Abstract

We use an analytical method based on the perturbation of effective index at fixed frequency to calculate optical forces between silicon waveguides. We use the method to investigate the mechanical Kerr effect in a coupled-waveguide system with bipolar forces. We find that a positive mechanical Kerr coefficient results from either an attractive or repulsive force. An enhanced mechanical Kerr coefficient several orders of magnitude larger than the intrinsic Kerr coefficient is obtained in waveguides for which the optical mode approaches the air light line, given appropriate design of the waveguide dimensions.

© 2011 OSA

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References

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  1. M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).
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2010 (2)

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).

J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97(15), 151102 (2010).

2009 (8)

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).

J. Ma and M. L. Povinelli, “Large tuning of birefringence in two strip silicon waveguides via optomechanical motion,” Opt. Express 17(20), 17818–17828 (2009).
[PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[PubMed]

P. T. Rakich, M. A. Popović, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17(20), 18116–18135 (2009).
[PubMed]

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95(12), 123507 (2009).

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[PubMed]

2008 (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(16), 165129 (2008).

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(7221), 480–484 (2008).
[PubMed]

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).

2007 (2)

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1(7), 416–422 (2007).

2005 (4)

2004 (1)

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).

2003 (1)

2001 (1)

1985 (1)

Antezza, M.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).

Baehr-Jones, T.

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(7221), 480–484 (2008).
[PubMed]

Baets, R.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[PubMed]

Capasso, F.

Carusotto, I.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).

Chan, C. T.

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[PubMed]

De Vlaminck, I.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[PubMed]

Eichenfield, M.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1(7), 416–422 (2007).

Fan, S.

Gondarenko, A.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[PubMed]

Hochberg, M.

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(7221), 480–484 (2008).
[PubMed]

Ibanescu, M.

Ippen, E. P.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).

Joannopoulos, J. D.

Johnson, S. G.

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(16), 165129 (2008).

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(16), 165129 (2008).

Lagae, L.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[PubMed]

Li, M.

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95(12), 123507 (2009).

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(7221), 480–484 (2008).
[PubMed]

Lin, Q.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).

Lin, Z.

Lipson, M.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[PubMed]

Loncar, M.

Lonèar, M.

Luo, C.

Ma, J.

J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97(15), 151102 (2010).

J. Ma and M. L. Povinelli, “Large tuning of birefringence in two strip silicon waveguides via optomechanical motion,” Opt. Express 17(20), 17818–17828 (2009).
[PubMed]

Maes, B.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[PubMed]

McCullen, J. D.

Meystre, P.

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(7), 416–422 (2007).

Mizrahi, A.

Ng, J.

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(16), 165129 (2008).

Painter, O.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1(7), 416–422 (2007).

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(7), 416–422 (2007).

Pernice, W. H. P.

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).

W. H. P. Pernice, M. Li, and H. X. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95(12), 123507 (2009).

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[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(7221), 480–484 (2008).
[PubMed]

Popovic, M. A.

P. T. Rakich, M. A. Popović, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17(20), 18116–18135 (2009).
[PubMed]

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).

Povinelli, M. L.

Rakich, P. T.

P. T. Rakich, M. A. Popović, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17(20), 18116–18135 (2009).
[PubMed]

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).

Recati, A.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).

Riboli, F.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).

Roels, J.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[PubMed]

Rosenberg, J.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).

Schächter, L.

Sheng, P.

Smythe, E.

Smythe, E. J.

Soljacic, M.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).

M. Soljacić, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28(8), 637–639 (2003).
[PubMed]

Tang, H. X.

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95(12), 123507 (2009).

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).

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(7221), 480–484 (2008).
[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(16), 165129 (2008).

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(16), 165129 (2008).

Van Thourhout, D.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[PubMed]

Vignes, E.

Wang, Z.

Wiederhecker, G. S.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[PubMed]

Wright, E. M.

Xiong, C.

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(7221), 480–484 (2008).
[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(16), 165129 (2008).

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(16), 165129 (2008).

Appl. Phys. Lett. (3)

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).

J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97(15), 151102 (2010).

W. H. P. Pernice, M. Li, and H. X. Tang, “A mechanical Kerr effect in deformable photonic media,” Appl. Phys. Lett. 95(12), 123507 (2009).

Eur. Phys. J. D (1)

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).

J. Opt. Soc. Am. B (1)

Nat. Nanotechnol. (1)

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[PubMed]

Nat. Photonics (5)

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1(7), 416–422 (2007).

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).

Nature (2)

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[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(7221), 480–484 (2008).
[PubMed]

Opt. Express (6)

Opt. Lett. (3)

Phys. Rev. B (1)

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(16), 165129 (2008).

Other (2)

S. Timoshenko, Theory of Elasticity (McGraw-Hill, 1934).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

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Figures (5)

Fig. 1
Fig. 1

(a) Two coupled silicon waveguides, each with cross section w × h, separated by a distance d and resting on a SiO2 substrate with suspended length L. (b) Dispersion relation for modes of the suspended section. The two lowest modes with electric field vectors primarily in the z-direction are plotted for separations d = 100 nm and d = 1000 nm. The cross section of each waveguide is 350 × 350 nm2. The gray region shows the light cone for air.

Fig. 2
Fig. 2

Schematic diagram showing a shift in the dispersion relation due to a change in separation between waveguides d. The solid and dashed lines represent the initial and shifted dispersion relation, respectively.

Fig. 3
Fig. 3

For fixed wavelength 1550 nm and waveguide dimensions h = w = 350 nm: (a) Symmetric mode: n eff as a function of d (black curve); F/(LP) calculated by the MST method (red curve) and by Eq. (8) (red triangles). (b) Antisymmetric mode: n eff as a function of d (black curve); F/(LP) calculated by the MST method (red curve) and by Eq. (8) (red triangles).

Fig. 4
Fig. 4

Symmetric mode: (a) Band structure. The gray region represents the air light cone. The red line shows fixed wavelength 1550 nm (frequency 193.55 THz). (b) Attractive force per length per unit power F/(LP). The initial separation d = 100 nm, suspended length L = 20 μm, optical power P = 10 mW, and the signal wavelength is 1550 nm. (c) Mechanical Kerr coefficient n 2 m .

Fig. 5
Fig. 5

Anti-symmetric mode: (a) Band structure. The gray region represents the air light cone. The red line shows fixed wavelength 1550 nm (frequency 193.55 THz). (b) Repulsive force per length per unit power F/(LP). The initial separation d = 100 nm, suspended length L = 20 μm, optical power P = 10 mW, and the signal wavelength is 1550 nm. (c) Mechanical Kerr coefficient n 2 m .

Equations (13)

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

F L P = 1 c n g ω ω d | k ,
F L P = 1 c n g n eff n eff d | k .
δ n eff | k = c k 0 ( 1 ω 1 1 ω 0 ) = c k 0 δ ω ( ω 0 + δ ω ) ω 0 ,
δ n eff | ω = c ω 0 ( k 2 k 0 ) = c ω 0 δ k ,
δ n eff | k δ n eff | ω = δ ω δ k k 0 ( ω 0 + δ ω ) δ ω δ k k 0 ω 0 .
δ n eff | k δ n eff | ω n eff n g .
F L P 1 c n g n eff δ n e ff δ d | k .
F L P = 1 c n eff d | ω ,
Δ n eff = Δ d × n eff d | ω ,
Δ d = 2 × ( F L 3 / 32 E h w 3 ) ,
Δ n eff = 1 c ( n eff d | ω ) 2 P L 4 16 E h w 3 ,
Δ n eff = c ( F L P ) 2 P L 4 16 E h w 3 .
n 2 m = Δ n eff I = c ( F L P ) 2 L 4 16 E h w 3 A .

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