## Abstract

On-chip waveguides on insulator with high stimulated Brillouin gain have wide potential application prospects in the field of nanophotonic structures. We propose a new on-chip hybrid silicon-chalcogenide slot waveguide structure consisting of a chalcogenide As_{2}S_{3} rectangle core with an air slot and a wrapping layer of silicon. In the new hybrid waveguide, the high radiation pressure and electrostriction force, determined by pump and Stokes optical waves, and the acoustic displacement, determined by acoustic wave, can be achieved by adjusting the dimensions of rectangle core, the thickness of wrapping layers and the width of air slot. Therefore, a strong optomechanical coupling between high radiation pressure and transverse acoustic displacement will be generated. In such a way, a nonlinear gain for backward stimulated Brillouin scattering can be theoretically achieved with a high gain coefficient of 2.88×10^{4} W^{−1}m^{−1}. The enhanced gain coefficient in the proposed waveguide is around 2.4 times as that in an on-chip silicon-chalcogenide hybrid slot waveguide on insulator without the wrapping layer. The Stokes amplification reaches 85.7 dB with the waveguide length of 2.5 cm. Therefore, this method provides a new idea to design nanophotonic waveguides for giant backward stimulated Brillouin scattering.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Stimulated Brillouin scattering (SBS), an important third-order nonlinear optical process, is generated from the interaction between propagating acoustic and optical waves. SBS is tailored by electrostrictive or thermoelastic material response in conventional systems such as single mode and multimode optical fibers [1,2]. Since 1964 [3], SBS has been widely applied in a number of optical fields, such as slow light [4,5], signal-processing techniques [6], sensors [7] and filters [8,9]. Recently, SBS in micro and nanoscale photonic devices has drawn much attention. Lots of microdevices and nanostructures have been theoretically or experimentally employed to enhance SBS effect, including subwavelength diameter optical fibers [10–11], photonic crystal fibers [12], suspended waveguides [13–17], and on-chip waveguides [19–21]. A variety of platforms have proved promising for on-chip Brillouin lasers, fast and slow light, signal-processing, and Brillouin Stokes amplification based on nonlinear materials including chalcogenide [22–25], silicon [26,27], hybrid silicon and chalcogenide [28], silicon nitride [29], CaF_{2} [30] and silica [31].

In the design of integrated photonics, silicon is the most ideal platform for nanophotonic devices, due to its compatibility with the complementary metal-oxide semiconductor (CMOS), which provides fabrication large-scale technology [32]. Compared with the suspended silicon waveguide structures [13–17], which have a relatively high SBS but low mechanical stability, silicon on insulator (SOI) provides a stable platform for on-chip nonlinear optical process and makes it efficient in the integrated systems. However, suffering from high stiffness in silicon, i.e., a high sound velocity, acoustic waves are difficult to be guided in a pure SOI waveguide [20]. That will be greatly suppress the interaction between photons and phonons and thus lead to a weak SBS in SOI. To excite strong photon-phonon interaction in SOI, a hybrid silicon-chalcogenide waveguide structure was proposed [19]. By partly replacing the silicon beams with soft chalcogenide As_{2}S_{3} glass (it possesses large photo-elastic constant corresponding to huge Brillouin gain coefficients and low stiffness), the trap of light and acoustic waves is simultaneously realized in the hybrid waveguide structure. Furthermore, it was suggested that an air slot added in the center of chalcogenide As_{2}S_{3} glass can induce substantial contribution of the Brillouin gain coefficient for radiation pressures, and the Brillouin gain coefficient is further enhanced [20]. However, the acoustic index contrast on top of air/As_{2}S_{3} boundaries is much higher than that on bottom of As_{2}S_{3}/SiO_{2} boundaries, driving an upward distribution for acoustic fields in proposed structures. At the same time, the relative vertical symmetry distribution for optical fields and their relating optical forces appears because of the lower optical index contrast on between the top and bottom boundaries. Thus, the interaction between photons and phonons is cut down, resulting in a lower SBS gain coefficient. To date, however, a more feasible way to acquire an enhanced SBS is desired, especially for backward Brillouin scattering in on-chip waveguide system with compact design.

In this paper, we further develop the hybrid on-chip waveguide system and propose a new design that consists of a chalcogenide As_{2}S_{3} glass rectangle core with an air slot and a wrapping layer of silicon, which can obtain an enhancement of the backward SBS mainly driven by radiation pressure. We present that the adding wrapping layer is a promise of vertical symmetry of distribution of acoustic fields by inducing the As_{2}S_{3}/Si interfaces on both top and bottom of the rectangle core. We show that waveguides with high components of shear acoustic displacement have the potential to yield high backward SBS mainly driven by radiation pressure. We prove that the reduction of backward SBS gain coefficient with the increase of air slot width should be attributed to the interaction between the other factors (such as optical group velocity, material quality factor, optical and acoustic energy flux) and the coupling between radiation pressure and *x*-component of acoustic displacement fields. We analyze the effect of optical losses on Stokes amplification with variation of waveguide length. We also provide a fabrication process and calculate the effect of the fabrication tolerance on backward SBS gains. In particular, for the waveguide system with optimizing profiles, i.e., geometric size and materials, the giant value of backward SBS gain coefficient is obtained as 2.88×10^{4} W^{−1}m^{−1} with the air slot width of 10 nm. It is at least as 2.4 times as that in the existing theoretical on-chip waveguide systems [19–21], which are 4500 W^{−1}m^{−1}, 12127 W^{−1}m^{−1} and 500 W^{−1}m^{−1}. To the best of our knowledge, it is the highest backward SBS gain coefficient achieved in on-chip-based traveling-wave waveguides on insulator. The Stokes amplification can reach 85.7 dB with the waveguide length of 2.5 cm. When considering fabrication tolerance of the proposed waveguide, the least Stokes amplification is 52.8 dB, which is comparable with that of the experimental on-chip systems with much longer waveguide length [24,25]. Our proposed waveguide system exhibits a new way to design on-chip waveguide systems with large backward SBS gain, which have potential applications in high-performance signal processing and sensing self-testing system on chip.

## 2. Principle

#### 2.1 SBS gain coefficient from optical and acoustic coupling

In SBS, interference between a pump optical wave with angular frequency of *ω _{p}* and a Stokes optical wave with angular frequency of

*ω*(

_{s}*ω*<

_{s}*ω*) generates a time-varying and spatially dependent optical force distribution, which drives appearance of Brillouin active phonons with angular frequency Ω [14]. During SBS, the relations of momentum and energy conservation satisfy frequency matching and phase matching conditions simultaneously:

_{p}*q*,

*k*and

_{p}*k*represent the wavenumbers of acoustic, pump and Stokes waves, respectively. Considering the propagating direction of optical waves, SBS can be categorized as forward SBS (FSBS) and backward SBS (BSBS). In the case of FSBS, the pump and Stokes waves co-propagate while they contra-propagate in BSBS. Optical waves enable to carry identical optical mode (intra-mode coupling) or different optical modes (inter-mode coupling) [14]. In this paper, we only concentrate on the case of BSBS with intra-mode coupling.

_{s}To begin with, we presume that a pump optical wave ${{\textbf {E}}_p}(r,t) = {\tilde{{\textbf {E}}}_p}(x,y) \cdot {e^{i({k_p} \cdot z - {\omega _p}t)}}$ and a Stokes optical wave ${{\textbf {E}}_s}(r,t) = {\tilde{{\textbf {E}}}_s}(x,y) \cdot {e^{i({k_s} \cdot z - {\omega _s}t)}}$ propagate in the forward and backward direction of *z*-axis, respectively. Applying the small signal approximation, i.e. assuming that pump power is larger than Stokes power at any position *z* throughout the waveguide, the governing equations for the power transfer between pump and Stokes waves are [19]:

*P*and

_{p}*P*are the powers of pump and Stokes waves, and

_{s}*α*is the linear loss coefficient of optical waves.

*β*and

*γ*are the nonlinear losses of two photon absorption (2PA) and 2PA induced free carrier absorption (FCA), respectively, which are detailed discussed in the later section. In Eq. (4),

*g*denotes the sum of SBS gain spectrum of all individual acoustic modes, which has Lorentzian shape and can be expressed as [2]:

*denotes the eigen-frequency of the eigen-equation of acoustic mode*

_{m}**u**

*in the absence of acoustic loss. Г*

_{m}*is the loss coefficient of acoustic mode when acoustic loss is involved, depending on the mechanical quality factor*

_{m}*Q*with the relation given as

_{m}*Q*=Ω

_{m}*/Г*

_{m}*[14]. Subscript*

_{m}*m*represents the m

^{th}acoustic mode (m = 1,2,3…).

Considering the acoustic loss, the peak value of the SBS gain coefficient spectrum of each **u*** _{m}* can be simplified as [14]:

*v*is the optical group velocity, and

_{g}*ɛ*and

*ρ*are the electrical conductivity and mass density, respectively.

**f**is the total optical force driven by pump and Stokes waves. We assume that ${\omega _p} \approx {\omega _s} = \omega$

*.*Besides, $\langle{{\textbf {A}},{\textbf {B}}} \rangle = \int {{{\textbf {A}}^\ast } \cdot {\textbf {B}}ds} $, in which the integral covers the waveguide cross section. The <

**f**,

**u**

*>, the overlap integral between the total optical forces and individual m*

_{m}^{th}acoustic eigen-mode, represents the intensity of optomechanical coupling (named in Ref. [17]) in the on-chip waveguide structure.

The acoustic displacement field is induced by total optical forces, which satisfies the phase matching conditions of Eq. (1) and Eq. (2). To calculate **u*** _{m}*, an ideal acoustic equation should be presented, in which an isotropic medium is considered with the elastic loss neglected [33]:

*c*denotes the elastic stiffness tensor,

_{ijkl}*u*and

_{i}*f*denote the field components of acoustic displacement and the total optical force, respectively. In this equation, ${\partial _j}$ shows the spatial derivative in the

_{i}*j*-th spatial direction along

*j*, in which $j \in \{{x,y,z} \}$ . The

*u*

_{mi}can be obtained when the driving force

*f*is absent in Eq. (7). Hybrid acoustic waves (HAW) including shear and longitudinal displacement components are excited in waveguide structures [11].

_{i}To further clearly describe Eq. (6), we rewrite Eq. (6) as:

where*Q*

_{Cm}_{ }= <

**f**,

**u**

*> is the optomechanical coupling and*

_{m}*C*

_{OTm}_{ }=

*C*is the other factors (such as optical group velocity, material quality factor, optical and acoustic energy flux) influencing

_{FVm}C_{EFm}*G*, where ${C_{FVm}} = \frac{{2\omega {Q_m}}}{{\Omega _m^2{v_{gp}}{v_{gs}}}}$ and ${C_{EFm}} = \frac{1}{{\langle{{{\textbf {E}}_p},\varepsilon {{\textbf {E}}_p}} \rangle \langle{{{\textbf {E}}_s},\varepsilon {{\textbf {E}}_s}} \rangle \langle{{{\textbf {u}}_m},\rho {{\textbf {u}}_m}} \rangle }}$. From the expression of above two parameters, we can see that in waveguide structures, the angular frequency and the group velocity of optical waves, the energy flux of the optical and acoustic waves, and the mechanical factor of the waveguide material are related to

_{m}*C*.

_{OTm}#### 2.2 Optical forces analysis

The optomechanical coupling in the on-chip waveguide structure, is the linear sum of all the overlap integrals between individual optical force **f** * ^{n}* and individual m

^{th}acoustic eigen-mode, which is ${Q_{Cm}} = \sum\limits_n {\langle{{{\textbf {f}}^n},{{\textbf {u}}_m}} \rangle } .$ given as [14]:

**f**

*=*

^{total}**f**

*+*

^{es}**F**

*. Electrostriction is a quadratic response of mechanical strain excited by an applied electric field. It can be observed in all materials [34]. The i*

^{RP}^{th}component of the electrostriction force is defined as [35]: where

*σ*is the component of the electrostriction stress that is given by [14]:

_{ij}*p*is the element of material photo-elastic tensor,

_{ijkl}*ɛ*and

_{r}*ɛ*are the relative permittivity in materials and the permittivity in vacuum, respectively.

_{0}Radiation pressure acts only on boundaries where the gradient of *ɛ _{r}* is not equal to zero [36]. It can be derived from Maxwell Stress Tensor (MST) between material 1 and 2, and is given by [14]:

**n**is the normal vector points from material 1 to material 2. Since magnetic fields are continuous in the direction of normal vector, we only consider the components of electric part of MST, which is

*T*and defined as [14]:

_{ij}#### 2.3 Waveguide with radiation-pressure- induced gain coefficient enhancement

Compared with the BSBS gain coefficient for limited electrostriction forces, by an appropriate material selection and a careful structure design, slot waveguides have potential to yield a huge total BSBS gain coefficient mainly by an enhancement in radiation pressure on slot boundaries, which is at least a factor (10^{2}) larger than electrostriction forces contribution [16,20]. In this paper, we consider to present a rectangle chalcogenide As_{2}S_{3} slot waveguide with a wrapping silicon layer on silica substrate, which mainly enhance the BSBS gain coefficient for radiation pressure because of high contrast in refractive index on air/As_{2}S_{3} boundaries, and the gain coefficient contribution from electrostriction forces can be neglected.

The cross section of the proposed hybrid on-chip structure is shown in Fig. 1. The As_{2}S_{3} rectangle core symmetrically wrapped in the silicon layer is placed on a SiO_{2} substrate. An air slot is placed in the center of the As_{2}S_{3} rectangle core. Air slot promises a strong and overwhelming contribution of radiation pressure for BSBS gain coefficient. SiO_{2} substrate provides the stability of the structure and makes fabrication more feasible.

From Eq. (8) and Eq. (9), a strong optomechanical coupling between radiation pressure and acoustic waves is a promise of a huge BSBS gain coefficient mainly driven by radiation pressure. Practically, we concentrate on describing the main effects of slot width *e* in **F*** ^{RP}* and

**u**

*, and the thickness of the silicon wrapping layer*

_{m}*h*in

**u**

*. Thus the optomechanical coupling can be described as:*

_{m}**F**

*and*

^{RP}**u**

*contribute to*

_{m}*Q*, we define parameter

_{Cm}^{RP}*S*to describe the spatial integral ratio of acoustic shear-to-total displacement of individual m

_{m}^{th}eigen-mode of acoustic wave, which is:

*a*,

*b*and

*c*,

*h*and

*e*are the dimension parameters of the waveguide cross section shown in Fig. 1. As can be seen, all these parameters are complex functions relating to the structure dimension. Here $U_m^{Trans.} = \sqrt {{{|{{u_x}} |}^2}\ +\ {{|{{u_y}} |}^2}} $ and $U_m^{Total} = \sqrt {{{|{{u_x}} |}^2}\ +\ {{|{{u_y}} |}^2} + {{|{{u_z}} |}^2}} $ are the acoustic transverse and total displacements for individual acoustic mode with fixed structure dimension parameters. Acoustic fields with high

*S*induce effective optomechanical coupling.

_{m}In waveguides cross section, the larger overlapping degree between the large intensity in optical forces and acoustic fields distributions, the stronger *Q _{Cm}^{RP}*. However, for most existing on-chip waveguides, such as in [19,20], displacements of acoustic fields in upper solid/air interfaces are more remarkable than that in bottom solid/solid interfaces because of the higher acoustic index contrast on solid/air interfaces. Acoustic fields tend to be large on the top part of the structure, inducing a vertical asymmetry in acoustic fields. However, optical fields and their corresponding optical forces present a relative vertical symmetry in the waveguide, because of the relative comparable refractive index contrast between the top (solid/air interface) and the bottom of the waveguide (solid/solid interface). As a result, the poor optomechanical coupling appears.

To improve *Q _{Cm}^{RP}*, a layer symmetrically wrapping the slot waveguide can be added to interrupt the vertical asymmetry in acoustic fields, by inducing solid/solid interfaces both on the top and the bottom of the waveguide. The thickness of the top and bottom wrapping layers

*h*effects the vertical symmetry of the acoustic fields. By regulating

*h*, the profile of

**u**

*can be controlled and strengthen the overlapping degree. The width of air gap also play an import role in waveguide design with high BSBS gain coefficients. By decrease the gap width, the BSBS gain coefficient is increased because*

_{m}**F**

*on slot boundaries can be effectively enhanced in [16,21]. However, as the slot width*

^{RP}*e*also remarkably effects the magnitude of acoustic displacement fields, the BSBS gain enhancement is a combined effect between the increased radiation pressure and acoustic displacements on gap boundaries. The influence will be further discussed in later section. Also, the

*C*is proportional to BSBS gain coefficient, a suitable air slot structure size contribute to a large

_{OTm}*C*and enhance BSBS gain coefficient. Consequently, an enhancement of BSBS gains for radiation pressure should satisfy four requirements: large

_{OTm}*C*, high ratio of acoustic shear-to-total displacement, perfect profile overlapping degree between huge intensity in radiation pressure and acoustic displacement fields, strong radiation pressure and acoustic displacement fields on slot boundaries.

_{OTm}## 3. Waveguide structure and material properties

In this section, we further describe some structural and material properties of the proposed waveguide, whose cross section is shown in Fig. 1. The value of the top and bottom silicon wrapping layers thickness *h* can be expressed as *h*= (*b*-*d*)/2. This is owing to the strongest optomechanical coupling occurs when the thickness of top and bottom wrapping silicon layers are same in the waveguide. Compared with the waveguides proposed in [19,20], where acoustic modes have an upward tendency of field distribution, our design can well support both guided acoustic and optical modes in the soft chalcogenide As_{2}S_{3} core wrapped by the silicon layer with a better vertical symmetry. Such a feature is due to the existence of the symmetrical wrapping silicon layer, which introduces the spatial symmetry of proposed waveguides boundaries on the top (air/As_{2}S_{3}) and the bottom (As_{2}S_{3}/SiO_{2}) section. Thus, the overlapping degree between optical forces and acoustic fields is improved, leading to a stronger optomechanical coupling.

To compute the BSBS gain coefficient of the hybrid structure, the optical and acoustic properties of the above mentioned materials (except air) should be considered and are listed in Table 1 [19]. The quality factor *Q* is a measure of the energy loss, which is a complex function related to the interaction between excited acoustic and thermal modes in waveguides [18]. Even in the same waveguide, the value of *Q* will change as the waveguide temperature varies [13]. From the Eq. (6), the high value of *Q* is corresponding to a high BSBS gain coefficient. Although the value of *Q* can be larger than 1000 with some special structure and material design, such as *Q** *= 4×10^{4} W^{−1}m^{−1} in [28], here we assume a low *Q*-factor of 1000, which has been used in [13–16,19–21], to compare our results with these related works.

## 4. Results and discussion

#### 4.1 BSBS gain coefficient calculation for an example waveguide

In this section, based on above theoretical formulas, we study the BSBS gain coefficient of the hybrid waveguide by using finite element solver.

At first, we calculate the optical and acoustic modes of the proposed waveguide in section 3. We consider the hybrid waveguide with the size of *a *= 450 nm, *b *= 365 nm, *c *= 180 nm, *d *= 340 nm, *e *= 10 nm, *f *= 1000 nm and *g* = 300 nm. The properties of all above media are listed in Table 1 [19]. We assume our pump optical wave at 1550 nm in the study. Since Ω is negligible compared with *ω*, the pump and Stokes waves are recognized as the same waveguide mode. Focusing on the fundamental mode with large intensity in the air slot, we obtain characteristic optical modes of the proposed waveguide and they are shown in Fig. 2. The high contrast of refractive index between As_{2}S_{3} rectangle core and air offers optical confinement in the air slot. The sketch of radiation pressure on boundaries of left part of chalcogenide As_{2}S_{3} rectangle core are also presented. Radiation pressure on boundaries are symmetry to the central air slot.

We calculate the eigen-modes of acoustic fields in the hybrid waveguide with free boundary conditions and the phase matching condition in Eq. (2). In the BSBS case, we assume that *k _{p}*=-

*k*=

_{s}*k*and

*q*= 2

*k*. The five lowest order hybrid acoustic modes (labeled by A1-A5) excited at wavelength 1550 nm are also shown in Fig. 3. As can be seen in Fig. 3, acoustic modes are confined mainly in the area of chalcogenide As

_{2}S

_{3}rectangle core.

The corresponding BSBS gain coefficient are described in Fig. 4. As that shown in Fig. 4(a), the linewidth of each acoustic mode is narrow and no overlap occurs in Brillouin gain spectrum calculated by Eq. (5). The peak value of BSBS gain coefficient spectrum illustrated in Fig. 4(b) is obtained by applying Eq. (8), which is also used in the following BSBS gain coefficients calculation in later sections. As we can see, only acoustic modes A1, A3 and A5 can match the correct selection rule for the interaction between acoustic and optical fields and thus generate non-zero BSBS gain coefficients [14]. The acoustic mode A5 has the largest BSBS gain coefficient of 2.78×10^{4} W^{−1}m^{−1}, which is mainly contributed by radiation pressure (2.71×10^{4} W^{−1}m^{−1}) at air slot boundaries. The gain coefficients of 2.19×10^{3} W^{−1}m^{−1} corresponding to acoustic mode A1 comes from a coherent combination of the electrostriction force gain coefficient of 5.66×10^{2} W^{−1}m^{−1} and radiation pressure gain coefficient of 5.29×10^{2} W^{−1}m^{−1}. While the total gain coefficient of 6.11×10^{3} W^{−1}m^{−1} corresponding to acoustic mode A3 comes from a coherent combination of the electrostriction force gain coefficient of 1.15×10^{3} W^{−1}m^{−1} and radiation pressure gain coefficient of 1.95×10^{3} W^{−1}m^{−1}. The BSBS total gain coefficient of A5 occupies around 77% of the sum of total gain coefficients of all of the 5 acoustic modes.

To illustrate the strong potential to enhance effective coupling, which can yield a huge enhancement of total BSBS gain coefficients, we further analyze gain coefficients from mode A1, A3 and A5. For radiation pressure, which can only interact with the transverse components in HAW, we calculate the *S* parameter of A1, A3 and A5, respectively. Specially, in our structure, since the narrow air slot extremely strengthen radiation pressure on air/As_{2}S_{3} boundaries (*F _{x}^{RP}*) (around 3.8 times larger than that on vertical As

_{2}S

_{3}/Si boundaries and 2.3 times larger than that on horizontal As

_{2}S

_{3}/Si boundaries),

*u*-components in transvers components of HAW mainly effect the

_{x}*Q*

_{Cm}

*. We define*

^{RP}*S*as the ratio between spatial integrals of the

_{m}^{x}*x*-component displacements and transverse displacements. The results of the calculated

*S*and

_{m}^{x}*S*are shown in Fig. 5. Both high values of

_{m}*S*(0.94) and

_{5}^{x}*S*(0.95) enable an efficient

_{5}*Q*

_{Cm}

*for mode A5, resulting in both the largest gain coefficient for radiation pressure and the highest total gain coefficient (which are an order of magnitude larger than that for A1 and A3). In contrast, for A1, although with a high*

^{RP}*S*(0.95), there is a small

_{1}*S*(0.09) consisting of the predominately

_{1}^{x}*u*in acoustic fields. That determines an inefficient optomechanical coupling on air slot boundaries, and thus yields the least gain coefficient for radiation pressure among three acoustic modes. Hence, in this case, considering the mainly optomechanical coupling between radiation pressure and

_{y}*x*-component acoustic displacements on air/As

_{2}S

_{3}boundaries, both high

*S*and

_{m}*S*are required to efficiently enhance the BSBS gain coefficient.

_{m}^{x}#### 4.2 Effect of the As_{2}S_{3} rectangle core dimension on BSBS gain coefficient

In this subsection, we explore the effect of the chalcogenide As_{2}S_{3} rectangle core dimension on BSBS gain coefficient of the proposed structure. As the maximum total BSBS gain coefficient mainly contributed from radiation pressure is generated in the acoustic mode of A5, we take it as an example for further study. The following subscript *m* (*m *= 5) of all parameters are omitted. We fix the silicon layer at *a *= 450 nm and *b *= 365 nm with the air slot width *e *= 10 nm. The chalcogenide As_{2}S_{3} core width *c* and core height *d* vary from 140 nm to 230 nm and from 300 nm to 365 nm, respectively. The corresponding total gain coefficients are presented in Fig. 6(a).

In Fig. 6(a), the region with huge BSBS gain coefficients is marked with the black dashed rectangular line frame. We obtain huge total BSBS gain coefficients in the region where *c* varies from 160 nm to 200 nm and *d* varies from 320 nm to 364 nm. Every total BSBS gain coefficient in this region is beyond 1.8×10^{4} W^{−1}m^{−1} and higher than 12124 W^{−1}m^{−1} in [20], which gets a maximum value of 2.88×10^{4} W^{−1}m^{−1} with size *c *= 180 nm and *d *= 352 nm. From the BSBS gain coefficient results shown in Fig. 6(a), we can see that the profile of the total BSBS gain coefficient is determined by the size of *c* and *d*. The top edge of the Fig. 6(a) shows the gain coefficient of structure without top and bottom wrapping silicon layers (*b** *= *d *= 365 nm and *h** *= 0), where the gain coefficient value is relative small (the largest value is only 2.17×10^{4} W^{−1}m^{−1}).

Notably, there are some areas with low BSBS gain coefficients in Fig. 6(a). The reason is that *S* parameter of acoustic modes is closely depended on the dimension of As_{2}S_{3} rectangle core. In areas with low BSBS gain coefficients, *S* is cut down. For example, *S* only reaches 0.75 when *c *= 220 nm and *d *= 310 nm with a gain coefficient of 5.10×10^{2} W^{−1}m^{−1}. While in areas surrounded by black dashed lines, every *S* is larger than 0.9.

Furthermore, we analyze the influence of the top and bottom thickness of wrapping silicon layers *h* on controlling distribution of acoustic displacement and the corresponding BSBS gain coefficients. Figure 6(b) shows that with a fixed *c *= 180 nm, the total BSBS gain coefficient value peaks when *h *= 6.5 nm (*d *= 352 nm) and then falls during *h* varies from 0 to 23.5 nm. The *S* parameter has a slight variation from 0.936 to 0.962.

The variation trend of BSBS total gain coefficient can be understood that the variation of *h* effects the vertical symmetry distributions of acoustic fields relating to the *Q _{C}^{RP}* when

*S*parameter varies slowly. From Fig. 6(c), we can see that compared with the distribution of

*u*acoustic fields with

_{x}*h*= 22.5 nm, it presents a slight movement upward towards air with

*h*= 2.5 nm, and leads to a vertical asymmetrical distributions of the acoustic fields. While the distribution of radiation pressure has little variation with the change of

*h*, and still remains an almost symmetrical distribution in vertical direction (slightly towards bottom). Thus, an appropriate

*h*can improve overlapping between acoustic and optical force fields. While the acoustic fields has presented a well symmetrical vertical distribution by selecting an appropriate

*h*(in this case

*h*= 6.5nm), the continue increase of

*h*corresponds to the decrease of the interaction boundaries, which impairs the

*Q*and thus leads to the decrease of the BSBS gain coefficient.

_{C}^{RP}Considering the fabrication tolerance, we choose the proposed waveguide with air slot width *e *= 10 nm as a typical structure in this paper, which yields a total gain coefficient of 2.88×10^{4}W^{−1}m^{−1} (calculated in Section 4.2). To compare the typical structure with the waveguide structure without air slot (similar to Ref. [19]), we considerate two comparable structures. In one case, air slot is absent, i.e., *a *= 450 nm, *b *= 365 nm, *c *= 180 nm, *d *= 352 nm, *h *= 6.5 nm and *e *= 0. In another one, both air slot and the silicon wrapping layers of top and bottom are removed, i.e., a = 450 nm, *b *= *d *= 365 nm, *c *= 180 nm, *e *= 0 and *h *= 0. In the first case, we obtain the highest BSBS total gain coefficient of 4.14×10^{3} W^{−1}m^{−1}, which is mainly contributed from electrostriction force (3.15×10^{3} W^{−1}m^{−1}) and is only 0.14 times of the total gain coefficient of the proposed typical waveguide. In the second case, the maximum total gain coefficient is 2.97×10^{3} W^{−1}m^{−1}, which is only 0.1 times of the total gain coefficient of the proposed typical waveguide. The total gain coefficient is the result of the destructive combine of electrostriction force and radiation pressure, which yields the electrostriction gain coefficient of 3.28×10^{3} W^{−1}m^{−1} and the radiation pressure gain coefficient of 7.89 W^{−1}m^{−1}. Theses comparisons also illustrate the effect of the air slot and the top and bottom wrapping silicon layers on BSBS enhancement.

Therefore, by adding the top and bottom wrapping silicon layers and choosing the size of the structure to get a high *S* and thus a strong photomechanical coupling, our design opens up a path to obtain a giant BSBS gain coefficient.

#### 4.3 Effect of the air slot dimension on BSBS gain coefficient

Next, we study the effect of the air slot width *e* on BSBS gain coefficient. We fix the geometric size and let *a *= 450 nm, *b *= 365 nm, *c *= 180 nm, *d *= 352 nm and *h *= 6.5 nm. Then we vary *e* from 5 nm to 35 nm and compute BSBS gain coefficients. The corresponding BSBS gain coefficients are depicted in Fig. 7(a). The results show that with the increase of the air slot width, BSBS gain coefficients for radiation pressure always guide the variation tendency of total gain coefficient and decreases when *e* increases. The maximum value of total gain coefficeint of 4.83×10^{4} W^{−1}m^{−1} appears when *e *= 5 nm. It’s reasonable to predict that the total gain coefficient can reach higher when *e* is less than 5 nm.

To further analyze the reason for the aforementioned result, we normalize the corresponding value of *C _{OT}*, |

*Q*|

_{C}^{RP}^{2}and radiation pressure gain coefficient in Eq. (8) into the same interval. As the results described in Fig. 7(b), the decrease of radiation pressure gain coefficient is the coherent interaction between

*C*and |

_{OT}*Q*|

_{C}^{RP}^{2}. As the simultaneous decline of the

*C*and

_{FV}*C*in Fig. 7(c) with the increase of

_{EF}*e*, the parameter

*C*is decreasing. For

_{OT}*Q*, the parameter is the sum of the overlapping integral between radiation pressure and acoustic displacement fields on every As

_{C}^{RP}_{2}S

_{3}rectangle core boundaries. Since radiation pressure and acoustic field are symmetrical about

*y*-axis respectively, we calculate the magnitude of overlapping integral between radiation pressure and acoustic displacement fields on boundaries (i)-(iv) (shown in Fig. 7(d)). The results show that the overlapping integral on boundary (ii) (i.e. air/As

_{2}S

_{3}interface) generates an overwhelming magnitude compared with those on other boundaries. The value of overlapping on boundary (ii) decreases with the increase of

*e*, leading the identical variation trend of |

*Q*| as well as |

_{C}^{RP}*Q*|

_{C}^{RP}^{2}.

Closer examine of the *Q _{C}^{RP}* on boundary (ii) appears that the effect of acoustic displacement variation with the decrease of

*e*contributes mainly to the decrease of |

*Q*| in Fig. 8. In Fig. 8, both magnitude variation of radiation pressure and

_{C}^{RP}*u*-component of acoustic fields on different

_{x}*y*position of boundary (ii) are described as

*e*= 5 nm,

*e*= 15 nm,

*e*= 25 nm, and

*e*= 35 nm, respectively. When

*e*= 5 nm, the magnitude of radiation pressure increases and then decreases as

*y*varies from −174 nm to 174 nm, which gets the peak value with

*y*=−20 nm and shows symmetry in Fig. 8(a). As

*e*increases, the peak value drops slowly and the symmetry of radiation pressure distribution is lowered. Values on both ends of boundary (ii) (

*y*is around −174 nm and 174 nm) tend to rise up. In Fig. 8(b), every distribution of magnitude of

*u*-component of acoustic field shows better symmetry and gets peak values as

_{x}*y*= 20 nm with every

*e*. Compared with the peak value of radiation pressure, the peak value of every

*u*-component of acoustic field falls rapidly with the increasing of

_{x}*e*and creates a simultaneously trend of |

*Q*| as well as |

_{C}^{RP}*Q*|

_{C}^{RP}^{2}.

#### 4.4 Effect of the optical losses on Stokes amplification

In this subsection, we discuss the effect of optical losses and calculate the Stokes amplification (corresponding to BSBS on-off gain in [24]) of the proposed waveguide. In semiconductor-based waveguides design, the effect of linear optical loss and some types of nonlinear losses such as 2PA and FCA should be involved. Linear loss mainly comes from the waveguide surface roughness [19]. 2PA involving corresponding FCA has been measured in bulk silicon [37]. Both of them affect the performance of silicon devices and activities of SBS.

Here, we consider the effect of linear loss and the nonlinear losses including 2PA and FCA in Eq. (3) and Eq. (4). To achieve Stokes amplification, the defining the figure of merit for SBS must be satisfied [19]:

Only for $\mathscr{F} >1$, a net Stokes amplification is possible to obtain in the waveguide where power growth with respect to BSBS gain partly compensate the power reduction due to optical losses. The Stokes amplification*A*is expressed as follows, which can by solving Eq. (4) numerically [38]: where

*L*is the waveguide length.

The Stokes amplification is also related to the input pump power. The optimum value of initial pump power *P _{opt}* can be calculated by [19]:

*L*can also be obtained by [19]:

_{opt}*P*

_{opt}and

*L*

_{opt}, respectively. The detailed derivation can be found in [19].

Next, we calculate nonlinear coefficents *β* and *γ*. We assume the value *α *= 11.5 m^{−1} (0.5 dB/cm), which is an achievable value for exist photonic waveguide [39]. The values of nonlinear losses coefficients *β* and *γ* are computed [38] as the change of air slot width in proposed waveguide (*a *= 450 nm, *b *= 365 nm, *c *= 180 nm, *d *= 352 nm and *h *= 6.5 nm) are presented in Fig. 9. Considering the fabrication possibility, the air slot width *e* is set in the range from 10 nm to 35 nm. As can be seen, the value of *β* decreases with the increase of air slot width, and the value of *γ* decreases with the air slot width increases in the range from 10 nm to 30 nm, but *γ* has a little tendency to increase as the air slot width increases in the range from 30 nm to 35 nm.

After obtain the nonlinear coefficients *β* and *γ*, we calculate the corresponding figure of merit $\mathscr{F}$ for BSBS in Eq. (16), and the results is shown in Fig. 10. The figure shows that in the presented interval of air slot width, the values of $\mathscr{F}$ are always larger than 1, which means that the positive net Stokes amplification can be obtained. $\mathscr{F}$ decreases with the air slot width increases, and the peak value of $\mathscr{F}$ = 16.7 occurs when air slot width *e *= 10 nm. Since greater $\mathscr{F}$ causes stronger Stokes amplifications, the proposed structure with air slot width *e *= 10 nm can be the best candidate for yielding a huge Stokes amplification.

Applying the Eq. (18) and Eq. (19), and combining *α*, *β*, *γ* and $\mathscr{F}$, we obtain the optimum value of initial pump power *P*_{opt }= 0.44 W and the optimum value of waveguide length *L*_{opt }= 0.3 m when the air slot width *e *= 10 nm. The change of Stokes amplification versus the waveguide length are calculated in the Fig. 11, with the initial pump power *P _{p}* (0)=

*P*

_{opt}=0.44 W. Figure 11 shows that as the waveguide length expends, the Stokes amplification increases quickly and then becomes saturated. The value of Stokes amplification peaks (188 dB) when waveguide length is

*L*

_{opt}. We assume our linear waveguide structure is invariant in optical propagation direction without deliberate bending design. Considering the fabrication feasibility, it’s unnecessary to fabricate a waveguide with so long length to obtain a huge Stokes amplification larger than 100 dB. The proposed waveguide yields a net Stokes amplification value of 85.7 dB with

*L*= 2.5 cm, which is larger than those giant gain SBS on modern photonic integrated platforms in [24,25], which are 52 dB with a waveguide length of 23 cm. The good performance of the Stokes amplification in the proposed waveguide shorten the waveguide length, thus it does not need to be a spiral waveguide with long effective length and high bending loss.

## 5. Fabrication feasibility and tolerance analysis

There are many fabrication challenges to obtain the hybrid chalcogenide-silicon waveguide structure with a 10 nm air slot. Fortunately, these challenges can be solved with existing advanced fabrication techniques. For example, recent work has presented silicon slot waveguides filled with chalcogenide glass by thermal evaporation after the process of electron beam lithography and reactive-ion etching [40]. Another work has realized a fabrication of narrow dielectric slots in silicon waveguides with a 10 nm slot width [41]. 6 nm silicon nanowires have been fabricated by anisotropic wet etching and oxidation [42]. Besides, the extreme ultraviolet (EUV) lithography has potential to fabricate nano-structure with a resolution of 7 nm [43]. Therefore, it’s reasonable to realize the proposed waveguide by using some kind of these technologies.

Here we provide a possible fabrication process to obtain the proposed hybrid waveguide with a 10 nm air slot, and try to complete it in our future research.

The five fabrication processes of the proposed structure are shown in Fig. 12. As it shown in Fig. 12(a), the first process is slot and strip waveguides fabrication. During this process, using suitable optical lithography and etching (such as in [41,42]) on the SOI wafer, a rectangle waveguide with the width of *a* and thickness of *d *+ *h* is fabricated firstly. Then two slots with the same width of (*c*-*e*)/2 and the depth of *d* are created symmetrically and smoothed to decrease optical loss in the rectangle waveguide. The distance between the left boundary of silicon waveguide and the left boundary of the left slot is (*a*-*c*)/2. The two slots will be filled with As_{2}S_{3} core in the third process. The central silicon beam is maintained and as a central silicon stripe. The second **s**tep is central SiO_{2} stripe fabrication. As it shown in Fig. 12(b), during the step, the central silicon stripe becomes SiO_{2} stripe by thermal oxidation. The extra SiO_{2} layer on other position such as the side walls of the slot waveguide should be removed by Hydrofluoric Acid (HF) as that in [41]. The third step is shown in Fig. 12(c), which is As_{2}S_{3} layer deposition. In the process, the As_{2}S_{3} layer is deposited to fill the slots via thermal evaporation. The fourth step is deposition of the top silicon wrapping layer, as that shown in Fig. 12(d). After removing extra As_{2}S_{3} layer covered on the top of silicon strips by iron milling step, the top silicon wrapping layer can be deposited at last, and the thickness of the layer is *h*. The last step is air slot generation. As is shown in Fig. 12(e), the last process is cleaning the central SiO_{2} stripe to obtain an air slot by using HF. It should be noted that, in the fabrication processes, we reduce the optical lithography and etching difficulty by replacing the fabrication of central air slot directly with two chalcogenide slots, which have aspect-ratio of 4 and can be achieved by modern silicon etching technology [44].

The sidewall roughness on the waveguide and the lithography tolerance lead to fabrication tolerance. We assume that the fixed size of *a *= 450 nm, *b *= 365 nm and *L *= 2.5 cm. When the tolerance is 8 nm [45], the air slot width *e* varies from 6 nm to 14 nm, the chalcogenide As_{2}S_{3} glass size *c* varies within the range of 176∼184 nm, and *d* changes within the range of 348∼356 nm. From Section 4 we know that the BSBS total gain coefficient decreases versus the increase of *e*. When *e* is fixed, the total gain coefficient weakens when *c* and *d* deviate to the optimized geometric size. The highest value of gain coefficient can be obtained among the combination of *c*, *d* and *e* in Table 2, which shows the effect of fabrication tolerance on BSBS gain. From Table 2, the fabrication tolerance lead to the total gain coefficient varies from 1.84×10^{4} W^{−1}m^{−1} to 4.38×10^{4} W^{−1}m^{−1}, and the Stokes amplification changes from 52.8 dB to 123 dB. Although the fabrication tolerance can lead to a variation of BSBS interaction on the proposed waveguide, for the worst case, the Stokes amplification of 52.8 dB with *L* = 2.5 cm is still beneficial to obtain a high gain within a short waveguide length. The Stokes amplification of 52.8 dB is comparable with the on-off gains of 52 dB in experimental on-chip systems with much longer waveguide length [24,25].

The tolerance of thickness of wrapping layer *h* can also induces the variations of BSBS gain coefficient and Stokes amplification. The tolerance of the top silicon layer thickness depends on deposition environment, and the tolerance of the bottom silicon layer thickness depends on etching process. Although the ideal *h* is 6.5 nm, we assume it varies from 2.5 nm to 10.5 nm, considering its tolerance is also 8 nm [45]. When the other geometric size is fixed at *a *= 450 nm, *b *= 365 nm and *c *= 180 nm, the effect of *h* variation on BSBS gain coefficient can be obtained from Fig. 6(b). When *h* is in the range of 2.5∼10.5 nm, the corresponding BSBS gain coefficient is in the range from 2.74×10^{4} W^{−1}m^{−1} to 2.88×10^{4} W^{−1}m^{−1}. The BSBS gain coefficient reaches the least value of 2.74×10^{4} W^{−1}m^{−1} when *h *= 2.5 nm. Therefore, the variation of BSBS gain coefficient due to the uncertainty in the thickness of the silicon wrapping layers is very small and can be neglected.

## 6. Conclusion

In conclusion, we have proposed a new-style on-chip hybrid silicon-chalcogenide slot waveguide to enhance BSBS scattering. We have analyzed the influence of transverse displacement components in HAW on the optomechanical coupling between radiation pressure and individual acoustic field. We have proved that the wrapping silicon layer has the ability to enable the vertical symmetry of acoustic fields with an appropriate top and bottom layer thickness. Besides, we have computed the BSBS gain coefficient variation with the change of air slot width. The effect of optical losses on BSBS gains is estimated, and the Stokes amplification value versus the waveguide length is calculated. The fabrication feasibility and the tolerance analysis of the proposed waveguide are also discussed. We find that the variation of acoustic displacement with different slot widths has the main influence on the gain coefficient driven by radiation pressure. By tailoring the high ratio of acoustic shear-to-total displacement, the distribution of acoustic and optical fields, the designed structure can achieve a significantly enhancement of BSBS gain coefficient mainly for radiation pressure. In the proposed structure, the highest Brillouin gain coefficient is 2.88×10^{4} W^{−1}m^{−1} with an air slot width of 10 nm, which is much larger than the value of Brillouin gain coefficient in conventional nonlinear fibers, some suspended theoretical silicon waveguides and calculated on-chip based traveling-wave waveguides on insulator. The Stokes amplification reaches the value of 85.7 dB with the wavelength of 2.5 cm. The least Stokes amplification is 52.8 dB when considering the fabrication tolerance, which is still comparable for those experimentally proved devices with giant on-off gains. It’s reasonable to predict that with the decrease of air slot width and the length extension of proposed waveguide, the Brillouin gain have potential to be larger. Our work opens a new path to design high-performance nanophotonic structures with giant stimulated Brillouin scattering.

## Funding

National Natural Science Foundation of China (61875086, 61377086); National Aerospace Science Foundation of China (2016ZD52042).

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