## Mechanical Kerr nonlinearities due to bipolar optical forces between deformable silicon waveguides |

Optics Express, Vol. 19, Issue 11, pp. 10102-10110 (2011)

http://dx.doi.org/10.1364/OE.19.010102

Acrobat PDF (1185 KB)

### 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

## 1. Introduction

14. 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]

17. 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]

## 2. Coupled waveguides exhibiting bipolar optical forces

*n*= 3.45) are separated by a distance

*d*. Both ends of the waveguides rest on a SiO

_{2}(refractive index

*n*= 1.5) substrate. The suspended section has length

*L*. Each waveguide has a cross section of dimensions

*w*×

*h*. From previous work [2

2. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. **30**(22), 3042–3044 (2005). [PubMed]

2. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. **30**(22), 3042–3044 (2005). [PubMed]

*z*-direction. Calculations were performed using the MIT Photonic Bands (MPB) package [21

21. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**(3), 173–190 (2001). [PubMed]

*d*. For the lowest mode, the electric field component

*E*is symmetric in the two waveguides. The symmetric mode corresponds to an attractive force. Decreasing separation shifts the dispersion relation to a larger

_{z}*k*value, in the direction shown by the blue, solid arrow. For the second lowest mode,

*E*is antisymmetric and the force is repulsive. Increasing separation also shifts the dispersion relation to larger

_{z}*k*value, in the direction shown by the blue, dashed arrow. Below, we relate the shift in dispersion relation at fixed frequency to the optical force.

*k*to

_{o}*k*If we consider a constant wave vector

_{2}.*k*, the dispersion relation shifts from

_{o}*ω*to

_{0}*ω*.

_{1}## 3. Calculation of optical force

2. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. **30**(22), 3042–3044 (2005). [PubMed]

*F/*(

*LP*) represents the force per length per unit power,

*n*is the group index, and

_{g}*c*is the speed of light in vacuum. Note that the derivative is taken at constant

*k*. Equation (1) has been shown to agree well with direct calculations via the Maxwell Stress Tensor (MST) [22] for the coupled waveguide system [2

**30**(22), 3042–3044 (2005). [PubMed]

*ω = ck/n*

_{eff}into Eq. (1), the force may alternately be expressed as a function of the effective index

*n*

_{eff}[23

23. 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]

*k*, a small change in separation

_{0}*δd*moves the eigenmode from (

*k*

_{0},

*ω*

_{0}) to (

*k*

_{0},

*ω*

_{1}), corresponding to a shift of

*n*

_{eff}:where

*δω = ω*

_{1}

*-ω*

_{0}. For fixed frequency

*ω*

_{0}, the shift of the dispersion relation changes the wave vector from

*k*

_{0}to

*k*

_{2}. The change of

*n*

_{eff}for fixed

*ω*is:where

*δk = k*

_{2}

*- k*

_{0}. By comparing Eq. (3) and (4), we find:

*δω*/

*δk*can be estimated by

*–c/n*. (The negative sign results from the fact that

_{g}*δω*=

*ω*

_{1}-

*ω*

_{0}is defined to be negative in Fig. 2.) Eq. (5) can then be rewritten as:

*n*

_{eff}(

*d*) at a fixed optical frequency. Negative values here correspond to attractive forces. Ref [13

13. 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]

*h*=

*w*= 350 nm. The derivative in Eq. (8) was approximated as a finite difference, using values of

*n*

_{eff}(

*d*) calculated with the MPB package. Figure 3(a) shows

*n*

_{eff}(

*d*) for the symmetric mode (black line). The force calculated using Eq. (8) is plotted in Fig. 3(a) (red triangles). It agrees well with the force calculated using the MST method (red curve). The MST was evaluated using the full vectorial eigenmodes obtained from MPB. As predicted by Eq. (8), the force is attractive (negative), corresponding to a decreasing

*n*

_{eff}(

*d*).

*n*

_{eff}(black curve) and

*F*/(

*LP*) for the antisymmetric mode. Forces calculated by Eq. (8) (red triangles) agree well with values calculated using the MST method (red curve). Note that the value of

*n*

_{eff}first decreases and then increases as a function of

*d*. Correspondingly, the derivative

*n*

_{eff}.

## 4. Calculation of Mechanical Kerr coefficient

*P*, and causes the waveguides to deform. For the symmetric mode, the force is attractive and the waveguides are pulled together. For the antisymmetric mode, the force is repulsive and the waveguides are pushed apart. The maximum deformation will be at the center of the waveguide, since the ends of the waveguides are fixed to the silica substrate. We consider the case where the deformation is small compared to the initial separation. The magnitude of the deformation is also much smaller than the length of the suspended region,

*L.*Under deformation, the effective index

*n*

_{eff}varies along the length of the waveguides, according to the local separation. Following Ref [10], we define the mechanical Kerr coefficient

*I*is the light intensity. Specifically, we take

*I = P/A*, where

*A*is the mode area, equal to

*n*

_{eff}decreases with

*d*. From Eq. (8), the force is negative (attractive) and pulls the waveguides closer together. The separation at the center of the suspended length decreases, resulting in an increase in

*n*

_{eff}. The mechanical Kerr coefficient

*n*is thus positive. For the antisymmetric mode, in the region where

_{2}^{m}*n*

_{eff}decreases with

*d*, the same argument shows that

*n*is positive. In the region where

_{2}^{m}*n*

_{eff}increases with

*d*, the force is positive (repulsive), which increases the separation at the center of the suspended region and increases

*n*

_{eff}. Again, a positive mechanical Kerr coefficient

*n*is obtained.

_{2}^{m}*d*is the change in separation, equal to twice the deformation of a single waveguide, which can be modeled as a double-clamped beam [24]:where

*E*is the Young's modulus of silicon. By manipulating Eq. (8), (9) and (10), we getorThe mechanical Kerr coefficient isNote that

*F*/(

*LP*) depends on the initial separation

*d*and the waveguide cross section parameters,

*h*and

*w*.

*h*,

*w,*and

*L*.

*n*depends on (

_{2}^{m}*F*/(

*LP*))

^{2}.

### 4.1. Symmetric mode

*h*and

*w*to enhance

*F*/(

*LP*) and

*λ*= 1550 nm,

*L*= 20 µm, initial separation

*d*= 100 nm, and

*P*= 10 mW.

*h*= 260 nm and

*w*= 350 nm, marked by the circle in Fig. 4(b),

*F/*(

*LP*) has a relatively large, negative value of −4.1 × 10

^{−3}N/m/W. The corresponding point in the dispersion relation is below and close to the air light line, as indicated by the circle in Fig. 4(a).

*h*= 540 nm and

*w*= 350 nm,

*F/*(

*LP)*is reduced to −2.7 × 10

^{−3}N/m/W. The mode in such a system is well below the air light line. The light is confined in the two waveguides, with minimal optical coupling between them, and the magnitude of the force is decreased.

^{−3}N/m/W. For example,

*F/*(

*LP*) on the waveguide with

*h*= 110 nm and

*w*= 350 nm (marked by the star) is on the order of 10

^{−5}N/m/W. Figure 4(a) shows that the mode is on the asymptote of the light line. The mode energy spreads into the surrounding air, giving rise to a small force. In order to achieve a larger attractive force, we can design the waveguide cross section to make the optical mode approach the air light line but not overlap with the light line, as indicated by the circle in Fig. 4(a).

^{−18}m

^{2}/W. The value of

*y*direction. For example,

*F/*(

*LP*) for a waveguide of

*h*= 400 nm and

*w*= 200 nm is −6.7 × 10

^{−3}N/m/W, which has the same order of magnitude as

*F/*(

*LP*) = −7.2 × 10

^{−3}N/m/W for a waveguide of

*h*= 400 nm and

*w*= 100 nm. However, by reducing

*w*from 200 nm to 100 nm, we can increase

^{−14}m

^{2}/W to 1.7 × 10

^{−13}m

^{2}/W.

### 4.2. Antisymmetric mode

*F/*(

*LP*) and

*F*/(

*LP*). The force is positive (repulsive). The bright region indicates the combination of waveguide heights and widths that result in relatively large optical force for 1550-nm light. The blue curve indicates the conditions for mode cutoff: a guided mode exists only above and to the right of the curve. Large forces can be obtained by choosing cross-sectional parameters on the upper, right boundary of the curve. For example, for

*h*= 320 nm and

*w*= 250 nm, marked by the circle in Fig. 5(b),

*F/*(

*LP*) is 9.5 × 10

^{−3}N/m/W. The corresponding mode in the dispersion relation lies close to and below the light line, as indicated by the circle in Fig. 5(a).

*h*= 540 nm and

*w*= 250 nm,

*F/*(

*LP)*is reduced to 5.7 × 10

^{−3}N/m/W. The mode energy is confined in the waveguides, and the light intensity in the air gap is reduced, decreasing the optical force. If the waveguide cross-sectional dimension is too small, as marked by the star in Fig. 5(b) for

*h*= 110 nm and

*w*= 250 nm, the mode becomes leaky.

*w*to make the waveguides more deformable. For example, if we choose

*h*= 540 nm and

*w*= 140 nm,

^{−13}m

^{2}/W, which is more than 150,000 times larger than the intrinsic Kerr coefficient of silicon.

### 4.3. Discussion

*F*and the waveguide displacement

*w*(Figs. 4(c) and 5(c)).

^{−13}m

^{2}/W. Comparing Figs. 4(c) and 5(c), we observe that the symmetric mode has a bigger, brighter region representing

^{−14}m

^{2}/W. In other words, to realize the same value of

14. 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]

## 5. Conclusion

## Acknowledgments

## References and links

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. |

2. | M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. |

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

4. | J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. |

5. | M. L. Povinelli, S. G. Johnson, M. Lonèar, M. Ibanescu, E. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators,” Opt. Express |

6. | 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 |

7. | 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 |

8. | D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics |

9. | F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D |

10. | M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics |

11. | G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature |

12. | 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 |

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

14. | J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. |

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

16. | J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics |

17. | J. Ma and M. L. Povinelli, “Large tuning of birefringence in two strip silicon waveguides via optomechanical motion,” Opt. Express |

18. | 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. |

19. | P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B |

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

21. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

22. | J. D. Jackson, |

23. | 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 |

24. | S. Timoshenko, |

25. | M. Soljacić, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(190.0190) Nonlinear optics : Nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 10, 2011

Revised Manuscript: April 30, 2011

Manuscript Accepted: May 3, 2011

Published: May 9, 2011

**Citation**

Jing Ma and Michelle L. Povinelli, "Mechanical Kerr nonlinearities due to bipolar optical forces between deformable silicon waveguides," Opt. Express **19**, 10102-10110 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10102

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### References

- 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).
- M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30(22), 3042–3044 (2005). [PubMed]
- A. Mizrahi and L. Schächter, “Mirror manipulation by attractive and repulsive forces of guided waves,” Opt. Express 13(24), 9804–9811 (2005). [PubMed]
- J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30(15), 1956–1958 (2005). [PubMed]
- M. L. Povinelli, S. G. Johnson, M. Lonèar, M. Ibanescu, E. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators,” Opt. Express 13(20), 8286–8295 (2005). [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).
- 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).
- D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
- 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).
- 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]
- 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]
- 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]
- 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).
- 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]
- 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).
- P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2(11), 1830–1840 (1985).
- 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).
- S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [PubMed]
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
- 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]
- S. Timoshenko, Theory of Elasticity (McGraw-Hill, 1934).
- 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]

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