Photonic integrated circuits have enabled progressively active functionality in compact devices with the potential for large-scale integration. To date the lowest loss photonic circuits are achieved with silica or silicon nitride-based platforms. However, these materials generally lack reconfigurability. In this work we present a platform for achieving active functionality in any dielectric waveguide via large-scale opto-electro-mechanical tuning of the effective refractive index (Δneff≈0.01-0.1) and phase (Δϕ>2π). A suspended microbridge weakly interacts with the evanescent field of a low-mode confinement waveguide to tune the effective refractive index and phase with minimal loss. Metal-coated bridges enable electrostatic actuation to displace the microbridge to dynamically tune nEFF. In a second implementation we place a non-metallized dielectric microbridge in a gradient electric field to achieve actuation and tuning. Both approaches are broadband, universally applicable to any waveguide, and pave the way for adding active functionality to many passive optical materials.
© 2016 Optical Society of America
Optical communications are essential in many areas. A significant advance that has made optical communications  possible is the development of low-loss passive optical fibers [2,3]. However, any dynamic functionality in a fiber-optic network generally requires discrete components, e.g. for switching, modulation, or tunable filtering. More recently, optical or photonic integrated circuits  have enabled active functionality in a compact chip-scale device. Although losses in semiconductor waveguides have been reduced substantially, the lowest loss is still achieved in silica  or silicon nitride  based photonic integrated circuits. These material systems, while enabling ultra-low losses, are generally not reconfigurable. Active functionality in silica or silicon nitride has been shown to a limited extent [7,8] using temperature tuning, although the thermo-optic effect is only of the order of Δn/ΔT = 10−5 which makes this approach  highly inefficient. While recent work has shown some electro-optic effects in strained silicon/silicon nitride [10,11], the origin and magnitude of these effects are still being studied [12,13]. Liquid crystals  on the other hand enable large Δn, although their response time is limited.
Recently, micro-electro-mechanical systems (MEMS)  and optical MEMS [16,17] have enabled reconfigurable photonic devices  including actuated-mirror phase shifters  and tunable filters . These approaches have relied on dynamically controlling the physical path length and have achieved large phase shifts up to 3π , although they require a substantial device footprint. A different approach relies on changing the optical or effective path length by changing a material’s effective refractive index (neff) via evanescent field interaction. This approach does not require a change in the physical path length and thus has the potential for compact footprint devices and large-scale integration. Initial demonstrations utilized off-chip atomic force microscope tips  or silica fiber probes  to perturb the effective index of micro-/nanophotonic devices. More recently, a chip-scale approach utilized a silicon nitride waveguide side-coupled to an electrostatically-actuated perturber . Actuation leads to a variable phase shift with modest effective index tuning (Δneff≈0.002) due to the tight optical mode confinement. In another implementation  an array of gold bridges suspended above a fixed gold layer serves as gap plasmon waveguides whose neff can be tuned by actuating the bridges and varying the air gap. The gap plasmons are probed with free-space-coupled light in order to measure the phase shift by imaging the interference pattern resulting in a measured Δneff≈0.03 . However, plasmonic structures are challenging in terms of loss  and coupling to dielectric-based waveguides is often difficult, which may limit the integration of such structures in photonic integrated circuits.
In this work, we present a fully-integrated waveguide-coupled opto-electro-mechanical phase shifter architecture enabling an exceptionally large Δneff. Our approach is broadband, low-loss, and provides a direct means for coupling to other components on-chip and is therefore suitable for large-scale integration in photonic integrated circuits. The device combines a thin-core low-mode confinement engineered waveguide  with a suspended microbridge fabricated directly above the waveguide surface in a compact footprint device. The evanescent field interaction between the guided mode and the suspended microbridge leads to a controllable induced phase shift. We use electrostatic actuation in a metal-coated bridge to modify the waveguide-bridge separation and to dynamically control the phase shift. We also take advantage of forces induced by gradient electric fields [28–31] to actuate an uncoated dielectric bridge (no metal) to tune the effective index and phase while eliminating metal-induced losses. Both approaches are widely-applicable to micro-/nanophotonic waveguides to enable active functionality via large effective index tuning and phase shifts in structures using otherwise passive optical materials.
Figure 1 shows the device principle of operation. A thin silicon nitride (Si3N4) rib waveguide confines and guides light. Owing to the large aspect ratio, the optical mode is strongly confined along the waveguide width (wrib = 2.5 μm with etch depth hrib = 55 nm) but only weakly confined along the thickness (tSi3N4 = 175 nm). The result is a large evanescent field at the waveguide surface that was maximized by choosing an appropriate Si3N4 layer thickness . The fundamental quasi-TE waveguide mode (TE00) has an effective index neff = 1.465 at λ = 1.5 μm, which is only slightly larger than the lower cladding index (nSiO2 = 1.444) and is indicative of the weak mode confinement. At the same time, the measured waveguide propagation loss is modest (as low as 0.7 dB/cm ). A 300nm thick tensile SiNX microbridge is suspended just above the waveguide surface with a nominal air gap of g = 190nm, as shown in Fig. 1(a). The microbridge is coated with a titanium/gold layer (Ti/Au = 5nm/35nm) and, together with an adjacent lower side electrode (Fig. 1(b)), forms a capacitive electrostatic actuator. By grounding the microbridge and applying a bias to the lower electrode the air gap separating the waveguide and microbridge perturber can be varied.
We simulate  the waveguide’s TE00 mode at λ = 1550 nm and find a strong dependence of the effective index on the air gap with neff increasing from 1.47 to 1.62 (Fig. 1(c)). The weak mode confinement results in an evanescent field that extends hundreds of nanometers above the waveguide surface and hence the waveguide’s neff is strongly affected by evanescent field interaction with the microbridge. The simulated loss induced by the Ti/Au metal layer on top of the microbridge is modest as shown in Fig. 1(c) (<0.08 dB for a L = 100 μm interaction length), likely because the thick SiNX microbridge keeps the fundamental mode far away from the Ti/Au for gaps>150 nm (Fig. 1(d)). Finally, we note that the structure in Fig. 1 also supports a microbridge optical mode that is confined largely in the suspended SiNX microbridge. However, this hybrid dielectric-metal plasmonic mode exhibits significant loss and is quasi-TM-polarized so that this mode is not expected to contribute to the device operation since we launch only horizontally-polarized light (see Appendix).
3. Experimental results
3.1 Device actuation
A fabricated phase shifter is shown in Fig. 2(a) (see Appendix for fabrication process). Suspended above the waveguide is the SiNx/Ti/Au microbridge; the waveguide is continuous underneath the microbridge. Although not apparent in Fig. 2(a), the lower electrode has collapsed toward the substrate after sacrificial release (Fig. 2(b)). We use a white light optical profilometer to measure the bridge profile as we increase the bias voltage on the lower side electrode while keeping the bridge grounded. The beam deflection is measured as a function of bias (Fig. 2(c)) revealing that the beam pull-in voltage  has a 1/LMEMS-dependence (Fig. 2(d)), where LMEMS is the suspended microbridge length. Although our actuator geometry is not strictly parallel plate, the actuation is predominantly in the vertical direction since the large width-to-height ratio (wMEMS/tMEMS = 3 μm/350 nm) results in a bridge that is approximately 10x more compliant in the vertical direction compared to the lateral direction.
3.2 Optical characterization
We measure the opto-electro-mechanical device response using the setup in Fig. 3(a). Lensed fibers couple light from a tunable laser to the device. The phase shift is measured using an unbalanced Mach-Zehnder interferometer (MZI) with a built-in path length imbalance of ΔLMZI = 100 μm on one arm. The laser wavelength is tuned and the device transmittance is measured by collecting the output signal using a second lensed fiber and sending the signal to a photodetector. The bias on the side electrode is increased while grounding the microbridge, and for each voltage the MZI response is measured. As the microbridge is actuated down toward the waveguide surface the evanescent field interaction becomes larger and the waveguide neff increases resulting in a broadband MZI fringe shift (Fig. 3(a), inset). The total phase is ϕMZI ± ϕMEMS, where ϕMZI is the built-in phase shift due to the path length imbalance (ΔLMZI) and ± ϕMEMS is the phase shift induced by the MEMS bridge (LMEMS). The sign of ϕMEMS depends on the location of ΔLMZI compared to LMEMS. We measured several devices, some of which had the microbridge (LMEMS) on the same waveguide segment as the built-in MZI path imbalance (ΔLMZI) as shown in Fig. 3(a) and some of which had LMEMS and ΔLMZI on opposite waveguide arms. As shown in Fig. 3(b) and 3(c), the fringe shift direction depends on the relative location of the microbridge (LMEMS) compared to the built-in MZI path imbalance (ΔLMZI). For each spectrum ϕMEMS is measured and Δneff is extracted (Fig. 4).
Figure 4(a) shows the measured MZI phase shift as a function of bias voltage for a device with LMEMS = 120 μm. A phase shift Δϕ>2π is measured for Vbias = 4.2 V. From the measured phase we can extract the change in effective index as Δϕ = 2πΔneffLMEMS/λ, where λ is the wavelength and the displacement calibration curves measured in Fig. 2 were used to convert the applied bias voltage into an effective air gap (see Appendix). This conversion allows us to calculate the expected Δneff and Δϕ. The measurements and calculations in Fig. 4(a) and 4(b) show excellent agreement indicating that the modeling in Fig. 1 accurately describes our device behavior. In Fig. 4(c) we show the measured and calculated Δneff for five phase shifters with bridge lengths LMEMS = 50-120 μm. The maximum extracted Δneff > 0.02, which is among the largest measured MEMS-based Δneff reported to date [24,25], although it is still significantly smaller than the maximum Δneff≈0.1 predicted by the simulation in Fig. 1(c); this is likely due to the fact that the g>100 nm for the measurements in Fig. 4(b) and 4(c). Our phase shifter figure of merit (FOM) is the measured transmittance vs. the induced phase shift, which we plot in Fig. 4(d). The transmittance is obtained by monitoring the peak transmission (i.e. at the MZI fringe maxima) as we increase the applied bias. The devices exhibit a FOM = 0.75 dB/π (1.5 dB/2π in Fig. 4(d)). We attribute the reduction in transmittance at larger phase shifts due to the strong coupling to the microbridge and the increased optical mode overlap with the Ti/Au metal layer at small air gaps (see Fig. 1(d)). Finally, we note that due to the nonlinear microbridge displacement with applied bias (see Fig. 2d) as well as the exponential decay of the evanescent optical field the devices exhibit a tradeoff between higher stability (smaller Δϕ) and larger Δϕ (at the expense of lower stability and higher phase noise).
3.3 Device actuation via gradient electric fields
In order to decrease potential metal-induced optical losses we fabricated and measured a MEMS phase shifter that relies on a different type of actuation mechanism that is similar to dielectrophoresis (DEP). Although DEP  is a technique commonly used to displace and sort microparticles and cells in microfluidics  (similar to the gradient optical force  used in optical tweezers ), it has only recently been used to actuate MEMS/NEMS devices [28–31]. Our actuator design consists of a suspended microbridge on top of a rib waveguide, identical to previous devices. However, the microbridge is now left uncoated (i.e. no metal). Two Ti/Au (5nm/55nm) side electrodes are patterned on both sides of the microbridge on top of the sacrificial SiO2 layer, as shown in Fig. 5(a) (see Appendix for fabrication details). Since the microbridge has been etched partially into the underlying SiO2, the side electrodes are beneath the microbridge. By applying a bias across the electrodes, the microbridge experiences a gradient electric field (Fig. 5(b)) and hence a vertical force that displaces the microbridge towards the waveguide surface (Fig. 5(c)). The gap between the two electrodes is designed to be 4 μm for ease of fabrication – significantly larger than the 1 μm actuation gap for the devices in Figs. 2-4. The electric field is therefore modest and we require significantly larger actuation voltages compared to devices using electrostatic actuation. It should be noted that simple design changes, e.g. reducing the side electrode separation, will lead to enhanced electric field gradients and a reduced actuation voltage.
The optical phase shift was measured using the same approach as before. Figure 6(a) shows a representative MZI fringe shift measured for different applied bias voltages (LMEMS = 100 μm). A second device with LMEMS = 40 μm was measured and the phase shift as a function of applied bias for both devices is shown in Fig. 6(b). The two devices achieve phase shifts of Δϕ≈3π/2 and Δϕ>π/2 at bias voltages of 36V and 60V, respectively. An accurate displacement vs. bias calibration could only be performed on a test bridge without a waveguide underneath (LMEMS = 40 μm). Nonetheless, the calculated Δϕ and Δneff show general agreement with measurements for the LMEMS = 40 μm device (Fig. 6(b) and 6(c)). More importantly, the phase shifters show substantially lower loss compared to the devices in Fig. 4 with a figure of merit FOM = 0.07dB/π (LMEMS = 100 μm, 0.1 dB/1.5 π in Fig. 6(d)) and FOM = 0.53dB/ π (LMEMS = 40 μm, 0.4 dB/0.75 π in Fig. 6(d)).The reduced loss is a direct consequence of the absence of metal in the suspended microbridge perturber and also results from the slightly thinner microbridge that minimizes coupling of optical power to the microbridge (see Appendix).
In terms of effective index tuning, our waveguide-coupled devices substantially exceed the Δneff measured in previous silicon nitride optomechanical phase shifters  and are comparable to previous work using gap plasmon waveguides . We attribute the improved Δneff to the thin waveguide core design, which maximizes the evanescent field at the surface . The large evanescent field maximizes the sensitivity to the presence of the dielectric microbridge perturber and enables a large Δneff to be achieved. In our measurements, the microbridge is actuated down towards the waveguide resulting in an increase in Δneff. We note that other actuator geometries can be envisioned in which the air gap separating the microbridge and waveguide is increased during actuation, for example by placing a side actuation electrode above the waveguide . The increased gap decreases Δneff so that bi-directional tuning can be envisioned by using both types of actuators in a single device.
The gradient electric field actuated devices in Figs. 5-6 have shown that metal-coated bridges are not required for actuation and phase shifting. The advantage with this approach is the strong reduction in optical loss at small gaps due to the absence of metal. However, for sufficiently thick microbridges the waveguide optical mode can still couple to the microbridge (Appendix) leading to power transfer and, effectively, loss. An additional improvement to the phase shifters would therefore be to adiabatically taper  the width of the microbridge along its length to prevent any significant power coupling to the microbridge.
The power requirements for electrostatic actuators are inherently low since they consume virtually no power in the steady-state. The energy required to tune the phase shifter can be estimated from the device capacitance as 0.5 × CMEMS × Vbias2, where CMEMS is capacitance of the MEMS electrostatic actuator. Using simple models based on microstrip transmission lines [40–42] we estimate a device capacitance of CMEMS≤1 pF. Assuming a worst-case operation at 10 MHz continuous switching, a device capacitance CMEMS = 1 pF and a 5 V switching voltage the switching energy is 12.5 pJ which gives an operating power of P = 12.5 pJ/100 ns = 125 μW/π-phase shift. This is still significantly lower power than thermo-optic phase shifters, which require in excess of 10 mW/π-phase shift . Additional design optimization to reduce the device capacitance will further reduce the switching power. Furthermore, many applications require lower switching rates leading to a further reduction in power. Finally, a key advantage compared to thermo-optic phase shifters is the minimal steady-state electrical power required in electrostatic actuators. In contrast, thermooptic approaches require electrical power both during switching and at steady-state.
Comparing the two actuation schemes, we find that the devices actuated by a gradient electric field require much larger actuation voltages (Figs. 5-6) compared to the electrostatically-actuated devices (Figs. 2-4). A detailed modeling study for the two actuation mechanisms is found in ref . The gradient electric force scales linearly with the microbridge thickness whereas the electrostatic force scales linearly with the microbridge width . In general, electrostatic actuation of micromechanical structures is therefore more efficient since the spring constant of a microbridge has a linear width-dependence but a highly non-linear dependence on the thickness. Nonetheless, we emphasize that the actuation voltage required for the gradient electric field devices can be lowered substantially by reducing the separation between the side electrodes in order to increase the electric field.
In addition to enabling lower-power phase shifting compared to thermo-optic approaches, an optomechanical phase shifter also potentially enables much faser device speeds. In Fig. 7(a) and 7(b) a LMEMS = 75 μm phase shifter similar to the devices in Fig. 4 is biased at Vdc = 4.5 V while a small-signal of amplitude Vac = 100 mV is used to excite the microbridge. For sinusoidal actuation the device response can exceed frequencies of 1 MHz (Fig. 7(a)), and actuation with a square-wave drive indicates rise- and fall-times of around τ = 400 ns (Fig. 7(b)). This is substantially faster than most thermo-optic phase shifters while doing so without the need for a high-Q optical cavity. Figure 7(c) shows the fundamental mechanical resonance measured with a network analyzer demonstrating a f0,m = 2.3 MHz and a mechanical quality factor Q0,m = 103 (in air). The rise-/fall-time is essentially dominated by the mechanical resonance, i.e. τ = 1/f0,m. Simulations show that by simply shortening the microbridge length the mechanical resonance frequency can exceed 10 MHz, which would enable switching speeds of τ<100 ns. By moving to other materials or device geometries, mechanical resonance frequencies approaching 1 GHz (i.e. nanosecond-range response times) may be possible in the future [29,44,45], although there will be a trade-off between higher-frequency operation and total phase shift due to the generally smaller microbridge displacements at higher frequencies.
As shown in Fig. 1, our weakly-confined waveguide design results in an evanescent field that extends hundreds of nanometers above the waveguide surface. This has implications not only for tuning neff but also for any optical forces that may arise. Although optical forces  in optomechanical structures are an active field of research, we expect the optical force to be modest for g>100 nm since the force is proportional to dneff/dg (see Appendix). The weak mode confinement design may therefore be advantageous in terms of device stability.
Although all of the measurements presented here were performed with the electric field in the wafer plane (TE-polarization), this is not a strict limitation of our optomechanical phase shifter architecture. For the metal-coated microbridge devices in Figs. 1-4 we use TE-polarization to prevent coupling to the hybrid metal-dielectric plasmonic mode (Appendix), which experiences higher loss. In contrast, the pure dielectric microbridges (Figs. 5 and 6) are not limited to a particular polarization and can be designed for operation with TM-polarized light.
In conclusion, we have demonstrated a broadband opto-electro-mechanical phase shifter enabling Δϕ>2π phase shifts via large-scale effective index tuning (Δneff = 0.01-0.1) in a LMEMS≈100 μm device. Our waveguide-integrated MEMS-based phase shifter not only enables a large tunable Δneff comparable to liquid crystal  and gap plasmon  based approaches, the vertical architecture results in an extremely small device footprint occupying an area only slightly larger than the waveguide itself. The evanescent coupling between a standard dielectric waveguide and a MEMS microbridge perturber coated with metal (as in Figs. 3 and 4) may also provide a scalable approach to coupling to plasmonic or hybrid metal-dielectric modes or potentially gap plasmons . Furthermore, actuation via gradient electric fields (as in Figs. 5 and 6) is versatile and can be implemented in any suspended dielectric structure. Both approaches may find wide application in many different areas to provide active functionality to many passive optical materials via broadband Δneff tuning at low electrical powers.
See Table 1 for variables.
A2. Device fabrication
We start with commercial Si(100) wafers of 325 μm thickness with 5,000 nm thermal oxide (SiO2) and 175 nm silicon nitride (LPCVD Si3N4). The waveguides and Mach-Zehnder interferometers (MZI’s) are defined by electron-beam lithography using MaN2403 resist followed by an ICP/RIE dry etch to a depth of 55 nm using SF6/C4F8 chemistry.
Next, the micro-electro-mechanical (MEMS) layers are deposited using PECVD (SiO2=230 nm and tensile SiNX=310 nm; thicknesses verified by ellipsometry) after which Ti/Au=5 nm/35 nm is evaporated. The tensile stress in the as-deposited SiNX is found to be σ≈100 MPa by performing test depositions and measuring the wafer curvature before and after deposition. We emphasize that the intrinsic stress calibration does not include the effects of the Ti/Au metal layers. For this reason, we also fabricated a number of test microbridges and measured their profile after sacrificial release. With an optimized process we can release microbridges up to LMEMS=300 μm with a flat profile.
The MEMS bridges and side electrodes are patterned using MaN2403 resist. For ease of alignment the MEMS bridges are designed to be 3.0 μm wide, which is slightly wider than the rib waveguides, which have Wrib=2.5 μm. The pattern is then transferred into the Ti/Au layer using Ar-milling and into the SiNX layer using an ICP/RIE etch with SF6/C4F8 chemistry. We etch 110 nm into the SiO2 sacrificial layer (measured by contact profilometry). The devices in Figs. 1-4 and Fig. 7 are cleaved to expose the waveguide facets and are then released using buffered-HF (BHF) followed by critical point drying. After sacrificial release the microbridges remain flat; however, the side electrode, which acts like a compliant cantilever, reproducibly collapses towards the substrate due to the built-in strain gradient  between the SiNX and Ti/Au layers. This enables the vertical electrostatic actuator to be realized using only a single and self-aligned lithography step. The actuator process results in a 230nm vertical separation (not considering the waveguide rib) and a 1 μm lateral gap between the microbridge and side electrode.
Before sacrificial etching, the devices in Figs. 5-6 have their Ti/Au layers removed using gold etch TFA and piranha clean (H2SO4:H2O2). The side electrodes are patterned using electron-beam lithography with ZEP-520A resist followed by Ti/Au=5 nm/55 nm evaporation and lift-off. The devices are then released in BHF followed by critical point drying. The bridge thickness is reduced to 254 nm±12 nm (measured using contact profilometry on collapsed cantilevers) since the SiNx is not protected by Ti/Au during the release etch.
A3. Data analysis
A LabVIEW program extracts the average phase over the wavelength scan and records the average fringe spacing or FSR (free-spectral range) for each bias voltage. The MZI’s BAR output is IBAR(λ)∝1+cos(ϕMZI ± ϕMEMS), where ϕMZI is the phase introduced by the MZI’s built-in path length imbalance and ϕMEMS is the phase contribution from the MEMS microbridge perturber. The sign of ϕMEMS depends on the location of the microbridge (LMEMS) relative to the built-in phase shift (ΔLMZI).
The effective index tuning (Δneff) is obtained from Δϕ=(2π/λ)∫x=0-L Δneff[m(x)]dx, where λ is the laser wavelength, L=LMEMS is the microbridge length, Δneff[m(x)] is the change in effective index due to the evanescent field interaction between the optical mode and the microbridge, and m(x) is the microbridge deflection or mode profile that represents the microbridge-waveguide gap (height) at position x. A typical microbridge profile during actuation is shown in Fig. 8a from which we find that a cosine gives a good approximation to the measured bridge profile. Therefore, Δϕ=2πΔneff(g)LMEMS/λavg, where λavg is the center wavelength of the MZI spectrum and Δneff(g) is the effective index tuning assuming a constant average air gap g (Fig. 8b). The assumption of a constant average air gap is consistent with previous work  and does not affect the accuracy of the extracted Δneff.
A4. Displacement vs. bias calibration
The microbridge displacement vs. bias calibration is performed using a white-light optical profilometer (Zygo). A bias voltage is applied to the respective MZI-MEMS bridge and the beam deflection is measured. The maximum deflection is then extracted from a linescan across the bridge surface to obtain the displacement vs. bias calibration. We measured the displacement vs. bias for a variety of MEMS actuator geometries. By performing a curve fit to the data, we can convert the applied bias to a microbridge displacement; this lets us plot both measured and simulated Δneff vs. air gap on the same plot (as in Fig. 4b and 4(c) and Fig. 6(c)). Similarly, we can convert from a simulated gap to an applied bias in order to plot the measured and simulated phase shift vs. bias on a single plot (as in Fig. 4(a) and Fig. 6(b)).
For the gradient electric field device (Figs. 5-6) it is difficult to measure the bridge deflection accurately since the high reflectivity of the two side electrodes, the bridge transparency, and the rib waveguide underneath the microbridge result in a reduced signal from the microbridge surface which makes optical profilometry a non-ideal method for calibrating the displacement in these devices. For this reason, the displacement calibration for the gradient electric field actuation devices (Fig. 6) could only be performed on a test bridge without a waveguide underneath. In contrast, the displacement calibration for the electrostatic devices (Fig. 4) were performed on MEMS bridges with waveguides (i.e. opto-electro-mechanical phase shifters).
A5. Optical simulations: other modes and device configurations
A5.1 Microbridge modes
The device in Fig. 1 also supports a mode that resides predominantly in the MEMS microbridge (Fig. 9). However, the loss is significantly higher than for the waveguide mode due to the larger mode overlap with the Ti/Au(5nm/35nm) layers. The field component is predominantly Ey which indicates that this is a TM-polarized mode. We therefore expect that this mode does not contribute substantially to the device operation since we perform our measurements with TE-polarization (launch horizontally-polarized light).
A5.2 Effect of microbridge thickness
We performed simulations for various microbridge thicknesses for the device in Fig. 5 (microbridge with no Ti/Au metal) to account for the partial etch of the microbridge during sacrificial release. We find that for thinner microbridges (e.g. tMEMS=200 nm) the effective index decreases compared to thicker bridges (e.g. 250 nm). Nonethless, there is still substantial effective index tuning (Fig. 10a). For bridges of 300 nm thickness the structure supports both waveguide and microbridge modes and the device more closely resembles a directional coupler. The simulations show that as the microbridge-waveguide separation (g) is decreased the waveguide mode neff decreases since this mode is pulled upwards and experiences a larger overlap with the air region above the waveguide; in contrast, the microbridge mode neff increases with decreasing g (Fig. 10b). Furthermore, for small g the waveguide mode is no longer supported and all the optical power resides in the microbridge mode. The simulations in Fig. 10b suggest that device configurations can be envisioned in which actuation of a microbridge perturber down towards the waveguide can lead to a decrease in effective index so that bi-directional tuning may be possible by using thin (as in Fig. 10a) or thick microbridges (as in Fig. 10b).
A6. Optical forces
Optical gradient forces are an active area of research. However, in an opto-mechanical phase shifter any optical force can lead to device instability or other undesirable device performance. Although we have not measured any optical forces in our devices, we can estimate the induced gradient optical force from : Foptical=(ngroup/neff)(LMEMS/c)(dneff/dg)P, where P is the optical power. In Fig. 11 we plot the calculated optical force per unit power and unit length for our device based on the dnEFF/dg simulations from Fig. 1. We find that the optical force is substantially smaller compared to similar waveguide structures made from silicon . We attribute the smaller force to the weak mode confinement in our Si3N4 waveguide design so that the evanescent field extends hundreds of nanometers away from the waveguide surface leading to a smaller dnEeff/dg and hence a smaller Foptical.
We thank NRL/ONR for funding this work under a base 6.1 program (ONR 61153N). We also thank the NRL Nanoscience Institute (NSI) staff for cleanroom access and fabrication assistance. MWP thanks J.B. Khurgin (JHU) for helpful discussions.
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