## Enhanced optical forces in integrated hybrid plasmonic waveguides |

Optics Express, Vol. 21, Issue 10, pp. 11839-11851 (2013)

http://dx.doi.org/10.1364/OE.21.011839

Acrobat PDF (1969 KB)

### Abstract

We demonstrate gradient optical forces in metal-dielectric hybrid plasmonic waveguides (HPWG) for the first time. The magnitude of optical force is quantified through excitation of the nanomechanical vibration of the suspended waveguides. Integrated Mach-Zehnder interferometry is utilized to transduce the mechanical motion and characterize the propagation loss of the HPWG. Compared with theory, the experimental results have confirmed the optical force enhancement, but also suggested a significantly higher optical loss in HPWG. The excessive loss is attributed to metal surface roughness and other non-idealities in the device fabrication process.

© 2013 OSA

## 1. Introduction

1. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science **321**(5893), 1172–1176 (2008). [CrossRef] [PubMed]

5. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature **459**(7246), 550–555 (2009). [CrossRef] [PubMed]

6. F. Marquardt and S. Girvin, “Optomechanics,” Physics **2**, 40 (2009). [CrossRef]

8. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. **107**(6), 063602 (2011). [CrossRef] [PubMed]

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

4. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature **462**(7273), 633–636 (2009). [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. K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. **11**(2), 791–797 (2011). [CrossRef] [PubMed]

11. M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. **6**(11), 726–732 (2011). [CrossRef] [PubMed]

12. H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun **3**, 1091 (2012). [CrossRef] [PubMed]

13. K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express **19**(16), 15098–15108 (2011). [CrossRef] [PubMed]

2. 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]

11. M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. **6**(11), 726–732 (2011). [CrossRef] [PubMed]

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]

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

15. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature **482**(7383), 63–67 (2012). [CrossRef] [PubMed]

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

5. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature **459**(7246), 550–555 (2009). [CrossRef] [PubMed]

13. K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express **19**(16), 15098–15108 (2011). [CrossRef] [PubMed]

16. C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. **14**(9), 095014 (2012). [CrossRef]

18. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics **4**(2), 83–91 (2010). [CrossRef]

19. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. **11**(2), 321–328 (2011). [CrossRef] [PubMed]

21. C. Huang and L. Zhu, “Enhanced optical forces in 2D hybrid and plasmonic waveguides,” Opt. Lett. **35**(10), 1563–1565 (2010). [CrossRef] [PubMed]

22. V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. **2**, 331 (2011). [CrossRef]

23. M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. **5**(8), 570–573 (2010). [CrossRef] [PubMed]

24. 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). [CrossRef] [PubMed]

26. 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]

*f*

_{o}is the optical force per unit length normalized to the optical power,

*ω*

_{o}is the optical angular frequency,

*q*is a generalized coordinate corresponding to the mechanical degree of freedom that is under consideration,

*n*

_{eff}(

*ω*

_{o},

*q*) is the effective index of optical mode and

*c*is the speed of light in vacuum. This expression reveals that any dispersive dependence of

*n*

_{eff}on a positional coordinate (

*q*) corresponds to an optical force along that coordinate direction. The stronger the dependence (i.e. ∂

*n*

_{eff}/∂

*q*), the larger the corresponding optical force is. Such a position dependence of optical mode index can be found in almost all nanophotonic structures—generally their optical modal profile are sensitive to mechanical displacement or deformation. Therefore, the existence of optical forces is ubiquitous in photonic systems. This is more the case in nanophotonic devices because the optical fields therein interact and couple more strongly in the near-field, in a way highly dependent on the distance between the coupled structures. This implies the existence of strong optical forces at the nanoscale. With this fundamental understanding of its mechanism, optical forces and optomechanical effects can be effectively generated in many nanophotonic systems, including both dielectric and metallic plasmonic devices, as well as the hybrids of them.

19. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. **11**(2), 321–328 (2011). [CrossRef] [PubMed]

21. C. Huang and L. Zhu, “Enhanced optical forces in 2D hybrid and plasmonic waveguides,” Opt. Lett. **35**(10), 1563–1565 (2010). [CrossRef] [PubMed]

22. V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. **2**, 331 (2011). [CrossRef]

27. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics **2**(8), 496–500 (2008). [CrossRef]

22. V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. **2**, 331 (2011). [CrossRef]

27. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics **2**(8), 496–500 (2008). [CrossRef]

19. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. **11**(2), 321–328 (2011). [CrossRef] [PubMed]

21. C. Huang and L. Zhu, “Enhanced optical forces in 2D hybrid and plasmonic waveguides,” Opt. Lett. **35**(10), 1563–1565 (2010). [CrossRef] [PubMed]

28. P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science **325**(5940), 594–597 (2009). [CrossRef] [PubMed]

28. P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science **325**(5940), 594–597 (2009). [CrossRef] [PubMed]

30. M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. **93**(11), 113110 (2008). [CrossRef]

## 2. Device structure and theoretical analysis

_{2}layer underneath and is free to move in-plane. The optical, mechanical and optomechanical properties of this HPWG structure will be theoretically analyzed in this section.

### 2.1 Hybrid plasmonic modes (HPM) and their optical force

*i*[31

31. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

*P*(

*x*) =

*P*(0)

*e*

^{−}

*, where*

^{αx}*x*axis is defined along the HPWG,

*P*(

*x*) is the optical power as a function of position

*x*and

*α*is the decay constant. Using the imaginary part of the HPM complex mode index

*n*

_{eff}, which is calculated by FEM simulation,

*α*can be expressed as

*α*= 2

*ω*

_{o}|Im(

*n*

_{eff})|/

*c*.

*p*(

*x*) as the optical mode propagates, as depicted in Fig. 3(a) . (See Table 1 for definitions of symbols, which are used consistently throughout the paper.) Such exponential decay of the optical force distribution

*p*(

*x*) leads to a nonlinear relationship between

*F*

_{n}, the total optical force normalized by the input optical power

*P*(0), and

*L*, the length of HPWG, as is shown in Eq. (2). In contrast, in low-loss dielectric waveguide, the normalized total optical force

*F*

_{n}is considered to be proportional to the waveguide length [24

24. 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). [CrossRef] [PubMed]

26. 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]

*F*

_{n}on the waveguide length is compared, for different waveguide structures reported in the literature [2

2. 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]

13. K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express **19**(16), 15098–15108 (2011). [CrossRef] [PubMed]

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]

**11**(2), 321–328 (2011). [CrossRef] [PubMed]

**35**(10), 1563–1565 (2010). [CrossRef] [PubMed]

25. 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*

_{n}for HPMs plateau while those for dielectric waveguides continue to increase linearly and eventually exceed the

*F*

_{n}in HPMs. Thus, HPWG is capable to generate large total optical force in a short distance and potentially advantageous for miniaturizing device footprint. For longer distance, HPWG may be too lossy to compete with dielectric waveguides in order to generate large optical forces.

*p*

_{n}depends non-monotonically on the Si waveguide width, which has an optimum that generates the maximum force. This optimal Si waveguide width is different for different order of HPM. The HPM0 and HPM1 curves in Fig. 1(d) are calculated with 220 nm thicknesses, 30 nm gaps and their respective optimal Si waveguide widths, which are 250 nm for HPM0 and 550 nm for HPM1. It turns out that the HMP0 is the most efficient to generate large optical force, so in our experiments, we only focused on HMP0. It is worth noting that the normalized local optical force

*p*

_{n}is only determined by the HPWG structure and the dispersion property of the pertinent HPM, as is described in Eq. (1), so it is the quantity that our experimental measurements aim to determine.

### 2.2 Multimode vibrational theory for doubly clamped beam (DCB)

34. M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. **81**(6), 2480 (1997). [CrossRef]

*E*is the Young’s Modulus,

*I*is the cross sectional area moment of inertia with respect to the neutral axis,

*ρ*is the density,

*A*is the cross-sectional area,

*t*is time,

*x*is the axis along the waveguide and

*u*is the DCB in-plane transverse displacement along the

*z*axis, which is defined in Fig. 3(a). The solution of Eq. (3) has the form of normal mode expansion which satisfies the doubly clamped boundary conditions: where

*ω*and

_{j}*ϕ*(

_{j}*x*) are the angular frequency and mode profile of the

*j*th mechanical mode, respectively, while

*λ*is a solution of

_{j}*λ*and

_{j}*ω*is given by

_{j}*q*(

_{j}*t*),

*p*(

_{j}*t*),

*m*,

_{j}*k*and

_{j}*Q*are the instantaneous amplitude, driving force, effective mass, spring constant and quality factor for the

_{j}*j*th mode, respectively. The expression for

*p*(

_{j}*t*) is given by Eq. (10), where

*p*(

*x*,

*t*) is the time varying force distribution. In the case of optical force excitation,

*p*(

*x*,

*t*) should be the time varying optical force distribution, which decays exponentially along the HPWG. The expressions for

*m*and

_{j}*k*in terms of the beam parameters are given by Eq. (11).

_{j}*p*(

*x*,

*t*) will excite more than one mode, unless it has the same profile as one of the normal modes

*ϕ*(

_{j}*x*), in which case only that single mode will be excited. Furthermore, different force distributions will excite different sets of modes and it is possible to purposely engineer the force distribution to selectively excite one or more specific modes only. In Fig. 3(b), three typical types of force distribution—point force at the middle of the beam, uniform force and exponential decaying force (with decay constant

*α*= 2/

*L*)—are compared. In each case, the overlap integral in Eq. (10) is evaluated and normalized with Eq. (12), where the results

*N*are plotted. In the fraction on the right-hand side of Eq. (12), the first term in the denominator normalizes the force distribution

_{j}*p*(

*x*,

*t*) in the overlap integral to the total force, which is the integral of the absolute value of the force distribution along the entire beam. Meanwhile, the mode profile

*ϕ*(

_{j}*x*) in the overlap integral is normalized by the second term in the denominator.

*q*, if

_{j}*Q*is high and

_{j}*j*is not very large,where

*k*

_{B}is the Boltzmann constant and

*T*is the temperature. The thermomechanical noise PSD is directly measurable and can be used to calibrate key parameters of the Si beam in the experiments.

### 2.3 Frequency response of the Si beam driven by optical force

*p*(

*x*) and

*P*(

*x*) is explicitly indicated. The resultant definite integral in Eq. (14) can be evaluated analytically with Eq. (15).Fourier transforming both sides of Eq. (9) and organizing it into the form of a transfer function, we obtain the full frequency response of the Si beam:where the sign “~” indicates the Fourier transform of the corresponding quantity and

*i*is the imaginary unit. At the mode resonance frequency, the amplitude of the transfer function in Eq. (16) reaches the resonance peak, which isSubstituting Eq. (14) into Eq. (17), we obtain the expression for normalized local optical force

*p*

_{n}shown in Eq. (18), which can be evaluated from experimental results, assuming

*Q*is high.

_{j}## 3. Device fabrication

36. M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. **51**, 06FC05 (2012). [CrossRef]

## 4. Optical characteristics

*ER*) of the MZI transmission spectra, using Eq. (19).

*ER*with increasing length—hence increasing total loss, in agreement with theory. These measurements were conducted without releasing the Si waveguide in the HPWG to avoid the uncertainty induced in the wet etching process. This approach is justified by simulation results, which suggest that the presence of the SiO

_{2}substrate only slightly perturbs the HPM and introduces insignificant change to the mode profile and optical loss. The linear dependence of total loss on the HPWG length is further shown in Fig. 6(b). However, the measured value of total loss is about 30 times higher than the theoretical expectation. In Fig. 6(c), the normalized loss is plotted against HPWG gap width, showing qualitatively the same trend as the theory, but again is about 30 times higher. We attribute this large discrepancy to the inevitable metal surface roughness and other non-idealities in the fabrication process.

## 5. Optomechanical characteristics

*p*

_{n}generated in the HPWG.

### 5.1 Transduction gain factor calibration by thermomechanical noise measurement

*G*, defined as the derivative of the photodetector output voltage with respect to the amplitude

_{j}*q*of the

_{j}*j*th mechanical mode, can be calibrated by measuring the thermomechanical noise of the Si beam, as described in reference [2

2. 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]

*ω*

_{1}and

*Q*

_{1}, by fitting the experimental curve with Eq. (13). From the width, thickness and resonance frequency of the Si beam, using Eq. (7), we can determine its actual length, which is difficult to be accurately measured by SEM imaging. Subsequently the mode spring constant

*k*

_{1}can be calculated with Eq. (11) and finally we use Eq. (13) again to derive the theoretical PSD of the thermomechanical noise and compare with the measured noise spectrum to calibrate the transduction gain

*G*

_{1}. The results from a typical device are shown in Fig. 7(a) , in which case the transduction gain is 4.82 V/nm.

### 5.2 Optical force measurement by driven response

*p*

_{n}except for the last fraction on the right hand side, which can be obtained from driven frequency response measurement.

*α*is calculated from the

*ER*and HPWG length,

**456**(7221), 480–484 (2008). [CrossRef] [PubMed]

## 6. Conclusion and discussion

28. P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science **325**(5940), 594–597 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science |

2. | 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 |

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

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

5. | M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature |

6. | F. Marquardt and S. Girvin, “Optomechanics,” Physics |

7. | A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. |

8. | A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. |

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

10. | K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. |

11. | M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. |

12. | H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun |

13. | K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express |

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

15. | E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature |

16. | C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. |

17. | S. A. Maier, |

18. | D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics |

19. | X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. |

20. | X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano |

21. | C. Huang and L. Zhu, “Enhanced optical forces in 2D hybrid and plasmonic waveguides,” Opt. Lett. |

22. | V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. |

23. | M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. |

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

25. | 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. |

26. | 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 |

27. | R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics |

28. | P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science |

29. | L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. |

30. | M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. |

31. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

32. | J. D. Jackson, |

33. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, |

34. | M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. |

35. | S. Timoshenko, |

36. | M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(220.4880) Optical design and fabrication : Optomechanics

(230.7370) Optical devices : Waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: January 22, 2013

Revised Manuscript: April 14, 2013

Manuscript Accepted: April 22, 2013

Published: May 7, 2013

**Citation**

Huan Li, Jong W. Noh, Yu Chen, and Mo Li, "Enhanced optical forces in integrated hybrid plasmonic waveguides," Opt. Express **21**, 11839-11851 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11839

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