1. Introduction

The gradient optical force has been recently explored in optomechanical devices of various geometries [1–8]. Promising applications including tunable photonics, backaction cooling and amplification and optomechanical signal processing have been proposed. However, these exciting applications rely on transduction with strong gradient force, which is only obtainable with narrow coupling gaps. In this short range regime, besides the forces imposed by real photons, there also exists optical force mediated by virtual photons – the Casimir force [11]. This virtual photon force has been known to affect the fabrication yield and dynamical performance of micro- and nano-mechanical devices [12–17]. The relative intensity of these two forces therefore has profound impact on the optomechanical performance of optical force driven tunable couplers [9,18,19] and tunable photonic filters [6,20].

The experimental demonstration of such photon-mediated force has largely benefited from initial theoretical investigations based on numerical simulations and analytical approaches [9,10]. The theoretical framework for the analysis of gradient optical force has been proven to be very efficient and rather accurate for the investigation of nanophotonic devices that carry guided optical modes. On the other hand,theoretical work on Casimir interactions until recently has been restricted to relatively simple geometries, such as parallel plates or combinations of elementary surfaces. New numerical methods capable of computing the Casimir force in arbitrary geometries have been developed to investigate the strong dependence of the force on geometry and led to a variety of interesting effects [21–23]. These time-domain simulation based methods are emerging as a convenient way to calculate Casimir forces because they allow for the integration of complex material properties into the simulation engine, which is of importance when dispersive materials are being considered. Such computational methods are also particularly suitable for the investigation of optical forces in nano-optomechanical devices, whose optical properties are nowadays routinely analyzed with finite-difference time-domain (FDTD) methods.

Free-standing nano-mechanical waveguides constitute fundamental optomechanical elements which can be mechanically actuated by a driving optical force. In coupled waveguides, both the magnitude and sign of the gradient optical force can be controlled [2]. When the separation between the waveguides is reduced to nanoscale dimensions the influence of the Casimir force has to be considered in addition to the gradient force.

In this article we determine real and virtual photon forces in coupled free-standing waveguides of typical nanophotonic dimensions, using the finite-difference time-domain and finite-element methods. Because the Casimir force is a broadband force, the chromatic dispersion of silicon is incorporated in the calculation. We find that, compared to metallic reference structures, the Casimir force is reduced by a factor of three in silicon for small waveguide separations and decreases super-exponentially with increasing gap between the two waveguides. Our results show that for gaps greater then 100nm substantial repulsive optical forces can be generated which dominate the Casimir force. When the gap between the waveguides is smaller than 100nm mW levels of optical power are needed to offset the Casimir force.

2. Casimir forces in coupled nanophotonic waveguides

We consider the geometry shown schematically in Fig. 1 . Two free-standing waveguides are separated by a gap g. The waveguides are of width w and height h and designed to be single-mode. The waveguides are surrounded by air, assuming that the substrate is sufficiently far away so that its effect can be neglected in the following calculations.

 

Fig. 1 a) The geometry used for the determination of both the gradient optical force and the Casimir force. Overlaid is the profile of the odd optical mode, responsible for the generation of repulsive gradient force.

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We calculate the Casimir force numerically using the time-domain methodology developed by Rodriguez et al. [24,25]. The numerical procedure provides the Casimir force between arbitrary structures at zero temperature from time-domain electro-magnetic field values. The FDTD method is used to obtain the electrical field distribution resulting from excitation of the coupled waveguides with current sources placed on the surface of the dielectric. This allows us to extract the Maxwell-Stress tensor and subsequently the Casimir force through surface integration of the field correlations in combination with the time-domain function introduced in reference [24].

The time-domain fields are sampled in close proximity to the waveguide surface. The geometry depicted in Fig. 1a) is surrounded by perfectly matched layers to absorb any electromagnetic fields scattered during the FDTD calculation, leaving sufficient space around the waveguides to avoid boundary effects. We use a fine grid resolution of 5nm to make sure that the results have converged to an average error below 1%. In order to avoid numerical artifacts, we perform three individual Casimir force calculations for each configuration, as suggested in reference [24]. We obtain the Casimir force for the coupled system, the left waveguide only and the right waveguide only, respectively. The forces resulting from the individual waveguides are then subtracted from the force of the coupled beams, in order to correct for discretization errors during the FDTD calculation.

To establish a reference, we first consider perfectly conducting metallic waveguides, with a square cross-section of a width varying from 100nm to 2000nm. The separation between the waveguides is considered at a scale relevant to nano-device fabrication. The calculated Casimir force is divided by the length of the waveguide, yielding a force per unit length in that direction. As shown in Fig. 2a ), the Casimir force decays exponentially with increasing gap, which is also predicted by the proximity force approximation (PFA) [26]. Fitting the results to a power law, we find a decrease of the Casimir force with waveguide separation as F~g^-4.1A, where A is the waveguide area, which agrees well with the PFA. The largest Casimir forces are obtained for small gaps, approaching 3pN/μm at 100nm separation. Due to the exponential decay, Casimir forces are most relevant in short range and thus in the regime corresponding to the preferred operation parameter space of optomechanical devices.

 

Fig. 2 a): The Casimir force calculated for square coupled nano-beams composed of perfectly conducting metal, in dependence of waveguide separation. The width of the waveguide is varied from 100nm (black) to 2000nm (purple). The Casimir force decays exponentially with increasing gap and linearly with increasing waveguide width. b) The equivalent calculation for waveguides made from pure silicon, taking into account the frequency dispersion of silicon. Compared to the metallic case, the Casimir force is reduced by a factor of 3 for small separations, increasing to a factor of 10-20 for large gaps.

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Because the Casimir force correlates with the interaction surface, larger Casimir forces are obtained for wider waveguides. As shown in Fig. 2a) and b), the Casimir force increases linearly with waveguide width for a given gap g in first-order approximation. For gaps approaching 100nm we obtain Casimir forces on the order of pN/μm. Such force values are comparable to gradient optical forces simulated numerically and measured experimentally [1,2,6,7]. Compared to the PFA with perfect metallic plates, the calculated Casimir forces in silicon are reduced by a factor of roughly ten, which is consistent with the results presented by Rodriguez et.al.

In the context of optomechanical devices we then consider two coupled silicon beams of square cross-section. The frequency dispersion of silicon is modeled by approximating the dielectric function with a the following Drude-Lorentz function [27]

Next we consider coupled beam waveguides of 220nm height, which is a typical thickness for conventional silicon waveguides fabricated from silicon-on-insulator substrates. As shown in Fig. 3a ), we keep the waveguide height constant and vary the width of the waveguides for gaps ranging from 100nm to 2000nm. As expected from Casimir force theory, for large gaps, the attractive Casimir force acting between the waveguides does not depend on the waveguide width because the interaction surface between the waveguides remains unchanged. In Fig. 3a), the upper and lower groups of curves represent calculation results for perfectly conducting waveguides and for silicon waveguides, respectively. Clearly visible is the super-exponential decay for the silicon waveguides towards greater separation gaps. As in the case of square waveguides, for small separations, the Casimir force between silicon beams is about a factor of three smaller than between perfectly conducting beams. This reduction factor increases to a factor of 100-200 for large separations. The reduction in Casimir force is due to the smaller refractive index contrast in silicon structures compared to the index contrast in perfectly conducting structures. The dielectric property of silicon allows the electro-magnetic vacuum fields to penetrate into the waveguides and thus leads to reduced modal confinement.

 

Fig. 3 a) The Casimir force for waveguides with a fixed height of 220nm in dependence of waveguide separation, for perfectly conducting and silicon materials. The upper curves correspond the metallic waveguides (larger Casimir force), while the lower curves correspond the silicon waveguides. As expected from theory, the Casimir force shows only weak dependence on the waveguide width. b) The Casimir force ratio between metal and silicon waveguides.

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The ratio between the Casimir force in the metal waveguides with respect to the silicon waveguides is shown in Fig. 3b). A sharp increase of the ratio for wide gaps is observed, approaching 160 for 200nm gap, which implies that the Casimir force in silicon waveguides is significantly reduced for large separations. Interpreting the numerical results obtained above, we find that the modal guiding properties of the free-standing waveguides can be engineered by tuning the beam width without significantly affecting the magnitude of the Casimir force, provided that the waveguide height is kept constant. This is a convenient situation for top-down fabrication of photonic waveguides since the waveguide height is usually fixed.

3. Gradient optical force in coupled waveguides

The coupled waveguide system described above supports two optical modes, termed the even and odd modes because of their respective symmetry properties with respect to the direction of propagation. When the distance between the waveguides is varied, the effective refractive mode index changes as well. The resulting change of the eigenenergy of the propagating mode gives rise to an optical gradient force, which was first calculated theoretically by Povinelli et al. [9].

When the even optical mode is excited in the coupled waveguide system, the effective mode index decreases with increasing gap and thus the resulting optical force is attractive. The situation is different for the odd mode, which experiences a minimal effective mode index for a cross-over gap g_c. As a result the optical force is attractive for gaps smaller than g_c and repulsive for gaps greater than g_c.

The optical force between two waveguides can be employed to efficiently actuate mechanical motion of the waveguides. Attractive optical forces pull the waveguides together and repulsive optical forces push the waveguides apart. While gradient optical forces are mediated by real photons, Casimir forces arise from virtual photons (or vacuum fluctuation). The Casimir force between two silicon waveguides in air is attractive and thus pulls the waveguides together, which can lead to stiction effects when the separation between the waveguides is very small. Nevertheless, the repulsive optical force generated by the odd-mode of coupled waveguides can be exploited to counteract the attractive Casimir force.

Turning to the optical gradient force, we determine the repulsive force for dielectric waveguides that support only a single mode in each beam. The optical force is obtained with the method suggested originally in reference [9], which involves determining the effective propagation index of the coupled waveguide modes in dependence of waveguide separation. The gap between the waveguides is varied from 50nm to 600nm, which correspond to relevant fabrication distances in coupled free-standing nano-photonic waveguides. The optical force is then proportional to the spatial derivative of the effective index. The effective index is calculated with a commercial finite-element simulation tool, taking into account the frequency dependence of the refractive index of silicon. Because the beam waveguides are intended for operation at telecom wavelengths, we obtain the gradient optical force for a representative wavelength of 1550nm. As shown in Fig. 4a ) we calculate the repulsive force as a function of gap for waveguide widths ranging from 300nm to 500nm.

 

Fig. 4 (a) The calculated repulsive optical force in dependence of waveguide separation and waveguide width, for waveguides of 220nm height. The corresponding Casimir force is shown by the dashed orange line. (b) The amount of optical power required for the cancellation of the Casimir force. The power is determined in dependence of waveguide separation and waveguide width.

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For reference, the expected Casimir force is also included with the dashed orange line. As mentioned above, the repulsive optical force emerges for gaps exceeding a cross-over gap g_c. This cross-over point depends on the waveguide width, where wider waveguides yield smaller g_c. At the same time, wider waveguides yield smaller repulsive forces because the decay of the effective index becomes more pronounced due to reduced evanescent mode tails. It is clear from Fig. 4a) that the repulsive optical force (normalized to optical input power and waveguide length) is able to overcome the Casimir force beyond a certain compensation gap. For the waveguide widths considered here, the smallest compensation gap is found to be near 85nm for the 400nm wide waveguides. Below the compensation gap, the optical force decays rapidly and eventually switches sign to attractive (not shown in Fig. 4a)). Both the optical force and the Casimir force decay exponentially for larger gaps, however the optical force remains stronger in this regime.

Building on the results in Fig. 4a) we determine the optical input power that is required to achieve repulsive gradient force equivalent to the Casimir force for various separation gaps. The results presented in Fig. 4b) show that the compensation of the Casimir force can be achieved with optical input powers below 1mW for gaps of 100nm or larger. The optical power necessary for the compensation increases with increasing waveguide width, because the repulsive gradient optical forces become weaker when the mode is more confined inside the waveguide. For larger gaps the power required for compensation decays more slowly, because the reduction of the optical force is slower than the Casimir effect.

In terms of optomechanical actuation, the Casimir force provides a constant force offset which will reposition the waveguides and thus change the working condition of the optomechanical devices. This effect is significant when the separation between the coupled waveguides is reduced below 100nm and therefore has to be taken into account in the design of functional optomechanical devices.

4. Conclusion

In conclusion we have compared Casimir and repulsive gradient optical forces in nano-optomechanical devices made from silicon. Our results show that in movable nano-photonic waveguides with separations below 100nm, the Casimir force can be stronger than obtainable repulsive gradient optical forces. Thus for efficient operation of optomechanical devices sufficient optical power at mW level is required to overcome the Casimir effect for nanoscale coupling gaps.