1. Introduction

Light and sound multiphysics interactions can be efficiently explored in dielectric devices by using stimulated Brillouin scattering (SBS) process [1–3]. In the last decades, the suppression and enhancement of Brillouin scattering have been extensively studied in optical fibers [1]. In conventional optical fibers, SBS effect is mainly generated by electrostrictive forces induced by light – electrostriction is a bulk effect that occurs in dielectric materials, related to the deformation of their original geometric shapes due to the application of an electric field [1]. Electrostriction effect is energetically linked to its reciprocal process called the photoelastic effect, which describes changes in the material optical properties due to a mechanical deformation [4–9]. However, besides the electrostrictive force, another kind of optical force becomes relevant to the SBS effect when the waveguides’ dimensions are reduced to the nanoscale – the radiation-pressure induced force [4–10]. Radiation pressure exerts forces on the waveguide boundaries due to the intrinsic physical nature of an optical mode, by means of multiple reflections (momentum exchange) on the internal dielectric interfaces [9]. This surface effect is also energetically linked to its reciprocal process called moving-boundary effect, which describes the change in the waveguide cross section due to the action of a transverse force [4–11]. In particular, Rakich et al. have shown that, in nanowaveguides, radiation pressure can achieve very high values and, therefore, become the dominant optical force in SBS effect, overcoming electrostriction effect [4,5,10]. Waveguides’ nanoscale dimensions and high-index contrast are responsible for the high confinement of light, resulting in a dramatic enhancement on the SBS gain, much far beyond of what is obtained solely through their intrinsic materials nonlinearities, opening new opportunities for acousto-optics interactions in nanophotonics [5,12]. Such theoretical predictions have been closely followed by experimental demonstrations of SBS effects on different nanophotonic and nano-optomechanical waveguides [13–20]. Furthermore, novel SBS-based on-chip signal-processing technologies have been introduced, including recent experimental demonstrations of receiver/emitters [21], memories [22], and lasers [23–25], among many others [26].

Here, we develop a new, simple, and general method to obtain the waveguide’s dimensional parameters (e.g., height and/or width) that maximizes the radiation pressure on its surface, for any order mode, polarization, and wavelength. We derive analytical expressions capable of computing this specific dimension using only one line of code, saving a tremendous amount of computational effort and time, commonly demanded during the optimization design, by employing either FDTD (Finite-Difference Time-Domain) or FEM (Finite-Element Method) numerical tools. We show that, for a given combination of materials’ refractive indexes (cladding and core) and wavelength, there is solely one specific waveguide set of dimensional parameters that corresponds to a unique (global) maximum point of radiation pressure, which changes accordingly with the optical mode order and polarization. As examples of our findings, we apply our method to two very distinct waveguide geometries: a rib silicon waveguide suspended in air and a strip chalcogenide waveguide buried in silicon dioxide, and exact results are obtained for both cases. Finally, we prove that the existence of a maximum point of radiation pressure on the waveguide surfaces is generated by the minimization of longitudinal momentum flow, which besides of explaining the accuracy of our method, also physically explain the origin of such optimal point – this approach, to the best of our knowledge, has not been presented before.

2. Radiation pressure in rectangular and planar dielectric waveguides

Consider the dielectric structures schematically shown in Fig. 1 (a)-(c). These structures are formed by a high-index material waveguide (${n_H}$) and surrounded by a low-index material medium (${n_L}$). Figure 1(a) shows a 3D view of the rectangular cross-section waveguide, with width w and height h, as illustrated in the panel (b), which is invariant along the propagation direction ($z$-direction) and delimited by the length L.

 

Fig. 1. Schematic views of the dielectric waveguides and diagram of forces. (a) 3D view of the dielectric waveguide, with a rectangular cross-section and composed by a high-index material (${n_H}$) core surrounded by a low-index material (${n_L}$). (b) cross-section (2D) view of the rectangular waveguide with width w and height h. (c) planar (1D) approximation by considering the structure’s invariance also in the x-direction. (d) and (e) show the schematics of radiation-pressure induced optical forces on the surfaces of the rectangular (${F_x},{F_y}$) and the planar (${{{\cal F}}_y}$) waveguides.

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Our analytical models are developed by considering the symmetric planar (slab) waveguide shown in Fig. 1(c), which is translational invariant in both the transverse $x$-direction and in the propagation $z$-direction. We assume that the planar waveguide is excited by a harmonic source with a fixed wavelength ${\lambda _0}$ and a vacuum wavevector of magnitude ${k_0}$, where ${k_0} = 2\pi /{\lambda _0}$. Maxwell's equations, under these assumptions, lead to two independent sets of solutions for the field polarization: Transverse Electric (TE), where the electric field is polarized along the $x$-axis (${E_x},{H_y},{H_z}$) or Transverse Magnetic (TM), where the magnetic field is polarized along the $x$-axis (${E_y},{E_z},{H_x}$). Due to the structure symmetry, with respect to the $x$-axis, the electromagnetic field distributions for each mode can be classified into symmetric/even or antisymmetric/odd. The well-known transcendental (characteristic) equations for the TE and TM symmetric eigenmodes is given, respectively, by

On the other hand, the spatial-averaged radiation pressure in a dielectric waveguide with a rectangular cross-section, as shown in Fig. 1(b), is given by [10]

From a mathematical point of view, due to its transcendental nature presented in (1) and (2), the modes’ effective indexes have no closed-form solution. However, by solving it numerically (or graphically), it is possible to notice that it has a sigmoid function behavior, which monotonically increases from one asymptote, bounded below by ${n_L}$, to the other, limited to ${n_H}$, as height goes from zero to infinity; this behavior is shown in Fig. 2(a) and (b), for the fundamental TE and TM modes (TE₀ and TM₀) on a silicon planar waveguide (${n_H} = 3.4764$) in air (${n_L} = 1.0003$), at a wavelength of ${\lambda _0} = 1550$ nm. This kind of curve has a unique non-stationary point of inflection, that occurs at a specific value of height, and its first derivative has a bell shape format with a maximum point that corresponds to the maximum value of optical force, accordingly to (4) or (5). By performing the derivative on the transcendental equations ((1) and (2)) with respect to the planar waveguide’s height, h, we obtain

 

Fig. 2. Optical forces distributions on the silicon waveguides in air and highlighted optimal height. (a) and (b) Optical forces on the horizontal boundaries (${{{\cal F}}_y}$) and their respective effective indexes for a silicon planar waveguide in air, for the fundamental TE and TM modes. (c) and (d) Intensity maps of y-component of optical forces (${F_y}$) on the horizontal boundaries of the silicon rectangular waveguide in air, as a function of its cross-section dimension, for the quasi-TE and the quasi-TM fundamental modes at ${\lambda _0} = 1550$ nm. The optimized heights are ${h^{TE}} = 52.04$ nm and ${h^{TM}} = 225.8$ nm

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3. Results and discussion

Figure 2(a) and (b) show the optical forces per unit length (${{{\cal F}}_y}$) applied over the horizontal interfaces of a silicon planar waveguide, as a function of its height, as well as their highlighted optimal heights (where the maximum forces occur), for both the fundamental TE and TM modes at 1550 nm for ${{\cal P}} = 1\; \textrm{mW}/\mu \textrm{m}$ . The optimized height for the TE polarization, ${h^{TE}} = 52.04$ nm, is lower than the TM one, ${h^{TM}} = 225.8$ nm, which is explained mainly due to the positive extra term in the numerator of (9) compared to that of (8), as well as to the fact that, for a given height, the effective indexes of the TM modes are lower than those for their TE counterparts. Furthermore, at these optimized force values, the TM polarization generates a stronger maximum optical force (${{{\cal F}}_y}({{h^{TE}}} )= 45.7$ pN/µm²/mW) than the TE one (${{{\cal F}}_y}({{h^{TM}}} )= 44.2$ pN/µm²/mW), due to the large discontinuity of the electric field component (${E_y}$) at the planar interface in the former polarization. However, it is interesting to notice that the TE polarization still generates a reasonably strong force, even though it has neither discontinuity in the electric field component (${E_x}$), nor it is orientated on the same direction of the optical force.

On the other hand, Fig. 2(c)-(d) show the intensity maps of the y-component of the optical forces (${F_y}$) applied over the horizontal boundaries of a silicon rectangular waveguide in air (Fig. 1(d)), as a function of its cross-section dimension, for both the quasi-TE (qTE) and the quasi-TM (qTM) fundamental modes at a wavelength of 1550 nm. These maps clearly indicate a strong convergence of the maximum optical forces to the same values of ${h^{TE}}$ and ${h^{TM}}$, also represented in Fig. 2(c)-(d). Although such tendencies could be expected in an extreme limit case, it is very surprising that they converge so quickly, taking into account the giant difference between their widths, where in the planar waveguide $w \to \infty $, whereas in the rectangular one, $w \cong 1$ µm. Another point we have noticed is that such a fast convergence has been achieved for an extreme high-index contrast case, as it is the case for silicon in air, where the vector nature of the electromagnetic fields plays an important role; therefore, even better results are expected in the cases of lower index contrast. Additionally, the intensity maps also show that few tens nanometers off their optimized heights may represent a difference of almost 50-fold in the optical force magnitude, which can be tailored either to enhance or to suppress the Brillouin gain.

To demonstrate the general applicability of the Eqs. (8) and (9) in optimizing the radiation pressure in realistic waveguides, we choose two very recent examples found in the literature. The first structure is composed by a silicon rib waveguide suspended in air, and the second one is a chalcogenide glass (As₂S₃) strip waveguide (${n_H} = 2.4440$) buried in a silicon dioxide (SiO₂) (${n_L} = 1.4440)$, as illustrated schematically in Fig. 3(a) and (b), respectively. Both structures have successful used to experimentally demonstrate the Brillouin effect and its applications on integrated waveguide platforms [13,16,18,19,22,23], including the first demonstration of a silicon guided-wave phonon laser based on this effect [24]. Figure 3(c) and (d) present the total optical forces (${F_x} + {F_y}$) for the silicon and the chalcogenide waveguides, respectively, for both polarizations at ${\lambda _0} = 1550$ nm. The results show that the optimized heights computed by (8) and (9) agree exactly with the points of maximum forces, which proves that in waveguides with a high-aspect ratio of the type $w > h$, the y-component is the major responsible for the total optical force. Moreover, Fig. 3(c) shows that for silicon waveguide in air with a height near ${h^{TM}} = 225.8$ nm, the total optical force is approximately equal to 60 pN/µm/mW. This value corresponds to a total spatial-averaged radiation pressure (${\bar{p}_x}$+${\bar{p}_y}$) of 20.1 × 10³ N/m²/W in the fundamental qTM mode, whereas it is about 5.4 × 10³ N/m²/W (optical force of 15 pN/µm/mW) in the quasi-TE counterpart, which represents an enhancement of almost four times in optical pressure (force), just by changing the polarization. Similarly, Fig. 3(d) shows that by reducing the chalcogenide core height, from 850 nm to ${h^{TE}} = 119.1$ nm, increases the total radiation pressure from 294.9 to 3.6 × 10³ N/m²/W, keeping the same optical mode and polarization (qTE₀₀). Despite of the fact that Brillouin effects in nanowaveguides have richer multiphysics, which involves the material photoelastic constants and mechanical modes dynamics, such results might represent significant improvements on the device’s performance. This is especially useful, if we take into account the fact that the Brillouin gain has a quadratic dependence on the optical force.

 

Fig. 3. Schematic of waveguides and their respective total optical forces for both polarizations at a wavelength of 1550 nm. (a) Silicon rib waveguide (${n_H} = 3.4764$) suspended in air (${n_L} = 1.0003$). (b) Chalcogenide strip waveguide (${n_H} = 2.405$) buried in silicon dioxide cladding (${n_L} = 1.4440$). (c) and (d) total optical forces on the silicon and the chalcogenide waveguides, respectively. The optimized heights for the silicon waveguide are ${h^{TE}} = 52.04$ nm (not shown) and ${h^{TM}} = 225.8$ nm and for the chalcogenide are ${h^{TE}} = 119.1$ nm and ${h^{TM}} = 308.7$ nm.

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The method developed here is based on a horizontal-oriented planar waveguide (x-direction invariance); therefore, the optimized height maximizes the y-component of the optical force (${F_y}$). The numerical simulations show that for dielectric waveguides of the type $w > h$, when $w \ge 1$ µm, the derived expressions give the exact waveguide’s height that maximizes the total optical force for any optical mode, polarization, and wavelength, covering most of the cases described in the literature. However, due to the waveguide symmetry and the orthogonality of the optical modes, orienting the planar waveguide in the vertical direction (y-direction invariance), our expressions may also be used to obtain its optimized width that now maximizes the x-component of the optical force (${F_x}$). In the same way, it also gives exact results for waveguides of the type $w < h$, given that $h \ge 1$ µm. In fact, if we make the following changes: $w \to h$, $h \to w$, $F_y^{qTE} \to F_x^{qTM}$, $F_y^{qTM} \to F_x^{qTE}$ and ${h^{TE,TM}} \to {w^{TM,TE}}$, in the Fig. 2(c) and (d), they remain perfectly valid. We test the robustness of our findings by considering the fundamental qTE mode of a silicon rectangular waveguide with a fixed height of $h = 315$ nm surrounded by air at a wavelength of 1550 nm [10]. The specific width where the radiation pressure is maximum in the rectangular waveguide is $w = 273$ nm, whereas the optimized one is ${w^{TM}} = {h^{TE}} = 225.8$ nm; This small difference, in the tens-of-nanometers range, shows that our method still provides a quite good approximation, taking into account that, in this case, it is approximately a square waveguide ($w\sim h$).

4. Fundamental relation between the radiation pressure and the momentum flow in the propagation direction

A fundamental question remains: why do these very specific dimensions maximize the optical force? In order to explain such behavior, we analyze the generalized version of (3) [10]

5. Conclusions

In conclusion, we have used a planar dielectric waveguide and radiation pressure’s theories to derive analytical expressions, capable of computing the exact waveguide’s height (or width) that corresponds to the maximum value of the radiation-pressure induced optical force on the waveguide’s boundaries, for any optical mode, polarization, and wavelength. We show that the origin of this point of maximum value of optical force (radiation pressure) is physically linked to the minimization of the momentum flow in the propagation direction. We believe that our findings are powerful and simple tools that can be directly applied to the optimization of Brillouin-based integrated waveguides and, therefore, this may have some impact on the designs of high-performance photonic-phononic technologies.