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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 14 — Jul. 5, 2010
  • pp: 14439–14453
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Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces

Peter T. Rakich, Paul Davids, and Zheng Wang  »View Author Affiliations


Optics Express, Vol. 18, Issue 14, pp. 14439-14453 (2010)
http://dx.doi.org/10.1364/OE.18.014439


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Abstract

Radiation pressure is known to scale to large values in engineered micro and nanoscale photonic waveguide systems. In addition to radiation pressure, dielectric materials also exhibit strain-dependent refractive index changes, through which optical fields can induce electrostrictive forces. To date, little attention has been paid to the electrostrictive component of optical forces in high-index contrast waveguides. In this paper, we examine the magnitude, scaling, and spatial distribution of electrostrictive forces through analytical and numerical models, revealing that electrostrictive forces increase to large values in high index-contrast waveguides. Similar to radiation pressure, electrostrictive forces increase quadratically with the optical field. However, since electrostrictive forces are determined by the material photoelastic tensor, the sign of the electrostrictive force is highly material-dependent, resulting in cancellation with radiation pressure in some instances. Furthermore, our analysis reveals that the optical forces resulting from both radiation pressure and electrostriction can scale to remarkably high levels (i.e., greater than 104(N/m2)) for realistic guided powers. Additionally, even in simple rectangular waveguides, the magnitude and distribution of both forces can be engineered at the various boundaries of the waveguide system by choice of material system and geometry of the waveguide.This tailorability points towards novel and simple waveguide designs which enable selective excitation of elastic waves with desired symmetries through engineered stimulated Brillouin scattering processes in nanoscale waveguide systems.

© 2010 Optical Society of America

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

1. A. Mizrahi and L. Schachter, “Optical Bragg accelerators,” Phys. Rev. E 70, 016505 (2004). [CrossRef]

]. 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

1. A. Mizrahi and L. Schachter, “Optical Bragg accelerators,” Phys. Rev. E 70, 016505 (2004). [CrossRef]

]. 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.

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

22. M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett.102, 113601 (2009). [CrossRef] [PubMed]

, 23

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

]. 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

9. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1, 416–422 (2007). [CrossRef]

, 13–16

13. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008). [CrossRef] [PubMed]

, 24

24. R. Olsson III, I. El-Kady, M. Su, M. Tuck, and J. Fleming, “Microfabricated VHF acoustic crystals and waveguides,” Sens. Actuators A 145, 87–93 (2008). [CrossRef]

, 25

25. I. El-Kady, R. Olsson III, and J. Fleming, “Phononic band-gap crystals for radio frequency communications,” Appl. Phys. Lett. 92, 233504 (2008). [CrossRef]

].

Fig. 1. (a) Schematic showing dimensions of silicon waveguide. (b) (c) and (d) show computed Ex, Ey, and Ez components of TE waveguide mode. (e) and (f) show the timeaveraged Txx and Tyy stress distributions. (g) and (h) show the time averaged x and y force densities. (i) and (j) are schematics showing the dominant forces seen in the plots of rpx and rpy.

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 Ex,Ey, and Ez 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

26. W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addision-Wesley, 1962).

]

Tij=εoε(x,y)[EiEj12δijE2]+μoμ[HiHj12δijH2].
(1)

Here, Ek (Hk) is the kth electric (magnetic) field component, εo (μo) 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 Tij as rpj = iTij [26

26. W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addision-Wesley, 1962).

, 28

28. J. Jackson, Classical Electrodynamics (Wiley, 1975).

]. 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 〈Txx〉 and 〈Tyy〉 are shown in Figs. 1(e) and 1(f). Figures 1(g) and 1(h) show the computed intensity maps of 〈rpy〉 and 〈rpx〉 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

29. R. Boyd, Nonlinear Optics, 3rd Edition (Academic Press, 2009).

]. 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

27. J. Stratton, Electromagnetic theory (McGraw-Hill, 1941).

, 31

31. A. Feldman, “Relations between electrostriction and the stress-optical effect,” Phys. Rev. B 11, 5112–5114 (1975). [CrossRef]

]. 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: σkl is the local material stress; Skl is the material strain; νk is the material displacement in the kth coordinate direction; Cklmn is the elastic compliance tensor; Ek and Dk are the kth electric and displacement field components respectively; εkl is the material dielectric tensor; and pijkl is the photoelastic (or elasto-optic) tensor. The phenomenological relations between these quantities are

Di=εoεijEj
(2)
Sij=12(jvi+ivj)
(3)
Skl=Cklmn[σmnrp+σmnes+σmnmech]
(4)
εij1(Skl)=εij1+Δ(εij1)=εij1+pijklSkl.
(5)

Above, Δ(ε−1ij) is defined as the strain-induced change in the inverse dielectric tensor, ε−1ij, and σesmn, σrpmn, and σmechmn 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 Skl = Cklmnσmechmn, the familiar relationship between stress and strain in elastic materials. The canonical relation which defines the electrostrictive tensor is

Skles=12γijklEiEj.
(6)

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

σijes=12εo[εijpjkmnεkl]ElEi.
(7)

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

σkles=12εo·n4·pijkl·EiEj.
(8)

To examine the effects of electrostriction in a silicon waveguide, and to compare electrostrictive effects with those of radiation pressure, we consider the σesxx and σesyy 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 Oh point-group) yields

σxxes=12εo·n4[p11Ex2+p12(Ey2+Ez2)],
(9)
σyyes=12εo·n4[p11Ey2+p12(Ex2+Ez2)].
(10)

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

33. D. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196–1199 (1974). [CrossRef]

]. Since the photoelastic coefficients change very little from 3.39 − 1.5µms [33

33. D. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196–1199 (1974). [CrossRef]

, 34

34. L. Hounsome, R. Jones, M. Shaw, and P. Briddon, “Photoelastic constants in diamond and silicon,” Phys. Stat. Solidi C 203, 3088–3093 (2006). [CrossRef]

], 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

33. D. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196–1199 (1974). [CrossRef]

]. 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 esx = −∂jσesjx and esy = −∂jσesjy. 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 σesxx and σesyy 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 esx and esy force densities. (e) and (f) are schematics illustrating the dominant forces found in the plots of esx and esy.

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 σesxx is positive while σesyy 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 Ex 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 p11Ex2 dictates that σesxx is positive, while the term p 12Ex2 dictates that σesyy is predominantly negative.

4. Quantitative comparison of radiation pressure and electrostrictive forces

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.

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

σ¯ij=1a·bwgσij·dxdy.
(11)

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

The significance of σ^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 Sx,x to S′x,x = Sx,x + δSx,x. For nonzero, σoptij, the virtual work done against optical forces in deforming the body is given by the integral of σoptxxδSxx over the volume of the waveguide segment, or

δUEM=σxxoptδSxxdV
(12)
=σ¯xxoptδSxx(a·b·L).
(13)

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

fxopt=1L·P(δUEMδa)=σ¯xxoptbP.
(14)

Here, foptx 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

fxrp=σ¯xxrpbP,
(15)
fyrp=σ¯yyrpaP.
(16)

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

fxes=σ¯xxesbP=εo·n42·Pi·awg[|Ex|2p11+(|Ey|2+|Ez|2)p12]dxdy,
(17)
fyes=σ¯xxesaP=εo·n42·Pi·bwg[|Ey|2p11+(|Ex|2+|Ez|2)p12]dxdy.
(18)

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

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 fx and fy 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.

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 Ex 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 fesx and fesy. 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.

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 foptx and fopty 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 (frpx), the electrostrictive component (fesx), and total optical force density (foptx) 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

4. M. Povinelli, M. Loncar, M. Ibanescu, E. Smythe, S. Johnson, F. Capasso, and J Joannopoulos, “Evanescentwave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005). [CrossRef] [PubMed]

]. 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 (frpy), the electrostrictive component (fesy), and total optical force density (fopty) 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 pij 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 (frpx), the electrostrictive component (fesx), and total optical force density (foptx) 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 (frpy), the electrostrictive component (fesy), and total optical force density (fopty) respectively, acting on the vertical waveguide boundary over an identical range of dimensions.

Table 1. Photoelastic coefficients for select materials.

table-icon
View This Table
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.

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 pij (e.g., Ref. [34

34. L. Hounsome, R. Jones, M. Shaw, and P. Briddon, “Photoelastic constants in diamond and silicon,” Phys. Stat. Solidi C 203, 3088–3093 (2006). [CrossRef]

]), 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.

Appendix A: Relation between electrostrictive stress and the photoelastic coefficients.

Here, we derive Eq. (7) through a general analysis of energetics where the mechanical degrees of freedom are assumed to be strain degrees of freedom, Sij. We use the following definitions: u is the energy per unit volume, T is temperature, s is entropy per unit volume, and Bk is the magnetic field. In terms of the defined thermodynamic variables, differential expansion of u yields

du=(us)Sijds+(uSij)sdSij
(19)
=T·ds+σij·dSij
(20)

From the above, we see that σij ≡ (∂u/∂Sij)s. To examine the electrostrictively induced stress, we take u to be u=uEM=12DkEk+12HkBk and expand uEM for a variation in Sij. Through a change in locally induced material strain, we assume that Sij becomes Sij = Sij + δSij, transforming Dj into Dj, while leaving all other field components unchanged. Expressing Dj and Dj in terms of the dielectric tensor, we have

Di=εoεijEj
(21)
Di=εo(εij+δεij)Ej.
(22)

Thus, the strain-induced change in uEM becomes δuEM=12εo(δεijEj)Ei . Through first order expansion of δεil one finds δεil = −[εij pjkmnεkl]δSmn [36

36. E. Dieulesaint and D. Royer, Elastic waves in solids II: Generation, acousto-optic interaction, applications (Springer, 2000).

], yielding

δuEM=12εo[εijpjkmnεklδSmn]ElEi.
(23)

Dividing both sides of Eq. (23) by δSmn and taking the limit as δSmn → 0, we have the following expression for the time-averaged electrostrictively induced stress:

σijes=12εo[εijpjkmnεkl]ElEi.
(24)

Acknowledgments

We acknowledge the generous support and encouragement of F. B. McCormick, M. Soljačić, Y. Fink and J. D. Joannopoulos. Thanks to Miloš A. Popović for use of his mode solver code and to Charles Reinke for careful reading of the manuscript. We are grateful to C. E. Rakich and W.J. Purvis for help in preparing this manuscript. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. This work was supported in part by the office of the Director of Defense Research and Engineering under Air Force contract FA8721-05-C-0002 and by a Seedling effort managed by Dr. Mike Haney of DARPA MTO.

References and links

1.

A. Mizrahi and L. Schachter, “Optical Bragg accelerators,” Phys. Rev. E 70, 016505 (2004). [CrossRef]

2.

M. Povinelli, M. Ibanescu, S. Johnson, and J. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85, 1466–1468 (2004). [CrossRef]

3.

M. Povinelli, S. Johnson, M. Loncar, M. Ibanescu, E. Smythe, F. Capasso, and J. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators,” Opt. Express 13, 8286–8295 (2005). [CrossRef] [PubMed]

4.

M. Povinelli, M. Loncar, M. Ibanescu, E. Smythe, S. Johnson, F. Capasso, and J Joannopoulos, “Evanescentwave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005). [CrossRef] [PubMed]

5.

A. Mizrahi and L. Schachter, “Mirror manipulation by attractive and repulsive forces of guided waves,” Opt. Express 13, 9804–9811 (2005). [CrossRef] [PubMed]

6.

M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett. 97, 23903 (2006). [CrossRef]

7.

A. Mizrahi and L. Schachter, “Electromagnetic forces on the dielectric layers of the planar optical Bragg acceleration structure,” Phys. Rev. E 74, 36504 (2006). [CrossRef]

8.

A. Mizrahi and L. Schachter, “Two-slab all-optical spring,” Opt. Lett. 32, 692–694 (2007). [CrossRef] [PubMed]

9.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1, 416–422 (2007). [CrossRef]

10.

P. Rakich, M. Popovic, M. Soljacic, and E. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1, 658–669 (2007). [CrossRef]

11.

A. Mizrahi, M. Horowitz, and L. Schaechter, “Torque and longitudinal force exerted by eigenmodes on circular waveguides,” Phys. Rev. A 78, 23802 (2008). [CrossRef]

12.

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78, 165129 (2008). [CrossRef]

13.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008). [CrossRef] [PubMed]

14.

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4, 377–382 (2009). [CrossRef] [PubMed]

15.

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009). [CrossRef] [PubMed]

16.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–553 (2009). [CrossRef] [PubMed]

17.

P. Rakich, M. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17, 18116–18135 (2009). [CrossRef] [PubMed]

18.

W. Pernice, M. Li, and H. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95, 123507 (2009). [CrossRef]

19.

M. Li, W. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008). [CrossRef] [PubMed]

20.

Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009). [CrossRef] [PubMed]

21.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 36–242 (2010). [CrossRef]

22.

M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett.102, 113601 (2009). [CrossRef] [PubMed]

23.

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

24.

R. Olsson III, I. El-Kady, M. Su, M. Tuck, and J. Fleming, “Microfabricated VHF acoustic crystals and waveguides,” Sens. Actuators A 145, 87–93 (2008). [CrossRef]

25.

I. El-Kady, R. Olsson III, and J. Fleming, “Phononic band-gap crystals for radio frequency communications,” Appl. Phys. Lett. 92, 233504 (2008). [CrossRef]

26.

W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addision-Wesley, 1962).

27.

J. Stratton, Electromagnetic theory (McGraw-Hill, 1941).

28.

J. Jackson, Classical Electrodynamics (Wiley, 1975).

29.

R. Boyd, Nonlinear Optics, 3rd Edition (Academic Press, 2009).

30.

Y. Shen, The principles of nonlinear optics (Wiley-Interscience, 1984).

31.

A. Feldman, “Relations between electrostriction and the stress-optical effect,” Phys. Rev. B 11, 5112–5114 (1975). [CrossRef]

32.

E. Dieulesaint and D. Royer, Elastic waves in solids I: Free and guided wave propagation. (Springer, 2000).

33.

D. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196–1199 (1974). [CrossRef]

34.

L. Hounsome, R. Jones, M. Shaw, and P. Briddon, “Photoelastic constants in diamond and silicon,” Phys. Stat. Solidi C 203, 3088–3093 (2006). [CrossRef]

35.

Z. Levine, H. Zhong, S. Wei, D. Allan, and J. Wilkins, “Strained silicon: A dielectric-response calculation,” Phys. Rev. B 45, 4131–4140 (1992). [CrossRef]

36.

E. Dieulesaint and D. Royer, Elastic waves in solids II: Generation, acousto-optic interaction, applications (Springer, 2000).

37.

A. Feldman, R. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589 (1978). [CrossRef]

38.

M. Gottlieb, “2.3 Elasto-optic Materials,” CRC Handbook of Laser Science and Technology: Optical Materials, Part 2: Properties, 319, (1986).

39.

R. Guenther, Modern Optics (Wiley, 1990).

40.

R. Galkiewicz and J. Tauc, “Photoelastic properties of amorphous As2S3,” Solid State Commun.10, 1261–1264 (1972).

OCIS Codes
(130.2790) Integrated optics : Guided waves
(200.4880) Optics in computing : Optomechanics

ToC Category:
Physical Optics

History
Original Manuscript: April 9, 2010
Revised Manuscript: June 6, 2010
Manuscript Accepted: June 16, 2010
Published: June 22, 2010

Citation
Peter T. Rakich, Paul Davids, and Zheng Wang, "Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces," Opt. Express 18, 14439-14453 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14439


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References

  1. A. Mizrahi and L. Schachter, “Optical Bragg accelerators,” Phys. Rev. E 70, 016505 (2004). [CrossRef]
  2. M. Povinelli, M. Ibanescu, S. Johnson, and J. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85, 1466–1468 (2004). [CrossRef]
  3. M. Povinelli, S. Johnson, M. Loncar, M. Ibanescu, E. Smythe, F. Capasso, and J. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators,” Opt. Express 13, 8286–8295 (2005). [CrossRef] [PubMed]
  4. M. Povinelli, M. Loncar, M. Ibanescu, E. Smythe, S. Johnson, F. Capasso, and J. Joannopoulos, “Evanescent wave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005). [CrossRef] [PubMed]
  5. A. Mizrahi and L. Schachter, “Mirror manipulation by attractive and repulsive forces of guided waves,” Opt. Express 13, 9804–9811 (2005). [CrossRef] [PubMed]
  6. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett. 97, 23903 (2006). [CrossRef]
  7. A. Mizrahi and L. Schachter, “Electromagnetic forces on the dielectric layers of the planar optical Bragg acceleration structure,” Phys. Rev. E 74, 36504 (2006). [CrossRef]
  8. A. Mizrahi and L. Schachter, “Two-slab all-optical spring,” Opt. Lett. 32, 692–694 (2007). [CrossRef] [PubMed]
  9. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1, 416–422 (2007). [CrossRef]
  10. P. Rakich, M. Popovic, M. Soljacic, and E. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1, 658–669 (2007). [CrossRef]
  11. A. Mizrahi, M. Horowitz, and L. Schaechter, “Torque and longitudinal force exerted by eigenmodes on circular waveguides,” Phys. Rev. A 78, 23802 (2008). [CrossRef]
  12. H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78, 165129 (2008). [CrossRef]
  13. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008). [CrossRef] [PubMed]
  14. M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4, 377–382 (2009). [CrossRef] [PubMed]
  15. J. Chan, M. Eichenfield, R. Camacho, and O. Painter, ““Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009). [CrossRef] [PubMed]
  16. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic crystal optomechanical cavity,” Nature 459, 550–553 (2009). [CrossRef] [PubMed]
  17. P. Rakich, M. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17, 18116–18135 (2009). [CrossRef] [PubMed]
  18. W. Pernice, M. Li, and H. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95, 123507 (2009). [CrossRef]
  19. M. Li, W. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008). [CrossRef] [PubMed]
  20. Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009). [CrossRef] [PubMed]
  21. Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010). [CrossRef]
  22. M. Tomes, and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102, 113601 (2009). [CrossRef] [PubMed]
  23. P. Dainese, P. Russell, N. Joly, J. Knight, G. Wiederhecker, H. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2, 388–392 (2006). [CrossRef]
  24. R. OlssonIII, I. El-Kady, M. Su, M. Tuck, and J. Fleming, “Microfabricated VHF acoustic crystals and waveguides,” Sens. Actuators A 145, 87–93 (2008). [CrossRef]
  25. I. El-Kady, R. OlssonIII, and J. Fleming, “Phononic band-gap crystals for radio frequency communications,” Appl. Phys. Lett. 92, 233504 (2008). [CrossRef]
  26. W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addision-Wesley, 1962).
  27. J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  28. J. Jackson, Classical Electrodynamics (Wiley, 1975).
  29. R. Boyd, Nonlinear Optics, 3rd Edition (Academic Press, 2009).
  30. Y. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).
  31. A. Feldman, “Relations between electrostriction and the stress-optical effect,” Phys. Rev. B 11, 5112–5114 (1975). [CrossRef]
  32. E. Dieulesaint and D. Royer, Elastic waves in solids I: Free and guided wave propagation. (Springer, 2000).
  33. D. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196–1199 (1974). [CrossRef]
  34. L. Hounsome, R. Jones, M. Shaw, and P. Briddon, “Photoelastic constants in diamond and silicon,” Phys. Stat. Solidi C 203, 3088–3093 (2006). [CrossRef]
  35. Z. Levine, H. Zhong, S. Wei, D. Allan, and J. Wilkins, “Strained silicon: A dielectric-response calculation,” Phys. Rev. B 45, 4131–4140 (1992). [CrossRef]
  36. E. Dieulesaint, and D. Royer, Elastic waves in solids II: Generation, acousto-optic interaction, applications (Springer, 2000).
  37. A. Feldman, R. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589 (1978). [CrossRef]
  38. M. Gottlieb, “2.3 Elasto-optic Materials,” CRC Handbook of Laser Science and Technology: Optical Materials, Part 2: Properties, 319, (1986).
  39. R. Guenther, Modern Optics (Wiley, 1990).
  40. R. Galkiewicz and J. Tauc, “Photoelastic properties of amorphous As2S3,” Solid State Commun. 10, 1261–1264 (1972).

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