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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 11839–11851
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Enhanced optical forces in integrated hybrid plasmonic waveguides

Huan Li, Jong W. Noh, Yu Chen, and Mo Li  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 11839-11851 (2013)
http://dx.doi.org/10.1364/OE.21.011839


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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

Nano-optomechanical system, or NOMS, is a burgeoning field that combines nanophotonic and nano-electromechanical (NEMS) devices seamlessly in an integrated system [1

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

5

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]

]. By utilizing near-field optical forces generated in nanophotonic structures, NOMS exploits the interaction between the mechanical degrees of freedom in NEMS and the optical functionalities of photonic devices. For fundamental research, the optomechanical effects represent a new form of light-matter interaction that can be rationally engineered in nanophotonic systems, leading to many unprecedented physical phenomena in both classical and quantum regimes [6

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

8

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

]. For practical applications, NOMS enables ultrahigh measurement sensitivity in NEMS-based sensors [3

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

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

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

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]

], opens up new possibilities to implement non-volatile mechanical memory operation [11

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]

], all-optical signal processing [12

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]

], tunable or reconfigurable optical circuits [13

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]

], and potentially brings even more novel device functionalities to nanophotonics.

During the past a few years, significant advances have been made in NOMS devices based on all-dielectric nanophotonic systems, such as those implemented with silicon [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]

,11

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

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]

], silica [3

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

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]

], silicon nitride [4

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

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

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]

] or aluminum nitride [16

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]

]. In the meantime, optical forces and optomechanical effects in another important category of nanophotonic devices, namely plasmonics [17

17. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

,18

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

], have become a new research focus which has been under active theoretical investigation [19

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

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

]. Nevertheless, few relevant experimental studies [22

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

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]

] have been published so far, primarily due to the challenges in the device fabrication and experimental characterization.

In this work we experimentally demonstrate optical forces in integrated metal-dielectric hybrid plasmonic waveguides (HPWG), which is one of the most widely investigated plasmonic structures [19

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

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

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

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]

]. As far as we are aware of, this is the first experimental characterization of optical forces in HPWG. Compared with all-metallic surface plasma polariton waveguides and all-dielectric waveguides, their hybrids, HPWG, exhibits lower loss than the former [22

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

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]

] and deeper sub-wavelength optical mode confinement than the latter, which leads to optical forces that are enhanced by an order of magnitude or more [19

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

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

]. This enhanced optical force has been experimentally confirmed and quantified in this work. The propagation loss in HPWG, however, is significantly higher than theoretical expectations, as a result of the roughness of metal surfaces [28

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]

] and other non-idealities in the fabrication processes. Nevertheless, many fabrication methods [28

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

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]

] have been reported to achieve ultra-smooth metal surfaces and they can be applied in HPWG to reduce scattering loss. Thus, the plasmonic optomechanical systems demonstrated here have a great potential in leading to optical force mediated adaptive photonic devices and sensors.

2. Device structure and theoretical analysis

The HPWG structure employed in this work is shown schematically in Fig. 1(a)
Fig. 1 HPWG structure, optical modes and optical force comparison. (a) HPWG structure illustration. (b) and (c) Transverse electric field component of the HPM0 and HPM1, respectively, with the overlaid green curves showing the cross-sectional profile of the transverse electric field. (d) Comparison of the dependence of the total optical force on the waveguide length in different waveguide structures from literatures and this work. Solid lines represent experimental work while the other line styles represent theoretical calculation.
. On a standard silicon-on-insulator (SOI) substrate, a metal patch is placed in parallel with a strip Si waveguide of equal thickness. The gap between the metal patch and the Si strip is less than 100 nm. A section of the Si waveguide is suspended by removing the SiO2 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

In this structure, the transverse electric (TE) optical modes, with electric field predominantly in the plane of Si waveguide and metal patch, exhibit the features of both the Si waveguide mode (SWM) and the surface plasma polariton mode, hence the name “hybrid plasmonic mode” (HPM).

The fundamental (or zeroth order) and first order TE modes of the HPWG (HPM0 and HPM1) are simulated with finite element method (FEM) and shown in Fig. 1(b) and Fig. 1(c), respectively. Here the Si waveguide is 450 nm wide. Both the Si waveguide and the metal are 220 nm thick. The gap between them is 50 nm wide. The vacuum optical wavelength is 1550 nm. The refractive indices used for Si and metal (gold) are 3.45 and 0.524 + 10.742i [31

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

], respectively. The simulation results clearly show the strongly enhanced transverse electric field in the gap region, compared with the field in the silicon waveguide. For a given HPWG thickness, the number of HPMs supported is primarily determined by the width of the Si waveguide. In the case of 220 nm thickness, when the Si waveguide is narrower than 350 nm, only HPM0 can be supported. If the Si waveguide is wider than 450 nm, second or higher order HPMs can also be supported. However, in the devices fabricated in the present experimental work, only the fundamental mode is excited, due to the following two reasons. First, the Si waveguides we fabricated are no more than 450 nm wide so second or higher order HPMs are not supported. Second, in our devices, HPMs are excited by launching the fundamental TE mode of silicon waveguide (TE SWM) into the hybrid region, as shown in Fig. 2(a)
Fig. 2 2D FDTD simulation of the mode evolution between fundamental TE SWM and HPM. (a) The overview of the simulated HPWG structure overlaid by the transverse electric field component. The red arrows indicate where the SWM enters and exits. (b) A close-up view of the enhanced transverse electric field component in the nano gap. (c) Mode evolution from SWM to HPM, and then back to SWM again. The red curves show the cross sections of the transverse electric field component in the evolving mode.
. Compared with the HPM0, the HPM1 has a negligible mode overlap with the fundamental TE SWM. Therefore the coupling efficiency for the HPM1 is negligibly small compared with that for the HPM0.

This qualitative analysis on mode evolution between SWM and HPM has been confirmed by 2D finite difference time domain (FDTD) simulations shown in Fig. 2. In this simulation, a Si slab waveguide approaches a metal wall gradually, becomes parallel with the metal wall with a nano-sized gap, and then gradually leaves the metal wall. The fundamental TE SWM enters into the Si slab waveguide on the left side, and exits from the right side. From the simulation result, it is evident that the fundamental TE SWM evolves predominantly into HPM0 only, while as expected, HPM1 cannot be excited. Although the HPWG and the silicon waveguide in practice are 3D structures, the 2D simulation shown here is sufficient to demonstrate the features of the mode evolution between them.

Because HPM is inevitably lossy due to Ohmic loss in the metal, the power of optical mode decays exponentially while propagating in HPWG: P(x) = P(0)eαx, where 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 neff, which is calculated by FEM simulation, α can be expressed as α = 2ωo|Im(neff)|/c.

Because the local optical force for a certain mode is always proportional to the local optical power, the lossy nature of HPWG results in exponential decay of the generated optical force p(x) as the optical mode propagates, as depicted in Fig. 3(a)
Fig. 3 HPWG local optical force and mechanical properties. (a) Local optical force and beam mechanical in-plane mode profile. (b) Comparison of the excitation of the mechanical modes by different force distributions.
. (See Table 1

Table 1. Definitions of the terms of optical forces

table-icon
View This Table
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 Fn, 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 Fn 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

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]

].

Fn=0L|p(x)|dxP(0)=pn0Leαxdx=pnα(1eαL).
(2)

This difference is demonstrated in Fig. 1(d), where the dependence of Fn 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

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

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]

,19

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

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

,25

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]

] and the HPWG in this work. Instead of Eq. (1), the method of Maxwell Stress Tensor (MST) [32

32. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999).

,33

33. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon Press, 1984).

] is used to calculate the optical forces in HPWG because it is accurate even when material loss is present, although MST does not provide as much insight as Eq. (1) does. When plotting Fig. 1(d), for theoretical curves, the maximum optical forces reported in the corresponding literature using a gap no less than 30 nm are chosen; while for experimental curves, the maximum demonstrated optical forces in the corresponding literature are used. Because gaps less than 30 nm are quite challenging to fabricate, so we have excluded these cases in the theoretical results when comparing with the experimental results. The two curves from this work are simulation results for HPM0 and HPM1. Compared with optical forces in dielectric systems, the optical forces for HPM0 and HPM1 can be significantly stronger over a short distance, thanks to the field enhancement effect. Over a longer distance, however, the normalized total optical forces Fn for HPMs plateau while those for dielectric waveguides continue to increase linearly and eventually exceed the Fn 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.

For given HPWG thickness and gap width, the optical force pn 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 pn 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)

The suspended Si waveguide in the HPWG structure can be modeled as a thin and long doubly clamped beam (DCB). Because the optical force distribution along the waveguide is not uniform, it is necessary to rigorously analyze the mechanical modes and the driven response of the waveguide in order to quantify the optical force. In the following, we employ the multimode theory as described in [34

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]

,35

35. S. Timoshenko, Vibration Problems in Engineering, 2nd ed. (D. Van Nostrand company, Inc., 1937).

] for such a purpose.

In absence of damping and loads and in the stress limit, where tension is negligible compared to bending rigidity, the equation of motion of the DCB is given by
EI4u(x,t)x4+ρA2u(x,t)t2=0,
(3)
where 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:
u(x,t)=j=1Cjsin(ωjt+δj)ϕj(x),with
(4)
ϕj(x)=[sin(λjL)sinh(λjL)][cos(λjx)cosh(λjx)][sin(λjx)sinh(λjx)][cos(λjL)cosh(λjL)],
(5)
where ωj and ϕj(x) are the angular frequency and mode profile of the jth mechanical mode, respectively, while λj is a solution of

cos(λjL)cosh(λjL)=1.
(6)

The profile of the first order mode is shown in Fig. 3(a) in the in-plane direction. The relationship between λj and ωj is given by

(λjL)4=ωj2ρAL4EI.
(7)

In order to study the dynamics of the DCB subjected to time varying forces (including damping), the motion of the beam is expanded with the normal modes as shown in Eq. (8). The equation of motion for each mechanical mode is given by Eq. (9), where qj(t), pj(t), mj, kj and Qj are the instantaneous amplitude, driving force, effective mass, spring constant and quality factor for the jth mode, respectively. The expression for pj(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 mj and kj in terms of the beam parameters are given by Eq. (11).

u(x,t)=j=1ϕj(x)qj(t).
(8)
mjq¨j(t)+kjmjQjq˙j(t)+kjqj(t)=pj(t).
(9)
pj(t)=0Lp(x,t)ϕj(x)dx.
(10)
mj=ρA0L(ϕj)2dxandkj=EI0L(ϕj)2dx.
(11)

It is worth noting that Eq. (10) indicates that generally a certain force distribution p(x,t) will excite more than one mode, unless it has the same profile as one of the normal modes ϕj(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 Nj are plotted. In the fraction on the right-hand side of Eq. (12), the first term in the denominator normalizes the force distribution 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 ϕj(x) in the overlap integral is normalized by the second term in the denominator.

Nj=pj(t)0L|p(x,t)|dx1L0Lϕj2(x)dx.
(12)

Due to symmetry, the point force and the uniform force only excite odd order modes, while the exponential force excites both the odd and even order modes. For the same total force, point force is much more efficient to excite higher order modes than the other two cases.

For analysis of thermomechanical noise of the DCB, application of equipartition theorem yields the Lorentzian expression of noise power spectral density (PSD) of qj, if Qj is high and j is not very large,
Sqjqj(ω)=kBTωj2Qjkj(ωωj)2+ωj24Qj2,
(13)
where kB 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

When the Si beam is subject to an exponentially decaying optical force distribution, the overlap integral in Eq. (10) can be evaluated as
pj(t)=0Lp(0,t)eαxϕj(x)dx=pnP(0,t)0Leαxϕj(x)dx,
(14)
where the time dependence of p(x) and P(x) is explicitly indicated. The resultant definite integral in Eq. (14) can be evaluated analytically with Eq. (15).
eαxϕj(x)dx=α3ϕj(x)+α2ϕj(x)+αϕj(x)+ϕj(x)λj4α4eαx.
(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:
Hj(ω)=q˜j(ω)p˜j(ω)=1ω2mj+iωkjmjQj+kj,
(16)
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 is
|Hj(ωj)|=|q˜j(ωj)||p˜j(ωj)|=Qjkj.
(17)
Substituting Eq. (14) into Eq. (17), we obtain the expression for normalized local optical force pn shown in Eq. (18), which can be evaluated from experimental results, assuming Qj is high.

pn=10Leαxϕj(x)dxkjQj|q˜j(ωj)||P˜(0,ωj)|.
(18)

3. Device fabrication

The fabrication process for the HPWG, which is illustrated in Fig. 4
Fig. 4 Fabrication process diagram of the HPWG device. All of the EBL processes are done with Vistec EBPG 5000 + system, with which 20 nm alignment precision can be routinely achieved.
, begins with a standard silicon-on-insulator (SOI) wafer with a 220 nm thick top Si layer and a 3 μm thick buried oxide layer. First, a recess in the top Si layer is created by electron beam lithography (EBL) and plasma etching and filled with gold by evaporation and lift-off. In order to reduce the surface roughness of the deposited gold, this step is done with diluted electron beam resist ZEP 520 which is developed with developer solution in a cold bath [36

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]

]. Then, another step of EBL with ultrahigh alignment precision is used to define the Si waveguides. The alignment precision is critical in determining the size of the gap between the waveguide and the gold patch. In the final step, the Si waveguide is plasma etched and released from the substrate by wet etching the buried oxide to form the suspended structure. All of the EBL processes are done with Vistec EBPG 5000 + system, with which 20 nm alignment precision can be routinely achieved.

The fabricated device is shown in Fig. 5
Fig. 5 Images of the fabricated HPWG device. (a) Optical microscope image showing the device overview. (b) Optical microscope image showing the HPWG with gold and suspended Si waveguide. (c) SEM image showing the HPWG with gold and suspended Si waveguide. (d) SEM image showing the gap in the HPWG.
. The device consists of a pair of grating couplers which couples light into and out of the device and a Mach-Zehnder interferometer (MZI) structure which is used to characterize the optical properties of the HPWG and transduce the HPWG motion. HPWG is fabricated in only one of the arms of the MZI. The suspended length of the Si waveguide in HPWG is varied from 10 μm to 30 μm. The typical size of the gap in HPWG varies from 20 nm to 100 nm, depending on the design and the precision of alignment. To achieve longer suspended waveguide or smaller gap is challenging due to built-in stress induced buckling of the Si waveguides or stiction during the wet releasing process. Using the cold development process can noticeably improve the smoothness of the gold structure. The root mean square (RMS) value of the gold surface line edge roughness (LER) is estimated to be about 15 nm.

4. Optical characteristics

In order to confirm the excitation of the HPM, we systematically measured the optical loss in HPWG with varying lengths and gap sizes using the MZI structures. Since the reference arm of the MZI is a low-loss Si waveguide, the optical loss (or the decay constant) in the HPWG arm can be derived from the extinction ratio (ER) of the MZI transmission spectra, using Eq. (19).

α=20Llog10(ER1ER+1)[dB/m].
(19)

In Fig. 6(a)
Fig. 6 Optical characteristics of the fabricated HPWGs with 450 nm wide Si waveguides. The measured HPWG loss is about 30 times higher than theoretical calculation. (a) MZI transmission spectra showing deceasing extinction ratio as the length of the HPWG is increased. The gap is 100 nm. (b) Experimental and theoretical results of the waveguide loss as a function of HPWG length. The gap is 100 nm. (c) Experimental and theoretical results of the waveguide loss normalized to unit HPWG length as a function of the HPWG gap.
, the MZI transmission spectra of HPWG with the same Si waveguide width and gap size but four different lengths are compared, showing decreasing 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 SiO2 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

In order to determine the generated optical force and demonstrate the force enhancement effect, for each device we conducted the thermomechanical noise measurement to calibrate the displacement transduction gain factor and subsequently, driven frequency response measurement to determine the normalized local optical force pn generated in the HPWG.

5.1 Transduction gain factor calibration by thermomechanical noise measurement

The MZI structure is used to transduce the motion of the Si beam in the HPWG. The in-plane motion of the Si beam changes the gap in the HPWG and hence both the real and imaginary parts of the effective index, which further leads to power modulation at the output of the MZI. For a given probe laser wavelength and mechanical mode, this transduction should be linear when the amplitude of the Si beam displacement is small. The transduction gain factor Gj, defined as the derivative of the photodetector output voltage with respect to the amplitude qj of the jth 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]

]. Here we only focus on the first in-plane mechanical mode because it is the most efficient to excite and detect.

5.2 Optical force measurement by driven response

Knowing all the key parameters of the Si beam and the transduction gain, we are ready to use Eq. (18) to measure the normalized local optical force pn except for the last fraction on the right hand side, which can be obtained from driven frequency response measurement.

The driven frequency response of the Si beam is measured with the well-known pump-probe scheme. In addition to the probe laser, a pump laser is sent into the device. The pump laser is power modulated using an electro-optical modulator with the modulation frequency swept near the resonance frequency of the Si beam. It generates a dynamic optical force to excite the in-plane vibration of the Si beam. Meanwhile the mode amplitude of the Si beam is measured by the probe laser at the same wavelength at which the transduction gain factor has been calibrated. A tunable Fabry-Perot filter is used in front of the photodetector to filter out the pump laser. The driven frequency response of the Si beam is thus measured with a network analyzer and converted into the Si beam mode amplitude using the calibrated transduction gain factor. Typical experimental results from three different devices are shown in Fig. 7(b), after normalization for ease of comparison. In the final step, the normalized local optical force is calculated by substituting the experimental results for all the parameters into Eq. (18), where α is calculated from the ER and HPWG length, q˜1(ω1) is extracted from the peak point of the measured driven frequency response and P˜(0,ω1) is carefully calibrated from the measured MZI transmission spectrum.

The measured normalized local optical forces from seven devices of different Si waveguide widths and gaps are plotted in Fig. 7(c), showing good agreement with the theoretical calculation. The horizontal error bars originate from the fact that the actual gaps are not uniform over the entire HPWG length, due to the alignment uncertainty in the EBL process and the slight buckling of Si beams induced by the built-in stress in the top Si layer of SOI wafer. The vertical error bars account for all the uncertainties in the measurement process, including the uncertainties of the actual optical power in the HPWG due to interference effects induced by reflections where mode conversion happens, and the non-uniform gap in the HPWG, which cannot be described by the model developed in this work. In the plot, the relative uncertainty introduced by the two reasons above is estimated to be about 40%. Even though this estimation may be crude because the gap width varies randomly, measurement of multiple devices can help reduce this uncertainty. Thus, the apparent agreement between theoretical and experimental results in Fig. 7(c) confirms the enhanced optical force in HPWG within the experimental precision. When the gap size is as low as 20 nm, the optical force per unit length is determined to be approximately 100 pN/μm/mW, which is 200 times larger than that in a silicon waveguide coupled to silicon dioxide substrate [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]

].

6. Conclusion and discussion

In this work we fabricated HPWG devices and for the first time experimentally characterized their optical and mechanical properties and optical forces. The experimental results confirmed the theoretically predicted optical force enhancement, although the loss is significantly higher than theoretical prediction due to the metal surface roughness and other non-idealities in the fabrication process. Future work can be focused on further reducing the metal surface roughness by techniques such as template stripping [28

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]

]. Despite of its lossy nature, HPWG is a potentially very attractive solution in applications where large optical force is desired in limited device footprint to change device structure and/or circuit topology, such as mechanical memory operation, all-optical signal processing, tunable or reconfigurable optical circuits.

Acknowledgments

This work is supported by the Young Investigator Program (YIP) of AFOSR (Award No. FA9550-12-1-0338). Parts of this work were carried out in the University of Minnesota Nanofabrication Center which receives partial support from NSF through NNIN program, and the Characterization Facility which is a member of the NSF-funded Materials Research Facilities Network via the MRSEC program.

References and links

1.

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

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]

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]

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. 97(24), 243905 (2006). [CrossRef] [PubMed]

8.

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

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.

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]

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]

17.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

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]

20.

X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano 5(4), 2831–2838 (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]

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]

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]

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]

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]

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. 5(7), 1399–1402 (2005). [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]

31.

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

32.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999).

33.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon Press, 1984).

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]

35.

S. Timoshenko, Vibration Problems in Engineering, 2nd ed. (D. Van Nostrand company, Inc., 1937).

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]

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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References

  1. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321(5893), 1172–1176 (2008). [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,” Nature456(7221), 480–484 (2008). [CrossRef] [PubMed]
  3. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009). [CrossRef]
  4. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(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,” Nature459(7246), 550–555 (2009). [CrossRef] [PubMed]
  6. F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009). [CrossRef]
  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.97(24), 243905 (2006). [CrossRef] [PubMed]
  8. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett.107(6), 063602 (2011). [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 Commun3, 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. Express19(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. Photonics3(8), 464–468 (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,” Nature482(7383), 63–67 (2012). [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]
  17. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  18. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(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]
  20. X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano5(4), 2831–2838 (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. Express17(20), 18116–18135 (2009). [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]
  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. Express17(3), 1806–1816 (2009). [CrossRef] [PubMed]
  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. Photonics2(8), 496–500 (2008). [CrossRef]
  28. P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science325(5940), 594–597 (2009). [CrossRef] [PubMed]
  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.5(7), 1399–1402 (2005). [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]
  31. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
  32. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999).
  33. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon Press, 1984).
  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]
  35. S. Timoshenko, Vibration Problems in Engineering, 2nd ed. (D. Van Nostrand company, Inc., 1937).
  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]

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