.
Under the phase-matching condition, we can express ∆E and ∆D as
ΔE=Δesapb*+Δepasb+c.c.,
ΔD=Δdsapb*+Δdpasb+c.c..
Then, we evaluate the contribution from the pump light to Eq. (13),
0=××(ases+Δesapb*)+μ0(asds+apb*Δds)t,
0=as(××es+μ0dst)+[x^×(×es)+×(x^×es)]asx2iωsμ0dsast+apb*(××Δes+μ02Δdst2)+h.o.t.+c.c.,
where “h.o.t.” stands for the higher-order terms in the perturbations and higher-order derivatives of the envelope functions. We can neglect “h.o.t.” under the slowly-varying-envelope assumption. We can also drop all the complex conjugate terms after projecting onto the mode es and averaging over a time interval much longer than the optical time scale. Multiplying (es)* on both sides of Eq. (17) and integrating it over one period in the x direction yields
iωsaPsasxiωsPsvsast=apb*x0x0+adx[(es)*(iω)2Δds(iω)2dsΔes]dydz.
Since the envelope functions as, ap, and b vary slowly and can be treated as constants over a period, they can be taken out of the integral in the above equation. Following the same steps, we can also obtain the equation for the envelope of the Stokes light which is similar to Eq. (18), so we arrive at a set of coupled equations for the envelope functions
vsasx+ast=iωsQOMsξsapb*,
vpapx+apt=iωpQOMpξpasb,
where
QOMp=x0x0+adx[(ep)*ΔdpdpΔep]dydz,
QOMs=x0x0+adx[(es)*ΔdsdsΔes]dydz.
On the other hand, the acoustic wave in the phoxonic crystal waveguide is governed by
ρ2Uit2+jmlj[cijml+ηijmlt]mUl=Fi,
where η is the viscosity tensor. The source term F on the right-hand side is the external driving force field per unit volume, through which the coupling to the electromagnetic fields is introduced. The driving force density F can be expressed as
F(r,t)=f(r,t)as*(x,t)ap(x,t)+c.c..
Substituting this ansatz into Eq. (23) and dropping higher-order terms eventually yields
iΩjml[(cixmlm+jcijxl)ulbx2iΩρuibt+jηijmlulmb+fias*ap]+c.c.=0.
After projecting onto the mode u under the phase-matching condition, we arrive at the equation for the acoustic envelope function
vbbx+αbb+bt=iΩQOMbξbas*ap,
where αb is the loss rate of the propagating acoustic wave and
QOMb=x0x0+adxu*fdydz.
As the acoustic frequency is orders of magnitude smaller than the optical frequency, we can treat the frequencies of the pump (ωp) and Stokes (ωs) light to be the same (ω), then we get the relationship if we ignore the irreversible optomechanical coupling in the phoxonic crystal waveguide [32]
QOMs=(QOMp)*=QOMb,
so that we can use a single parameter QOM to characterize the optomechanical coupling [34]. In the steady state, all the time derivatives (∂/∂t) become zero. Equations (19), (20), and (26) reduce to a set of coupled equations for the time-independent envelope functions of the pump, Stokes, and acoustic waves in the phoxonic crystal waveguide:

vsas(x)x=iωsQOM*ξsap(x)b*(x),
vpap(x)x=iωpQOMξpas(x)b(x),
vbb(x)x+αbvbb(x)=iΩQOM*ξbas*(x)ap(x).

The optomechanical coupling is the sum of the coupling coefficients from the photoelastic effect QPE [35] and from the moving-boundary effect QMB [36]. The photoelastic effect refers to the variation of a material’s electric susceptibility (or dielectric constant) due to the strain. The photoelastic contribution can be derived from a first-order perturbation theory [37]

QPE=ε0Vd3rijmlεr2(Eip)*EjspijmlSml,
where εr is the relative permittivity (dielectric constant) of the periodic waveguide and ε0 is the vacuum permittivity, p is the rank-four photoelastic tensor, and S is the strain tensor. The moving-boundary effect refers to the frequency variation of an optical mode due to the displacement of structural surfaces. As an acoustic wave propagates along a waveguide, it induces a strong change to the optical fields over a very small area at the waveguide surfaces. The moving-boundary contribution can be expressed as [19]
QMB=Sd2r(u*n^)[(εrεair)ε0(n^×Ep)*(n^×Es)(εr1εair1)ε01(n^Dp)*(n^Ds)],
where n^ is the unit vector of surface normal pointing outward, εair is the relative permittivity of air, and dp and ds are the basis electric displacement fields of the pump and Stokes light. The optomechanical coupling parameter QOM depends on the magnitudes of the optical fields and the mechanical displacement of the acoustic field. In the community of cavity optomechanics, the optomechanical coupling coefficient gOM is defined as the frequency shift of the optical mode per unit length of mechanical displacement. Therefore, the gOM in our case can be defined as
gOM=ωQOMsQOMp/(ξsξp),
where the normalized acoustic modal field [max(|u|) = 1] should be used.

Lastly, the SBS gain coefficient in units of W−1 m−1 under the phase-matching condition can be expressed as [32]

G=Ω|gOM|2ωvpvsvbαb(ξb/a),
where ξb should also be evaluated by using the normalized acoustic modal field. According to Eq. (35), the SBS gain is inversely proportional to the group velocities of both the pump (vp) and Stokes (vs) waves. Therefore, the SBS gain will be enhanced by slowing down both the pump and Stokes light’s speeds in the waveguide. Additionally, acoustic waves of a slower group velocity will induce a larger strain in the core of the waveguide than that in the slab, leading to stronger modal confinement to the waveguide core with a higher modal amplitude [24]. As a result, the SBS gain will also be enhanced by slowing down the group velocity of the acoustic wave.

3. Engineering the phoxonic crystal waveguide for obtaining simultaneous slow light and slow sound

A two-dimensional phoxonic crystal structure can be formed by periodic arrangement of cross-shaped openings in a silicon slab, with the unit cell depicted in Fig. 1(a). Among various lattice structures, we choose the square lattice because the triangular lattice cannot achieve a sizable phononic bandgap and the hexagonal lattice’s photonic bandgap is unsuitable for practical use [27]. The geometric parameters of the unit cell are the thickness h = 340 nm, the array pitch a = 490 nm, and the cross-shaped slot size 0.2a × 0.8a. We use the following material parameters for silicon: refractive index n = 3.47, Young’s modulus E = 170 GPa, Poisson’s ratio ν = 0.28, and density ρ = 2329 kg/m3. We also assume that the [100], [010], and [001] crystalline directions are aligned with the x, y, and z axis respectively, where the photoelastic tensor pijkl in the contracted notation is (p11, p12, p44) = (−0.09, 0.017, −0.051) [15]. Figures 1(b) and 1(c) show the quasi-TE photonic and phononic band diagrams calculated from MPB [38] and COMSOL Multiphysics, respectively, where the bandgaps are marked in red and the light cone is marked in gray.

 

Fig. 1 (a) Unit cell of a two-dimensional silicon phoxonic crystal structure. The structural parameters are h = 340 nm and a = 490 nm. (b, c) Calculated quasi-TE photonic (b) and phononic (c) band diagrams of the phoxonic crystal structure with the unit cell shown in (a). The bandgaps are marked in red and the light cone is marked in gray.

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By removing a row of cross-shaped slots along the Γ–X direction from the above two-dimensional phoxonic crystal structure we create a W1 phoxonic crystal waveguide as shown in Fig. 2(a). This phoxonic crystal waveguide has three symmetry planes x = 0, y = 0, and z = 0, which should be aligned with the respective crystalline planes of the material so that its anisotropic optical, elastic, and photoelastic constants have simple expressions. We calculated the photonic [Fig. 2(b)] and phononic [Fig. 2(c)] band diagrams along the x direction, where the defect-guided optical and acoustic modes (marked in red) are clearly identified in the respective bandgaps. The right panel of Fig. 2(a) shows the modal profiles of the defect-guided optical (upper two) and acoustic (lower two) modes. Figure 2(d) shows the calculated group velocities of the defect-guided optical mode (in units of the speed of light in vacuum, c) and acoustic mode (in units of m/s). Both the light and sound speeds are slowed down remarkably in the defect phoxonic crystal waveguide band and diminish to zero at the edge of the Brillouin zone where the propagating waves reduce to standing waves.

 

Fig. 2 (a) Left: Schematic of the W1 phoxonic crystal waveguide structure. Right: Profiles of the defect-guided optical mode (|Ey|2) and acoustic mode (Uy). (b) Photonic band diagram of the W1 phoxonic crystal waveguide structure. The cyan-shaded regions denote the slab-guided band continuum, where the modes in the dielectric (air) band reside mostly in the material (air holes). The black and red lines in the bandgap denote the defect-guided optical modes which do not cross each other in the entire kx range. (c) Phononic band diagram of the W1 phoxonic crystal waveguide structure. The light blue regions denote the phononic bandgaps. The red line denotes the defect-guided acoustic mode. (d) Group velocities of the defect-guided optical (vl,g, in red) and acoustic (vs,g, in blue) modes propagating in the phoxonic crystal waveguide.

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The SBS gain in a phoxonic crystal waveguide can be greatly enhanced if the three requirements are satisfied: (i) simultaneous slow optical and acoustic propagating waves, (ii) phase matching among the pump light, the Stokes light, and the acoustic wave, and (iii) strong optomechanical coupling between the slow optical and acoustic modes. First, let us consider a forward SBS process where a defect-guided pump light wave (angular frequency ωp, wavevector kp) interacts with a defect-guided acoustic wave (Ω, –q), producing a scattered Stokes light wave (ωp – Ω, kp + q). Since the acoustic frequency Ω is typically orders of magnitude smaller than ωp, the scattered Stokes light mode is in the same defect-guided band as the pump light. In addition, due to the substantially different slopes of the dispersion curves, the acoustic wavevector q which matches the difference between the wavevectors of the pump light and Stokes light should approach zero. Therefore, all the three waves, the pump light, the Stokes light, and the acoustic wave, are in the respective slow-light and slow-sound regions, simultaneously.

Next, we explain how the phase-matching condition can be achieved among the pump light, the Stokes light, and the acoustic wave. As shown in Fig. 3, the pump light and Stokes light are both in the slow-light region of the defect-guided optical mode. In the right zoomed-in diagram, the green (red) line denotes the dispersion curve of the defect-guided optical (acoustic) mode traveling in the x direction of the phoxonic crystal waveguide. We superimpose the phononic band diagram onto the photonic band diagram by aligning the origin of the former to the operating point of the pump light in the latter. Since the frequency variation of an optical mode is orders of magnitude larger than that of the acoustic mode with the same wavevector, the dispersion curve of the optical mode should be much steeper than that of the acoustic mode, as shown in the right zoomed-in diagram of Fig. 3. Therefore, an intersection point between the photonic and phononic dispersion curves should always exist, which sets the phase-matching condition for the Stokes light.

 

Fig. 3 Illustration showing the phase matching among the pump light, the Stokes light, and the acoustic wave in the engineered phoxonic crystal waveguide. The pump light and Stokes light are both in the slow-light region of the defect-guided mode. In the right zoomed-in diagram, the green (red) line denotes the dispersion curve of the defect-guided optical (acoustic) mode traveling along the x direction in the phoxonic crystal waveguide. The phononic band diagram is superimposed onto the photonic band diagram with the origin of the former aligned to the operating point of the pump light in the latter.

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Third, a large optomechanical coupling coefficient gOM can be obtained with all the three waves confined laterally and propagating slowly in the phoxonic crystal defect waveguide because of a large overlap between their modal fields. Actually, those defect-guided light modes can be categorized into “yeven/yodd (zeven/zodd)” according to the symmetry of their fields with respect to the y = 0 (z = 0) plane. As seen from the right panel of Fig. 2(a), the dominant component of the electric field |Ey|2 is both yeven and zeven. Since the pump light and the Stokes light share similar modal field patterns, the resulting optical force distribution is also symmetric with respect to planes y = 0 and z = 0. In addition, the right panel of Fig. 2(a) also shows the acoustic modal profile Uy (mechanical displacement relative to the structural surface) of the phoxonic crystal waveguide. It is easy to find that the acoustic mode shares the same field symmetry with the optical force distribution, which renders a strong optomechanical coupling coefficient in the phoxonic crystal waveguide.

4. Calculation of the SBS gain in the engineered phoxonic crystal waveguide

In order to calculate the SBS gain coefficient in Eq. (35), we need all the six parameters gOM, ξb, vp, vs, vb, and αb. Owing to the intrinsic material loss [39], the acoustic wave is attenuated as it propagates along the waveguide. According to the measurement data from bulk-mode acoustic resonators, it is reasonable to assume a dissipation rate αbvb of 1.63 × 107 rad/s [40]. In typical cases of optical fibers and waveguides, the propagation loss of the pump and Stokes waves is negligible over the acoustic attenuation length αb−1 [33, 34]. In the case of phoxonic crystal waveguide considered here, the per-unit-length loss rates of optical and acoustic waves are both enhanced due to the slow-light and slow-sound effects. We calculated the group velocities vp, vs, and vb in the slow-light region. Since the frequency difference between the pump and Stokes light is quite small compared with their optical frequencies, the pump light and Stokes light can be regarded to share the same group velocity. Figure 4(a) shows the group velocities of the pump and Stokes light vl,g and acoustic wave vs,g with the pump light wavelength varying in 1520–1565 nm under the phase-matching condition. It is clear that the three waves, the pump and Stokes light as well as the acoustic wave, all exhibit a great amount of slowing effect in the entire C band. It should be noted that the group velocity of light maintains larger than 0.005c from 1520 to 1565 nm, indicating that the optical slow-down factor is less than 100. In contrast, the acoustic wave under the phase-matching condition is slowed down by more than 1000 times. In this regard, it is still valid to assume negligible propagation loss of the pump and Stokes waves over the acoustic attenuation length αb−1.

 

Fig. 4 (a) Calculated group velocities of the pump and Stokes light vl,g (red) and acoustic wave vs,g (blue) at the phase-matching condition with the pump light wavelength in the C band. (b) Calculated SBS gain coefficient (red) and optomechanical coupling coefficient gOM (blue) at the phase-matching condition with the pump light wavelength in the C band.

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Due to their small frequency difference, the pump light and Stokes light can also be regarded to share the same modal fields. The blue line in Fig. 4(b) shows the calculated gOM with the pump light wavelength varying in 1520–1565 nm under the phase-matching condition, where the red line shows the forward SBS gain coefficient. It is easy to find that as the pump wavelength moves from 1520 to 1565 nm, the gOM increases by a factor of two while the SBS gain coefficient is enhanced by near two orders of magnitude, which confirms that the giant enhancement of the SBS gain results predominantly from the simultaneous slow-light and slow-sound effects.

It should be noted that the advantage of engineering simultaneous slow light and slow sound in a phoxonic crystal waveguide is to enhance the spatial interaction of light and sound, so as to increase the Brillouin scattering rate per unit length. The scattering rate is not enhanced in the time domain. Assuming a forward evolution direction (+x) of the acoustic wave, we solve Eqs. (29)–(31) by the Green’s function method [32],

b(x)=iΩgOM0+dx[as*(xx)ap(xx)exp(αbx)]=igOMas*(x)ap(x)αbvb.
Actually, the factor αbvb in Eq. (36) which characterizes the acoustic loss per unit time is a constant for a certain acoustic mode. Therefore, |b(x)| in the final steady state depends only on the optical fields as*ap and the optomechanical coupling strength gOM, while the slow group velocity vb does not play a role. Here, the advantage of the slow sound lies in the reduction of the device size that is required for the acoustic wave to reach its final steady state. The acoustic wave needs a length to build up from zero to its steady-state amplitude which is set by the optical fields as*ap. We solved b(x) numerically based on Eqs. (29)–(31) with initial ap and as (|ap| >> |as|). Figure 5 plots the evolution amplitude |b(x)| for acoustic waves of slow (vg = 1.08 m/s) and normal (vg = 5,330 m/s) group velocities. It is clear that they increase at the same rate in the end, but the length required to reach the final steady rate depends on the group velocity. Therefore, it is in this way that the engineered slow sound can effectively enhance the SBS gain and reduce the device size.

 

Fig. 5 Evolution of the amplitude of the acoustic wave |b(x)| along the propagation direction, where red solid and blue dash-dotted lines correspond to acoustic waves of slow (vg = 1.08 m/s) and normal (vg = 5,330 m/s) group velocities, respectively.

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5. Conclusion

We have engineered a suspended phoxonic crystal waveguide structure for giant enhancement of the forward stimulated Brillouin scattering. The designed structure can provide tight lateral confinement for both optical and acoustic modes to copropagate within a guided line defect thus enabling strong optomechanical coupling between them. We have also engineered the photonic and phononic band structures to achieve simultaneous slow light and slow sound. The combination of these two effects results in significantly enhanced SBS nonlinearity over that in conventional nonlinear fibers, suspended silicon waveguides, and hybrid Si/Si3N4 waveguides, etc. The calculated SBS gain coefficient is greater than 3 × 104 W−1 m−1 in the entire C band with the highest value beyond 106 W−1 m−1. This is at least an order of magnitude higher than those in the existing demonstrations in silicon. The giant SBS gain will enable high-performance Brillouin photonic devices with substantially reduced footprint and open up applications of photonics-assisted RF and microwave signal generation, amplification, and processing.

Funding

Hong Kong Research Grants Council Early Career Scheme (24208915); National Natural Science Foundation of China (NSFC) / Research Grants Council (RGC) of Hong Kong Joint Research Scheme (N_CUHK415/15).

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17. H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson III, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013). [CrossRef]   [PubMed]  

18. E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016). [CrossRef]  

19. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010). [CrossRef]   [PubMed]  

20. V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011). [CrossRef]   [PubMed]  

21. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009). [CrossRef]   [PubMed]  

22. A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014). [CrossRef]   [PubMed]  

23. R. Zhang, G. Chen, and J. Sun, “Analysis of acousto-optic interaction based on forward stimulated Brillouin scattering in hybrid phononic-photonic waveguides,” Opt. Express 24(12), 13051–13059 (2016). [CrossRef]   [PubMed]  

24. J. M. Escalante, A. Martínez, and V. Laude, “Design of single-mode waveguides for enhanced light-sound interaction in honeycomb-lattice silicon slabs,” J. Appl. Phys. 115(6), 064302 (2014). [CrossRef]  

25. F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012). [CrossRef]  

26. Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010). [CrossRef]  

27. S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18(9), 9164–9172 (2010). [CrossRef]   [PubMed]  

28. V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530. [CrossRef]  

29. Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012). [CrossRef]  

30. S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013). [CrossRef]  

31. T. Søndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61(23), 15688–15696 (2000). [CrossRef]  

32. C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92(1), 013836 (2015). [CrossRef]  

33. W. Qiu, P. T. Rakich, M. Soljacic, and Z. Wang, “Stimulated brillouin scattering in slow light waveguides,” arXiv:1210.0738 (2012).

34. C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016). [CrossRef]  

35. D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32(21), 1196–1199 (1974). [CrossRef]  

36. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002). [CrossRef]   [PubMed]  

37. J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012). [CrossRef]  

38. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef]   [PubMed]  

39. S. A. Chandorkar, M. Agarwal, R. Melamud, R. N. Candler, K. E. Goodson, and T. W. Kenny, “Limits of quality factor in bulk-mode micromechanical resonators,” in 2008 IEEE 21st International Conference on Micro Electro Mechanical Systems(2008), pp. 74–77. [CrossRef]  

40. G. Chen, R. Zhang, J. Sun, H. Xie, Y. Gao, D. Feng, and H. Xiong, “Mode conversion based on forward stimulated Brillouin scattering in a hybrid phononic-photonic waveguide,” Opt. Express 22(26), 32060–32070 (2014). [CrossRef]   [PubMed]  

References

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  1. R. W. Boyd, Nonlinear Optics (Academic, 2003).
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  3. P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
    [Crossref]
  4. M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
    [Crossref]
  5. P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2(1), 011008 (2012).
    [Crossref]
  6. J. Wang, Y. Zhu, R. Zhang, and D. J. Gauthier, “FSBS resonances observed in a standard highly nonlinear fiber,” Opt. Express 19(6), 5339–5349 (2011).
    [Crossref] [PubMed]
  7. D. Braje, L. Hollberg, and S. Diddams, “Brillouin-enhanced hyperparametric generation of an optical frequency comb in a monolithic highly nonlinear fiber cavity pumped by a cw laser,” Phys. Rev. Lett. 102(19), 193902 (2009).
    [Crossref] [PubMed]
  8. T. Schneider, M. Junker, and D. Hannover, “Generation of millimetre-wave signals by stimulated Brillouin scattering for radio over fibre systems,” Electron. Lett. 40(23), 1500–1502 (2004).
    [Crossref]
  9. T. Sakamoto, T. Yamamoto, K. Shiraki, and T. Kurashima, “Low distortion slow light in flat Brillouin gain spectrum by using optical frequency comb,” Opt. Express 16(11), 8026–8032 (2008).
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  10. M. J. Damzen and M. H. R. Hutchinson, “High-efficiency laser-pulse compression by stimulated Brillouin scattering,” Opt. Lett. 8(6), 313–315 (1983).
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  11. N. Olsson and J. V. D. Ziel, “Characteristics of a semiconductor laser pumped brillouin amplifier with electronically controlled bandwidth,” J. Lightwave Technol. 5(1), 147–153 (1987).
    [Crossref]
  12. T. Tanemura, Y. Takushima, and K. Kikuchi, “Narrowband optical filter, with a variable transmission spectrum, using stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27(17), 1552–1554 (2002).
    [Crossref] [PubMed]
  13. B. C. Rodgers, T. H. Russell, and W. B. Roh, “Laser beam combining and cleanup by stimulated Brillouin scattering in a multimode optical fiber,” Opt. Lett. 24(16), 1124–1126 (1999).
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  14. C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
    [Crossref] [PubMed]
  15. J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
    [Crossref]
  16. R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
    [Crossref]
  17. H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
    [Crossref] [PubMed]
  18. E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
    [Crossref]
  19. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010).
    [Crossref] [PubMed]
  20. V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
    [Crossref] [PubMed]
  21. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
    [Crossref] [PubMed]
  22. A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
    [Crossref] [PubMed]
  23. R. Zhang, G. Chen, and J. Sun, “Analysis of acousto-optic interaction based on forward stimulated Brillouin scattering in hybrid phononic-photonic waveguides,” Opt. Express 24(12), 13051–13059 (2016).
    [Crossref] [PubMed]
  24. J. M. Escalante, A. Martínez, and V. Laude, “Design of single-mode waveguides for enhanced light-sound interaction in honeycomb-lattice silicon slabs,” J. Appl. Phys. 115(6), 064302 (2014).
    [Crossref]
  25. F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
    [Crossref]
  26. Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
    [Crossref]
  27. S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18(9), 9164–9172 (2010).
    [Crossref] [PubMed]
  28. V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
    [Crossref]
  29. Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
    [Crossref]
  30. S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
    [Crossref]
  31. T. Søndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61(23), 15688–15696 (2000).
    [Crossref]
  32. C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92(1), 013836 (2015).
    [Crossref]
  33. W. Qiu, P. T. Rakich, M. Soljacic, and Z. Wang, “Stimulated brillouin scattering in slow light waveguides,” arXiv:1210.0738 (2012).
  34. C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016).
    [Crossref]
  35. D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32(21), 1196–1199 (1974).
    [Crossref]
  36. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
    [Crossref] [PubMed]
  37. J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
    [Crossref]
  38. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001).
    [Crossref] [PubMed]
  39. S. A. Chandorkar, M. Agarwal, R. Melamud, R. N. Candler, K. E. Goodson, and T. W. Kenny, “Limits of quality factor in bulk-mode micromechanical resonators,” in 2008 IEEE 21st International Conference on Micro Electro Mechanical Systems(2008), pp. 74–77.
    [Crossref]
  40. G. Chen, R. Zhang, J. Sun, H. Xie, Y. Gao, D. Feng, and H. Xiong, “Mode conversion based on forward stimulated Brillouin scattering in a hybrid phononic-photonic waveguide,” Opt. Express 22(26), 32060–32070 (2014).
    [Crossref] [PubMed]

2016 (3)

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
[Crossref]

R. Zhang, G. Chen, and J. Sun, “Analysis of acousto-optic interaction based on forward stimulated Brillouin scattering in hybrid phononic-photonic waveguides,” Opt. Express 24(12), 13051–13059 (2016).
[Crossref] [PubMed]

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016).
[Crossref]

2015 (4)

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92(1), 013836 (2015).
[Crossref]

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
[Crossref]

2014 (3)

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

J. M. Escalante, A. Martínez, and V. Laude, “Design of single-mode waveguides for enhanced light-sound interaction in honeycomb-lattice silicon slabs,” J. Appl. Phys. 115(6), 064302 (2014).
[Crossref]

G. Chen, R. Zhang, J. Sun, H. Xie, Y. Gao, D. Feng, and H. Xiong, “Mode conversion based on forward stimulated Brillouin scattering in a hybrid phononic-photonic waveguide,” Opt. Express 22(26), 32060–32070 (2014).
[Crossref] [PubMed]

2013 (2)

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

2012 (4)

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2(1), 011008 (2012).
[Crossref]

Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
[Crossref]

F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
[Crossref]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
[Crossref]

2011 (2)

2010 (4)

2009 (3)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

D. Braje, L. Hollberg, and S. Diddams, “Brillouin-enhanced hyperparametric generation of an optical frequency comb in a monolithic highly nonlinear fiber cavity pumped by a cw laser,” Phys. Rev. Lett. 102(19), 193902 (2009).
[Crossref] [PubMed]

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
[Crossref]

2008 (1)

2006 (1)

P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
[Crossref]

2004 (1)

T. Schneider, M. Junker, and D. Hannover, “Generation of millimetre-wave signals by stimulated Brillouin scattering for radio over fibre systems,” Electron. Lett. 40(23), 1500–1502 (2004).
[Crossref]

2002 (2)

T. Tanemura, Y. Takushima, and K. Kikuchi, “Narrowband optical filter, with a variable transmission spectrum, using stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27(17), 1552–1554 (2002).
[Crossref] [PubMed]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref] [PubMed]

2001 (1)

2000 (1)

T. Søndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61(23), 15688–15696 (2000).
[Crossref]

1999 (1)

1987 (1)

N. Olsson and J. V. D. Ziel, “Characteristics of a semiconductor laser pumped brillouin amplifier with electronically controlled bandwidth,” J. Lightwave Technol. 5(1), 147–153 (1987).
[Crossref]

1983 (1)

1974 (1)

D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32(21), 1196–1199 (1974).
[Crossref]

Adibi, A.

Baets, R.

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
[Crossref]

Bahl, G.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Benchabane, S.

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

Beugnot, J. C.

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

Beugnot, J.-C.

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

Biegelsen, D. K.

D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32(21), 1196–1199 (1974).
[Crossref]

Braje, D.

D. Braje, L. Hollberg, and S. Diddams, “Brillouin-enhanced hyperparametric generation of an optical frequency comb in a monolithic highly nonlinear fiber cavity pumped by a cw laser,” Phys. Rev. Lett. 102(19), 193902 (2009).
[Crossref] [PubMed]

Brenn, A.

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
[Crossref]

Camacho, R.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2(1), 011008 (2012).
[Crossref]

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

Chan, J.

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

Chen, G.

Chiu, C.-C.

F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
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P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2(1), 011008 (2012).
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Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
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V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
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Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
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V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
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Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
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C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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Eggleton, B. J.

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92(1), 013836 (2015).
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M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
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S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
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Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
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J. M. Escalante, A. Martínez, and V. Laude, “Design of single-mode waveguides for enhanced light-sound interaction in honeycomb-lattice silicon slabs,” J. Appl. Phys. 115(6), 064302 (2014).
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S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
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P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
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Fu, W.

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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Gauthier, D. J.

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Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
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C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
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A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
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J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
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Hollberg, L.

D. Braje, L. Hollberg, and S. Diddams, “Brillouin-enhanced hyperparametric generation of an optical frequency comb in a monolithic highly nonlinear fiber cavity pumped by a cw laser,” Phys. Rev. Lett. 102(19), 193902 (2009).
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F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
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F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
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F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
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Ibanescu, M.

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H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
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Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
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Johnson, S. G.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
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P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
[Crossref]

Junker, M.

T. Schneider, M. Junker, and D. Hannover, “Generation of millimetre-wave signals by stimulated Brillouin scattering for radio over fibre systems,” Electron. Lett. 40(23), 1500–1502 (2004).
[Crossref]

Kang, M. S.

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
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Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
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S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18(9), 9164–9172 (2010).
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P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
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Kikuchi, K.

Kim, J.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

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E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
[Crossref]

Knight, J. C.

P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
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Kuyken, B.

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Kuzyk, M. C.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Laude, V.

J. M. Escalante, A. Martínez, and V. Laude, “Design of single-mode waveguides for enhanced light-sound interaction in honeycomb-lattice silicon slabs,” J. Appl. Phys. 115(6), 064302 (2014).
[Crossref]

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
[Crossref]

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
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Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
[Crossref]

Makhoute, A.

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

Martinez, A.

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

Martínez, A.

J. M. Escalante, A. Martínez, and V. Laude, “Design of single-mode waveguides for enhanced light-sound interaction in honeycomb-lattice silicon slabs,” J. Appl. Phys. 115(6), 064302 (2014).
[Crossref]

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

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A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
[Crossref]

Mohammadi, S.

Nazarkin, A.

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
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N. Olsson and J. V. D. Ziel, “Characteristics of a semiconductor laser pumped brillouin amplifier with electronically controlled bandwidth,” J. Lightwave Technol. 5(1), 147–153 (1987).
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H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
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S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
[Crossref]

Painter, O.

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
[Crossref]

A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010).
[Crossref] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

Papanikolaou, N.

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

Pennec, Y.

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
[Crossref]

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
[Crossref]

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

Poulton, C. G.

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92(1), 013836 (2015).
[Crossref]

Qiu, W.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

Rakich, P. T.

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
[Crossref]

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2(1), 011008 (2012).
[Crossref]

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18(14), 14439–14453 (2010).
[Crossref] [PubMed]

Reinke, C.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2(1), 011008 (2012).
[Crossref]

Rodgers, B. C.

Roh, W. B.

Rolland, Q.

Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
[Crossref]

Russell, P. S. J.

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
[Crossref]

P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
[Crossref]

Russell, T. H.

Safavi-Naeini, A. H.

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016).
[Crossref]

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
[Crossref]

A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010).
[Crossref] [PubMed]

Sakamoto, T.

Sarabalis, C. J.

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016).
[Crossref]

Schneider, T.

T. Schneider, M. Junker, and D. Hannover, “Generation of millimetre-wave signals by stimulated Brillouin scattering for radio over fibre systems,” Electron. Lett. 40(23), 1500–1502 (2004).
[Crossref]

Shen, Z.

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Shin, H.

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
[Crossref]

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

Shiraki, K.

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref] [PubMed]

Søndergaard, T.

T. Søndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61(23), 15688–15696 (2000).
[Crossref]

Starbuck, A.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

Steel, M. J.

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92(1), 013836 (2015).
[Crossref]

Sun, J.

Takushima, Y.

Tanemura, T.

Vahala, K. J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

Van Laer, R.

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
[Crossref]

Van Thourhout, D.

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
[Crossref]

Vasseur, J. O.

Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
[Crossref]

Wang, H.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Wang, J.

Wang, Z.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2(1), 011008 (2012).
[Crossref]

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18(14), 14439–14453 (2010).
[Crossref] [PubMed]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref] [PubMed]

Wiederhecker, G. S.

P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
[Crossref]

Wolff, C.

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92(1), 013836 (2015).
[Crossref]

Xie, H.

Xiong, H.

Yamamoto, T.

Zhang, R.

Zhang, Y.-L.

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Zhu, Y.

Ziel, J. V. D.

N. Olsson and J. V. D. Ziel, “Characteristics of a semiconductor laser pumped brillouin amplifier with electronically controlled bandwidth,” J. Lightwave Technol. 5(1), 147–153 (1987).
[Crossref]

Zou, C.-L.

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

APL Photonics (1)

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016).
[Crossref]

Appl. Phys. Lett. (3)

Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
[Crossref]

F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
[Crossref]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
[Crossref]

Electron. Lett. (1)

T. Schneider, M. Junker, and D. Hannover, “Generation of millimetre-wave signals by stimulated Brillouin scattering for radio over fibre systems,” Electron. Lett. 40(23), 1500–1502 (2004).
[Crossref]

J. Appl. Phys. (1)

J. M. Escalante, A. Martínez, and V. Laude, “Design of single-mode waveguides for enhanced light-sound interaction in honeycomb-lattice silicon slabs,” J. Appl. Phys. 115(6), 064302 (2014).
[Crossref]

J. Lightwave Technol. (1)

N. Olsson and J. V. D. Ziel, “Characteristics of a semiconductor laser pumped brillouin amplifier with electronically controlled bandwidth,” J. Lightwave Technol. 5(1), 147–153 (1987).
[Crossref]

Nat. Commun. (2)

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

Nat. Photonics (2)

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
[Crossref]

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
[Crossref]

Nat. Phys. (3)

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
[Crossref]

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
[Crossref]

Nature (1)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

Opt. Express (9)

R. Zhang, G. Chen, and J. Sun, “Analysis of acousto-optic interaction based on forward stimulated Brillouin scattering in hybrid phononic-photonic waveguides,” Opt. Express 24(12), 13051–13059 (2016).
[Crossref] [PubMed]

S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18(9), 9164–9172 (2010).
[Crossref] [PubMed]

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18(14), 14439–14453 (2010).
[Crossref] [PubMed]

T. Sakamoto, T. Yamamoto, K. Shiraki, and T. Kurashima, “Low distortion slow light in flat Brillouin gain spectrum by using optical frequency comb,” Opt. Express 16(11), 8026–8032 (2008).
[Crossref] [PubMed]

J. Wang, Y. Zhu, R. Zhang, and D. J. Gauthier, “FSBS resonances observed in a standard highly nonlinear fiber,” Opt. Express 19(6), 5339–5349 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010).
[Crossref] [PubMed]

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001).
[Crossref] [PubMed]

G. Chen, R. Zhang, J. Sun, H. Xie, Y. Gao, D. Feng, and H. Xiong, “Mode conversion based on forward stimulated Brillouin scattering in a hybrid phononic-photonic waveguide,” Opt. Express 22(26), 32060–32070 (2014).
[Crossref] [PubMed]

Opt. Lett. (3)

Phys. Rev. A (1)

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92(1), 013836 (2015).
[Crossref]

Phys. Rev. B (2)

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

T. Søndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61(23), 15688–15696 (2000).
[Crossref]

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32(21), 1196–1199 (1974).
[Crossref]

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

D. Braje, L. Hollberg, and S. Diddams, “Brillouin-enhanced hyperparametric generation of an optical frequency comb in a monolithic highly nonlinear fiber cavity pumped by a cw laser,” Phys. Rev. Lett. 102(19), 193902 (2009).
[Crossref] [PubMed]

Phys. Rev. X (1)

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2(1), 011008 (2012).
[Crossref]

Surf. Sci. Rep. (1)

Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
[Crossref]

Other (4)

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

W. Qiu, P. T. Rakich, M. Soljacic, and Z. Wang, “Stimulated brillouin scattering in slow light waveguides,” arXiv:1210.0738 (2012).

R. W. Boyd, Nonlinear Optics (Academic, 2003).

S. A. Chandorkar, M. Agarwal, R. Melamud, R. N. Candler, K. E. Goodson, and T. W. Kenny, “Limits of quality factor in bulk-mode micromechanical resonators,” in 2008 IEEE 21st International Conference on Micro Electro Mechanical Systems(2008), pp. 74–77.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Unit cell of a two-dimensional silicon phoxonic crystal structure. The structural parameters are h = 340 nm and a = 490 nm. (b, c) Calculated quasi-TE photonic (b) and phononic (c) band diagrams of the phoxonic crystal structure with the unit cell shown in (a). The bandgaps are marked in red and the light cone is marked in gray.
Fig. 2
Fig. 2 (a) Left: Schematic of the W1 phoxonic crystal waveguide structure. Right: Profiles of the defect-guided optical mode (|Ey|2) and acoustic mode (Uy). (b) Photonic band diagram of the W1 phoxonic crystal waveguide structure. The cyan-shaded regions denote the slab-guided band continuum, where the modes in the dielectric (air) band reside mostly in the material (air holes). The black and red lines in the bandgap denote the defect-guided optical modes which do not cross each other in the entire kx range. (c) Phononic band diagram of the W1 phoxonic crystal waveguide structure. The light blue regions denote the phononic bandgaps. The red line denotes the defect-guided acoustic mode. (d) Group velocities of the defect-guided optical (vl,g, in red) and acoustic (vs,g, in blue) modes propagating in the phoxonic crystal waveguide.
Fig. 3
Fig. 3 Illustration showing the phase matching among the pump light, the Stokes light, and the acoustic wave in the engineered phoxonic crystal waveguide. The pump light and Stokes light are both in the slow-light region of the defect-guided mode. In the right zoomed-in diagram, the green (red) line denotes the dispersion curve of the defect-guided optical (acoustic) mode traveling along the x direction in the phoxonic crystal waveguide. The phononic band diagram is superimposed onto the photonic band diagram with the origin of the former aligned to the operating point of the pump light in the latter.
Fig. 4
Fig. 4 (a) Calculated group velocities of the pump and Stokes light vl,g (red) and acoustic wave vs,g (blue) at the phase-matching condition with the pump light wavelength in the C band. (b) Calculated SBS gain coefficient (red) and optomechanical coupling coefficient gOM (blue) at the phase-matching condition with the pump light wavelength in the C band.
Fig. 5
Fig. 5 Evolution of the amplitude of the acoustic wave |b(x)| along the propagation direction, where red solid and blue dash-dotted lines correspond to acoustic waves of slow (vg = 1.08 m/s) and normal (vg = 5,330 m/s) group velocities, respectively.

Equations (36)

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E p ( x , y , z , t ) = a p ( x , t ) e p ( x , y , z , t ) + c . c . ,
E s ( x , y , z , t ) = a s ( x , t ) e s ( x , y , z , t ) + c . c . ,
e p ( x , y , z , t ) = e ˜ p ( x , y , z ) exp [ i ( ω p t k p x ) ] ,
e s ( x , y , z , t ) = e ˜ s ( x , y , z ) exp [ i ( ω s t k s x ) ] .
P p = 2 x ^ { [ e p ( x , y , z , t ) ] * × h p ( x , y , z , t ) } d y d z ,
P s = 2 x ^ { [ e s ( x , y , z , t ) ] * × h s ( x , y , z , t ) } d y d z ,
ξ p = 2 0 a d x ε ( x , y , z ) [ e p ( x , y , z , t ) ] * e p ( x , y , z , t ) d y d z ,
ξ s = 2 0 a d x ε ( x , y , z ) [ e s ( x , y , z , t ) ] * e s ( x , y , z , t ) d y d z ,
U ( x , y , z , t ) = b ( x , t ) u ( x , y , z , t ) + c .c . ,
u ( x , y , z , t ) = u ˜ ( x , y , z ) exp [ i ( Ω t q x ) ] .
P b = 2 i Ω j m l c x j m l u j * u l m d y d z ,
ξ b = 2 Ω 2 0 a d x ρ | u | 2 d y d z ,
× × ( E + Δ E ) + μ 0 ( D + Δ D ) t = 0.
Δ E = Δ e s a p b * + Δ e p a s b + c .c .,
Δ D = Δ d s a p b * + Δ d p a s b + c .c ..
0 = × × ( a s e s + Δ e s a p b * ) + μ 0 ( a s d s + a p b * Δ d s ) t ,
0 = a s ( × × e s + μ 0 d s t ) + [ x ^ × ( × e s ) + × ( x ^ × e s ) ] a s x 2 i ω s μ 0 d s a s t + a p b * ( × × Δ e s + μ 0 2 Δ d s t 2 ) + h .o .t . + c .c . ,
i ω s a P s a s x i ω s P s v s a s t = a p b * x 0 x 0 + a d x [ ( e s ) * ( i ω ) 2 Δ d s ( i ω ) 2 d s Δ e s ] d y d z .
v s a s x + a s t = i ω s Q OM s ξ s a p b * ,
v p a p x + a p t = i ω p Q OM p ξ p a s b ,
Q OM p = x 0 x 0 + a d x [ ( e p ) * Δ d p d p Δ e p ] d y d z ,
Q OM s = x 0 x 0 + a d x [ ( e s ) * Δ d s d s Δ e s ] d y d z .
ρ 2 U i t 2 + j m l j [ c i j m l + η i j m l t ] m U l = F i ,
F ( r , t ) = f ( r , t ) a s * ( x , t ) a p ( x , t ) + c .c ..
i Ω j m l [ ( c i x m l m + j c i j x l ) u l b x 2 i Ω ρ u i b t + j η i j m l u l m b + f i a s * a p ] + c .c . = 0.
v b b x + α b b + b t = i Ω Q OM b ξ b a s * a p ,
Q OM b = x 0 x 0 + a d x u * f d y d z .
Q OM s = ( Q OM p ) * = Q OM b ,
v s a s ( x ) x = i ω s Q OM * ξ s a p ( x ) b * ( x ) ,
v p a p ( x ) x = i ω p Q OM ξ p a s ( x ) b ( x ) ,
v b b ( x ) x + α b v b b ( x ) = i Ω Q OM * ξ b a s * ( x ) a p ( x ) .
Q PE = ε 0 V d 3 r i j m l ε r 2 ( E i p ) * E j s p i j m l S m l ,
Q MB = S d 2 r ( u * n ^ ) [ ( ε r ε air ) ε 0 ( n ^ × E p ) * ( n ^ × E s ) ( ε r 1 ε air 1 ) ε 0 1 ( n ^ D p ) * ( n ^ D s ) ] ,
g OM = ω Q OM s Q OM p / ( ξ s ξ p ) ,
G = Ω | g OM | 2 ω v p v s v b α b ( ξ b / a ) ,
b ( x ) = i Ω g OM 0 + d x [ a s * ( x x ) a p ( x x ) exp ( α b x ) ] = i g OM a s * ( x ) a p ( x ) α b v b .

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