## Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials |

Optics Express, Vol. 20, Issue 20, pp. 22372-22382 (2012)

http://dx.doi.org/10.1364/OE.20.022372

Acrobat PDF (1886 KB)

### Abstract

Here we demonstrate that giant transverse optical forces can be generated in nanoscale slot waveguides of hyperbolic metamaterials, with more than two orders of magnitude stronger compared to the force created in conventional silicon slot waveguides, due to the nanoscale optical field enhancement and the extreme optical energy compression within the air slot region. Both numerical simulation and analytical treatment are carried out to study the dependence of the optical forces on the waveguide geometries and the metamaterial permittivity tensors, including the attractive optical forces for the symmetric modes and the repulsive optical forces for the anti-symmetric modes. The significantly enhanced transverse optical forces result from the strong optical mode coupling strength between two metamaterial waveguides, which can be explained with an explicit relation derived from the coupled mode theory. Moreover, the calculation on realistic metal-dielectric multilayer structures indicates that the predicted giant optical forces are achievable in experiments, which will open the door for various optomechanical applications in nanoscale, such as optical nanoelectromechanical systems, optical sensors and actuators.

© 2012 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]

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]

4. 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]

5. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high-Q
double-layer cavities of photonic crystal slabs,” Phys. Rev.
Lett. **97**(2), 023903 (2006). [CrossRef] [PubMed]

6. S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal
resonators,” Nano Lett. **10**(1), 99–104
(2010). [CrossRef] [PubMed]

7. A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in
sub-wavelength slot waveguides,” Nature **457**(7225), 71–75
(2009). [CrossRef] [PubMed]

*Q*optical resonators where the circulating optical power is considerably amplified due to the long photon lifetime [8

8. M. L. Povinelli, S. G. Johnson, M. Lonèar, 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]

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

10. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void
nanostructure,” Opt. Lett. **29**(11), 1209–1211
(2004). [CrossRef] [PubMed]

11. 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]

12. 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]

13. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high
index of refraction,” Phys. Rev. Lett. **94**(19), 197401 (2005). [CrossRef] [PubMed]

15. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive
index,” Nature **470**(7334), 369–373
(2011). [CrossRef] [PubMed]

16. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: far-field imaging beyond the diffraction
limit,” Opt. Express **14**(18), 8247–8256
(2006). [CrossRef] [PubMed]

19. Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh
refractive indices,” J. Opt. Soc. Am. B **29**(9), 2559–2566
(2012). [CrossRef]

11. 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]

20. W.-P. Huang, “Coupled-mode theory for optical waveguides: an
overview,” J. Opt. Soc. Am. A **11**(3), 963–983
(1994). [CrossRef]

21. A. Ashkin, “Acceleration and trapping of particles by radiation
pressure,” Phys. Rev. Lett. **24**(4), 156–159
(1970). [CrossRef]

22. S. Chu, “Laser manipulation of atoms and particles,”
Science **253**(5022), 861–866
(1991). [CrossRef] [PubMed]

23. H. Cai, K. J. Xu, A. Q. Liu, Q. Fang, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Nano-opto-mechanical actuator driven by gradient optical
force,” Appl. Phys. Lett. **100**(1), 013108 (2012). [CrossRef]

24. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical
oscillators,” Nat. Phys. **5**(12), 909–914
(2009). [CrossRef]

## 2. Deep mode confinement and giant optical force

*L*= 40 nm and height

_{x}*L*= 30 nm are separated with a nanoscale air gap

_{y}*g*along the

*y*direction. In each waveguide, the metamaterial is constructed with alternative thin layers of silver (Ag) and germanium (Ge). The multilayer metamaterial can be regarded as a homogeneous effective medium and the principle components of the permittivity tensor can be determined from the effective medium theory (EMT) [25

25. J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical
anisotropy,” Appl. Phys. Lett. **89**(26), 261102 (2006). [CrossRef]

*f*

_{m}is the volume filling ratio of silver,

*ε*

_{d}and

*ε*

_{m}are the permittivity of germanium and silver, respectively.

*ε*

_{d}= 16, and

*ε*

_{m}(

*ω*) =

*ε*

_{∞}-

*ω*

_{p}

^{2}/(

*ω*

^{2}+ i

*ωγ*) from the Drude model, with a background dielectric constant

*ε*

_{∞}= 5, plasma frequency

*ω*

_{p}= 1.38 × 10

^{16}rad/s and collision frequency

*γ*= 5.07 × 10

^{13}rad/s. Figure 1(b) shows the dependence of the permittivity tensor on the silver filling ratio

*f*

_{m}at the telecom wavelength

*λ*

_{0}= 1.55 μm. All the components of the permittivity tensor will grow in magnitude as the filling ratio increases. For instance, the permittivity tensors of hyperbolic metamaterial are

*ε*= 29.2 + 0.12i,

_{y}*ε*=

_{x}*ε*= −39.8 + 2.1i for

_{z}*f*

_{m}= 0.4, and

*ε*= 76.3 + 1.4i,

_{y}*ε*=

_{x}*ε*= −81.7 + 3.6i for

_{z}*f*

_{m}= 0.7, respectively. It should be noted that the wavelength

*λ*

_{0}= 1.55 μm is merely used as an example throughout the paper, and this design can actually work in a broadband frequency range due to the non-resonant nature of hyperbolic metamaterials [26

26. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium
made of metallic nanowires in the visible region,” Opt.
Express **16**(20), 15439–15448
(2008). [CrossRef] [PubMed]

*M*

_{s}) and the other is the anti-symmetric mode (denoted by

*M*

_{a}). Both of the two modes are TM-like (in which

*H*,

_{x}*E*and

_{y}*E*components are dominant), so an incident light with

_{z}*E*polarization is necessary to efficiently excite these two modes. The effective indices

_{y}*n*

_{eff}

*≡*

_{,z}*k*

_{z}/

*k*

_{0}and the propagation length

*L*

_{m}≡ 1/2Im(

*k*) corresponding to the two eigenmodes are obtained by finite-element method (FEM) with the software package COMSOL (where

_{z}*k*

_{0}is the wave vector in free space and

*k*is the wave vector along the propagation direction

_{z}*z*). Figure 1(c) and (d) show that the dependences of

*n*

_{eff}

*and*

_{,z}*L*

_{m}on the gap sizes are distinct for the two eigenmodes. As the gap size

*g*shrinks,

*n*

_{eff}

*grows dramatically for mode*

_{,z}*M*

_{s}but decreases slightly for mode

*M*

_{a}. In fact, the magnitude of the effective index variation is equivalent to the mode coupling strength between two identical waveguides [12

12. 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]

*M*

_{s}and a weak coupling strength for mode

*M*

_{a}. Furthermore, the opposite effective index variations for two eigenmodes indicate an attractive force for mode

*M*

_{s}and a repulsive force for mode

*M*

_{a}. The propagation length

*L*

_{m}decreases for mode

*M*

_{s}and increases for mode

*M*

_{a}as the gap size gets narrower, due to the tradeoff between the optical mode confinement and the propagation loss. The effective index and the propagation length for the unperturbed mode of the individual waveguide are shown in Fig. 1(c) and (d) for comparison (denoted by

*M*

_{0}).

*E*,

_{y}*H*and

_{x}*S*for slot waveguides with

_{z}*g*= 10 nm are shown in Fig. 2 . As can be seen from Fig. 2(a), distinct behaviors in electric field

*E*are obtained for modes

_{y}*M*

_{s}and

*M*

_{a}, where a strong (weak) electric field is localized in the gap region for

*M*

_{s}(

*M*

_{a}) mode. It has been proposed that the optical field could be tightly confined and greatly enhanced in the nanoscale slot region due to the large discontinuity of normal electric fields (

*E*in our case) at the high-index-contrast interface [10

_{y}10. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void
nanostructure,” Opt. Lett. **29**(11), 1209–1211
(2004). [CrossRef] [PubMed]

*n*

_{eff,}

*and the optical forces through the integration of Maxwell’s stress tensor for both the symmetric mode and the anti-symmetric mode.*

_{z}*n*

_{eff,}

*for the symmetric mode increases noticeably as the gap size shrinks [see Fig. 3(a)], implying a strong mode coupling strength between the two waveguides. Accordingly, the attractive optical force for the symmetric mode grows dramatically with the decreased gap sizes, resulting in optical forces up to 8 nNμm*

_{z}^{−1}mW

^{−1}for

*L*= 30 nm and 4 nNμm

_{y}^{−1}mW

^{−1}for

*L*= 80 nm [see Fig. 3(b)], over two orders of magnitude larger than that in a dielectric slot waveguide [11

_{y}11. 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]

*n*

_{eff,}

*for the anti-symmetric modes show negligible variation with gap sizes [see Fig. 3(c)], so that the repulsive optical forces for the anti-symmetric modes just increase slightly when the gap size shrinks [see Fig. 3(d)]. As a result, optical forces for the anti-symmetric mode are much weaker than that of symmetric modes, in sharp contrast to the case in dielectric slot waveguides, where the optical forces for the symmetric mode and the anti-symmetric mode are comparable in magnitude [4*

_{z}4. 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]

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

28. W. H. P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating lightwaves
in parallel 3D waveguides,” Opt. Express **17**(18), 16032–16037
(2009). [CrossRef] [PubMed]

## 3. Mode coupling analysis

*n*

_{eff,}

*=*

_{z}*k*/

_{z}*k*

_{0}>> 1, we obtain the following analytical expression for the optical forces in the slot waveguides of hyperbolic metamaterials:

*c*is the speed of light in vacuum. Figure 3 also plots the analytically derived effective indices

*n*

_{eff,}

*and optical forces*

_{z}*f*

_{opt}, which match the numerical FEM simulation results quite well. The critical dependence of the optical forces on the gap size

*g*can be explained using Eq. (4). The constructive (destructive) interference of exponentially decayed evanescent optical fields in the slot region gives rise to hyperbolic-sine-function (hyperbolic-cosine-function) dependence on gap sizes, resulting in strong (weak) transverse optical forces for the symmetric (anti-symmetric) modes as the gap size gets small. When the gap sizes becomes larger (

*g*> 20 nm), the two waveguides can only couple weakly with each other through the tails of the evanescent optical fields, so that

*f*

_{opt}becomes quite weak for both eigenmodes. According to Eq. (4), the waveguide height

*L*, the mode indices

_{y}*n*

_{eff,}

*and the permittivity*

_{z}*ε*and

_{y}*ε*also affect the magnitude of optical forces.

_{z}*f*

_{m}of hyperbolic metamaterials for slot waveguides of

*L*= 30 nm with different gap sizes

_{y}*g*= 1 nm and

*g*= 3nm. The effective indices

*n*

_{eff,}

*at different filling ratios are shown in Fig. 4(a) and (c) for the symmetric modes and the anti-symmetric modes, respectively. It is noted that ultra-high refractive indices can be achieved for both low*

_{z}*f*

_{m}(< 0.3) and high

*f*

_{m}(> 0.75) in the metamaterial slot waveguides. A high

*f*

_{m}leads to a large

*ε*, and a low

_{y}*f*

_{m}gives a large ratio of

_{εy/|εz|}in hyperbolic metamaterials, both of which can result in high refractive indices along the propagation direction. However, the optical forces turn out to be large for metamaterial slot waveguides with low

*f*

_{m}, for both the symmetric and anti-symmetric modes, as shown in Fig. 4(b) and (d). This phenomenon can also be explained with Eq. (4), where optical forces are more sensitive to |

*ε*| (determined by the filling ratio

_{z}*f*

_{m}) other than the effective indices

*n*

_{eff,}

*, due to the hyperbolic-function terms. Figure 4 shows that the optical forces for the symmetric modes are always significantly higher than those for the anti-symmetric modes. Moreover, a smaller gap size (*

_{z}*g*= 1 nm) corresponds to a stronger optical force compared with the case of a larger gap size (

*g*= 3nm). All these results are consistent with the previous calculation in Fig. 3.

## 4. Broadband operation

*λ*

_{0}from 1 μm to 2 μm, which verify the broadband nature of our design. The optical forces for the anti-symmetric modes are relatively weak and drop very fast as the operating wavelength increases [Fig. 6(b)].

## 5. Realistic multilayer structures

*f*

_{m}= 0.4. The comparison between the multilayer structures and the ideal effective medium is shown in Fig. 7 . With the height of

*L*= 80 nm, the slot waveguides of multilayer structures can reproduce the results of the effective indices

_{y}*n*

_{eff,}

*and the optical forces*

_{z}*f*

_{opt}calculated based on the waveguides of ideal effective media, for both the symmetric modes and the anti-symmetric modes. While with

*L*= 30 nm, both

_{y}*n*

_{eff,}

*and*

_{z}*f*

_{opt}become smaller than the results of the effective media calculation, which is due to that the large wave vector along the

*y*direction becomes close to the Brillouin zone of periodic multilayer structures, so that the waveguide mode profiles begin to deviate from those predicted from the effective medium theory.

*f*

_{opt}(

*z*) =

*f*

_{opt}(

*z*= 0)∙exp[-2Im(

*n*

_{eff,z})

*k*

_{0}

*z*], where the optical force is normalized to the incident optical power at

*z*= 0 [12

12. 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]

## 6. Conclusions

^{−1}mW

^{−1}in nanoscale slot waveguides of hyperbolic metamaterials, due to the strong optical field confinement and extreme optical energy compression within the air slot region. The influences of the waveguide geometries and the metal filling ratios of the hyperbolic metamaterials on both the symmetric modes and the anti-symmetric modes are studied numerically using the Maxwell stress tensor integration method, together with an analytical approach using a 2D approximation of the 3D coupled waveguides. Furthermore, the relation between the transverse optical forces and the waveguide mode coupling strength is derived from the coupled mode theory, revealing the mechanism of optical force enhancement in slot waveguide system. Finally, it is shown that the predicted giant optical force is achievable in realistic metal-dielectric multilayer structures. The strongly enhanced optical forces in slot waveguides of hyperbolic metamaterial will open a new realm in many exciting nanoscale optomechanical applications.

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

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

5. | M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high-Q
double-layer cavities of photonic crystal slabs,” Phys. Rev.
Lett. |

6. | S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal
resonators,” Nano Lett. |

7. | A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in
sub-wavelength slot waveguides,” Nature |

8. | M. L. Povinelli, S. G. Johnson, M. Lonèar, 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 |

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

10. | V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void
nanostructure,” Opt. Lett. |

11. | 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. |

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

13. | J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high
index of refraction,” Phys. Rev. Lett. |

14. | J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective
refractive index over a broad bandwidth,” Phys. Rev. Lett. |

15. | M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive
index,” Nature |

16. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: far-field imaging beyond the diffraction
limit,” Opt. Express |

17. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited
objects,” Science |

18. | J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of
indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. |

19. | Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh
refractive indices,” J. Opt. Soc. Am. B |

20. | W.-P. Huang, “Coupled-mode theory for optical waveguides: an
overview,” J. Opt. Soc. Am. A |

21. | A. Ashkin, “Acceleration and trapping of particles by radiation
pressure,” Phys. Rev. Lett. |

22. | S. Chu, “Laser manipulation of atoms and particles,”
Science |

23. | H. Cai, K. J. Xu, A. Q. Liu, Q. Fang, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Nano-opto-mechanical actuator driven by gradient optical
force,” Appl. Phys. Lett. |

24. | G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical
oscillators,” Nat. Phys. |

25. | J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical
anisotropy,” Appl. Phys. Lett. |

26. | Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium
made of metallic nanowires in the visible region,” Opt.
Express |

27. | J. D. Jackson, |

28. | W. H. P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating lightwaves
in parallel 3D waveguides,” Opt. Express |

29. | 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 |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(160.3918) Materials : Metamaterials

(250.5403) Optoelectronics : Plasmonics

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: June 15, 2012

Revised Manuscript: August 29, 2012

Manuscript Accepted: August 30, 2012

Published: September 14, 2012

**Citation**

Yingran He, Sailing He, Jie Gao, and Xiaodong Yang, "Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials," Opt. Express **20**, 22372-22382 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22372

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### References

- T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321(5893), 1172–1176 (2008). [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,” Nature456(7221), 480–484 (2008). [CrossRef] [PubMed]
- 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]
- M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3(8), 464–468 (2009). [CrossRef]
- M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett.97(2), 023903 (2006). [CrossRef] [PubMed]
- S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal resonators,” Nano Lett.10(1), 99–104 (2010). [CrossRef] [PubMed]
- A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature457(7225), 71–75 (2009). [CrossRef] [PubMed]
- M. L. Povinelli, S. G. Johnson, M. Lonèar, 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. Express13(20), 8286–8295 (2005). [CrossRef] [PubMed]
- G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009). [CrossRef] [PubMed]
- V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett.29(11), 1209–1211 (2004). [CrossRef] [PubMed]
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