## Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces |

Optics Express, Vol. 18, Issue 14, pp. 14439-14453 (2010)

http://dx.doi.org/10.1364/OE.18.014439

Acrobat PDF (2330 KB)

### 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 10^{4}(*N*/*m*^{2})) 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

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

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

^{4}(

*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.

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]

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. 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. 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]

## 2. Anatomy of radiation pressure in suspended silicon waveguides

*E*,

_{x}*E*, and

_{y}*E*electric field distributions corresponding to a silicon waveguide of width,

_{z}*a*= 400

*nm*, height,

*b*= 250

*nm*, and optical wavelength,

*λ*=1550

*nm*. 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]

*E*(

_{k}*H*) is the

_{k}*k*electric (magnetic) field component,

^{th}*ε*(

_{o}*μ*) is the electric permittivity (magnetic permeability) of free space, and

_{o}*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*as

_{ij}*ℱ*=

^{rp}_{j}*∂*[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 〈…〉.

_{i}T_{ij}*T*〉 and 〈

_{xx}*T*〉 are shown in Figs. 1(e) and 1(f). Figures 1(g) and 1(h) show the computed intensity maps of 〈

_{yy}*ℱ*〉 and 〈

^{rp}_{y}*ℱ*〉 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.

^{rp}_{x}## 3. Anatomy of electrostrictive forces within rectangular silicon waveguides

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

*σ*is the local material stress;

_{kl}*S*is the material strain;

_{kl}*ν*is the material displacement in the

_{k}*k*coordinate direction;

^{th}*C*is the elastic compliance tensor;

_{klmn}*E*and

_{k}*D*are the

_{k}*k*electric and displacement field components respectively;

^{th}*ε*is the material dielectric tensor; and

_{kl}*p*is the photoelastic (or elasto-optic) tensor. The phenomenological relations between these quantities are

_{ijkl}*ε*) is defined as the strain-induced change in the inverse dielectric tensor,

^{−1}_{ij}*ε*, and

^{−1}_{ij}*σ*,

^{es}_{mn}*σ*, and

^{rp}_{mn}*σ*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

^{mech}_{mn}*S*=

_{kl}*C*, the familiar relationship between stress and strain in elastic materials. The canonical relation which defines the electrostrictive tensor is

_{klmn}σ^{mech}_{mn}*ε*, reduces to

_{ij}*ε*=

_{ij}*ε·δ*=

_{ij}*n*

^{2}·

*δ*. In this case, Eq. (7) becomes

_{ij}*σ*and

^{es}_{xx}*σ*tensor components where the

^{es}_{yy}*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*point-group) yields

_{h}*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 3.39

*µm*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]

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

*µm*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]

*ℱ*=

^{es}_{x}*−∂*and

_{j}σ^{es}_{jx}*ℱ*=

^{es}_{y}*−∂*. For clarity, the dominant electrostrictive forces are diagrammatically illustrated in Figs. 2(e) and 2(f).

_{j}σ^{es}_{jy}*σ*is positive while

^{es}_{xx}*σ*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

^{es}_{yy}*x*− and

*y*−directed body forces can be understood by observing that the

*E*field component of the TE-like mode is dominant in Eqs. (9) and (10), and

_{x}*p*

_{11}and

*p*

_{12}are of opposite sign. Therefore, the term containing

*p*∣

_{11}*E*∣

_{x}^{2}dictates that

*σ*is positive, while the term

^{es}_{xx}*p*

_{12}∣

*E*∣

_{x}^{2}dictates that

*σ*is predominantly negative.

^{es}_{yy}## 4. Quantitative comparison of radiation pressure and electrostrictive forces

^{4}(

*N*/

*m*

^{2})) for realistic guided powers.

*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,*σ ^

_{ij}*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*to

_{x,x}*S′*=

_{x,x}*S*+

_{x,x}*δS*. For nonzero,

_{x,x}*σ*, the virtual work done against optical forces in deforming the body is given by the integral of

^{opt}_{ij}*σ*over the volume of the waveguide segment, or

^{opt}_{xx}δS_{xx}*δS*as

_{xx}*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

*f*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.

^{opt}_{x}*E*field-component (which carries the most of mode energy) interacts through photoelastic coefficient

_{x}*p*

_{11}instead of

*p*

_{12}.

*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.

*f*and

_{x}*f*optical force densities for silicon waveguides of conventional dimensions. As can be seen in Fig. 4, both components of the optical force (units of

_{y}*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*= 315

*nm*, and waveguide widths,

*a*, ranging between 100

*nm*and 500

*nm*.

*nm*. 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 100

*mW*approaches 1.6 × 10

^{4}(

*N*/

*m*

^{2}) at waveguide dimensions of

*a*= 280

*nm*and

*b*= 315

*nm*.

*a*= 240

*nm*. 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*field-component. Given this, one can see from Eq. (18) that the sign difference between

_{x}*p*

_{12}and

*p*

_{11}is responsible for the sign difference between

*f*and

^{es}_{x}*f*. 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).

^{es}_{y}*a*= 290

*nm*, 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: .

*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*and

^{opt}_{x}*f*for waveguides of width,

^{opt}_{y}*a*, and height,

*b*, ranging between 100

*nm*and 500

*nm*. 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.

*f*), the electrostrictive component (

^{rp}_{x}*f*), and total optical force density (

^{es}_{x}*f*) respectively, acting in the lateral waveguide boundary, for waveguides width,

^{opt}_{x}*a*, and height,

*b*, ranging between 100

*nm*and 500

*nm*. 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]

*pN*/

*µm*/

*mW*.

*f*), the electrostrictive component (

^{rp}_{y}*f*), and total optical force density (

^{es}_{y}*f*) 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

^{opt}_{y}*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

*p*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.

_{ij}*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*(e.g., Ref. [34

_{ij}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]

_{2}S

_{3}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)].

_{2}S

_{3}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.

## 7. Conclusions

^{4}(

*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.

*S*. We use the following definitions:

_{ij}*u*is the energy per unit volume,

*T*is temperature,

*s*is entropy per unit volume, and

*B*is the magnetic field. In terms of the defined thermodynamic variables, differential expansion of

_{k}*u*yields

*σ*≡ (

_{ij}*∂u*/

*∂S*)

_{ij}_{s}. To examine the electrostrictively induced stress, we take

*u*to be

*u*for a variation in

_{EM}*S*. Through a change in locally induced material strain, we assume that

_{ij}*S*becomes

_{ij}*S*′

_{ij}=

*S*+

_{ij}*δS*, transforming

_{ij}*D*into

_{j}*D*′

_{j}, while leaving all other field components unchanged. Expressing

*D*and

_{j}*D*′

_{j}in terms of the dielectric tensor, we have

*u*becomes

_{EM}*δε*one finds

_{il}*δε*= −[

_{il}*ε*]

_{ij}p_{jkmn}ε_{kl}*δS*[36], yielding

_{mn}*δS*and taking the limit as

_{mn}*δS*→ 0, we have the following expression for the time-averaged electrostrictively induced stress:

_{mn}## Acknowledgments

## References and links

1. | A. Mizrahi and L. Schachter, “Optical Bragg accelerators,” Phys. Rev. E |

2. | M. Povinelli, M. Ibanescu, S. Johnson, and J. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. |

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 |

4. | M. Povinelli, M. Loncar, M. Ibanescu, E. Smythe, S. Johnson, F. Capasso, and J Joannopoulos, “Evanescentwave bonding between optical waveguides,” Opt. Lett. |

5. | A. Mizrahi and L. Schachter, “Mirror manipulation by attractive and repulsive forces of guided waves,” Opt. Express |

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. |

7. | A. Mizrahi and L. Schachter, “Electromagnetic forces on the dielectric layers of the planar optical Bragg acceleration structure,” Phys. Rev. E |

8. | A. Mizrahi and L. Schachter, “Two-slab all-optical spring,” Opt. Lett. |

9. | M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics |

10. | P. Rakich, M. Popovic, M. Soljacic, and E. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics |

11. | A. Mizrahi, M. Horowitz, and L. Schaechter, “Torque and longitudinal force exerted by eigenmodes on circular waveguides,” Phys. Rev. A |

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 |

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 |

14. | M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. |

15. | J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express |

16. | M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature |

17. | P. Rakich, M. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express |

18. | W. Pernice, M. Li, and H. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. |

19. | M. Li, W. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature |

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. |

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 |

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. |

24. | R. Olsson III, I. El-Kady, M. Su, M. Tuck, and J. Fleming, “Microfabricated VHF acoustic crystals and waveguides,” Sens. Actuators A |

25. | I. El-Kady, R. Olsson III, and J. Fleming, “Phononic band-gap crystals for radio frequency communications,” Appl. Phys. Lett. |

26. | W. Panofsky and M. Phillips, |

27. | J. Stratton, |

28. | J. Jackson, |

29. | R. Boyd, |

30. | Y. Shen, |

31. | A. Feldman, “Relations between electrostriction and the stress-optical effect,” Phys. Rev. B |

32. | E. Dieulesaint and D. Royer, |

33. | D. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. |

34. | L. Hounsome, R. Jones, M. Shaw, and P. Briddon, “Photoelastic constants in diamond and silicon,” Phys. Stat. Solidi C |

35. | Z. Levine, H. Zhong, S. Wei, D. Allan, and J. Wilkins, “Strained silicon: A dielectric-response calculation,” Phys. Rev. B |

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. |

38. | M. Gottlieb, “2.3 Elasto-optic Materials,” |

39. | R. Guenther, |

40. | R. Galkiewicz and J. Tauc, “Photoelastic properties of amorphous As2S3,” |

**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

Sort: Year | Journal | Reset

### References

- A. Mizrahi and L. Schachter, “Optical Bragg accelerators,” Phys. Rev. E 70, 016505 (2004). [CrossRef]
- 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]
- 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]
- 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]
- A. Mizrahi and L. Schachter, “Mirror manipulation by attractive and repulsive forces of guided waves,” Opt. Express 13, 9804–9811 (2005). [CrossRef] [PubMed]
- 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]
- 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]
- A. Mizrahi and L. Schachter, “Two-slab all-optical spring,” Opt. Lett. 32, 692–694 (2007). [CrossRef] [PubMed]
- 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]
- 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]
- A. Mizrahi, M. Horowitz, and L. Schaechter, “Torque and longitudinal force exerted by eigenmodes on circular waveguides,” Phys. Rev. A 78, 23802 (2008). [CrossRef]
- 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]
- 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]
- M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4, 377–382 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- W. Pernice, M. Li, and H. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95, 123507 (2009). [CrossRef]
- 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]
- 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]
- 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]
- M. Tomes, and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102, 113601 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- I. El-Kady, R. OlssonIII, and J. Fleming, “Phononic band-gap crystals for radio frequency communications,” Appl. Phys. Lett. 92, 233504 (2008). [CrossRef]
- W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addision-Wesley, 1962).
- J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
- J. Jackson, Classical Electrodynamics (Wiley, 1975).
- R. Boyd, Nonlinear Optics, 3rd Edition (Academic Press, 2009).
- Y. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).
- A. Feldman, “Relations between electrostriction and the stress-optical effect,” Phys. Rev. B 11, 5112–5114 (1975). [CrossRef]
- E. Dieulesaint and D. Royer, Elastic waves in solids I: Free and guided wave propagation. (Springer, 2000).
- D. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196–1199 (1974). [CrossRef]
- L. Hounsome, R. Jones, M. Shaw, and P. Briddon, “Photoelastic constants in diamond and silicon,” Phys. Stat. Solidi C 203, 3088–3093 (2006). [CrossRef]
- 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]
- E. Dieulesaint, and D. Royer, Elastic waves in solids II: Generation, acousto-optic interaction, applications (Springer, 2000).
- A. Feldman, R. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589 (1978). [CrossRef]
- M. Gottlieb, “2.3 Elasto-optic Materials,” CRC Handbook of Laser Science and Technology: Optical Materials, Part 2: Properties, 319, (1986).
- R. Guenther, Modern Optics (Wiley, 1990).
- R. Galkiewicz and J. Tauc, “Photoelastic properties of amorphous As2S3,” Solid State Commun. 10, 1261–1264 (1972).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.