## A lateral optical equilibrium in waveguide-resonator optical force |

Optics Express, Vol. 21, Issue 21, pp. 25257-25270 (2013)

http://dx.doi.org/10.1364/OE.21.025257

Acrobat PDF (1675 KB)

### Abstract

We consider the lateral optical force between a resonator and a waveguide, and study the possibility of an equilibrium that occurs solely from the optical force in such system. We prove analytically that a single-resonance system cannot give such an equilibrium in the resonator-waveguide force. We then show that two-resonance systems can provide such an equilibrium. We provide an intuitive way to predict the existence of an equilibrium, and give numerical examples.

© 2013 OSA

## 1. Introduction

1. D. Van Thourhout and J. Roels, “Optomechnical device actuation through the optical gradient force,” Nat. Photonics **4**, 211–217 (2010). [CrossRef]

18. G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express **19**, 2782–2790 (2011). [CrossRef] [PubMed]

4. W.H.P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating light-waves in parallel 3D waveguides,” Opt. Express **7**, 16032–16037 (2009). [CrossRef]

9. V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett. **98**, 241104 (2011). [CrossRef]

10. A. Einat and U. Levy, “Analysis of the optical force in the micro ring resonator,” Opt. Express **19**, 20405–20419 (2011). [CrossRef] [PubMed]

13. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. **103**, 223901 (2009). [CrossRef]

14. M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators,” Opt. Express **13**, 8286–8295 (2005). [CrossRef] [PubMed]

18. G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express **19**, 2782–2790 (2011). [CrossRef] [PubMed]

7. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. **30**, 3042–3044 (2005). [CrossRef] [PubMed]

9. V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett. **98**, 241104 (2011). [CrossRef]

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

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

18. G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express **19**, 2782–2790 (2011). [CrossRef] [PubMed]

7. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. **30**, 3042–3044 (2005). [CrossRef] [PubMed]

9. V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett. **98**, 241104 (2011). [CrossRef]

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

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

**19**, 2782–2790 (2011). [CrossRef] [PubMed]

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

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

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

**19**, 2782–2790 (2011). [CrossRef] [PubMed]

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

**98**, 241104 (2011). [CrossRef]

13. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. **103**, 223901 (2009). [CrossRef]

19. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express **15**, 17172–17205 (2007). [CrossRef] [PubMed]

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

7. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. **30**, 3042–3044 (2005). [CrossRef] [PubMed]

**98**, 241104 (2011). [CrossRef]

**1**, 416–422 (2007). [CrossRef]

**1**, 658–665 (2007). [CrossRef]

**19**, 2782–2790 (2011). [CrossRef] [PubMed]

**1**, 658–665 (2007). [CrossRef]

**1**, 658–665 (2007). [CrossRef]

*y*-direction), but we will also briefly discuss the force along the

*x*-direction, and, for an extension of three-dimensional system, the force along the

*z*-direction. We find that a lateral optical equilibrium cannot occur if the resonator supports only a single resonance in the vicinity of the operating frequency. Instead, in order to have an equilibrium, the system has to have at least two resonances that overlap in frequency. We provide detailed discussion of the requirements of these resonances in order to create an optical equilibrium in the waveguide-resonator systems.

## 2. The lack of an optical equilibrium in single-resonance systems

### 2.1. Theory

*ω*

_{0}. The light, at a frequency

*ω*, enters the system through the waveguide, which couples to the resonator with a coupling constant

*γ*. The light can exit from the resonator to the output port of the waveguide. The light can also dissipate through any loss mechanism such as material loss or radiation loss, with the loss rate

_{e}*γ*

_{0}. We will focus on the optical force between the input waveguide and the resonator. By using the coupled-mode theory for a travelling-wave resonance [21], the field enhancement factor

*η*(the ratio between the resonator and input field) and the transmission coefficient

*t*(the ratio between the output and input field) are found to be where Δ =

*ω*−

*ω*

_{0},

*t*is the round trip time in the resonator, and

_{r}*θ*

_{1}and

*θ*

_{2}are arbitrary phase factors depending on the positions along the waveguide where the fields are measured.

*ω*, a stable equilibrium in the lateral direction occurs if there exists a resonator-waveguide separation

*d*

_{0}such that 1) the lateral force is zero at

*d*

_{0}, 2) the force is attractive for

*d*>

*d*

_{0}, and 3) the force is repulsive for

*d*<

*d*

_{0}. Therefore, to study the possibility for a stable optical equilibrium, we need to study how the sign of the force changes in

*d*at a fixed frequency

*ω*. We will show analytically that in fact, at a fixed

*ω*, the force cannot change the sign as

*d*varies. As a result, neither a stable nor unstable equilibrium point can be obtained from a single resonance.

*d*, we need to know how

*γ*and

_{e}*ω*

_{0}explicitly vary with

*d*. Because the resonator and the waveguide couple evanescently, we expect

*γ*and

_{e}*ω*

_{0}to vary with

*d*as follows: where

*κ*is the decaying constant. is a positive constant.

*ω*

_{∞}is the resonance frequency in the absence of the waveguide. Ω is a constant that can be positive, zero, or negative, depending on the resonance’s type, mode, or polarization. Eqs. (12) and (13) will be verified numerically in the next section.

*γ*< 0, hence the first term of Eq. (9), which is antisymmetric with respect to the resonant frequency, shows an attractive force on the lower-frequency side and a repulsive force on the higher-frequency side. However,

_{om}*g*can be either positive or negative, so the second term of Eq. (9), which is symmetric around the resonant frequency, is repulsive if

_{om}*g*< 0 and attractive if

_{om}*g*> 0. As a result, the total force has a lineshape that is asymmetric, with the dominant sign of force depending on the sign on

_{om}*g*. This is illustrated in Fig. 2 where we assume an over-coupling regime (

_{om}*γ*≫

_{e}*γ*

_{0}) to achieve a large resonance contribution to the optical force [10

10. A. Einat and U. Levy, “Analysis of the optical force in the micro ring resonator,” Opt. Express **19**, 20405–20419 (2011). [CrossRef] [PubMed]

13. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. **103**, 223901 (2009). [CrossRef]

*ω*

_{0}nor

*ω*

_{∞}.

*d*, we now solve for the frequency

*ω*where the force vanishes. Using Eqs. (9) and (12)–(15), and after a few lines of algebra, we obtain a remarkable result: In another word, independent of the waveguide-resonator separation, the force always vanishes at the frequency

_{z}*ω*

_{∞}, which is the resonance frequency of the resonator in the absence of the waveguide. Moreover, since

*ω*is independent of

_{z}*d*, at each frequency, the force never changes sign as a function of

*d*. Therefore, one cannot achieve an optical equilibrium in this system. Also, while the results here are derived for a travelling-wave single mode resonator, we note that the form of the force spectrum, i.e., Eq. (9), and the dependency of various resonance parameters on

*d*, i.e., Eqs. (12)–(13), apply to a single-mode standing-wave resonator as well. Therefore, the main conclusion here, that one cannot achieve an optical equilibrium in the waveguide-resonator lateral force using a single-mode resonator, should hold in general.

*γ*and

_{e}*ω*

_{0}on the separation

*d*. This consideration of the lateral force is usually sufficient for on-chip systems because the systems are usually restricted to move only in the lateral direction, while an out-of-plane motion (in the

*z*-direction) and a longitudinal motion (in the

*x*-direction) are negligible or impossible [7

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

**98**, 241104 (2011). [CrossRef]

**1**, 416–422 (2007). [CrossRef]

*y*-direction is the main focus of our paper. However, using the same formalism we can also provide a study of an optical equilibrium in the

*z*-direction. For this purpose, one can rewrite Eqs. (12) and (13) as where

*z*is the out-of-plane relative position between the waveguide and the resonator, and

*d*is the lateral waveguide-resonator separation at

_{f}*z*= 0.

*γ*and

_{om}*g*are then the

_{om}*z*-derivative of

*γ*and

_{e}*ω*

_{0}respectively. Combining these results with Eq. (9) gives the force in the

*z*-direction (

*F*) of This shows that

_{z}*z*= 0 is always an equilibrium position. And the

*z*= 0 position is stable if

*ω*>

*ω*

_{∞}, and unstable if

*ω*<

*ω*

_{∞}. For the longitudinal force in the

*x*-direction, the momentum of the incident photons is transferred to the resonator, so the force on the resonator is in the +

*x*-direction due to the optical scattering force.

### 2.2. Numerical example

*ε*= 12.1 and are surrounded by air with

_{Si}*ε*= 1. Both the waveguide and the ring has a width of 0.2

_{air}*μ*m, which makes the waveguide single-mode. The radius of the semi-circular part of the resonator is 1.5

*μ*m and the straight section is 1.1

*μ*m long. The separation

*d*between the resonator and the waveguide will be varied in the range of 0.1

*μ*m – 0.5

*μ*m. The operating optical frequency corresponds to a free space wavelength near 1.55

*μ*m. We normalize the frequency to 2

*πc/a*, where

*c*is the light speed in vacuum, and

*a*= 1

*μ*m. Hence a free space wavelength of 1.55

*μ*m corresponds to an angular frequency of 0.65 × 2

*πc/a*. For simplicity, we consider a 2D system in TM mode (out-of-plane electric field and in-plane magnetic field). The simulation of this system is done using a finite-element frequency-domain method with a commercial software Comsol [23

23. www.comsol.com

*T*= |

*t*|

^{2}as a function of frequency

*ω*for several

*d*in Fig. 3(a). We can see that the transmission spectrum has a symmetric dip as expected. We can also deduce that

*g*is negative for this system because the resonance frequency decreases as

_{om}*d*increases.

*γ*,

_{e}*γ*

_{0}and

*ω*

_{0}from the transmission spectrum using Eq. (2) for each value of

*d*. We plot these parameters with respect to

*d*in Fig. 3(b). We see that

*γ*

_{0}is constant in

*d*, and

*γ*and |

_{e}*ω*

_{0}−

*ω*

_{∞}| decay exponentially with approximately the same decay constant (21.8/

*a*and 21.3/

*a*respectively), which verify the assumption of an evanescent coupling in Eqs. (12) and (13). We also found

*ω*

_{∞}= 0.655

*×*2

*πc/a*.

*F*is normalized to the unit of 1/

*c*. We can see that the force spectra are indeed asymmetric, with a small attractive force on the lower-frequency side and a large repulsive force on the higher-frequency side, as expected from Eq. (9) and the sign of

*g*. Also, while the frequency where the maximum force occurs shifts significantly as

_{om}*d*varies, we see that the force zero

*ω*does not change as

_{z}*d*changes, as expected from our theory. Note that the numerical values of the force should remain approximately the same if we consider instead the three-dimensional case where both the waveguide and the resonator have an equal finite thickness in the

*z*-direction. This is because the

*z*-dependence of the local force and the local power density should be the same, so this

*z*-dependence cancels out in the calculation of the force per input power.

*F*as a function of

*ω*and

*d*in Fig. 3(d). We can see that in the regime where the resonance lineshape is nearly perfect Lorentzian, and hence the coupled mode theory for a single resonance is valid, we indeed observe near independence of the zero-force frequency as a function of

*d*, in spite of the significant variation of the resonant frequency as a function of

*d*. As a result, no equilibrium exists. At a smaller

*d*, however,

*ω*changes slightly as

_{z}*d*changes. This occurs because at a small

*d*, the linewidth of the resonance is sufficiently large such that there is an additional contribution from the adjacent resonances, and as a result our assumption of having a single resonance no longer applies. Even in these cases where

*ω*does vary as a function of

_{z}*d*, this deviation in

*ω*is much smaller than the linewidth at those

_{z}*d*, and also smaller than the change in

*ω*

_{0}as

*d*changes, because the adjacent resonances are far away in frequencies. The analysis here therefore indicates that in order to achieve optical equilibrium in the waveguide-resonator system, one needs the resonator to support at least two resonances that are in a close proximity to each other in frequency.

## 3. Creating an optical equilibrium using two resonances

*i*are the parameters for each resonance. This expression assumes that the coherent interaction between the two resonances are weak such that we can neglect the cross-term when the power is calculated, which is an approximation. Nevertheless, as we will see in the numerical simulation, this expression does provide a reasonable explanation of the two-resonance case that we consider here. As a result, the exact formula of the force in a two-resonance case is very complicated and difficult to understand intuitively. Therefore, the goal of this section is to give an intuitive understanding of the lineshape of the force spectrum for each resonance. Guided by this intuition, we then design structures that can achieve optical equilibrium in waveguide-resonator systems.

### 3.1. Theory

*g*< 0. We assume that the two resonances are close to each other in frequency. For each resonance, the frequency

_{om}*ω*where the force vanishes for each resonance does not vary as

_{z}*d*changes, and the force is attractive for

*ω*<

*ω*and repulsive for

_{z}*ω*>

*ω*. Therefore, by assuming that the total force is the sum of the contributions from the two resonances, we see that the optical equilibrium can only occur at the frequencies between

_{z}*ω*’s of the two resonances (shaded region in Fig. 4), where the contributions from the two resonances are in opposite directions. Next, as shown in Fig. 2, a

_{z}*g*< 0 resonance, which has an asymmetric lineshape in its force spectrum, has a repulsive side with a larger peak amplitude, and an attractive side with a smaller peak amplitude. Also, as

_{om}*d*increases, this repulsive peak narrows significantly, while the attractive peak changes less significantly in its width. With the arrangement in Fig. 4(a), the force between the two

*ω*’s is a sum of the contributions of the repulsive side of the lower frequency resonance, and the attractive side of the higher frequency resonance. Therefore, the way each resonance varies with

_{z}*d*as described above can lead to a change in the sign of force. The total force should change from repulsive to attractive as

*d*increases, creating a stable optical equilibrium. With a similar argument, we can see that a system with two

*g*gt; 0 resonances, as shown in Fig. 4(b), results in an unstable optical equilibrium.

_{om}### 3.2. First example: a bumped ring resonator

*μ*m, at the top of the resonator as shown in Figs. 5(a)–(b). The ring resonator in Sec. 2 supports a pair of degenerate travelling-wave resonances. An incident wave in the waveguide from one side couples only to one of these two travelling wave resonances, and as a result the system can be described as a single-mode resonator. In the presence of the bump, the two travelling wave resonances in the ring couple to each other to form two standing wave resonances, with intensity patterns shown in Fig. 5(a) and Fig. 5(b). We see that one of the standing wave resonances have an intensity node, while the other has an intensity anti-node, at the position of the bump. Both resonances now couple to the waveguide, as can be seen in the transmission spectra for a range of the waveguide-resonator separation

*d*at Fig. 5(c). All these spectra exhibit two resonance dips. Since both resonances shift to smaller frequencies as

*d*increases, we deduce that

*g*< 0 for both resonances. Therefore, the resonances in this system have the characteristics of what is required in Fig. 4(a).

_{om}*d*= 0.18

*μ*m, and

*d*= 0.25

*μ*m, were shown in Fig. 5(d). We see that around the frequency

*ω*= 0.6549 × 2

_{e}*πc/a*(marked with the vertical line), the system indeed exhibit a stable optical equilibrium, with the force changes from repulsive to attractive, at a fixed frequency, as

*d*increases. To emphasize this fact, we plot the force as a function of

*d*at the frequency

*ω*in Fig. 5(e). This clearly shows a stable optical equilibrium at

_{e}*d*

_{0}= 0.21

*μ*m; at this frequency, as

*d*moves away from

*d*

_{0}, the optical force always points towards

*d*=

*d*

_{0}. We also plot the sign of

*F*as a function of

*ω*and

*d*in Fig. 5(f), which shows that there exists a stable optical equilibrium over the frequency range of 0.6546 × 2

*πc/a*to 0.6550 × 2

*πc/a*(shown as the bracket in the figure).

*ω*, at smaller

_{e}*d*, the repulsive force contribution from the lower-frequency resonance dominates over the attractive force contribution of the higher-frequency resonance. On the other hand, as

*d*increases, the contribution from the lower-frequency resonance decays faster than the higher-frequency resonance. As a result, the force changes from repulsive to attractive as

*d*increases, creating a stable optical equilibrium. In this case, the fast decay of the contribution of the lower-frequency resonance comes not only from the fact that it has

*g*< 0 and therefore has a fast-narrowing repulsive peak as indicated in Fig. 4(a), but also that in this case, the lower-frequency resonance enters the under-coupling regime (where the force is small [10

_{om}10. A. Einat and U. Levy, “Analysis of the optical force in the micro ring resonator,” Opt. Express **19**, 20405–20419 (2011). [CrossRef] [PubMed]

**103**, 223901 (2009). [CrossRef]

*d*than the higher-frequency resonance. This fact can be deduced from the plot of

*T*in Fig. 5(c) because in an under-coupling regime, the dip in

*T*spectrum becomes shallower as

*d*increases.

### 3.3. Second example: using two families of traveling-wave modes in a single ring resonator

*μ*m and an outer radius of 2.15

*μ*m). This resonator can support two families of travelling-wave modes, with the intensity patterns shown in Fig. 6(a)–(b). The first-order modes (Fig. 6(a)) have no intensity node inside the ring. The second-order modes (Fig. 6(b)) have an intensity node inside the ring. By studying the force lineshape of each mode (not plotted), we found that the first-order modes have

*g*≈ 0 while the second-order modes have

_{om}*g*< 0. The positions of these two families of modes in frequency are shown in Fig. 6(c), where the ratio of the energy inside the ring to input power (

_{om}*E*), in the unit of

*a/c*, is plotted as a function of frequency at

*d*= 0.18

*μ*m. We can see that in the two frequency ranges of 0.642 × 2

*πc/a*to 0.648 × 2

*πc/a*and 0.663 × 2

*πc/a*to 0.668 × 2

*πc/a*, there are two modes, one from each of the two families, that overlap in frequency. Based on the arguments above we will therefore seek to find an optical equilibrium in these ranges. Moreover, the relative positions in the frequency of the two modes are different in each range; in the first range, the second-order mode is at a higher frequency, but in the second range, the first-order mode is at a higher frequency. Therefore, these two ranges of frequency provide an interesting contrast of how different resonances affect the creation of an optical equilibrium.

*πc/a*to 0.668×2

*πc/a*, where the second-order mode is at a lower frequency than the first-order mode as shown in Fig. 7(a). In this system, the

*g*< 0 resonance has the lineshape that is very asymmetric with a very high repulsive peak; therefore, the contribution from this repulsive peak near

_{om}*ω*= 0.665 × 2

_{e}*πc/a*decreases significantly as

*d*increases. This drop in the repulsive force, combining with the contribution from the attractive side in the lineshape of the high-frequency first-order mode, creates a stable optical equilibrium. This is shown in Fig. 7(a) where, near

*ω*, the force changes from repulsive to attractive as

_{e}*d*increases. The plot of

*F*as a function of

*ω*and

*d*in Fig. 7(b) also shows a stable optical equilibrium occurring in the frequency range shown by the bracket.

*πc/a*to 0.648×2

*πc/a*, where the second-order mode is at a higher frequency than the first-order mode (Fig. 8). In this case, the

*g*< 0 resonance (second-order mode) is at a higher frequency. Because the strong repulsive contribution of this resonance does not lie between the two resonances, this system lacks the large swing in the force that is required for the change in the sign of the force. As a result, an optical equilibrium does not occur in this system. This is shown in Fig. 8(a) and 8(b), where no change in the sign of the force is observed.

_{om}*g*, which is the derivative of the coupling

_{om}*γ*with respect to

_{e}*d*, the resonator with low intrinsic loss

*γ*

_{0}is still desired even though

*γ*

_{0}does not directly affect the existence of an optical equilibrium. This is because the magnitude of the optical force decays quickly in

*d*if

*d*is large enough that

*γ*

_{0}>

*γ*(under-coupling regime) [11

_{e}11. V. Intaraprasonk and S. Fan, “Enhancing the waveguide-resonator optical force with an all-optical on-chip analog of electromagnetically induced transparency,” Phys. Rev. A **86**, 063833 (2012). [CrossRef]

*γ*

_{0}should be low enough that the system is still in the over-coupling regime at the equilibrium distance.

## 4. Summary and conclusions

## Acknowledgments

## References and links

1. | D. Van Thourhout and J. Roels, “Optomechnical device actuation through the optical gradient force,” Nat. Photonics |

2. | P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express |

3. | V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express |

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

5. | J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. |

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

9. | V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett. |

10. | A. Einat and U. Levy, “Analysis of the optical force in the micro ring resonator,” Opt. Express |

11. | V. Intaraprasonk and S. Fan, “Enhancing the waveguide-resonator optical force with an all-optical on-chip analog of electromagnetically induced transparency,” Phys. Rev. A |

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

13. | M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. |

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

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

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

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

18. | G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express |

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

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

21. | H. A. Haus, |

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

23. |

**OCIS Codes**

(230.4555) Optical devices : Coupled resonators

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

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: July 15, 2013

Revised Manuscript: September 22, 2013

Manuscript Accepted: September 22, 2013

Published: October 15, 2013

**Citation**

Varat Intaraprasonk and Shanhui Fan, "A lateral optical equilibrium in waveguide-resonator optical force," Opt. Express **21**, 25257-25270 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25257

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### References

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