## Abstract

We study slot waveguide geometries, comprising a combination of soft glasses and high-index guiding structures, for enhancing stimulated Brillouin scattering (SBS). We show that strong optical and acoustic mode confinement in these waveguides can lead to a substantial increase in SBS gain, comparable to or greater than recently proposed suspended silicon nanowire structures. We compute the optimal parameters of the structure and examine the physics of optical and acoustic confinement within slot waveguides. Finally, we compute the effects of linear and nonlinear loss mechanisms on optimum pump/Stokes powers and waveguide lengths.

© 2016 Optical Society of America

## 1. Introduction

Stimulated Brillouin Scattering (SBS) is a nonlinear interaction that occurs in optical waveguides where light is inelastically scattered by acoustic waves [1, 2]. In recent years SBS has generated much interest in the realm of nanophotonic structures. SBS has been used in tunable all-optical delay systems [3–5], optical synthesizers [6], sensors [7] and filters [8]. SBS has been demonstrated in a number of nanophotonic structures and systems, including multistructured fibres [9–11], micro-cavities [12, 13] and chip-scale waveguides constructed of highly nonlinear chalcogenide glasses [14].

A major challenge in SBS is the simultaneous confinement of both the acoustic and optical fields. Moreover, for strong SBS coupling, both light and acoustic waves should be highly localized within a small area that possesses large photoelastic constants. Ideally, one requires a waveguide with a high refractive index core and a low stiffness in order to simultaneously confine both optical and acoustic fields. In practice, however, high index materials are not necessarily mechanically compliant. As an approach for ensuring trapping of acoustic waves in high refractive index waveguides, some groups have proposed completely [15–17] or partially [18, 19] suspended structures to prevent escape of the acoustic waves while confining light in the waveguide. However, these structures are not ideal for mass fabrication, potentially suffering from issues of mechanical stability and sensitivity to environmental conditions.

A promising approach to co-localize optical and acoustic waves is to use multimaterial structures, whereby the mechanical or optical strengths of one material compensate for the corresponding weakness of the other. Within this catergory of multimaterial structures, slot waveguides, in which a large electric field can be concentrated within in a small area containing a material with a high nonlinearity and a relatively low refractive index, are particularly promising candidates [20]. Recent reports have shown that strong SBS gain, driven by radiation pressure, can theoretically be achieved in suspended silicon slot waveguides with an air gap between them [21]. Non-suspended slot waveguides, i.e. placed on a substrate, have previously been proven to significantly enhance nonlinearities in nanophotonic waveguides [21], have a stable mechanical configuration and can be fabricated with current technology. An open question is whether material systems or parameters can be found such that useful SBS gain can be achieved in a non-suspended slot waveguide.

In this paper, we study numerically and analytically SBS in hybrid silicon-chalcogenide slot waveguides. Chalcogenides are soft glasses with high nonlinearity and good acousto-optical properties that have been used in a number of applications [22]. We show that strong optical and acoustic mode confinement in a silicon-chalcogenide slot waveguide leads to a substantially increased SBS gain relative to suspended silicon structures [16, 21]. We optimize the waveguide dimensions for maximal gain and show that gains of up to 3300 *W*^{−1}*m*^{−1} can be achieved in these geometries that is comparable to the gain of suspended silicon nanowire waveguides [16]. Additionally, we demonstrate that under certain conditions radiation pressure has no net effect on the gain. We show that increasing the structure symmetry can lead to further confinement of acoustic waves in the slot gap, thereby both increasing the SBS gain and leading to greater stability and protection against environmentally-induced losses. We analyze the conditions under which acoustic waves leak to the substrate, and examine this effect on the parameters available for SBS waveguide design. Furthermore, we examine the effect of losses on the SBS Stokes amplification, including the linear loss as well as nonlinear losses of two photon absorption (TPA) and TPA-induced free carrier absorption (FCA). We finally determine optimum pump and Stokes powers for initializing the SBS process as well as the optimum length for the designed waveguide.

## 2. Formalism

In the following we outline the formalism for computing SBS gain, in which the optomechanical coupling term is formulated using optical forces. A full discussion of the computation of SBS gain for high-contrast waveguides can be found in [23].

#### 2.1. Definitions and gain

In SBS, pump photons with angular frequency of *ω** _{p}* are red-shifted to Stokes photons with angular frequency

*ω**(*

_{s}

*ω**<*

_{s}

*ω**) and simultaneously phonons with an angular frequency Ω are generated. Momentum and energy conservation determine how optical/acoustic frequencies and wavenumbers are interrelated:*

_{p}*q*,

*β*

_{p}and

*β*

_{s}are the acoustic, optical pump and Stokes wavenumbers, respectively [21].

The pump and Stokes waves can co-propagate (Forward SBS, FSBS)- or contra-propagate (Backward SBS, BSBS). They can also carry different optical modes (intermode coupling) or identical modes (intramode coupling) [16]. In the following, we choose the case of BSBS as an example. The governing equations for the optical powers, taking into account optical loss, are [24]

*P*and

_{p}*P*are the pump and Stokes powers, respectively, and Γ is the SBS gain. The coefficients

_{s}*α*,

_{p}*α*,

_{s}*γ*and

*β*correspond to linear and nonlinear optical losses. According to Eqs. (3) and (4) the pump decreases continuously along the direction of propagation (the z-axis), while the Stokes wave grows in the opposite direction. In practice, however, the Stokes amplification is limited by optical losses. Therefore, the SBS gain should be large enough to compensate losses and amplify Stokes. The SBS gain is given by [23]

**is the angular optical frequency (**

*ω*

*ω*

_{p}≈

*ω**=*

_{s}**) and**

*ω**α*is acoustic decay parameter. The pump, Stokes and acoustic modes are assumed to be normalized to the unit powers ${\mathcal{P}}^{\mathrm{p}}$, ${\mathcal{P}}^{\mathrm{s}}$ and ${\mathcal{P}}^{\mathrm{ac}}$, respectively. Here we adopt the convention (also used in [24]) that the Stokes power is normalised to a negative value in order to reflect the fact that the Stokes wave is counter-propagating. The overlap integral

_{ac}*Q*expresses the interaction between optical force

_{c}**F**and acoustic displacement fields

**u**.

*Q*is given by [23] in which the integral extends over the waveguide cross section.

_{c}#### 2.2. Optical parameters of SBS gain

Two dominant forces co-exist in nonmagnetic integrated waveguides; electrostriction and radiation pressure [15]. These forces can have similar magnitudes depending on the profiles of the electromagnetic modes, and hence on the waveguide geometry. Both forces depend quadratically on the electric field. Electrostriction is the quadratic response of mechanical strain to an applied electric field and can be observed in all materials [25]. The *i ^{th}* component of the electrostrictive force can be found by [26]

*σ*is the electrostrictive stress given by

_{i}j*ε*and the fourth rank photoelastic tensor

_{r}*p*. The sum extends over all Cartesian field components. The electric field

_{ijkl}*E*in Eq. (8) is of the form

_{k}*k*component of pump and Stokes electric field modes and

^{th}*a*(

_{p}*z,t*) and

*a*(

_{s}*z,t*) are the corresponding optical mode envelopes [23]. The optical properties of materials used in the proposed waveguides of our paper are given in Table 1 [26].

Radiation pressure acts only on boundaries where electric/magnetic properties change. Unlike electrostriction – which appears in the bulk – radiation pressure is an edge effect. The components of this force are expressed as [16]

where*nj*is the normal vector of the interface from material 1 to material 2 and

*T*are the components of Maxwell’s stress tensor

_{ij}**E**and

**H**are the total electric and magnetic fields, respectively. For translationally invariant waveguides, only the transverse components of this force with respect to the waveguide direction contribute to SBS.

It may be expected that waveguides with high refractive index materials provide strong optical forces that lead to high SBS gains [Eq. (8)]. However, the situation is more complex. First, dielectric materials with high refractive index do not necessarily exhibit a strong photoelastic effect. Second, optical forces might cancel each other on waveguide boundaries. This situation might occur because electrostriction and radiation pressure are of similar order of magnitude in integrated waveguides. Third, a large overlap integral (*Q _{c}*) is the result of constructive contribution of both optical forces and acoustic displacement fields. Therefore, even if the forces are strong, weak acoustic fields lead to a small

*Q*. Consequently, the acoustic properties of the propagation medium and the acoustic mode profiles play a significant role in enhancing SBS interactions.

_{c}The acoustic waves in waveguides are excited within SBS by optical forces that satisfy the phase matching condition (Eqs. (1) and (2)). The corresponding waves satisfy the acoustic wave equation [27]:

*u*,

_{i}*S*and

_{kl}*F*are the components of acoustic displacement fields, the mechanical strain tensor and the driving force, respectively and

_{i}*ρ*,

*C*and

_{ijkl}*η*denote the mass density, the stiffness and the viscosity coefficients of the propagation medium [27]. The viscosity damping defines the linewidth of the SBS response. The displacement fields therefore attenuate with the damping coefficient

_{ijkl}*α*throughout the waveguide length where ${\mathcal{E}}^{\mathrm{ac}}$ is the energy of the acoustic mode per unit length of waveguide and

_{ac}*Q*is the mechanical quality factor. For acoustic waves traveling longitudinally in an axially invariant waveguide Eq. (13) can be simplified to

*α*=

_{ac}*q/Q*where we have assumed that acoustic dispersion is linear. The mechanical quality factor in complicated waveguide structures is frequency dependent and can be approximately measured, for example by observing the spectra of Stokes waves of a chip-scale structure [14]. We assume a Q-factor of 1000, which has been used in [16, 28], to compare our results with these related works. The critical acoustic properties of materials used in our proposed waveguides are listed in Table 2.

## 3. Slot waveguide on substrate

We begin by describing the main characteristics of a hybrid silicon-chalcogenide glass slot waveguide. As can be seen in Fig. 1, the slot is composed of two identical silicon beams placed on a SiO_{2} substrate. The gap is filled with As_{2}S_{3} glass. We assume that the waveguide is axially infinite; we will apply limitations on power and waveguide length in Section 5.

The high index of silicon provides optical confinement in the slot. To provide acoustic confinement, one should add a less stiff material than silica in the slot or the acoustic waves will leak into the substrate. For this purpose, we choose *As*_{2}*S*_{3} glass which has low stiffness constants (see Table 2), however other choices of soft glass are possible. The silica substrate makes the structure stable and realistic. It should be noted that the presence of a substrate prevents the confinement of standing acoustic phonons required for forward SBS.

Among several optical modes that can propagate in the structure, the fundamental mode with the profile shown in Fig. 1 is interesting as it has comparatively large intensity in the gap. This results in a strong overlap between the optical and acoustic fields. To find the optical and acoustic modes in the waveguide we have used the finite element solver COMSOL. We here focus on the situation of intramode coupling between the pump and Stokes waves where both waves are in the fundamental mode.

At the optical wavelength *λ* = 1550 nm several acoustic modes can be excited. The lowest order acoustic modes are shown in Fig. 1. As can be seen in the figure, the acoustic modes are mainly localized in the gap. It is possible to excite any acoustic modes that satisfy the phase matching conditions and have appropriate symmetry [29]. However, it should be noted that only a few acoustic modes can produce large amount of gain. In Fig. 2 the SBS gain of the slot waveguide shown in Fig. 1 is depicted for the acoustic modes. According to the gain results, one can see that although the geometry, pump and Stokes powers and optical frequency are identical for all 3 acoustic modes, SBS gain is severely affected by the mode profiles. The first acoustic mode, appearing at 5.53 GHz is a dominantly transverse mode, and so, because the electrostrictive force is mostly longitudinal the net SBS coupling is weak. In contrast, the second acoustic mode is predominantly a longitudinal mode which accordingly produces strong coupling. Finally, the coupling for the third acoustic mode does not obey the correct selection rule for coupling between the acoustic and electromagnetic fields [29] and hence the gain is exactly zero.

We first study the effect of gap size on the SBS gain. We assume that the dimensions of the silicon beams are fixed at a = 250 nm and b = 190 nm and the gap width varies. The SBS gain is shown in Fig. 3(a). In the figure, the total SBS gain that includes the contribution of both electrostriction and radiation pressure is shown as a red curve for gap widths ranging from 90 nm to 260 nm. The SBS gain due to the contribution of the optical forces are depicted for comparison.

We see in Fig. 3(a) that the overall gain in the slot waveguide is mainly the result of electrostrictive forces. The gain due to radiation pressure is relatively small in this case, but in combination with electrostriction, modifies the total gain by up to 18% due to the quadratic dependence of the gain on the overlap integral *Q _{c}* in Eq. (5). The total gain rises steeply from 965 W

^{−1}m

^{−1}for

*c*= 85 nm to 3020 W

^{−1}m

^{−1}for

*c*= 200 nm then it falls off gradually as the gap width is further increased. This decreasing trend can be understood by noticing that the optical/acoustic field intensities decrease as the gap width is increased and hence, reduces the overlap integral

*Q*. At

_{c}*c*= 200 nm the gain has a value comparable with the values reported for silicon nanowires [16].

We observe a cutoff in the gain as the gap width reduces to c = 85 nm. This happens as a result of hybridization of the acoustic mode with Rayleigh surface waves. To understand the impact of the silica substrate through mode hybridization it is helpful to study how the acoustic frequency changes as the gap width is reduced.

The blue curve in Fig. 3(b) shows the frequency of the Rayleigh surface waves corresponding to the acoustic wave number assuming the substrate to be semi-infinite. The red curve represents the frequency of guided acoustic modes. As the gap width decreases the guided mode approaches the Rayleigh waves’ dispersion curve until both curves intersect at 85 nm. In practice the meeting point would be slightly offset since the dispersion of surface waves is modified by the presence of the slot. This means that the acoustic mode inside the gap hybridizes with the acoustic surface mode and hence SBS gain drops heavily [Fig. 3]. Hybridization of confined acoustic modes in the gap with the acoustic surface waves also prevents FSBS occurring in the slot structure shown in Fig. 1.

#### 3.1. Cancellation of radiation pressure

Figure 3 shows that the gain due to the radiation pressure falls dramatically at *c ≈* 130 nm where the total gain becomes equal to the gain due to electrostriction alone. This is interesting, because it indicates that the effect of either electrostriction or radiation pressure can be removed while preserving the SBS gain in the structure. We now examine the mechanism of this cancellation. According to Eq. (6), the SBS gain due to the radiation pressure will vanish when the part of the overlap integral *Q _{c}* that incorporates radiation pressure forces (given by the vector

**F**in Eq. (10)) becomes zero. To illustrate how this occurs in the slot waveguide considered here, we first consider the components of the Maxwell stress tensor

*T*and

_{xx}*T*, shown in Fig. 4(a). These components have opposite sign regardless of the phase of the electromagnetic mode — this indicates that when the radiation pressure force points inward to the slot on the horizontal slot walls the force on the vertical boundaries will point outward, and

_{yy}*vice-versa*(as shown in Fig. 4(b)). In contrast, the acoustic mode component

**u**is directed inward on all slot boundaries (or outward on all boundaries, depending on the phase), in accordance with the longitudinal character of the acoustic slot mode. The result is that the horizontal and vertical contributions to the overlap integral (corresponding to the quantities ∫

**F**·

**u***d

*x*and ∫

**F**·

**u***d

*y*respectively) have opposite signs regardless of the slot width.

We now examine the situation that occurs when the gap width increases [Fig. 4(b) right] as the power in both the pump and Stokes remains constant. Because the geometrical size of the waveguide increases while the modal power remains the same, both the radiation pressure and the acoustic displacement fields decrease in magnitude at each point on the boundary. This causes the magnitude of the overlap integrals along the vertical boundaries to decrease steadily as the gap width increases. However, as the gap width grows, the length of the horizontal boundaries also increases, and so the magnitude of the overlap integrals along the top and bottom of the slot does not change appreciably. Thus at a particular gap width (*c* = 130 nm in this structure), the values of the overlap integrals on the horizontal and vertical boundaries exactly cancel.

#### 3.2. Geometry optimization

Thus far, we have assumed that the silicon beams in our structure have fixed dimensions. We now seek the optimum size of the waveguide to deliver maximum SBS gain. We keep the configuration symmetry (i.e similar silicon beams) so that only three free parameters remain for the optimization: silicon width, height and the gap width. We concentrate on two situations: First, the waveguide height is fixed at 220 nm (a realizable dimension with current fabrication technology) and we vary the beam and gap. Second, we fix the gap width at 200 nm and seek optimum dimensions for the silicon beams. The results are shown in Figs. Fig. 5(a) and Fig. 5(b). One can see that high SBS gains up to about 3300 W^{−1}m^{−1} can be achieved from the slot waveguide.

## 4. Slot waveguides with silica cover

We have seen (Fig. 3) that the gain in the slot waveguides mainly comes from the effect of electrostriction. Since the relevant acoustic mode is largely longitudinal the term
${F}_{z}{u}_{z}^{*}$ forms the dominant contribution to the overlap integral *Q _{c}* in Eq. (5). This suggests a way to enhance the overlap integral and hence increase the gain. As we see in Fig. 6,

*F*is symmetric and is mainly concentrated in the middle of the gap. That is because the small difference between the refractive index of silica and air causes the optical fields and their corresponding forces to be distributed almost symmetrically -with a slight tendency toward the substrate. In contrast, a considerable vertical asymmetry in acoustic fields is observable (Fig. 6). Since the acoustic field intensity is large on open boundaries

_{z}*u*is concentrated mainly toward the upper gap boundary. However this component (

_{z}*u*) interacts with an optical force component that is not strong on top of the gap. A similar weak interaction happens between the electrostrictive force and displacement field in the middle of gap.

_{z}The contribution to *Q _{c}* can be increased by adding a silica layer on top of the slot. The displacement fields then shift down toward the gap center, thereby increasing the interaction (Fig. 6). In Fig. 7 one can see that the gain is increased markedly by increasing the covering layer thickness. The maximum gain is increased from about 3020 W

^{−1}m

^{−1}for an ordinary slot waveguide to roughly 4500 W

^{−1}m

^{−1}for a slot with a silica layer with 150 nm thickness. It should however be noted that the increasing rate of the gain becomes small as the layer thickness increases. Figure 7(b) shows the gain variation in the slot with cover thickness in the slot with

*a*=250 nm,

*b*=190 nm and

*c*=160 nm. From the figure, the overall SBS gain shows an upward trend as the cover layer becomes thicker until it levels off for

*t*> 200 nm in which the modal symmetry is fulfilled.

## 5. Impact of optical losses

Several types of linear and nonlinear optical losses are important in semiconductor-based waveguides. Linear loss originates mainly from waveguide surface roughness (in bulky waveguides, intrinsic loss constitutes the linear loss term). In waveguides composed of materials with strong odd order optical susceptibilities — such as silicon — multi-photon absorption mechanisms including two photon absorption (TPA) affect SBS interactions [24]. TPA also creates free carriers if the photon’s energy exceeds half the material band gap [30] thereby TPA-induced free carrier absorption (FCA) can contribute significantly to the overall optical loss of the waveguide.

Here, we discuss the effect of linear and nonlinear losses on SBS. By applying the small signal approximation, i.e assuming that *P _{p}*(

*z*) ≫

*P*(

_{s}*z*) throughout the waveguide those terms containing higher orders Stokes power as well as the terms with negligible coefficients can be ignored from Eqs. (3) and (4) [24, 31]. As a result, the coupled equations for a BSBS process with intramode coupling in the presence of linear, TPA and FCA losses are simplified to

According to Eq. (15) to achieve Stokes amplification the following condition must be satisfied

This inequality, specifies upper and lower limits for pump powers leading to Stokes amplifications

Since the pump power is a real valued quantity, $\left(\mathrm{\Gamma}-2\beta >2\sqrt{\alpha \gamma}\right)$ is a necessary requirement for Stokes amplification. This condition has been used previously for defining the figure of merit for SBS [31]:

From Eq. (18) one can also see that as long as ${P}_{p}^{\mathrm{min}}<{P}_{p}<{P}_{p}^{\mathrm{max}}$, the Stokes wave is amplified. Since the pump amplitude decreases along the waveguide length, the optimum value for the initial pump power is

Optical losses give us criteria to find the optimum waveguide length. If $\beta \ll 2\sqrt{\alpha \gamma}$ then the optimal length of the structure can be obtained by

#### 5.1. Computation of nonlinear losses in slot waveguides

We now examine the impact of nonlinear optical losses on the Stokes amplification in hybrid slot waveguides. Since loss causes reduction in the Stokes signal, only slot configurations with large SBS gains maintain Stokes amplification. We study two geometries: An ordinary slot waveguide and a slot with a silica cover with thickness of *t* = 150 nm. In both waveguides, we choose Si beams with square cross section (*a* = *b* = 220 nm). To have a rough idea of the approximate amounts of the optimum pump power and the waveguide length in the slot waveguides we specify some values for the linear loss in our case studies. We assume the linear loss to take the values *α* = 2.3 m^{−1} (*≈* 0.1 dBcm^{−1}) and *α* = 11.5 m^{−1} (*≈* 0.5 dBcm^{−1}). These represent achievable loss figures for modern nanophotonic waveguides [32,33]. We computed [24] the loss coefficients *β* and *γ* for both structures and plot them in Fig. 8. The figure shows that slots with larger gap widths are better candidates since the loss absorption coefficients decrease in both structures.

To obtain Stokes amplification the power growth due to SBS gain must compensate the power reduction caused by optical loss. The figure of merit Eq. (19) is calculated for the two geometries and is shown in Fig. 9(a). The figure illustrates that although both structures produce fairly large amounts of SBS gain, the optical losses prevent Stokes amplification for a range of gap widths. We have seen in Section 5 that a net stokes amplification is only possible for ℱ > 1; greater values of ℱ lead to stronger Stokes amplifications. According to Fig. 9(a) Stokes is not amplified in the slot waveguide with gap widths smaller than 170 nm if *α* = 11.5 m^{−1}. It is noteworthy that the figure of merit is dominated by loss coefficients rather than the SBS gain. Although the SBS gain decreases for both waveguides at larger gap widths, this is overcompensated by an even faster decrease in *β* and *γ*. Finally we study the maximally realizable Stokes amplification. By solving Eq. (15) numerically [24] or analytically the Stokes amplification can be computed

*L*is the waveguide length. Stokes amplification for both covered and non-covered structures are depicted in Fig. 9(b). These results indicate that Stokes amplification in the slot structure can be considerably enhanced by using larger gap widths. It is worth noting that the Stokes amplifications shown in Fig. 9(b) correspond to the slot structures with optimal lengths. In the non-covered slot structure with

*α*= 0.5 dBcm

^{−1}for instance (shown with dashed blue line in Fig. 9(b)) the optimum length is found to be 7.97 cm at

*c*= 270 nm. According to Eq. (21) as the figure of merit increases, the optimum length grows. Therefore, the realization of the slot waveguides with large figures of merit becomes problematic. In this regard, exploring Stokes amplification level in shorter length structures is certainly worth investigating.

If the waveguide length *L* becomes smaller than *L*_{opt} the Stokes amplification is reduced. To preserve consistent growth of the Stokes power along the structure the initial pump power is fixed at *P*_{opt}; but the lower limit of pump power at *z* = *L*; *P _{p}* rises from
${P}_{p}^{\mathrm{min}}$ — expressed by Eq. (17) — though it should satisfy the condition
${P}_{p}^{\mathrm{min}}$ <

*P*<

_{p}*P*

_{opt}. Figure 10 shows the variation of the Stokes amplification versus the waveguide length for the two slot waveguides introduced in this Section at the gap width

*c*= 270 nm.

As shown in Fig. 10, the net Stokes amplification for a 4 cm long slot reaches 1.75 dB. Such a length is within the reach of current fabrication technology [18]. With further improvement in the figure of merit in the two waveguide — either by reducing the linear loss or increasing the SBS gain — we can reach up to 9.6 dB Stokes amplification in the covered slot with *α* = 0.1 dBcm^{−1} (shown with solid red line). The rise of the net Stokes amplification in longer waveguide lengths highlights the advantage of longer slot structures.

It should be noted that the amplification found in Figs. 9(b) and 10 are within a small signal approximation. This indicates that the amplified Stokes power should not exceed range of applicability of small signal approximation shown in [24]. For *P _{s}*(

*z*) ≦ 10

^{−4}

*P*(

_{p}*z*) throughout the waveguide, initial Stokes levels above

## 6. Conclusion

We investigated silicon chalcogenide slot waveguides for enhancing stimulated Brillouin scattering. We showed that strong optical and acoustic mode confinement in a silicon-chalcogenide slot waveguide leads to levels of SBS gain that are promising for on-chip SBS applications. To investigate this we studied how the acoustic fields, as well as the optical forces contribute to the overall SBS gain. We found that the radiation pressure contribution to the overall gain switches from destructive to constructive as the gap size is increased with a cross-over point around *c* = 130 nm. Slot waveguides with silica cover were introduced to further improve the SBS gain. Finally, we studied the impact of optical loss in the structure that is used in determining the Stokes amplification and we showed that by increasing the gap size, we achieve considerable improvement in the Stokes amplification.

There are a number of fabrication challenges involved in constructing the proposed slot waveguides. Infiltration of the slot is potentially difficult, however recent work has demonstrated complete infiltration of silicon slot waveguides with chalcogenide glass [34]. Clearly, there are further challenges such as the minimization of linear optical loss and hindering the soft glass from covering the silicon beams. However, these issues appear to be within the reach of current fabrication capabilities; an in-depth discussion of the fabrication process is however beyond the scope of this theoretical study.

## 7. Appendix: The derivation of the optimum waveguide length

In the following we derive the optimum waveguide length presented in Eq. (21) by the coupled equations of Eqs. (3) and (4). The solution of (3) takes the following form

By assuming that $\beta \ll 2\sqrt{\alpha \gamma}$, the second term in the left hand side of the above equation becomes negligible. Therefore, the equation takes the form

_{1}and k

_{2}are constants and are decided by the initial boundary conditions. The pump power varies from ${p}_{p}^{\mathrm{max}}$ at the beginning of the waveguide, to ${p}_{p}^{\mathrm{min}}$ at the end of the waveguide.

Since the above equations are roots of the Eq. (16), then we can simplify them:

By diving these two expression, we finally arrive at (21).

## Acknowledgments

The authors acknowledge support from the Australian Research Council via discovery Grant DP130100832, its Laureate Fellowship (Prof. Eggleton, FL120100029) program and the ARC Center of Excellence CUDOS (CE110001018). We also acknowledge support of the ATN Industry Doctoral Training Centre (IDTC).

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