## Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate

Optics Express, Vol. 17, Issue 3, pp. 1806-1816 (2009)

http://dx.doi.org/10.1364/OE.17.001806

Acrobat PDF (2689 KB)

### Abstract

We present a study of transverse optical forces arising in a freestanding silicon nanowire waveguide. A theoretical framework is provided for the calculation of the optical forces existing between a waveguide and a dielectric substrate. The force is evaluated using a numerical procedure based on finite-element simulations. In addition, an analytical formalism is developed which allows for a simple approximate analysis of the problem. We find that in this configuration optical forces on the order of pN can be obtained, sufficient to actuate nano-mechanical devices.

© 2009 Optical Society of America

## 1. Introduction

01. S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett. **92**, 3112–3114 (2008). [CrossRef]

2. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express **15**, 17,172–17,205 (2007). [CrossRef]

9. 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**, 3042–3044 (2005). [CrossRef] [PubMed]

10. M. Povinelli, S. Johnson, M. Lonèar, 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]

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

12. H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Physical Review B (Condensed Matter and Materials Physics) **78**, 165,129 (2008). [CrossRef]

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

14. 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]

## 2. Numerical calculation of the optical force

### 2.1. The effective index method

*μ*m thickness. A freestanding waveguide is created by etching into the silicon oxide layer, thus removing the dielectric material under the waveguide.

*g*. When the waveguide is brought closer to the substrate, the waveguide mode will be coupled evanescently to the dielectric. With decreasing gap, the propagation constant of the fundamental mode decreases. This is demonstrated in the dispersion diagram obtained from the finite-element simulations as shown in Fig.2(a). Above cut-off, the dispersion curves corresponding to varying gap sizes show a local maximum before converging towards the propagation constant of the free waveguide. From the dispersion diagram we are able to derive the optical force, which acts on the waveguide. Following Povinelli [9

9. 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**, 3042–3044 (2005). [CrossRef] [PubMed]

*U*=

*PLn*/

_{g}*c*, where

*P*is the optical power,

*L*is the length of the beam and

*n*is the group index of the mode, we get the optical force normalized to length and power as

_{g}*k*=

_{n}*k*

_{0}

*w*/2

*π*, where

*k*

_{0}is the free space wave vector. From the negative sign it is apparent, that the force is attractive, thus pulling the beam towards the substrate. When the gap size is zero, the force reaches its maximum value of ≈ 13

*pN*/

*μm*/

*mW*. It decreases mono-tonically towards zero, when the gap is increased towards infinity. Based on the calculations, we find a typical strength of the force on the order of

*pN*when assuming device dimensions in the micrometer range. Therefore given the high sensitivity and small mass of the free standing waveguide [14

14. 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]

15. D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature **430**, 329–332 (2004). [CrossRef] [PubMed]

16. K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. **76**, 061,101 (2005). [CrossRef]

### 2.2. Image waveguide approximation

*. Here we essentially look at the interaction between the dipoles in the waveguide and their image dipoles induced by the dielectric substrate. We restrict ourselves to the symmetric mode and thus solve for the attractive lateral force. The concept is illustrated in Fig.2.2. The calculation is carried out as described by Povinelli and as elucidated in the preceding section. The force between coupled waveguides depends on the gap size in a fashion similar to the results in Fig.2(b). It is stronger than the force to the substrate, because the waveguides are assumed to be surrounded by air. To take the effect of the substrate into account, we introduce a shielding factor*

_{g}*α*into Eq.(3) as

## 3. A one-dimensional analytical method for determination of the optical force

18. F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D **46**, 157–164 (2008). [CrossRef]

20. A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett. **8**, 63–65 (1983). [CrossRef] [PubMed]

21. P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. **19**, 2848–2855 (1980). [CrossRef] [PubMed]

### 3.1. Calculation of the electrical fields

*n*of 3.5 is embedded in a host material with

_{c}*n*of 1.0. The waveguide is separated from the substrate with a refractive index

_{a}*n*of 1.5 by a distance

_{s}*g*. The slab is assumed to extend infinitely in the

*x*and

*z*directions. In this configuration the electric and magnetic fields are given by

*E*,

_{y}*E*and

_{z}*H*are 0 and therefore the field vector is given as

_{x}^{T}denotes the matrix transpose. Using the freespace wavevector

*x*component of the electric field can be expressed as

*a*in above Eq. are obtained by requiring the impedance

_{i}*ϕ*leads to the dispersion relation for the structure as

*m*is the mode number of the waveguide. Above Eq. does not have an analytical solution, so the values of

*β*corresponding to a given wavelength have to be obtained numerically.

*a*

_{i}*M*

### 3.2. Approximation of the propagation constant

*β*

_{0}is the propagation constant of the free waveguide. For small values of g we can introduce the following approximations

*β*

_{1}by solving the above Eq. for

*g*= 0. Since we assume

*β*

_{0}to be much larger than

*β*1, we can use

*β*

_{1}

*g*= ln(2). Then we use this value in the transcendental Eq. and find for σ the following expression:

*n*with the simple exponential decay from Eq.(27) is a valid simplification.

_{eff}### 3.3. Calculation of the optical force using the Maxwell stress tensor

*T*value. This is given for TE polarization as

_{yy}*T*in the gap to calculate the optical force. This leads to the following expression:

_{yy}*P*. The field amplitude is then given as

*A*is the mode area of the waveguide. In order to compare the analytical solution for the slab with simulation results for the cross section of a real waveguide we need to adjust the analytical formula to take the two-dimensionality of the geometry into account. This is done by defining an effective width of the waveguide slab in the 1d model. This effective width is driven by the fact, that the 1d field amplitude will only correspond to the mode pattern in the 2d waveguide in the center. Further away from the center, the mode pattern is concentrated in a smaller area. Therefore we define the effective width of the 1d slab as

_{wg}## 4. Conclusion

*pN*/

*μm*/

*mW*and thus large enough to efficiently actuate nanoscale devices. We have used a numerical approach to deduce the force from finite-element calculations and also formulated an approximate analytical formulation. Results from both calculations are in good agreement. For larger gap values, the comparison with the coupled-waveguide system shows good agreement, when an empirical correction factor is introduced.

## Acknowledgement

## References and links

01. | S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett. |

2. | T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express |

3. | H. Rokhsari, T. Kippenberg, T. Carmon, and K. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express |

4. | B. Kemp, T. Grzegorczyk, and J. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express |

5. | M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express |

6. | J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A |

7. | M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express |

8. | R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express |

9. | 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. |

10. | M. Povinelli, S. Johnson, M. Lonèar, 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 |

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

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,” Physical Review B (Condensed Matter and Materials Physics) |

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

14. | 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 |

15. | D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature |

16. | K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. |

17. | J. D. Jackson, |

18. | F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D |

19. | E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. |

20. | A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett. |

21. | P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(230.4685) Optical devices : Optical microelectromechanical devices

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 28, 2008

Revised Manuscript: January 26, 2009

Manuscript Accepted: January 27, 2009

Published: January 29, 2009

**Citation**

W. H. P. Pernice, Mo Li, and H. X. Tang, "Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate," Opt. Express **17**, 1806-1816 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1806

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### References

- S. S. Verbridge, H. G. Craighead, and J. M. Parpia, "A megahertz nanomechanical resonator with room temperature quality factor over a million," Appl. Phys. Lett. 92, 3112-3114 (2008). [CrossRef]
- T. J. Kippenberg and K. J. Vahala, "Cavity Opto-Mechanics," Opt. Express 15, 17,172-17,205 (2007). [CrossRef]
- H. Rokhsari, T. Kippenberg, T. Carmon, and K. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005). [CrossRef] [PubMed]
- B. Kemp, T. Grzegorczyk, and J. Kong, "Ab initio study of the radiation pressure on dielectric and magnetic media," Opt. Express 13, 9280-9291 (2005). [CrossRef] [PubMed]
- M. Mansuripur, "Radiation pressure and the linear momentum of light in dispersive dielectric media," Opt. Express 13, 2245-2250 (2005). [CrossRef] [PubMed]
- J. P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8, 14-21 (1973). [CrossRef]
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express 12, 5375-5401 (2004). [CrossRef] [PubMed]
- R. Loudon and S. M. Barnett, "Theory of the radiation pressure on dielectric slabs, prisms and single surfaces," Opt. Express 14, 11,855-11,869 (2006). [CrossRef]
- 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, 3042-3044 (2005). [CrossRef] [PubMed]
- M. Povinelli, S. Johnson, M. Lonèar, 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]
- 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, 658 -665 (2007). [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, 165,129 (2008). [CrossRef]
- 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]
- 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]
- D. Rugar, R. Budakian, H. Mamin, and B. Chui, "Single spin detection by magnetic resonance force microscopy," Nature 430, 329-332 (2004). [CrossRef] [PubMed]
- K. L. Ekinci and M. L. Roukes, "Nanoelectromechanical systems," Rev. Sci. Instrum. 76, 061,101 (2005). [CrossRef]
- J. D. Jackson, Classical electrodynamics, (J. Wiley and Sons, New York, 1975).
- F. Riboli, A. Recati, M. Antezza, and I. Carusotto, "Radiation induced force between two planar waveguides," Eur. Phys. J. D 46, 157-164 (2008). [CrossRef]
- E. A. J. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2102 (1969).
- A. Kumar, K. Thyagarajan, and A. K. Ghatak, "Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach," Opt. Lett. 8, 63-65 (1983). [CrossRef] [PubMed]
- P. Yeh and H. F. Taylor, "Contradirectional frequency-selective couplers for guided-wave optics," Appl. Opt. 19, 2848-2855 (1980). [CrossRef] [PubMed]

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