## Modeling of the optical force between propagating lightwaves in parallel 3D waveguides

Optics Express, Vol. 17, Issue 18, pp. 16032-16037 (2009)

http://dx.doi.org/10.1364/OE.17.016032

Acrobat PDF (231 KB)

### Abstract

We present a rigorous analysis of the optical gradient force between coupled single-mode waveguides in three dimensions. Using eigenmode expansion we determine the optical mode patterns in the coupled system. In contrast to previous work, the sign and amplitude of the optical force is found to vary along the waveguide with a characteristic beating length. Our results establish design principles for optomechanically tunable directional couplers.

© 2009 OSA

## 1. Introduction

1. 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). [CrossRef] [PubMed]

3. 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). [CrossRef]

4. M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. 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). [CrossRef] [PubMed]

6. 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). [CrossRef]

1. 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). [CrossRef] [PubMed]

7. 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). [CrossRef] [PubMed]

8. 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). [CrossRef] [PubMed]

9. M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. **4**(6), 377–382 (2009). [CrossRef] [PubMed]

10. G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. **5**(5), 554–557 (1993). [CrossRef]

11. P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. **14**(2), 164–166 (2002). [CrossRef]

13. A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antenn. Propag. **55**(6), 1495–1505 (2007). [CrossRef]

## 2. The force distribution in coupled waveguides using eigenmode expansion

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

15. U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. **30**(5), 406–408 (1994). [CrossRef]

*L*and

*R*). After passing through the coupling region of length

*l*, their new amplitudes will be denoted by

*L’*and

*R’*.

*L*and

*R*are the eigenmodes of the uncoupled system, i.e. the eigenmodes of the incoming waveguides. We assume that no optical or material loss will occur during the coupling.

*E*and

*O*, labeled by considering their symmetry properties with respect to the direction of propagation. We consider two parallel silicon waveguides of length

*l*and thickness

*h*as shown in Fig. 1(b). The waveguides are of not necessarily equal width

*w*,

_{1}*w*. Each individual waveguide is assumed to support only a single fundamental mode.

_{2}*E(x,y,d)*and

*O(x,y,d)*for the cross-sectional profiles of the even and odd modes, respectively. The modes are assumed to be normalized so that

*z*direction with propagation constants

*β*and

_{e}*β*.

_{o}

*Φ**. When both modes are excited with equal amplitude, for identical waveguides two dominant cases can be identified: both waveguide modes are in phase (symmetric) or both modes are 180 degrees out of phase (anti-symmetric). In order to avoid confusion, these situations are clarified in Fig. 2 .*

_{0}*L*) or the right waveguide (

*R*) is excited, the mode profile at position z=0 (the beginning of the coupling region) can be expressed in terms of the even and odd modes of the coupled system using a projection matrix

*M*as

*M*contains the decomposition coefficients into the even and odd mode. Because of the orthogonality of the eigenmodes, these coefficients can be obtained by evaluating Eq. (2) as

*L,R*system, a combination of even and odd modes are excited, which propagate along the coupler with their respective propagation constants.

*w*and

_{1}=300nm*w*. The thickness of the beams is assumed to be

_{2}=310nm*h = 220nm*and the beam length is

*10μm*. These are typical devices parameters for practical silicon devices. Because of the difference of the waveguide cross sections, the even and odd modes are not uniform. Thus the symmetric/anti-symmetric waveguide modes will excite both the even and odd modes. Due to the different propagation constants of the even and odd modes a typical beating pattern will occur along the length of the coupler. This is illustrated in Fig. 3 for two representative cases, the excitation of the symmetric and anti-symmetric modes from the left.

*6.4μm*. After one beating period the power from both waveguides is transferred completely from one waveguide to the other. Changing the input phase difference moves the beating pattern along the coupler in the

*z*direction.

## 3. Optical force distribution along the coupled beams due to optical mode beating

*T*is given as

*i,j*are counted over the three field components. Once the field vectors have been obtained, the optical force can be calculated by computing the time average of the surface integral of the MST as

*C*is the contour path along the surface of a beam. Plugging Eq. (1) into Eq. (3) we find the optical force for an input phase difference

*Φ*aswhere we have used the following expressions for the three force contributions in the above equation as

_{0}*w*

_{e},

*w*

_{o}) remains constant over the length of the directional coupler. The

*z*dependence of the force distribution is thus a result of the beating term (with a modulated weight of

*w*).

_{b}*g*between the waveguides. Results are presented in Fig. 4 , where the optical force has been normalized to the waveguide length and the optical input power.

*F*resulting from the even mode is the dominant term for all separations. This force contribution is always attractive, as also shown by Povinelli et al [1

_{e}1. 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). [CrossRef] [PubMed]

*F*results from the odd mode. For gaps smaller than the crossover gap

_{o}*g*(130nm), this force

_{c}*F*is also attractive, but then changes sign to be repulsive for larger gaps [14

_{o}14. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics **3**(8), 464–468 (2009). [CrossRef]

*F*is only noticeable at small distances, but decays quickly with increasing gap. However, for asymmetric waveguides, this beating term can be significant due to the large difference between the propagation constants.

_{b}## 4. Tunability of the net optical force and directional coupling

*g*. Thus from Eq. (4) it is obvious, that the force can be changed to be net repulsive or net attractive depending on the phase difference

_{c}*Φ*. This is illustrated in Fig. 5(a) for calculated force values of the geometry outlined above.

_{0}*L*mode, magenta curve), or excited from both waveguides. For the single-waveguide excitation a static tuning power of 0.95mW is required to achieve 100% switching. When the coupler is excited simultaneously in both waveguides, the switching power can be reduced to 0.86mW when operating at a phase difference of

*π*(blue line), and further to 0.44mW when operating at a phase difference of 0. If at the same time the force is phase tuned from attractive to repulsive to pull and push the waveguides, the full dynamic range of the opto-mechanical device can be exploited. This situation is shown by the green line in Fig. 5(b). In this case the switching power can be as low as 0.28mW, providing considerable improvement compared to the single waveguide excitation.

## 5. Conclusion

## Acknowledgements

## References and links

1. | 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. |

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

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

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

5. | 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 |

6. | 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 |

7. | 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 |

8. | 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 |

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

10. | G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. |

11. | P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. |

12. | A. Yariv, “Quantum Electronics,” Wiley, New York (1989). |

13. | A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antenn. Propag. |

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

15. | U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. |

**OCIS Codes**

(220.4880) Optical design and fabrication : Optomechanics

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

(130.3990) Integrated optics : Micro-optical devices

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: June 25, 2009

Revised Manuscript: August 18, 2009

Manuscript Accepted: August 20, 2009

Published: August 26, 2009

**Citation**

W. H. P. Pernice, Mo Li, King Yan Fong, and Hong X. Tang, "Modeling of the optical force between propagating lightwaves in parallel 3D waveguides," Opt. Express **17**, 16032-16037 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-16032

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### References

- 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). [CrossRef] [PubMed]
- A. Mizrahi and L. Schächter, “Two-slab all-optical spring,” Opt. Lett. 32(6), 692–694 (2007). [CrossRef] [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). [CrossRef]
- M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. 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). [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(11), 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(16), 165129 (2008). [CrossRef]
- 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). [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(7221), 480–484 (2008). [CrossRef] [PubMed]
- M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009). [CrossRef] [PubMed]
- G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. 5(5), 554–557 (1993). [CrossRef]
- P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002). [CrossRef]
- A. Yariv, “Quantum Electronics,” Wiley, New York (1989).
- A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antenn. Propag. 55(6), 1495–1505 (2007). [CrossRef]
- M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009). [CrossRef]
- U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. 30(5), 406–408 (1994). [CrossRef]

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