Introduction

A nanomechanical gap plasmon (GP) phase modulation principle has been recently proposed and experimentally verified [1]. Such modulators were predicted to scale down to below a 1 μm² footprint without loss of performance, and applications in miniaturized optical switching were proposed, an approach alternative to electro-optic plasmonic modulation [2,3] and switching [4]. Silicon photonic [5] and micromechanical switches using dielectric waveguides have been previously studied [6–8] and large port count, high-performance switch fabrics have been recently realized to address challenging data center circuit switching requirements [9]. In contrast to dielectrics, plasmonic optical components, while challenged by intrinsic optical losses in metals, have unrivaled miniaturization potential [10], motivating further exploration of their scaling and performance limits with an eye toward future on-chip reconfigurable optical communication requirements. In addition to small footprint and the resulting potential for compactly realizing complex optical reconfiguration, low voltage actuation and extremely small steady state power consumption make these devices appealing.

In order to realize such scaled down nanomecanical devices, multiple optical components have to be integrated. A 2x2 Mach Zehnder switch, schematically shown in Fig. 1(c), is a simple prototypical example, where two 3 dB couplers and input and output ports are implemented and incorporated with two arms, at least one of which is phase modulated. One possibility is to couple the plasmonic modulators to a more commonly used dielectric waveguide technology, such as silicon on insulator, where such components are routinely implemented. Low-loss coupling techniques between dielectric waveguides and GPs, as well as low-loss coupling to very narrow gaps have been reported [11,12], with losses as low as 1.3 dB per coupler reported [4,11]. However, such conventional dielectric waveguides and couplers cannot be further downscaled, hindering realization of the full potential for miniaturization inherent in the plasmonic approach.

 

Fig. 1 Nanomechanical gap plasmon 2x2 optical switch. (a) Perspective views and dimensions. Color scale – vertical displacement of the actuated modulator arm. Inset – input/output port GP modes calculated by FEM: electric field norm. (b) Side view with the vertically actuated modulator arm (c) top view and schematic showing the input and output ports, 3 dB couplers, the modulator arm and the reference arm.

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It is therefore desirable to realize such miniaturized components directly for GPs and tightly integrate them into fully plasmonic switches. Proposed in this work and numerically validated is a design for such an integrated 2x2 switch. First, two-dimensional (2D) finite-element modeling (FEM) is used for modal analysis of propagating, confined GP modes. The results are then used to derive geometrical parameters for the switch. Finally, using a full-vector three-dimensional (3D) frequency-domain FEM, the length of the switch’s 3 dB coupler is further optimized and its expected performance is predicted. With full-3D verification the GP phase modulator function is found to be in agreement with [1]. Note that the π phase modulators in this work have a smaller footprint than [1], 0.5 μm² (0.25 μm x 2 μm), including the lateral air gaps on both sides. All FEM is implemented and carried out via a commercial finite-element solver. Remarkably, modest loss and high extinction ratio can be expected from such small switches.

Switch design

The switch model consists of two stacked, initially identical, gold parts with an air gap in between, Fig. 1. Such a device is feasible to fabricate, for example, by first depositing gold, thin silicon dioxide or other dielectric sacrificial layer and then another layer of gold. The device shape can be formed by electron beam or modern high resolution stepper lithography (100 nm lines and spaces are required) with subsequent ion beam milling. The sacrificial layer, removed by an isotropic etch, forms an air gap. The device can be actuated, for example electrostatically [1, 13] and may have a response time in the 10 ns to 100 ns range [1]. An additional structural layer (not shown) is envisioned on top of the device, as is common in multilayer surface micromachining [13]. This layer is structured in such a way that only the movable modulator arm is allowed to move, while the coupler and reference arms are held rigidly in place. Following [1] we use a parabolic approximation for the deformation shape of the movable arm (vertical deformation is shown via color scale in Fig. 1). While a complete switch design would necessarily include both electrostatic and mechanical actuation analysis in full detail, as well as a suitable fabrication sequence, the goal of this work is to study and verify the expected optical performance of such devices, for which a parabolic approximation of deformation is adequate.

Figure 1(a,b) shows perspective and side views of the switch with only one arm actuated down. The confined GP modes are seen in the Fig. 1(a) inset. As seen in Fig. 1(c), bottom to top, the switch consists of two input GP ports, a 3 dB (50/50) coupler region, two separate GP waveguide arms, one of which is actuated to shift the GP phase by δϕ, another 3 dB coupler and two output ports. The calculated modes for each of the two single-mode GP input ports are shown graphically in the inset to Fig. 1(a). The optical fields are well-confined in the gap and the effective index is approximately 2 for a 17 nm gap with 100 nm wide ports. The ports are separated by a 100 nm lateral air gap, as are the two arms of the switch.

In this work, the optical wavelength in vacuum is λ₀ = 780 nm and gold is the plasmonic metal (dielectric constant = −22.4476 + 1.36505i at this frequency), so that the modulator design from [1] can be directly followed. However, due to the broadband nature of GPs, with an appropriate choice of sizes, the same type of GP switch can be made for any wavelength which is sufficiently below the surface plasmon cutoff wavelength and where no strong absorption otherwise exists in the metal of choice. Specifically the same approach can be followed to make a switch at the telecommunication wavelength bands, where the plasmonic losses in the metal will be even lower (estimated 1.6 times longer propagation distance), however somewhat larger mechanical modulation or increased length may still be required to achieve the needed phase range at the lower optical frequency.

2.1 Initial geometrical parameter estimates

The initial gap and length for the phase modulator and reference arms are g₀ = 17 nm and L_arm = 2 μm. These parameters have been selected because they result in a very compact modulator while providing sufficient phase modulation and a reasonable insertion and excess losses: a π phase modulation with 1/e (4.3 dB) insertion loss is provided by a change in the gap at the midpoint of approximately Δgap = 0.28g₀ = 4.76 nm [1]. The width of the individual arms is chosen to be much wider than g₀, so that the effective index is close to that of an infinitely wide GP and the phase modulation theory in [1] remains quantitatively valid. On the other hand, they should be narrow enough for the GP to remain single mode. An arm width of 100 nm satisfies these requirements. A narrower width may be possible, but this will reduce confinement, requiring larger separation. It may also reduce modulation strength, requiring longer arms and leading to higher optical loss. Arms of width 100 nm ensure that the GP is well confined and that the two arms can be reasonably close together without excessive mixing of their modes.

The inter-arm distance has to be large enough such that the modes of each arm remain isolated, but without unnecessarily increasing the device lateral size. To choose the inter-arm distance, the optical modes are modeled for the two arms together for several inter-arm separations. The closely spaced GP modes form symmetric and antisymmetric supermodes, Fig. 2(a), having slightly different effective indexes (n₁ and n₂). The index difference becomes larger with smaller separations and stronger mixing. If light is initially injected into only one of the arms, the energy becomes equally distributed between both arms when a π/2 phase difference is accumulated between the modes, requiring the travel length of L_3dB^arm = (λ₀/4)·(n₁^arm - n₂^arm)⁻¹. Ideally, this distance should be much larger than the arm length. For an inter-arm separation of 100 nm, the calculated indexes are n₁^arm = 2.066, n₂^arm = 2.000, and L_3dB^arm = 3 μm. The 100 nm width and 100 nm separation are chosen for the arms as well as for the input and output ports.

 

Fig. 2 Gap plasmonic mode FEM calculations. (a) Symmetric and antisymmetric supermodes from 100 nm wide GP arms spaced apart 100 nm (b) First and second modes of a 300 nm wide GP in the coupler region. Numbers are real parts of the calculated refractive indexes. Lower panel is an illustration of the respective device crossections.

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A similar calculation is used to design a 3 dB coupler. The modes and effective indexes are calculated for a 300 nm wide GP waveguide that only has two modes. Visual comparison of the mode shapes in Fig. 2(a) with those in Fig. 2(b), and taking into account their indexes, leads to the conclusion that a reasonably good mode overlap and therefore acceptable scattering and reflection losses can be expected, even for an abrupt transition between the two separate arms and the 300 nm slab. The 300 nm slab is sufficiently narrow, so that there is a sizable difference between the refractive indexes of its two modes. Given that the calculated n₁^slab = 2.074 and n₂^slab = 1.680, the slab length should be about L_3dB^slab = (λ₀/4)·(n₁^slab - n₂^slab)⁻¹ = 495 nm to work well as a 3 dB coupler.

However, given the L_3dB^arm = 3 μm calculated above, some coupling is expected to continue in the arms. For an unactuated device, the aim is to couple all the power from port 1 to port 4 and from port 2 to port 3. Therefore we need the total phase shift between the symmetric and antisymmetric modes accumulated by traversing the entire switch to be π which is seen below:

An abrupt transition between the 3 dB coupler region and the switch arms is left in place, while a more gradual transition is made between the ports and the couplers using a 100 nm diameter semi-circular cut. These transition shapes can be further improved in the future to minimize the switch insertion loss.

2.2 Coupler length optimization

What follows is a full 3D finite element model implementation of the switch. First used are symmetric arms (no mechanical deformation). Calculation of the S parameters of the switch uses numerical solutions for the 2D port modes on each of the four port faces [Fig. 1(a) inset] as boundary conditions for the 3D problem, implementing the ports numerically. The port boundaries use mode overlap integrals to model the incoming optical mode energy (e.g. on either port 1 or port 2), while absorbing the outgoing energy in the particular modes. In this implementation, the energy not matching the mode was reflected, which is reasonable for single-mode ports. The GP can then be injected through either port 1 or port 2 to calculate the transmission to port 3 (S31, S32), and port 4 (S41, S42), as well as reflection (S11, S22) and cross-coupling S12. While full complex S-parameters are calculated (including phase), only transmitted and reflected power is reported, in units of dB.

S31 and S41 are calculated as a function of the coupler length (the same for both couplers). The coupler length is defined as the distance along the length of the switch (direction normal to the port faces) from the end of the semi-circular gap between the ports (50 nm from port faces) to the start of the arms. Figure 3 shows that the highest port 4 power rejection (S41 minimum) occurs for a coupler length slightly more than 200 nm. This is somewhat shorter than the initial guess of 327 nm. Furthermore, by observing the time-harmonic oscillating optical field in the switch, it is evident that some amount of reflection occurs from the transition regions, and therefore the fields in the switch are somewhat more complicated than the simple analysis based on unidirectional travel would suggest. Only a weak reflection is observed (a small apparent oscillation of intensity of traveling waves through the switch, see Media 1, Media 2, Media 3, and Media 4), indicating no strong resonances and therefore broadband operation is expected, to be confirmed by a future study.

 

Fig. 3 Coupler length optimization in the OFF state. Optimal coupler length is near the high extinction point.

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Using an optimal coupler length of 200 nm, the total length of the switch is 2500 nm, including 50 nm for ports on each end. The physical width of the switch is 300 nm, however a more adequate number for the footprint is 500 nm, which is the same as the width of the model. This includes 100 nm air margins on each side to ensure that the fields have space to decay appropriately (the estimated longest 1/e power decay rate is L_dec⁻¹ ≈2 (n_eff – 1)^1/2 k₀ = (62 nm)⁻¹). This results in a switch footprint area of 0.5 μm x 2.5 μm = 1.25 μm².

Switch performance

Switch functionality is modeled by deforming the FEM mesh of the modulator arm and the surrounding air to approximate actuation. The mesh elements of the arm are moved down by an explicitly prescribed displacement. The mesh in the air outside the arm is moved down by the same distance, while the deformation in the gap is linearly interpolated, such that it is zero at the gap’s lower surface, so that the mesh remains continuous and is not inverted by deformation. The air mesh deformation between the arms is also linearly interpolated such that it is zero at the vertical symmetry mid-plane parallel to the arms. The deformation is parabolic along the arm, zero at each arm end, and is parameterized by the maximum downward displacement, Δgap, occurring at the center of the arm.

Unity optical power is injected in either port 1 or port 2, and time-harmonic full-vector 3D Maxwell’s equations are solved in the model’s volume, which includes all metal, air gaps, and the 100 nm wide empty regions on both sides of the switch, as previously described. For illustration, Fig. 4 shows the horizontal plane bisecting the switch through the middle of the gap and plots the instantaneous values of the in-plane magnetic-field component normal to the direction of GP propagation.

 

Fig. 4 Instantaneous values of the horizontal magnetic field component normal to the direction of GP propagation, in the mid-plane of the gap in the switch ON (Δgap = 4.6 nm) and OFF states under port 1 and port 2 excitation. Transmission to each of the output ports is marked for each case. The simulated worst case insertion loss and extinction ratio is −8.5 dB and 25.3 dB respectively. Animated images Media 1,Media 2,Media 3, and Media 4.

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Solving the model and calculating the switch S-parameters for each of the two excitation ports and for a series of Δgap values [Fig. 5] gives the following results: after optimization of the 3 dB coupler, in the OFF state, there is a modest insertion loss of S41 = S32 = −8 dB, and low crosstalk of S41 - S31 = 27.6 dB and S32 - S42 = 28.9 dB. The ON state is reached at Δgap = 4.6 nm, where the total loss S31 = −8.4 dB and S42 = −8.5 dB is achieved. The crosstalk in the ON state is low, since S31-S41 = 26.2 dB and S42-S32 = 24.8 dB. By comparing the ON and OFF states for each input and output port combination, it is evident that the worst case extinction ratio is S32_OFF - S32_ON = 25.3 dB. The switching Δgap = 4.6 nm is in good agreement with the expected 0.28·g₀ = 4.76 nm from the initial parameter estimates based on [1].

 

Fig. 5 Transmission to each of the output ports from each or the input ports as a function of the gap change. Switch operation is nearly symmetric w.r.t. input port.

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We note that in the ON state, ports 1 and 2 are not, in principle, equivalent, because only one arm of the switch is deformed, but this asymmetry is small. Finally, the reflections (S11 and S22) and input-port cross-coupling S21 = S12 are below −17 dB for both switch states.

Based on a 1/e (4.3 dB) insertion loss through the 2 μm phase modulator [1], the lower loss limit for a 2.5 μm MIM is about 5.4 dB, due to losses in the gold. It is likely that the additional 2.6 dB to 3.1 dB loss in the modeled switch results from scattering in the transition regions between the ports and the coupler, and the coupler and the modulator arms. To provide further evaluation, for the case of unactuated device with port 2 illumination, we have carried out surface and volume integrations to evaluate the power loss. Absorption in the gold is about 75% and scattering, primarily at the input, is about 7%, out of which the power reflected back into the input ports was under about 3%. This, together with the oscillations visible in Media 1,Media 2,Media 3, and Media 4, confirms backscattering at the 3 dB couplers. The gold absorption loss dominates, at over 7 dB, which is at least 1.6 dB larger than would be expected from the direct MIM transmission. The backscattering from 3 dB couplers partially traps the GP inside the device, increasing the plasmonic absorption losses. Further loss improvement may be achieved there.

We have varied the FEA mesh density and the number of mesh elements in the various parts of the model to estimate the uncertainty in our calculated numbers. The transmission for the pass ports varied by less than 0.5 dB for all reasonably dense meshes, while the blocked port transmission varied by less than 1 dB. The optimal length of the coupler varied by less than approximately 10% (20 nm), while the amount of gap change for switching of the optimized switch varied by approximately 0.1 nm. We however point out that these numerical errors are small compared to the differences one may encounter in the experiment due to the surface finish of the metal, variation of the dielectric constant with deposition parameters and the effects due to geometrically imperfect structure shape. Thus we see our work only as a stepping stone toward experimental realization of such devices.

Conclusion

A small nanomechanical gap-plasmon 2x2 switch design is presented with performance numerically verified. The device takes advantage of the tight confinement and strong opto-mechanical phase modulation possible in MIM structures. While the gap is only 17 nm, a reasonable 8.5 dB insertion loss is made possible by the short device length of only 2.5 μm. The switch footprint is only 0.5 μm wide for a total area of 1.25 μm².