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/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 OSA

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  36. E. Dieulesaint and D. Royer, Elastic waves in solids II: Generation, acousto-optic interaction, applications (Springer, 2000).
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2010 (1)

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]

2009 (6)

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]

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]

2008 (6)

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]

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

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]

2007 (3)

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]

2006 (4)

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]

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]

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

2005 (3)

2004 (2)

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]

1992 (1)

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]

1978 (1)

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

1975 (1)

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

1974 (1)

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

Allan, D.

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]

Baehr-Jones, T.

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]

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]

Biegelsen, D.

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

Boyd, R.

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

Briddon, P.

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

Camacho, R.

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]

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]

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]

Capasso, F.

Carmon, T.

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

Chan, J.

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]

Chang, D.

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]

Dainese, P.

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]

Dieulesaint, E.

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

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

Eichenfield, M.

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]

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]

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, 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]

El-Kady, I.

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

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]

Feldman, A.

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

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

Fleming, J.

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

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]

Fragnito, H.

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]

Galkiewicz, R.

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

Gottlieb, M.

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

Guenther, R.

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

Hochberg, M.

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]

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]

Horowitz, D.

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

Horowitz, M.

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

Hounsome, L.

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

Ibanescu, M.

Ippen, E.

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]

Jackson, J.

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

Jiang, X.

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]

Joannopoulos, J

Joannopoulos, J.

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. 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]

Johnson, S.

Joly, N.

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]

Jones, R.

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

Khelif, A.

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]

Knight, J.

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]

Kuga, T.

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]

Kuramochi, E.

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. 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]

Laude, V.

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]

Levine, Z.

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]

Li, M.

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

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. 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]

Lin, Q.

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]

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]

Loncar, M.

Michael, C. P.

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]

Mitsugi, S.

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]

Mizrahi, A.

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

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

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

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

Notomi, M.

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. 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]

Olsson III, R.

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]

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

Painter, O.

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]

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]

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]

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, 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]

Panofsky, W.

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

Perahia, R.

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]

Pernice, W.

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]

Pernice, W. H. P.

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

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]

Phillips, M.

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

Popovic, M.

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]

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]

Povinelli, M.

Rakich, P.

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]

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]

Rosenberg, J.

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]

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]

Royer, D.

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

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

Russell, P.

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]

Schachter, L.

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

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

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

Schaechter, L.

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

Shaw, M.

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

Shen, Y.

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

Smythe, E.

Soljacic, M.

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]

Stratton, J.

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

Su, M.

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]

Tang, H.

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]

Tang, H. X.

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

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]

Taniyama, H.

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. 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]

Tauc, J.

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

Tomes, M.

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

Torii, Y.

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]

Tuck, M.

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]

Vahala, K.

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]

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]

Vahala, K. J.

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]

Wang, Z.

Waxler, R.

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

Wei, S.

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]

Wiederhecker, G.

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]

Wilkins, J.

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]

Xiong, C.

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]

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]

Yamamoto, T.

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]

Yoshikawa, Y.

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]

Zhong, H.

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]

Appl. Phys. Lett. (3)

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]

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

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

J. Appl. Phys. (1)

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

Nat. Nanotechnol. (1)

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

Nat. Photonics (3)

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]

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]

Nat. Phys. (1)

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]

Nature (3)

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]

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]

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]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (1)

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

Phys. Rev. B (3)

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]

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

Phys. Rev. E (2)

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

Phys. Rev. Lett. (3)

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]

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

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]

Phys. Stat. Solidi C (1)

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

Sens. Actuators A (1)

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]

Other (11)

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

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

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

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).

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).

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Figures (6)

Fig. 1.
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 rp x and rp y .

Fig. 2.
Fig. 2.

(a) and (b) show intensity colormaps of the time averaged σes xx and σes 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 esx and es y force densities. (e) and (f) are schematics illustrating the dominant forces found in the plots of es x and es y .

Fig. 3.
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.

Fig. 4.
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.

Fig. 5.
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 (frp x ), the electrostrictive component (fes x ), and total optical force density (fopt 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 (frp y ), the electrostrictive component (fes y ), and total optical force density (fopt y ) respectively, acting on the vertical waveguide boundary over an identical range of dimensions.

Fig. 6.
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.

Tables (1)

Tables Icon

Table 1. Photoelastic coefficients for select materials.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

T ij = ε o ε ( x , y ) [ E i E j 1 2 δ ij E 2 ] + μ o μ [ H i H j 1 2 δ ij H 2 ] .
D i = ε o ε ij E j
S ij = 1 2 ( j v i + i v j )
S kl = C klmn [ σ mn rp + σ mn es + σ mn mech ]
ε ij 1 ( S kl ) = ε ij 1 + Δ ( ε ij 1 ) = ε ij 1 + p ijkl S kl .
S kl es = 1 2 γ ijkl E i E j .
σ ij es = 1 2 ε o [ ε ij p jkmn ε kl ] E l E i .
σ kl es = 1 2 ε o · n 4 · p ijkl · E i E j .
σ xx es = 1 2 ε o · n 4 [ p 11 E x 2 + p 12 ( E y 2 + E z 2 ) ] ,
σ yy es = 1 2 ε o · n 4 [ p 11 E y 2 + p 12 ( E x 2 + E z 2 ) ] .
σ ¯ ij = 1 a · b wg σ ij · dxdy .
δ U EM = σ xx opt δ S xx dV
= σ ¯ xx opt δ S xx ( a · b · L ) .
f x opt = 1 L · P ( δ U EM δ a ) = σ ¯ xx opt b P .
f x rp = σ ¯ xx rp b P ,
f y rp = σ ¯ yy rp a P .
f x es = σ ¯ xx es b P = ε o · n 4 2 · P i · a wg [|Ex | 2 p 11 +(| E y | 2 +|Ez | 2 ) p 12 ] dxdy ,
f y es = σ ¯ xx es a P = ε o · n 4 2 · P i · b wg [|Ey | 2 p 11 +(| E x | 2 +|Ez | 2 ) p 12 ] dxdy .
du = ( u s ) S ij ds + ( u S ij ) s dS ij
= T · ds + σ ij · dS ij
D i = ε o ε ij E j
D i = ε o ( ε ij + δ ε ij ) E j .
δ u EM = 1 2 ε o [ ε ij p jkmn ε kl δ S mn ] E l E i .
σ ij es = 1 2 ε o [ ε ij p jkmn ε kl ] E l E i .

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