1. Introduction

Recent studies of radiation pressure have revealed that large forces can be generated in nanometer-scale waveguides and cavities as a consequence of the high confinement and tremendous field enhancements generated within such systems [1–21]. In many cases, such optical forces can significantly impact the mechanics of bodies at micro and nanoscales, yielding useful mechanical transduction and actuation at micro to milliwatt power levels [1–21]. In highconfinement silicon photonics, the technological possibilities of such systems have been explored through numerous theoretical and experimental studies of optical forces, focusing on gradient forces generated by compound modes of coupled silicon waveguides.

While radiation pressure is known to scale to large values in the context of micro and nanoscale photonic waveguide systems, little attention has been paid to electrostrictively induced forces. In what follows, we examine the impact of electrostrictive forces and their complex nature, revealing that they can more than double the optical forces in high-index contrast waveguides when taken into account. Both electrostrictive and radiation pressure induced forces scale quadratically with electromagnetic field. Furthermore, electrostrictive forces scale with the fourth power of the waveguide refractive index, making them of increasing importance in high index-contrast waveguide systems.

In this paper, we perform analytical and numerical analyses of the electrostrictive forces generated within high-index contrast optical waveguides (consisting of core materials such as silicon), revealing that the electrostriction forces coexist with radiation pressure at a comparable magnitude. More importantly, the electrostrictive force can be independently controlled by material selection and structural design. We show that the magnitude of electrostrictive forces can scale to remarkably high levels (i.e., greater than 10⁴(N/m ²)) for realistic guided powers,exceeding that of radiation pressure in some instances. Even in simple geometries, such as a rectangular waveguide, the interplay between the electrostriction and the radiation pressure can be quite complex, yielding more design degrees of freedom than in the case of radiation pressure alone. Through analysis of the Maxwell stress tensor and the derived electrostrictive stress tensor, we compute the spatial distributions of the various forces within the silicon waveguide.We show that the induced electrostrictive stress corresponds to nontrivial spatial force distributions within the waveguide whose net effect can be to constructively add to or destructively interfere with radiation pressure. Through analysis of the material- and geometry-dependent design degrees of freedom associated with the electrostrictive as well as the radiation pressure induced forces, we show that the magnitude and distribution of such forces can be tailored along the principle axes of the waveguide system. Through a parametric study of rectangular single-mode silicon waveguides–of all aspect ratios–we map out the optimal waveguide dimensions to maximize total optically induced forces in the waveguide system for the purpose of optomechanical transduction of elastic waves. Furthermore, through design of the spatial force distributions, novel and simple waveguide designs could be employed to enable selective excitation of elastic waves with particular symmetries through stimulated Brillouin scattering processes.

Deeper understanding of the optical forces generated within nanoscale waveguide systems is important to the field of optomechanics, as it could provide a means of engineering novel stimulated Brillouin scattering processes in nanoscale waveguide systems. Examples of stimulated Brillouin processes recently reported in micro-scale silica systems can be found in Refs. [22, 23]. Through improved understanding of the optical forces generated within nanoscale waveguide systems, stimulated Brillouin processes similar to these could be engineered in highly integrated nanoscale waveguide systems. By way of efficient stimulated Brillouin scattering processes, conversion of optical signals to and from the phononic domain over broad bandwidths could be possible, enabling a host of powerful new hybrid photonic-phononic technologies for the purposes of delay and filtering of both radio frequency (RF) and optical signals[9, 13–16, 24, 25].

 

Fig. 1. (a) Schematic showing dimensions of silicon waveguide. (b) (c) and (d) show computed E_x, E_y, and E_z components of TE waveguide mode. (e) and (f) show the timeaveraged T_xx and T_yy stress distributions. (g) and (h) show the time averaged ℱ_x and ℱ_y force densities. (i) and (j) are schematics showing the dominant forces seen in the plots of ℱ^rp_x and ℱ^rp_y.

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2. Anatomy of radiation pressure in suspended silicon waveguides

Since the electrostrictive forces are dictated by the electrical field distribution, and exhibit a similar power dependence as radiation pressure, we start with a brief overview of the mode profile in a suspended silicon waveguide and the forces associated with radiation pressure for comparison. We choose to focus on a suspended rectangular waveguide structure without a substrate, where the optical forces are entirely determined by the core material and its geometry. In the absence of electrostrictive effects, optical forces generated by the fundamental mode of a silicon waveguide occur at the discontinuous dielectric boundaries of the waveguide system and can be attributed to radiation pressure. Example force distributions generated by the TE-like mode within a rectangular silicon waveguide can be seen in Fig. 1. Figures 1(a)–1(d) show the waveguide cross-section and computed E_x,E_y, and E_z electric field distributions corresponding to a silicon waveguide of width, a = 400nm, height, b = 250nm, and optical wavelength, λ =1550nm. From these mode field distributions, the optically induced force and stress distribution within the dielectric waveguide can be computed using the proper form of the Maxwell stress tensor in dielectric media, which has been shown to be [26–28]

Here, E_k (H_k) is the k^th electric (magnetic) field component, ε_o (μ_o) is the electric permittivity (magnetic permeability) of free space, and ε(x,y) (μ) is the relative electric permittivity (magnetic permeability). The body force (force per unit volume acting on the body) generated from radiation pressure and gradient forces are computed from T_ij as ℱ^rp_j = ∂_iT_ij [26, 28]. Since mechanical systems cannot respond to forces at time scales corresponding to an optical cycle, one generally seeks the time averaged Maxwell stress tensor, and body force, which we denote with 〈…〉.

The computed spatial distributions of 〈T_xx〉 and 〈T_yy〉 are shown in Figs. 1(e) and 1(f). Figures 1(g) and 1(h) show the computed intensity maps of 〈ℱ^rp_y〉 and 〈ℱ^rp_x〉 components of the power normalized body force (force per unit volume) respectively, revealing that the dominant optical forces within the system (i.e., radiation pressure) act on the boundaries of the waveguide system. For clarity, the sign and orientation of the dominant forces are diagrammatically illustrated in Figs. 1 (i) and 1(j). Note, while MST computations of the type seen in Fig. 1 are very useful, and widely applied, they do not take into account the optical forces generated through electrostriction.

3. Anatomy of electrostrictive forces within rectangular silicon waveguides

With the knowledge of the electrical field distribution and the radiation pressure, we proceed to calculate the electrostrictive force exerted by the optical field. In the context of nonlinear optics, electrostrictively induced refractive index changes are responsible for large third order nonlinearities (or Kerr nonlinearities), giving rise to self focusing and stimulated Brillouin scattering [29–31]. Similar to forces generated by radiation pressure, electrostrictive forces are proportional to optical power, scaling in an identical manner as the optical power is increased. Electrostriction, and the associated electrostrictive forces, produce material contraction (or expansion)which is quadratic with electromagnetic field, and results primarily from the strain dependence of the dielectric constant [27, 31]. Electrostriction, not to be confused with piezoelectricity, occurs independent of material symmetry, and increases with the fourth power of material refractive index (making its effects larger for high refractive index materials such as silicon).

Through our discussion of optical forces, we use the following definitions: σ_kl is the local material stress; S_kl is the material strain; ν_k is the material displacement in the k^th coordinate direction; C_klmn is the elastic compliance tensor; E_k and D_k are the k^th electric and displacement field components respectively; ε_kl is the material dielectric tensor; and p_ijkl is the photoelastic (or elasto-optic) tensor. The phenomenological relations between these quantities are

Above, Δ(ε⁻¹_ij) is defined as the strain-induced change in the inverse dielectric tensor, ε⁻¹_ij, and σ^es_mn, σ^rp_mn, and σ^mech_mn represent the electrostrictive, radiation pressure, and mechanically induced stresses of the system respectively. Thus, in the limit of vanishing optical fields, the optically induced stress components vanish, reducing Eq. (4) to S_kl = C_klmnσ^mech_mn, the familiar relationship between stress and strain in elastic materials. The canonical relation which defines the electrostrictive tensor is

Here, E_iE_j are the amplitudes of the rapidly oscillating optical fields. Since materials generally cannot respond mechanically at time scales of the carrier frequency, we also express electrostriction in terms of the time-averaged quantities. Above, 〈E_iE_j〉 is the time-averaged field product, which is related to the induced time-averaged electrostrictive strain, 〈S^es_kl〉 through the electrostrictive tensor, γ_ijkl. Following the energetics analyses of Ref. [31], one can show that the electrostrictive tensor is expressible in terms of the material photoelastic coefficients, in the limit when Kerr effects can be neglected (a reasonable approximation here).

Through the analysis presented here, we seek to understand the forces corresponding to electrostrictive effects. Therefore it is useful to instead relate the optical fields to the electrostrictively induced stress as

Derivation of this expression can be found in Appendix A. For simplicity of notation, we will no longer use 〈…〉 to denote time-averaged quantities. From this point forth, it is assumed that all stresses (σ_ij), force related quantities, and field products (E_iE_j), are averaged over an optical cycle. Note, while forces due to boundary effects (such as radiation pressure and gradient forces) are often included in the formulation of the electrostrictive stress, we need not include them in this formulation, as they are accounted for through evaluation of the Maxwell stress tensor. Just as with the Maxwell stress tensor, the electrostrictively induced body force acting on the waveguide can be found from the divergence of the stress distribution. However, it is important to note that the body stress defined through elastic theory, σ_ij, adopts a different sign convention in defining the stress tensor than is typical with the Maxwell stress tensor,T_ij. In elastic theory, the body force (or force density per unit volume exerted on the body) is given by ℱ_j = −∂_iσ_ij [32]. Therefore, consistency between the two formalisms is maintained if σ^rp_ij is defined as σ^rp_ij = −T_ij. Thus, the total electromagnetically induced stress on the waveguide is given by σ^opt_ij = σ^rp_ij + σ^es_ij = −T_ij + σ^es_ij, and the total body force is given by ℱ^opt_j = −∂_iσ^opt_ij = ∂_iT_ij − ∂_iσ^es_ij.

For isotropic materials (e.g., amorphous glasses) and for cubic crystals (e.g., silicon, germanium), the dielectric tensor, ε_ij, reduces to ε_ij = ε·δ_ij = n ²·δ_ij. In this case, Eq. (7) becomes

To examine the effects of electrostriction in a silicon waveguide, and to compare electrostrictive effects with those of radiation pressure, we consider the σ^es_xx and σ^es_yy tensor components where the x-direction coincides with the [100] crystal symmetry direction. Evaluation of Eq. (8) using the photoelastic tensor of silicon (having cubic crystal symmetry, corresponding to the O_h point-group) yields

Here, p ₁₁ and p ₁₂ are the material photoelastic coefficients, expressed in contracted notation where 11 → 1, 22 → 2, 33 → 3, 23,32 → 4, 13,31 → 5, and 12,21 → 6. The photoelastic coefficients of silicon have been measured to be p ₁₁ = −0.09 and p ₁₂ = +0.017 at 3.39µm wavelengths [33]. Since the photoelastic coefficients change very little from 3.39 − 1.5µms [33, 34], these values provide a reasonable approximation of the photoelastic properties of silicon in the vicinity of 1.5µm wavelengths. For more details on photoelastic properties of silicon, see Refs [33–36]. Using Eq. (9) and (10), the electrostrictively induced stress distribution in the silicon waveguide under consideration can be computed from the field distributions in Fig. 1. Figs. 2(a) and 2(b) show the spatial distribution of the electrostrictively generated stress, while Figs. 2(c) and 2(d) show electrostrictively induced force densities ℱ^es_x = −∂_jσ^es_jx and ℱ^es_y = −∂_jσ^es_jy. For clarity, the dominant electrostrictive forces are diagrammatically illustrated in Figs. 2(e) and 2(f).

 

Fig. 2. (a) and (b) show intensity colormaps of the time averaged σ^es_xx and σ^es_yy component of the stress distribution (units N/m ₂/mW) induced through electrostriction. The boundary of the waveguides is outlined with a dotted rectangle. (c) and (d) show the time averaged ℱ^es_x and ℱ^es_y force densities. (e) and (f) are schematics illustrating the dominant forces found in the plots of ℱ^es_x and ℱ^es_y.

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Note that the electrostrictive stress distribution is not uniform, but is instead more localized within the center of the waveguide, as it more closely follows the energy density of the optical mode. It is also noteworthy that σ^es_xx is positive while σ^es_yy is negative for all values in space. These stress densities correspond to electrostrictive force-densities (or body-forces) seen in Figs. 2(c) and 2(d). The force densities reveal that optical forces localized near the waveguide boundaries act to push the vertical boundaries apart while simultaneously acting to pull the horizontal boundaries inward. In in Figs. 2(e) and 2(f) the dominant forces resulting from electrostriction are illustrated. This sign difference between the x− and y−directed body forces can be understood by observing that the E_x field component of the TE-like mode is dominant in Eqs. (9) and (10), and p ₁₁ and p ₁₂ are of opposite sign. Therefore, the term containing p₁₁∣E_x∣² dictates that σ^es_xx is positive, while the term p ₁₂∣E_x∣² dictates that σ^es_yy is predominantly negative.

4. Quantitative comparison of radiation pressure and electrostrictive forces

In the previous section, we examined the origin, spatial distribution, and scaling of electrostrictive forces through analytical and computational means. With the understanding that, electrostrictive forces are determined by the material photoelastic tensor, it becomes clear that the sign of the electrostrictive force is highly material-dependent resulting in electrostrictive and radiation pressure-induced forces can be of opposing sign. (The material dependence of electrostrictive forces will be discussed in Section 6). Next, we examine the magnitude of electrostrictive forces in comparison to radiation pressure, revealing that in some instances cancellation of electrostrictive and radiation pressure-induced forces can occur. Through analysis of waveguides of various dimensions, we will show that even in simple rectangular waveguides, the optical forces resulting from both radiation pressure and electrostriction can scale to remarkably high levels (i.e., greater than 10⁴(N/m ²)) for realistic guided powers.

 

Fig. 3. (a) Rectangular waveguide segment of length, L, width, a, and height, b. (b) The right schematic shows the same waveguide after laterally strained by an and amount δa to a new dimension of a′ = a+δa. The waveguide height (b) and length (L) are held fixed.

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Figures 1 and 2 show that both radiation pressure and electrostriction result in nontrivial spatial force distributions within a silicon waveguide. However, their net effect in deforming the waveguide cross-section can be more simply examined through definition of an equivalent surface force. Using the quantities defined in Sections 2 and 3, we seek to compare the magnitude of electrostrictive forces with those of radiation pressure through an effective surface force.We show that a useful first-order estimate of the aggregate forces which act to deform the waveguide (resulting from either electrostriction or radiation pressure) can be obtained from the spatial averaged stress, which we define as

Here, integration is taken over the waveguide cross-section sketched in Fig. 3.

The significance of $\widehat{\sigma }$_ij can be understood through a simple virtual work formulation of the aggregate forces acting to deform the body. We consider a virtual displacement of the waveguide boundaries, δa, and the change in total energy (or virtual work), δU, associated with this displacement. Through a virtual displacement of the type illustrated in Fig. 3, the waveguide height (b) and length (L) are held fixed, while the waveguide width (a) is varied. In other words, uniaxial tensile strain is applied to the waveguide along the x-axis, transforming a to a′ = a + δa and S_x,x to S′_x,x = S_x,x + δS_x,x. For nonzero, σ^opt_ij, the virtual work done against optical forces in deforming the body is given by the integral of σ^opt_xxδS_xx over the volume of the waveguide segment, or

Expressing δS_xx as δS_xx = (δa/a), the principle of virtual work can be used to define the effective aggregate force density acting to deform the waveguide as

Here, f^opt_x represents the power normalized force per unit length acting along the outward normal on the lateral waveguide boundary [as illustrated by the dotted arrows in Fig. 3(b)], which we refer to as the effective linear force density.

From Eq. (14), the effective linear force density corresponding to radiation pressure induced forces acting on the vertical and horizontal boundaries of the waveguide can be expressed as

The analogous linear force density produced by electrostrictive effects is expressible as

Note, the primary difference between Eqs. (17) and (18) is that the E_x field-component (which carries the most of mode energy) interacts through photoelastic coefficient p ₁₁ instead of p ₁₂.

The effective linear force densities, defined in Eqs. (15)–(18), are useful in comparing the net effect of both types of forces in deforming the waveguide, since one can compare the force densities from electrostriction and radiation pressure through a similarly defined quantity. It is important to note that these force quantities provides only an approximate measure of the forces acting to deform the waveguide when the waveguide is treated as a lumped element system. This enables us to transform a nontrivial electrostrictive body force into a uniform surface force with approximately equivalent effect in deforming the waveguide. In this section and in Section 5 we will discuss the electrostrictive forces in terms of this effective surface force, in order to gain a better understanding of the magnitude of electrostrictive forces in comparison to radiation pressure induced forces. A complete model for the treatment of the waveguide deformation resulting from optical forces would require the use of the exact expressions for the electrostrictive stress and Maxwell stress presented in the previous sections.

Using the above relations, a full-vectorial mode solver can now be used to evaluate Eqs. (17) and (18) in a straightforward manner, yielding an approximate comparison between the components of the optical forces resulting from both electrostriction and radiation pressure. We begin by examining the f_x and f_y optical force densities for silicon waveguides of conventional dimensions. As can be seen in Fig. 4, both components of the optical force (units of pN/µm/mW) were computed as a function of waveguide width, a. The electrostrictive linear force density (dashes), radiation pressure induced linear force density (dots), and total linear force (red) are shown in Fig. 4 for a fixed waveguide height of b = 315nm, and waveguide widths, a, ranging between 100nm and 500nm.

From Fig. 4(a), we see that electrostrictive forces (dashes) and radiation pressure (dots) add constructively, effectively resulting in a larger total outward optical force (red) on the lateral boundaries of the waveguide. Upon further examination of Fig. 4(a), we see that total optical force takes on a maximum value for a waveguide widths of ~ 280nm. The observed peak in radiation pressure is a consequence of the modal expansion at these waveguide dimensions. Remarkably, the effective total force per unit area exerted on the lateral boundary of the waveguide for an incident power of 100mW approaches 1.6 × 10⁴(N/m ²) at waveguide dimensions of a = 280nm and b = 315nm.

Similarly, the forces exerted on the vertical boundaries of the waveguide can be evaluated through use of Eq. (18), yielding plotted curves seen in Fig. 4(b). From these data, we see that the electrostrictive force (dashed) is opposite in sign to the radiation pressure (dot), causing the total force on the vertical boundary change from attractive to repulsive at a waveguide width of a = 240nm. This change in sign of the electrostrictive force can be understood by examining Eq. (18), and noting that the majority of the electric field-energy in the guided mode resides in the E_x field-component. Given this, one can see from Eq. (18) that the sign difference between p ₁₂ and p ₁₁ is responsible for the sign difference between f^es_x and f^es_y. As a consequence, the electrostrictive component of the force acts to pull the vertical waveguide boundaries inward instead of pushing them outward (as they do for the lateral waveguide boundaries).

 

Fig. 4. (a) Plot of the linear optical force density (pN/µm/mW) produced by the TE-like waveguide mode on the lateral boundary of a rectangular waveguide as function of waveguide dimension. The total linear force density (red), and components due to electrostriction (dashes) and radiation pressure (dots) are plotted as a function of waveguide width (a) for a fixed waveguide height of b = 315nm. (b) Plot of the linear optical force density produced by the same mode on the vertical boundary of a rectangular waveguide, as waveguide dimension is varied in an identical manner.

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Finally, it is noteworthy that for waveguide widths of a = 290nm, the optical forces acting on the horizontal boundary vanishes, while those on the vertical boundaries remain quite large. This points to the possibility that, through proper choice of waveguide dimensions, combined forces from electrostrictive and radiation pressure forces could be optimized to achieve the selective excitation of either elastic waves of differing symmetries.

5. Optical force versus waveguide aspect ratio: .

From the computational example of Section 4, it is clear that the magnitude of optical total force exerted on the waveguide boundaries is highly sensitive to waveguide dimensions. In some instances the electrostrictive forces were found to cancel those of radiation pressure, while in others, they add constructively. However, for the purposes of transduction in novel optomechanical systems it might be interesting to determine the waveguide dimensions which provide a maximum of optical force. Alternately, one might be interested in arriving at a waveguide geometry which is optimized for the transduction of longitudinal waves, and generates virtually no shear motion. In this section, we use the results of Section 4 to perform a computational study of the total optical force exerted to waveguide boundaries by the fundamental TE-like waveguide mode over a large range of waveguide aspect ratios. Through the same methods described in Section 4, we compute the various contributions to both f^opt_x and f^opt_y for waveguides of width, a, and height, b, ranging between 100nm and 500nm. The computed components of the linear force density exerted on the lateral and vertical boundaries as a function of waveguide geometry are show as intensity maps in Fig. 5.

Figures 5(a), 5(b), and 5(c) are intensity maps showing the radiation pressure component of linear force density (f^rp_x), the electrostrictive component (f^es_x), and total optical force density (f^opt_x) respectively, acting in the lateral waveguide boundary, for waveguides width, a, and height, b, ranging between 100nm and 500nm. Figure 5(a) reveals that the radiation pressure exerted by the TE-like waveguide mode on the lateral boundary reaches a maximum for waveguides which are narrow and tall (e.g., an aspect ratio of ~ 250 : 500nm). In this case, the peak value of the radiation pressure contribution to the force-density approaches 48 pN/µm/mW, which is comparable with the largest forces predicted through evanescent-wave bonding in compound waveguide systems [4]. Computation of electrostrictive forces over the same range of dimensions, as shown in Fig. 5(b), reveals that the force exerted on the lateral boundary are positive in sign for all aspect ratios, and monatonically increasing in magnitude for waveguides of increasing width. The result is in an overall increase in the total linear optical force density, as seen in Fig. 5(c) with a maximum linear optical force density on the lateral boundary of 58 pN/µm/mW.

Figures 5(d), 5(e), and 5(f) are intensity maps showing radiation pressure component of linear force density (f^rp_y), the electrostrictive component (f^es_y), and total optical force density (f^opt_y) respectively, acting on the horizontal waveguide boundary over an identical range of dimensions. From Fig. 5(d) one finds that the radiation pressure exerted by the TE-like waveguide mode on the vertical boundary reaches a maximum for waveguides which are wide and short (e.g., an aspect ratio of ~ 500 : 100nm). In this case, the peak value of the radiation pressure contribution to the force-density approaches 38 pN/µm/mW. However, Fig. 5(e) reveals that the electrostrictive forces exerted on the vertical boundary are negative in sign for all aspect ratios, and are monotonically increasing for waveguides of increasing width. Since the electrostrictive forces add destructively with the radiation pressure-induced forces, the total linear optical force density is zero, and dips to negative values, as seen in Fig. 5(f).

6. Material dependence of electrostrictive forces

Thus far, we have computed the optically induced forces generated within silicon waveguides (aligned with the [001] crystal orientation) by the TE-like optical mode. However, the sign and magnitude of the electrostrictive component of the optical force can vary a great deal depending on: (1) the photoelastic tensor (i.e., crystal symmetry, and magnitude p_ij elements) for the material of study, (2) the crystal orientation of the waveguide material, and (3) the field distribution of the optical mode under consideration.

 

Fig. 5. Plots showing components of the linear optical force density (pN/µm/mW) produced by the TE-like waveguide mode on the lateral and vertical boundaries of a rectangular waveguide as function of waveguide dimension. (a), (b), and (c) are intensity maps showing the radiation pressure component of force density (f^rp_x), the electrostrictive component (f^es_x), and total optical force density (f^opt_x) respectively, acting in the lateral waveguide boundary, for waveguides width, a, and height, b, ranging between 100nm and 500nm. (d), (e), and (f) are intensity maps showing the radiation pressure component of force density (f^rp_y), the electrostrictive component (f^es_y), and total optical force density (f^opt_y) respectively, acting on the vertical waveguide boundary over an identical range of dimensions.

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Table 1. Photoelastic coefficients for select materials.

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Fig. 6. (a) Schematic of TE-like guided mode under consideration. (b), (c) and (d) show rough schematics showing the orientation of electrostrictive forces generated by the TE-like mode in GaAs (Ge), Si, and As₂S₃ (Silica) respectively, through examination of their Photoelastic coefficients.

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Through analysis of the silicon waveguide in Section 4, the electrostrictive forces on the horizontal and vertical boundaries were shown to have opposite sign due to differing signs of photoelastic coefficients p ₁₁ and p ₁₂. The electrostrictive forces exerted by the TE-like mode on the lateral boundary were shown to push outward, adding to the effects of radiation pressure, and that on the horizontal boundary pushes inward, acting to cancel the effects of radiation pressure. A sketch illustrating the orientation of the electrostrictive forces in the silicon waveguide can be seen in Fig. 6(c). Interestingly, one finds that the electrostrictive force distribution changes significantly with crystal orientation. For instance, a rotational transformation of crystal orientation from [001] to [111] changes the elements of p_ij (e.g., Ref. [34]), drastically modifying the resulting electrostrictive forces on both boundaries of the waveguide.

As can be seen in Table 1, the magnitude and sign of the photoelastic coefficients of materials can also vary a great deal. For instance, both As₂S₃ and silica have positive p ₁₁ and p ₁₂ which will lead to inward electrostrictively induced body-forces in both the x− and y−directions [as illustrated by Fig. 6(d)], which acts counter to the radiation pressure-induced forces of the TE-mode. Interestingly, however, GaAs and Ge possesses p ₁₁ and p ₁₂ coefficients which are negative in sign. As a consequence, we can expect the outward electrostrictively induced bodyforces in both the x− and y−directions, adding to the effects of radiation pressure [as illustrated by Fig. 6(b)].

Due to the tensor nature of the electrostriction coefficients, the electrostrictive forces can behave quite differently in various materials systems even if they have similar refractive indices. The material dependence of the electrostrictive force component could therefore offer a very useful new means of controlling and tailoring force profiles in optomechanically active waveguide systems. As we have already discussed in Section 4, the cancellation of electrostrictive forces on the horizontal boundary of the waveguide may enable the selective excitation of elastic waves of particular symmetries. Further tailoring of the electrostrictive forces could be made through use of waveguides with single or multiple cladding materials, as the different materials interacting with the optical field through the cladding materials will have nontrivial contribution to the total optical force generated through electrostriction.

We note that, through this study, we have chosen to examine the electrostrictive and radiation pressure induced forces in a silicon waveguide since: (1) silicon is the most widely used material in high confinements photonic circuits, and (2) its photoelastic properties are very well known. However, based on the photoelastic properties listed in Table 1, one can easily see that Ge, GaAs, and As₂S₃ are far more favorable for the generation of large electrostrictive forces, since all of these materials possess much larger photoelastic coefficients, and both p ₁₁ and p ₁₂ are of the same sign for all of these materials. Thus, in contrast to silicon, both terms in Eq. (9) and (10) will constructively add to produce forces which are several times larger than that of silicon. For example, in the case of a Ge waveguide, the electrostrictive forces would scale to values which are ~ 10 × larger than those of silicon, meaning that electrostriction would become the dominant force in this system.

Finally, while photoelastic properties of Si and silica are very well known due to the extensive use of these materials in optics, it should be noted that the photoelastic properties of most other materials are very poorly understood in comparison. Thus, if electrostrictive forces are to be optimally exploited for the generation of large forces and large optomechanical couplings in novel optomechanical systems, further study of the photoelastic properties of high refractive index media is warranted.

7. Conclusions

In developing a more complete model of the optical forces in dielectric waveguides, we have demonstrated that one must consider the effects of both radiation pressure and electrostriction within high index-contrast waveguide systems. Both electrostrictive and radiation pressure induced forces scale quadratically with electromagnetic field. However, since electrostrictive forces scale with the fourth power of the material refractive index, they are of increasing importance in high index-contrast waveguide systems. Through analysis of the Maxwell stress tensor and the electrostrictive stress tensor, we have computed the complex spatial distributions of the various forces within the silicon waveguide.We have shown that the induced electrostrictive stress corresponds to a nontrivial spatial force distributions within the waveguide whose net effect can be to constructively add to or destructively interfere with radiation pressure. Analysis of the optical forces generated in rectangular silicon waveguides revealed that the optical forces resulting from both radiation pressure and electrostriction are highly tailorable, and can scale to remarkably high levels (i.e., greater than 10⁴(N/m ²)) for realistic guided powers. Through analysis of the material- and geometry-dependent design degrees of freedom associated with the electrostrictive as well as the radiation pressure induced forces, we have shown that the magnitude and distribution of such forces can be tailored along the principle axes of the waveguide system, leading to novel and simple waveguide designs which enable selective excitation of elastic waves with particular symmetries through engineered stimulated Brillouin scattering processes in nanoscale waveguide systems. Through a parametric study of rectangular single-mode silicon waveguides, of all aspect ratios, we have mapped out the optimal waveguide dimensions for maximization of the total optically induced forces in a silicon waveguide system.