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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Grover Swartzlander
  • Vol. 30, Iss. 3 — Mar. 1, 2013
  • pp: 736–742
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Coupled-mode theory analysis of optical forces between longitudinally shifted periodic waveguides

Yue Sun, Thomas P. White, and Andrey A. Sukhorukov  »View Author Affiliations


JOSA B, Vol. 30, Issue 3, pp. 736-742 (2013)
http://dx.doi.org/10.1364/JOSAB.30.000736


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Abstract

We develop a coupled-mode theory that describes the dependence of optical gradient forces between side-coupled periodic waveguides on the longitudinal shift between the waveguides. Our approach is fully applicable to waveguides with a strong refractive-index modulation and in the regime of slow-light enhancement of optical forces, associated with the group-velocity reduction at the photonic band edge. Our method enables fast calculation of both the transverse and longitudinal forces for all longitudinal shifts, based on numerical simulations of mode profiles only at particular shift values. We perform a comparison with direct numerical simulations for photonic-crystal nanowire waveguides and demonstrate that our approach provides very accurate results for the slow-light enhanced transverse and longitudinal forces, accounting for the key features of force suppression and sign reversal at critical shift values.

© 2013 Optical Society of America

1. INTRODUCTION

Direct numerical calculation of the optical forces is a computationally intensive task. Analytical and semi-analytical approaches can allow faster force calculation under the presence of structure deformations, additionally providing an important insight into the properties and generic features of the optical forces. One powerful analytical approach is based on the relation between the optical force and the deformation-induced optical mode energy change [9

9. 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]

,10

10. P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17, 18116–18135 (2009). [CrossRef]

], which can be proved under the conditions of energy and photon-number conservation. Recently, this approach has been generalized to determine a relation between the optical force and the optical mode’s scattering properties of the photonic structure [11

11. Z. Wang and P. Rakich, “Response theory of optical forces in two-port photonics systems: a simplified framework for examining conservative and non-conservative forces,” Opt. Express 19, 22322–22336 (2011). [CrossRef]

]. However, this semi-analytical approach requires the knowledge of the optical response defined through the scattering matrix at a range of frequencies. Therefore, this approach cannot be directly applied to the calculation of force dependence on the structural parameters, such as the longitudinal shift between side-coupled modulated waveguides.

In this work, we develop a coupled-mode theory (CMT) to describe the optical modes and the associated gradient forces between side-coupled periodic waveguides. The key challenge that we address is that strong optomechanical effects can be observed for modulated nanoscale waveguides with arrays of perforated holes, where the periodic refractive-index modulation is accordingly very strong. However, the CMT is conventionally formulated for the case of a low-contrast modulation, such as Bragg grating waveguides [12

12. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 1988).

]. We develop a method for determining the effective CMT parameters based on numerical dispersion calculations for a single longitudinal shift, and then use CMT to describe the modes for arbitrary shifts. This allows very fast calculation of forces for all longitudinal shifts. Because the developed coupled-mode equations are very general, we essentially demonstrate that the effect of the longitudinal shift on the optical forces has a universal nature due to the hybridization of odd and even modes. We demonstrate that CMT provides very accurate results by performing a direct comparison with numerical results reported in [7

7. Y. Sun, T. P. White, and A. A. Sukhorukov, “Slow-light enhanced optical forces between longitudinally shifted photonic-crystal nanowire waveguides,” Opt. Lett. 37, 785–787(2012). [CrossRef]

].

The paper is organized as follows. In Section 2, we develop a CMT for side-coupled periodic waveguides with strong refractive-index modulation and present a method for determining the effective coupled-mode coefficients in the slow-light regime. Then in Section 3, we use the CMT to determine the dependence of transverse and longitudinal forces acting on the waveguides in terms of the coupled-mode amplitudes and fixed force coefficients calculated for specific longitudinal shifts. We show that the CMT correctly predicts the nontrivial force dependence on the longitudinal waveguide shift, in excellent agreement with direct numerical simulations. Finally, we present conclusions in Section 4.

2. MODE DISPERSION AND CMT

In this section we formulate a CMT to describe the optical modes of the coupled nanowire waveguides shown in Fig. 1 as a function of the longitudinal shift, Δx. We begin with describing the coupling of the even and odd supermodes of the structure, then change into the basis of the individual waveguide modes to apply a previously derived CMT formulation for low-contrast Bragg gratings [13

13. S. Ha and A. A. Sukhorukov, “Nonlinear switching and reshaping of slow-light pulses in Bragg-grating couplers,” J. Opt. Soc. Am. B 25, C15–C22 (2008). [CrossRef]

]. A similar approach was introduced previously to calculate the coupling coefficients between the forward modes of homogeneous nanowire waveguides [14

14. C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A 78, 033818(2008). [CrossRef]

,15

15. C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express 17, 5879–5884 (2009). [CrossRef]

]; however, in our case we have to consider a more complex situation due to the grating-induced coupling between the forward and backward modes. Whereas we have previously described the mode dispersion in coupled periodic waveguides using an asymptotic perturbation theory [16

16. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B 25, C65–C74 (2008). [CrossRef]

], such an approach produces quantitatively correct results only in the limit of weak coupling between the waveguides. Here, we develop an approach to obtain accurate description of mode properties even for strongly coupled waveguides.

Fig. 1. (a) 3D sketch of the longitudinally shifted side-coupled photonic-crystal waveguides. The waveguides are identical, and each is patterned with a regular array of rectangular airholes of period a. (b) Top view of the waveguide geometry.

As a starting point, we consider nanowires with no relative shift (Δx=0). In this case the structure is symmetric with respect to the reflection transformation yy, and its supermodes should satisfy the symmetry relations [17

17. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

]
(ExEyEzHxHyHz)(x,y,z;ω)=±(ExEyEzHxHyHz)(x,y,z;ω),
(1)
where the upper sign “+” corresponds to even and lower “” to odd mode, which in the following we will denote by superscripts “(e)” and “(o),” accordingly.

Our goal is to describe the optical modes in the slow-light regime, which is realized close to the photonic band edge [18

18. J. B. Khurgin and R. S. Tucker, eds., Slow Light: Science and Applications (Taylor & Francis, 2009).

]. We consider the lowest-order band edges, which for unshifted nanowires appear at the edge of the Brillouin zone, with wavenumber Kb=π/a and corresponding frequencies ωb(e) and ωb(o). A powerful method to describe the modes close to photonic band edges is based on the CMT, where the optical field is represented as a superposition of forward- and backward-propagating modes [12

12. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 1988).

]. Whereas this approach is rigorously derived for periodic structures with a weak refractive-index modulation such as Bragg gratings, we find that CMT can be successfully applied to nanowires where the modulation is large.

We first discuss the CMT formulation for unshifted nanowires, where even and odd modes can be considered separately. Due to the structure periodicity, the mode profiles satisfy the Bloch theorem such that
{E,H}(x+a,y,z;ω)={E,H}(x,y,z;ω)exp(iKx),
(2)
where K is the Bloch wavenumber. Alternatively, in CMT the fields are conventionally represented as a superposition of counterpropagating modes with wavenumbers offset by a reciprocal lattice vector 2π/a=2Kb,
{E,H}(x,y,z;K)=exp[i(KKb)x][A(f)(K){E(f),H(f)}(x,y,z)exp(iKbx)+A(b)(K){E(b),H(b)}(x,y,z)exp(iKbx)],
(3)
where the superscripts “(f)” and “(b)” correspond to the forward and backward modes, respectively, and A(f,b) are the mode amplitudes. Within the framework of CMT, the mode profiles {E(f,b),H(f,b)} are assumed to be fixed (frequency and wavenumber independent), which is justified within a sufficiently narrow frequency range, in particular close to a bandgap. Because the nanowires possess a double reflection symmetry xx and yy, the forward and backward modes are transformed into each other at the Brillouin zone edge (K=Kb). Applying the general symmetry relations [17

17. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

], we conclude that the forward and backward modes should satisfy
(Ex(b)Ey(b)Ez(b)Hx(b)Hy(b)Hz(b))(x,y,z)=(Ex(f)Ey(f)Ez(f)Hx(f)Hy(f)+Hz(f))(x,y,z).
(4)

Then, by performing a Taylor expansion close to the photonic band edge, one can match the mode profiles according to Eqs. (2) and (3) and determine {E(f),H(f)}. However, we omit this derivation, as for our analysis the explicit form of {E(f),H(f)} is not required.

Because we are considering linear modes, Eq. (3) can be transformed from a stationary to a time-dependent formulation assuming a slow variation of the mode amplitudes A(x,t),
{E,H}(x,y,z;t)=A(f)(x,t){E(f),H(f)}(x,y,z)exp(iKbx)+A(b)(x,t){E(b),H(b)}(x,y,z)exp(iKbx).
(5)

We now proceed to the main goal of formulating a CMT when a longitudinal shift between the nanowires is introduced, that is, Δx0. In this case the structure is no longer symmetric with respect to the reflection transformation yy, and the modes would have neither even nor odd symmetry. Indeed, the effect of a shift is analogous to tilted Bragg gratings [19

19. T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996). [CrossRef]

], leading to a hybridization between the even and odd modes. Therefore, we seek to approximate the modes of shifted nanowires as a superposition of odd and even modes of the unshifted structure. To do this, we generalize Eq. (5), taking into account Eq. (4), to obtain
(ExEyEzHxHyHz)=(Ex(e)Ex(e)Ex(o)Ex(o)Ey(e)Ey(e)Ey(o)Ey(o)Ez(e)Ez(e)Ez(o)Ez(o)Hx(e)Hx(e)Hx(o)Hx(o)Hy(e)Hy(e)Hy(o)Hy(o)Hz(e)Hz(e)Hz(o)Hz(o))(A(ef)eiKbxA(eb)eiKbxA(of)eiKbxA(ob)eiKbx),
(6)
where the superscripts “(o),” “(e),” “(f),” and “(b)” refer to odd, even, forward, and backward modes, respectively. Here {E,H}(o,e) are the field profiles of forward-propagating even and odd modes of the unshifted nanowires defined in the context of CMT according to Eq. (3).

We note that upon a transformation wjwjexp(iφ), Eqs. (7) remain the same with a modified grating coefficient ρρexp(iφ). Then, by choosing the free transformation parameter φ=arg(ρ), the grating coupling coefficient ρ can be made real and positive; we will consider such parameter values with no loss of generality.

All possible solutions of coupled-mode Eqs. (7) can be found as a linear superposition of supermodes of the form
u1,2(x,t)=U1,2ei(KKb)xi(ωω0)t,w1,2(x,t)=W1,2ei(KKb)xi(ωω0)t,
(10)
where ω0 is the bandgap center and {U,W}1,2 are the amplitudes of forward and backward modes of individual waveguides. Substituting Eq. (10) into (7), we obtain a linear eigenmode problem, which provides a dispersion relation between the frequency and wavenumber,
(ωω0)2c2=(Kπ/a)2n02+C02+|ρ|2±2C0(Kπ/a)2n02+|ρ|2cos2(ϕ/2).
(11)

There appear four branches of the dispersion dependence, consistent with the four-wave coupling described by Eqs. (7).

For half a period shift we have ϕ=π, and Eq. (11) can be written in a more compact form as
(ωω0)2c2=[(Kπ/a)/n0±C0]2+|ρ|2.
(12)

We then determine the position of the photonic band edge associated with the maximum (minimum) frequency of the second (third) dispersion branch as
Ke=π/aC0n0,ωe=ω0±|ρ|c.
(13)

According to these expressions, ω0 is the frequency in the middle of bandgap and
ρ=Δω/(2c),
(14)
where Δω is the bandgap width. We determine ω0 and Δω from numerically calculated dispersion curves using the fully vectorial planewave expansion software MIT photonic bands [22

22. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef]

]; see Fig. 2(a). We also match the frequency value ωb for the second band at the Brillouin zone edge,
K=Kb=π/a,ωb=ω0cC02+|ρ|2,
(15)
which we use to determine the value of C0,
C0=1c(ωbω0)2|Δω|24.
(16)

Fig. 2. (a) Dispersion relation for the TM bands near the band edge for a half-period longitudinal shift (ϕ=π), calculated using a fully vectorial plane-wave method. Labels show the bandgap center frequency (ω0), lower band edge frequency (ωe), second-lowest band’s frequency at the edge of the first Brillouin zone (ωb), and bandgap width (Δω). (b)–(g) Comparison between plane-wave simulation results (circles) and CMT prediction (solid curves) for the second lowest band [shown in red in (a)] at different longitudinal shifts ϕ: (b) 0, (c) 0.2π, (d) 0.4π, (e) 0.6π, (f) 0.8π, and (g) π. For all the plots, the structure geometrical parameters are w=7a/6, d=a/6, wx=0.445a, wy=2a/3; the waveguide height is h=2a/3; and the refractive index of the waveguides is 2.

Finally, we can determine n0 from Eq. (13),
n0=(π/aKe)/C0.
(17)

For the structure parameters corresponding to Fig. 2(a), we determine the coupled-mode coefficients as C0=0.1048/a, n0=1.7511, and ρ=0.2129/a.

Fig. 3. Normalized amplitudes of (a) even-forward, (b) odd-forward, (c) even-backward, and (d) odd-backward mode at the lower band edge (Ke, ωe) as a function of the longitudinal shift Δx. The curves were calculated using the CMT model with coefficients C0=0.1048/a, ρ=0.2129/a, and n0=1.7511.

3. OPTICAL GRADIENT FORCE ANALYSIS

In this section we combine the CMT results derived in Section 2 with optical force analysis to develop a coupled-mode formulation for the optical forces. In the framework of CMT, at each longitudinal shift Δx, the mode profile is expanded into a basis of four modes: even-forward, even-backward, odd-forward, and odd-backward modes. We substitute this expansion into the Maxwell stress tensor expression and derive expressions for the transverse and longitudinal forces acting on the waveguides, expressed in terms of the amplitudes of forward and backward modes determined from the CMT.

According to the conservation law of momentum, the optical force can be calculated as [23

23. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

]
Fα=βSβTαββ^dSβ,
(18)
where Tαβ is the time-averaged Maxwell stress tensor, βSβ is a surface enclosing the object on which the force is acting, and β^ is the unit normal vector to the surface. In order to derive an expression for the optical force acting over one period of a nanowire waveguide, we define the surface S as a rectangular box enclosing a section of the waveguide of length a, with sides parallel to the coordinate system axes. Therefore, α, β=x, y, z, and the three pairs of parallel surfaces are labeled Sx, Sy, and Sz, where the subscripts indicate the orientation of the surface normal vector β^. Consider first the integral over Sx, which consists of two surfaces perpendicular to the periodicity direction and separated by one period. Due to the periodicity of the Bloch modes, the tensor fields on each surface are the same while the unit vectors are in opposite directions so that the contributions cancel and the integral is zero. For the integral over surface Sz, we observe that because the electromagnetic field at z=± is zero, the tensor Tαz makes no contribution. For the integral over surface Sy, there is no contribution from plane y=±, where the field is zero. As a result, the only contribution to the force comes from the integral over the surface Sy|y=0, and accordingly we can simplify Eq. (18) as
Fα=SyTαyy^dSy|y=0.
(19)

In the stationary (continuous-wave) regime, the electromagnetic fields have the form {E,H}(x,y,z;t)={E,H}eiωt+c.c., where c.c. stands for the complex conjugate of the preceding term. Then, the time-averaged tensors Tαy can be written as
Txy=ϵ(Ex*Ey+ExEy*)+μ(Hx*Hy+HxHy*),
(20a)
Tyy=ϵ(|Ey|2|Ex|2|Ez|2)+μ(|Hy|2|Hx|2|Hz|2).
(20b)

We can then derive an expression for the optical forces acting on the waveguide using the CMT expressions for the optical modes by substituting Eq. (6) and (20) into Eq. (19). The longitudinal force in the x direction is found as
fx=(|A(ef)|2|A(eb)|2)fx(e)+(|A(of)|2|A(ob)|2)fx(o)+[(A(ef)A(of)*A(eb)A(ob)*)fx(oe)+c.c.],
(21)
where
fx(e)=Sy[ϵ(Ex(e)*Ey(e)+Ex(e)Ey(e)*)+μ(Hx(e)*Hy(e)+Hx(e)Hy(e)*)]y^dSy|y=0,
(22)
fx(o)=Sy[ϵ(Ex(o)*Ey(o)+Ex(o)Ey(o)*)+μ(Hx(o)*Hy(o)+Hx(o)Hy(o)*)]y^dSy|y=0,
(23)
and
fx(oe)=Sy[ϵ(Ex(e)Ey(o)*+Ex(o)*Ey(e))+μ(Hx(e)Hy(o)*+Hx(o)*Hy(e))]y^dSy|y=0.
(24)

The transverse force in the y direction is
fy=(|A(ef)|2+|A(eb)|2)fy(e)+(|A(of)|2+|A(ob)|2)fy(o)+[(A(ef)A(of)*A(eb)A(ob)*+c.c.)fy(oe)+c.c.],
(25)
where
fy(e)=Sy[ϵ(|Ey(e)|2|Ex(e)|2|Ez(e)|2)+μ(|Hy(e)|2|Hx(e)|2|Hz(e)|2)]y^dSy,
(26)
fy(o)=Sy[ϵ(|Ey(o)|2|Ex(o)|2|Ez(o)|2)+μ(|Hy(o)|2|Hx(o)|2|Hz(o)|2)]y^dSy|y=0,
(27)
and
fy(oe)=Sy[ϵ(Ey(e)Ey(o)*Ex(e)Ex(o)*Ez(e)Ez(o)*)+μ(Hy(e)Hy(o)*Hx(e)Hx(o)*Hz(e)Hz(o)*)]y^dSy|y=0.
(28)

We can further simplify the force expressions by taking into account that the field profiles {E,H}(o,e) correspond to odd and even modes of an unshifted structure (with Δx=0). Then, due to the orthogonality of odd and even modes, we have fy(oe)0. On the other hand, by using the general symmetry considerations [7

7. Y. Sun, T. P. White, and A. A. Sukhorukov, “Slow-light enhanced optical forces between longitudinally shifted photonic-crystal nanowire waveguides,” Opt. Lett. 37, 785–787(2012). [CrossRef]

], it can be shown that pure odd or even modes in an unshifted structure produce zero longitudinal forces acting on each waveguide, that is, fx(e)=fx(o)0. Then, the final expressions for the forces are
fx=(A(ef)A(of)*A(eb)A(ob)*)fx(oe)+c.c.
(29)
and
fy=(|A(ef)|2+|A(eb)|2)fy(e)+(|A(of)|2+|A(ob)|2)fy(o).
(30)

We point out that the values fx(oe), fy(e), and fy(o) are constants, and the force dependence on the longitudinal shift is fully defined through the CMT mode amplitudes.

Remarkably, by fitting the force values only at specific shifts, we can very accurately predict the longitudinal and transverse forces for any shift, as illustrated in Fig. 4. This prediction of the force variation is based on the CMT amplitude dependencies. Because the CMT equations are general, our approach provides a flexible tool to quickly and accurately determine the optical forces between almost arbitrarily structured periodic waveguides with any longitudinal shift.

Fig. 4. Comparison of the optical forces acting on the waveguides as a function of the longitudinal shift, calculated using CMT (solid curves) and full numerical simulations (circles). (a) Longitudinal force. (b) Transverse force. Shown are the force values per unit length per unit energy density.

4. CONCLUSIONS

In conclusion, we have developed the CMT approach for the calculation of transverse and longitudinal optical forces between parallel periodic waveguides. This method can accurately predict the dependence of slow-light enhanced forces at the photonic band edge on the longitudinal shift between the waveguides. Importantly, this approach requires the direct numerical calculation of modes only at specific values of the longitudinal shifts, making it computationally efficient. Furthermore, the CMT approach can be generalized to calculate the optical forces between multiple coupled waveguides and to incorporate the dependence of coupled-mode coefficients on the transverse waveguide separation, enabling precise and efficient modeling of nonlinear optomechanical dynamics.

ACKNOWLEDGMENTS

This work was supported by the Australian Research Council programs (including Future Fellowship FT100100160 and Discovery Project DP130100086) and the supercomputing infrastructure of the Australian NCI National Facility.

REFERENCES

1.

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

2.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009). [CrossRef]

3.

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]

4.

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, 726–732 (2011). [CrossRef]

5.

J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97, 151102 (2010). [CrossRef]

6.

A. Oskooi, P. A. Favuzzi, Y. Kawakami, and S. Noda, “Tailoring repulsive optical forces in nanophotonic waveguides,” Opt. Lett. 36, 4638–4640 (2011). [CrossRef]

7.

Y. Sun, T. P. White, and A. A. Sukhorukov, “Slow-light enhanced optical forces between longitudinally shifted photonic-crystal nanowire waveguides,” Opt. Lett. 37, 785–787(2012). [CrossRef]

8.

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a ‘zipper’ photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009). [CrossRef]

9.

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]

10.

P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17, 18116–18135 (2009). [CrossRef]

11.

Z. Wang and P. Rakich, “Response theory of optical forces in two-port photonics systems: a simplified framework for examining conservative and non-conservative forces,” Opt. Express 19, 22322–22336 (2011). [CrossRef]

12.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 1988).

13.

S. Ha and A. A. Sukhorukov, “Nonlinear switching and reshaping of slow-light pulses in Bragg-grating couplers,” J. Opt. Soc. Am. B 25, C15–C22 (2008). [CrossRef]

14.

C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A 78, 033818(2008). [CrossRef]

15.

C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express 17, 5879–5884 (2009). [CrossRef]

16.

A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B 25, C65–C74 (2008). [CrossRef]

17.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

18.

J. B. Khurgin and R. S. Tucker, eds., Slow Light: Science and Applications (Taylor & Francis, 2009).

19.

T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996). [CrossRef]

20.

S. Ha, A. A. Sukhorukov, and Yu. S. Kivshar, “Slow-light switching in nonlinear Bragg-grating couplers,” Opt. Lett. 32, 1429–1431 (2007). [CrossRef]

21.

Y. J. Tsofe and B. A. Malomed, “Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift,” Phys. Rev. E 75, 056603 (2007). [CrossRef]

22.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef]

23.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

OCIS Codes
(200.4880) Optics in computing : Optomechanics
(230.7370) Optical devices : Waveguides
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Optical Devices

History
Original Manuscript: November 21, 2012
Revised Manuscript: January 21, 2013
Manuscript Accepted: January 21, 2013
Published: February 28, 2013

Citation
Yue Sun, Thomas P. White, and Andrey A. Sukhorukov, "Coupled-mode theory analysis of optical forces between longitudinally shifted periodic waveguides," J. Opt. Soc. Am. B 30, 736-742 (2013)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-3-736


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References

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  2. J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009). [CrossRef]
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