Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate

W.H.P. Pernice; Mo Li; H.X. Tang

doi:10.1364/OE.17.001806

1. Introduction

With recent advances in the fabrication of nano-sized devices [1], optical effects that are too small to have a measurable impact on a macroscopic scale are attracting increasing interest. Among these is the optical force exerted by photons that arises when structures are illuminated by electromagnetic radiation. So far, past research has focussed primarily on the optical force that is a result of momentum transfer from photons to matter, the so-called radiation pressure [2–8]. Radiation pressure is a force normal to the device surface and thus usually requires reflective mirrors to be significant. However, an alternative lateral force has also been recently predicted by Povinelli [9, 10] in a coupled waveguide system, by Rakich et al. for coupled ring resonators [11] and Taniyama et al. for coupled photonic crystal cavities [12]. This force can show either attractive or repulsive behavior depending on the supermode excited in the waveguides. The physical origin of this optical force lies in the excitation of optical dipoles in dielectric media when radiation is applied. This was also confirmed experimentally by [13]. As a consequence the magnitude of the force is proportional to the dipole strength and therefore to the intensity of the radiation.

Recently, we have experimentally demonstrated the transverse optical force in an integrated photonic circuit [14]. The optical force was utilized to actuate a nano-mechanical beam clamped between two multi-mode interference couplers. This setup provides a very generic platform for the actuation of nano-mechanical devices. For optimization of future devices it is thus essential to develop an accurate model of the origin of the transverse optical force.

Here we provide a detailed analysis of optical forces arising in this general geometry. We calculate the transverse force between a silicon nano-wire and a dielectric substrate. We show that at moderate laser powers considerable optical forces are obtainable. The force values are calculated numerically using the finite element method and are also confirmed by applying an analytical approach based on the Maxwell stress tensor. A phenomenological model is presented that puts the substrate-coupled force into the context of coupled waveguides.

2. Numerical calculation of the optical force

In order to determine the magnitude of the transverse optical force we apply two separate approaches. First we deduce the force from finite element simulations. In a second approach we develop an analytical model and determine the force as a function of gap from the evaluation of the Maxwell stress tensor. The two approaches show good agreement and thus provide physical insight into this novel optical force generation mechanism.

2.1. The effective index method

In this article we consider a structure as presented in Fig. 1(a). A silicon waveguide of 500 nm width is realized on a SOI wafer. The thickness of the waveguiding silicon layer is 110 nm and the silicon layer is separated from the bulk silicon wafer by a silicon dioxide layer of 3 μm thickness. A freestanding waveguide is created by etching into the silicon oxide layer, thus removing the dielectric material under the waveguide.

Fig. 1. Illustration of the device. a) Render view of the freestanding silicon waveguide suspended over a SiO ₂ substrate. Overlaid in color is the mode pattern of a propagating wave with a wavelength of 1.55 μm. The nano-mechanical resonator is released by removing the oxide under the silicon waveguide by isotropic etching. b) A cross section view of the device in the center of the groove detailing the dimensions used in the force calculations.

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In order to evaluate the optical force with the finite element method we consider a cross section of the waveguide as shown in Fig.1(b). The fundamental mode of the waveguide is excited in TE polarization. Neglecting scattering losses, the fundamental mode will propagate along the beam without changing its profile. Therefore the investigation of a two-dimensional cross-section is a valid simplification. The waveguide is separated from the substrate by a gap of size g. When the waveguide is brought closer to the substrate, the waveguide mode will be coupled evanescently to the dielectric. With decreasing gap, the propagation constant of the fundamental mode decreases. This is demonstrated in the dispersion diagram obtained from the finite-element simulations as shown in Fig.2(a). Above cut-off, the dispersion curves corresponding to varying gap sizes show a local maximum before converging towards the propagation constant of the free waveguide. From the dispersion diagram we are able to derive the optical force, which acts on the waveguide. Following Povinelli [9] we calculate the optical forces by considering the frequency shift of the optical mode with decreasing gap distance as

F = - \frac{1}{ω} \frac{\partial ω}{\partial g} U

The frequency ω is related to the effective refractive index of the waveguide as $ω = \frac{ω_{0}}{n_{eff}}$ , where ω₀ is the free space frequency. Because we are measuring the effective index of the fundamental mode of the waveguide rather than the frequency above Eq. can be more conveniently expressed as

F = \frac{1}{n_{eff}} \frac{\partial n_{eff}}{\partial g} U

Assuming that the energy on the beam is given by U = PLn_g/c, where P is the optical power, L is the length of the beam and n_g is the group index of the mode, we get the optical force normalized to length and power as

Fig. 2. Illustration of the device. a) The dispersion relation of the device in Fig 1(a) obtained from finite-element calculations. b) The optical force calculated from the dispersion relation using Eq.(3). The force is normalized to optical power and beam length and thus given in pN/μm/mW.

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F_{n} = \frac{n_{g}}{c n_{eff}} \frac{\partial n_{eff}}{\partial g}

The results are presented in Fig.2(b).The force is plotted as a function of the normalized wavevector k_n = k ₀ w/2π, where k ₀ is the free space wave vector. From the negative sign it is apparent, that the force is attractive, thus pulling the beam towards the substrate. When the gap size is zero, the force reaches its maximum value of ≈ 13 pN/μm/mW. It decreases mono-tonically towards zero, when the gap is increased towards infinity. Based on the calculations, we find a typical strength of the force on the order of pN when assuming device dimensions in the micrometer range. Therefore given the high sensitivity and small mass of the free standing waveguide [14, 15] this force value is significant for the actuation the nanomechanical motion of the waveguide [16].

2.2. Image waveguide approximation

In addition to the numerical method outlined above, the optical force between the waveguide and the substrate can be approximated by considering the force existing between two coupled waveguides. This approach is motivated from the well-known method of mirror charges in electrostatics which is used to determine the electrical field distribution of a charged object with respect to a metal surface [17]. In analogy, we therefore calculate the optical force between two identical silicon nano-wires separated by a distance 2_g. Here we essentially look at the interaction between the dipoles in the waveguide and their image dipoles induced by the dielectric substrate. We restrict ourselves to the symmetric mode and thus solve for the attractive lateral force. The concept is illustrated in Fig.2.2. The calculation is carried out as described by Povinelli and as elucidated in the preceding section. The force between coupled waveguides depends on the gap size in a fashion similar to the results in Fig.2(b). It is stronger than the force to the substrate, because the waveguides are assumed to be surrounded by air. To take the effect of the substrate into account, we introduce a shielding factor α into Eq.(3) as

Fig. 3. The optical force acting on the waveguide can be understood in analogy to the concept of a mirror charge in electro-statics. An image waveguide is assumed to be present on the other side of the substrate-air interface. The symmetric mode of the coupled waveguides is the origin of the attractive optical force.

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F_{coupled} = α \frac{n_{g}}{c n_{eff}} \frac{\partial n_{eff}}{\partial g}

This factor is used as a fitting parameter to match the force obtained for the coupled waveguide setup to the values obtained with Eq.(3). It accounts for the fact, that the optical path in the dielectric substrate is different from the free-space solution and thus the mode pattern in the substrate differs from the mode of the free waveguide. The results are shown in Fig.2.2. For the above device we use a shielding factor of 0.65. The two methods are in good agreement, when the gap values are sufficiently large, so that the coupling between waveguide and substrate is weak. In the regime of strong mode coupling the agreement breaks down as would also be expected for coupled-mode theories. However, the shielding factor is only an empirical number, thus the analogy considered in this section is not intended as an alternative formulation. Rather, we attempt to provide the reader with an phenomenological model that elucidates the origin of the force and provides further physical insights.

Fig. 4. The optical force calculated using Eq.(4) with a shielding factor of 0.65. These results agree well when the gap values are large enough, that the coupling between the waveguide and the substrate is weak.

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3. A one-dimensional analytical method for determination of the optical force

The methods outlined in the previous section are purely numerical. In this section we present an analytical formulation, that we found useful in providing a tool for quick evaluations of the force in the simple geometry considered above. The approach closely follows the method outlined in [18]. We calculate the optical force in the waveguide by using the Maxwell stress tensor. This approach requires the mode patterns of the waveguide. However, analytical expressions are not available for rectangular dielectric waveguides. Approximate methods, such as Marcatili’s method [19] or first-order perturbation corrections made to it [20, 21] are also not accurate, because the refractive index contrast in the waveguide system is large. We therefore restrict our formulation to a one-dimensional slab model and attempt to generalize to higher dimensions from the result.

In order to calculate the force resulting from the asymmetry of the field distribution in the waveguide, we perform the following steps: i) The waveguide mode is assumed to be excited in TE-polarization and is propagating as a plane wave along the z-direction. For this configuration the modes are obtained with their corresponding propagation constant. The propagation constant is the solution of a transcendental Eq., so an explicit solution is not available. Therefore an approximate solution is obtained to express the effective index as a function of distance between the waveguide and the substrate. ii) The optical force is then calculated by integrating over the Maxwell stress tensor in the gap between waveguide and substrate.

3.1. Calculation of the electrical fields

The silicon waveguide of width h with a refractive index n_c of 3.5 is embedded in a host material with n_a of 1.0. The waveguide is separated from the substrate with a refractive index n_s of 1.5 by a distance g. The slab is assumed to extend infinitely in the x and z directions. In this configuration the electric and magnetic fields are given by

E (r, t) = E (y) e^{i (ωt - βz)}

H (r, t) = H (y) e^{i (ωt - βz)}

For TE polarization, the components E_y, E_z and H_x are 0 and therefore the field vector is given as

ψ = {1, - \frac{β}{ωμ} \frac{i}{ωμ}}^{T} E_{x}

where ^T denotes the matrix transpose. Using the freespace wavevector $k = \frac{2 π}{λ}$ we define the following wavevectors corresponding to the three different material regions as

k_{s} = \sqrt{β^{2} - k^{2} n_{s}^{2}}

k_{a} = \sqrt{β^{2} - k^{2} n_{a}^{2}}

k_{c} = \sqrt{k^{2} n_{c}^{2} - β^{2}}

Then the x component of the electric field can be expressed as

E_{x} = A {\begin{matrix} a_{1} e^{k_{s} y} y < 0 \\ a_{2} e^{- k_{a} y} + a_{3} e^{k_{a} y} 0 \leq y < g \\ \cos (kc (y - g) + ϕ) g \leq y < g + h \\ a_{4} e^{- k_{a} (y - g - h)} y \geq g + h \end{matrix}

The coefficients a_i in above Eq. are obtained by requiring the impedance $\frac{E_{x}}{H_{z}}$ to be continuous throughout the domain. Solving for ϕ leads to the dispersion relation for the structure as

k_{c} h = \tan^{- 1} (\frac{k_{a}}{k_{c}}) + \tan^{- 1} (\frac{k_{a}^{2} \sinh (k_{a} g) + k_{a} k_{s} \cosh (k_{a} g)}{k_{s} k_{c} \sinh (k_{a} g) + k_{c} k_{a} \cosh (k_{a} g)}) - m π

where m is the mode number of the waveguide. Above Eq. does not have an analytical solution, so the values of β corresponding to a given wavelength have to be obtained numerically.

Solving for the field amplitudes gives the following expressions for the coefficients a_i

a_{1} = 2 a_{3} \frac{k_{a}}{k_{a} + k_{s}} a_{2} = a_{3} \frac{k_{a} - k_{s}}{k_{a} + k_{s}}

a_{3} = \frac{k_{a} + k_{s}}{2 M} a_{4} = \frac{k_{c}}{\sqrt{k_{a}^{2} + k_{s}^{2}}}

where we used the following expression for M

M = \sqrt{{(\frac{k_{a}}{k_{c}})}^{2} {[k_{a} \sinh (k_{a} g) + k_{s} \cosh (k_{a} g)]}^{2} + {[k_{s} \sinh (k_{a} g) + k_{a} \cosh (k_{a} g)]}^{2}}

The above relations fully characterize the electromagnetic fields in the slab waveguide. In order to obtain the mode profiles for a given gap value, we thus need to solve the transcendental dispersion relation and can calculate the fields subsequently. This will still involve a numerical tool, therefore we derive an approximate solution to Eq.(12) in the following section.

3.2. Approximation of the propagation constant

Because an analytical solution to the transcendental Eq. for the propagation constant does not exist, we look instead for an approximate expression of the form

β (g) = β_{0} + β_{1} e^{- σg}

where β ₀ is the propagation constant of the free waveguide. For small values of g we can introduce the following approximations

k_{a} \approx α_{a} + \frac{β_{0} β_{1}}{α_{a}} e^{- σg}

k_{s} \approx α_{s} + \frac{β_{0} β_{1}}{α_{s}} e^{- σg}

k_{c} \approx α_{c} - \frac{β_{0} β_{1}}{α_{c}} e^{- σg}

where the wavevectors for the free waveguide are given as

α_{a} = \sqrt{β_{0}^{2} - k^{2} n_{a}^{2}}

α_{s} = \sqrt{β_{0}^{2} - k^{2} n_{s}^{2}}

α_{c} = \sqrt{k^{2} n_{s}^{2} - β_{0}^{2}}

We rewrite the Eq. for the dispersion relation as follows

\tan (k_{c} h) = \frac{k_{a}}{k_{c}} \frac{1 + Q}{1 - {(\frac{k_{a}}{k_{c}})}^{2} Q}

where Q is defined as

Q = \frac{k_{a} \sinh (k_{a} g) + k_{s} \cosh (k_{a} g)}{k_{s} \sinh (k_{a} g) + k_{a} \cosh (k_{a} g)}

First we determine the amplitude β ₁ by solving the above Eq. for g = 0. Since we assume β ₀ to be much larger than β1, we can use

\tan (k_{c} h) \approx \tan (α_{c}) \equiv P

and $Q (g = 0) = \frac{k_{a}}{k_{c}} .$ We then need to solve the following Eq.

(P - \frac{h}{α_{c}} β_{0} β_{1}) (1 - {(\frac{k_{a}}{k_{c}})}^{2} \frac{k_{s}}{k_{a}}) - \frac{k_{a}}{k_{c}} (1 + \frac{k_{s}}{k_{a}}) (1 + P \frac{h}{α_{c}} β_{0} β_{1}) = 0

After some lengthy but straightforward algebra we arrive at the following cubic Eq. for the amplitude β ₁

[P \frac{h}{α_{c}^{2}} (\frac{1}{α_{s}} + \frac{1}{α_{a}}) - \frac{h}{α_{c}} (\frac{1}{α_{c}^{2}} + \frac{1}{α_{a} α_{s}})] β_{0}^{3} β_{1}^{3}

The standard formula for the roots of a cubic polynomial can be applied to obtain the correct solution for the amplitude. In order to obtain the exponential factor σ of the approximation, we first determine the value for the gap at which the propagation constant has a value of $β (g) = β_{0} + \frac{β_{1}}{2}$ which is equivalent to requiring that σg = ln(2). Then we use this value in the transcendental Eq. and find for σ the following expression:

σ = 2 k_{a} In (2) - In {P \frac{(k_{a} - k_{s}) (k_{a}^{2} + k_{c}^{2})}{(k_{a} + k_{s}) (k_{a}^{2} - k_{c}^{2}) P + 2 k_{a} k_{c}}}

Thus the effective index is given as

n_{eff} (g) = k (β_{0} + β_{1} e^{- σg}) = n_{0} + n_{1} e^{- σg}

The results of the above approximation are shown in Fig.5. Both the approximate solution and the direct solution of the transcendental dispersion Eq. are displayed. Good agreement is found between the two for the gap values of interest. Thus approximation of n_eff with the simple exponential decay from Eq.(27) is a valid simplification.

Fig. 5. Comparison of the approximate solution for the effective index and the numerical solution of the dispersion Eq. for the effective index. The approximation of n_eff with the exponential decay from Eq.(27) is very close to the direct solution of Eq.(12).

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3.3. Calculation of the optical force using the Maxwell stress tensor

Using the above Eq.s we are now able to calculate the optical force. This is achieved by evaluating the Maxwell stress tensor in the gap between waveguide and substrate. The relevant component of the stress tensor is the T_yy value. This is given for TE polarization as

T_{yy} = - \frac{ε_{0}}{4} {{∣ E_{x} ∣}^{2} + c^{2} μ^{2} ({∣ H_{z} ∣}^{2} - {∣ H_{y} ∣}^{2})}

= - \frac{ε_{0}}{4} {{∣ E_{x} ∣}^{2} (1 - \frac{β^{2}}{k^{2}}) + \frac{1}{k^{2}} {∣ \partial_{y} E_{x} ∣}^{2}}

Evaluating the above Eq. in the host material with a refractive index of 1.0 we find that the stress tensor is 0 outside the waveguide because

T_{yy} = - \frac{ε_{0}}{4} {∣ E_{x} ∣}^{2} (1 - n_{a}^{2})

Therefore we only need to consider T_yy in the gap to calculate the optical force. This leads to the following expression:

F_{optical} = - T_{yy} = \frac{ε_{0}}{M^{2}} A^{2} (k^{2} - β^{2}) (n_{s}^{2} - n_{a}^{2})

Calculating the optical force therefore requires knowledge of the field amplitude. We can use the expression derived earlier in Eq.(13) and Eq.(14). However, in the derivation it was assumed that the optical power is applied over the complete slab, which is not the case. Hence we can alternatively use the standard Eq. for the field amplitude of a plane wave with optical power P. The field amplitude is then given as $A = \sqrt{\frac{2 P}{A_{wg} cn ε_{0}}},$ where A_wg is the mode area of the waveguide. In order to compare the analytical solution for the slab with simulation results for the cross section of a real waveguide we need to adjust the analytical formula to take the two-dimensionality of the geometry into account. This is done by defining an effective width of the waveguide slab in the 1d model. This effective width is driven by the fact, that the 1d field amplitude will only correspond to the mode pattern in the 2d waveguide in the center. Further away from the center, the mode pattern is concentrated in a smaller area. Therefore we define the effective width of the 1d slab as

w_{eff} = h \frac{\int_{A_{wg}} {(E_{y}^{wg})}^{2} dxdy}{\int_{A_{wg}} {(E_{y}^{analytical})}^{2} dxdy}

Using Eq.s to approximate the optical force. A comparison of the force calculated with the above expression and the effective index method for the 2d cross section is shown in Fig.6. When using the effective width approach, good agreement is obtained between the effective index method and the analytical model. Therefore, the one-dimensional model can be used as a good approximation to the complex two-dimensional case.

Fig. 6. The calculated force in comparison with the effective index method. When the effective width is used in the calculations (red markers), both methods are in good agreement. Strong deviations are found when the actual width of 110 nm is used in the analytical expressions.

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4. Conclusion

In conclusion, we have presented a numerical and analytical analysis of the transverse optical force in silicon nanowire waveguides. The force arising from evanescent coupling between the wire waveguide and a dielectric substrate is on the order of pN/μm/mW and thus large enough to efficiently actuate nanoscale devices. We have used a numerical approach to deduce the force from finite-element calculations and also formulated an approximate analytical formulation. Results from both calculations are in good agreement. For larger gap values, the comparison with the coupled-waveguide system shows good agreement, when an empirical correction factor is introduced.

The transverse optical force is expected to have a considerable impact on novel optomechanical nano-scale devices. It allows for all-optical operation of devices, providing an efficient actuation and sensing scheme. The methods presented in this article are easily applicable to alternative geometries and are thus a useful tool to design next generation opto-mechanical devices.

Acknowledgement

W.H.P. Pernice gratefully acknowledges the support of the Alexander-von-Humboldt foundation through a postdoctoral fellowship. H.X. Tang acknowledges a young faculty startup grant from Yale University.

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Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate

Abstract

1. Introduction

2. Numerical calculation of the optical force

2.1. The effective index method

2.2. Image waveguide approximation

3. A one-dimensional analytical method for determination of the optical force

3.1. Calculation of the electrical fields

3.2. Approximation of the propagation constant

3.3. Calculation of the optical force using the Maxwell stress tensor

4. Conclusion

Acknowledgement

References and links

Cited By

Figures (6)

Equations (38)

Optics Express