Optical bistability with a repulsive optical force in coupled silicon photonic crystal membranes

We demonstrate actuation of a silicon photonic crystal membrane with a repulsive optical gradient force. The extent of the static actuation is extracted by examining the optical bistability as a combination of the optomechanical, thermo-optic and photo-thermo-mechanical e ↵ ects using coupled-mode theory. Device behavior is dominated by a repulsive optical force which results in displacements of ⇡ 1 nm/mW. By employing an extended guided resonance which e ↵ ectively eliminates multi-photon thermal and electronic nonlinearities, our silicon-based device provides a simple, non-intrusive solution to extending the actuation range of MEMS devices.

Rapid developments in the field of optomechanics have opened up avenues for fundamental research on quantum state manipulation with macroscopic structures 1 and show promise for novel optomechanical sensors 2 and technologies for both radiofrequency 3 and telecom applications. 4 While most attention has been devoted to compact structures featuring low (picogram) mass and ultrahigh-frequency (gigahertz) mechanical modes, 5,6 the technological implications of static deformation due to optical forces have been less explored. 7 In coupled photonic waveguide geometries, 8,9 bonding and anti-bonding optical modes are supported and the corresponding attractive and repulsive optical forces exerted on a pliant structure (low mechanical frequency) could serve to broaden the range of motion of integrated microelectromechanical devices. This translates to improvement in the detection range of pressure and displacement sensors and the actuation range of electrostatic actuators. In particular, the pull-in limit of electrostatic actuators could be extended by increasing the plate separation with a repulsive optical force. Additionally, novel schemes for preventing stiction, which occurs when attractive forces like the Casimir force and electrostatic force become overwhelmingly large compared to the mechanical restoring force, have been proposed 10 using a real-time monitoring of the structure's displacement and a counteracting feedback repulsive force (of the order of nano-Newtons and linear with excitation power). In this paper, we demonstrate nanometer-pulling of a thin silicon photonic crystal (PhC) membrane under low vacuum with a repulsive optical gradient force and an attractive photo-thermo-mechanical force. Furthermore, optical bistability induced by optical forces and thermo-optic e↵ect is observed with large excitation powers while minimizing multi-photon nonlinearities.
Our devices -one of which is described in Fig. 1(a) and pictured in Fig. 1(c) -consist of a square silicon PhC slab suspended by four support arms ⇡ 250 nm above a Silicon-on-Insulator (SOI) substrate. They are fabricated from a double-SOI platform, formed by oxide-oxide bonding of two thermally oxidized SOI wafers.
A sacrificial silicon dioxide layer between the two silicon layers is s 0 = 265 nm thick. Electron-beam lithography is performed on a layer of resist (ZEP-520A) to define the pattern. To combat the strong buckling of the silicon device layer by the compressive stress and upward turning moments of the oxide layer underneath, novel stress management techniques 11,12 were incorporated to obtain structures with lithographically determined membrane-substrate gaps. After developing, a fluorinebased reactive-ion etch is employed to transfer the patterns to the top silicon layer.
The device is then released by undercutting the patterned silicon layer with the vaporphase hydrofluoric acid etch. Finally, an annealing step was performed at 500 C for 1 hour in a nitrogen environment to limit surface losses and maximize optical and mechanical quality factors. The height profiles of the released membranes from the substrate are characterized by a confocal microscope (Olympus LEXT OLS-4000).
The structure was designed to support an optical antibonding mode in the wavelength range of 1480-1680nm that results from the hybridization of waveguide modes in the membrane and substrate 12,13 . The precise spectral location of the resonance is determined by the optomechanical coupling between the two modes, the strength of which is defined as g OM ⌘ d!/ds. The distribution of the x-component of the electric field in the top membrane is out-of-phase from that in the bottom membrane, as depicted in the simulation results of the whole structure in Fig. 1(b), which corresponds to the generation of a repulsive gradient force. Additionally, the field symmetries along the x-z and y-z planes indicate that we are operating with a "dark" mode 14,15 , which theoretically does not couple to normally incident light because of mismatch in field symmetry. However, by breaking the periodicity of the full structure, we can couple to the dark mode and achieve high Q opt . Such devices have been the subject of numerous theoretical and experimental investigations on subjects ranging from the lowering of the laser thresholds 16 to increasing the sensitivity of photonic-crystal-based sensors 17 . Here, the dark mode is made accessible A white-light source and output from a near-IR laser are combined and sent through a 50-50 beam splitter, sending half of the signal to an IR power meter and half through a 20x objective placed above a vacuum chamber. The reflected signal is sent back through the beam splitter and can be directed onto a CCD camera allowing us to carefully align the laser spot to the membrane and to a photodetector (PD) to collect optical spectra via the DAq board and mechanical spectra via the real-time spectrum analyzer (RSA).
due to the finite size of the membrane and slight fabrication imperfection. The high Q opt of the dark mode, together with the mode's large optomechanical coupling coe cient g OM = 2⇡ ⇥ 23 GHz/nm (at s 0 = 220.6nm), boosts the strength of the optical force and hence the range of actuation.
As previously described 10 , the potential of a mechanical harmonic oscillator with equilibrium position s 0 , when perturbed by the potential of an optical "spring" 18 centered at s l for a laser wavelength l can create a multi-well potential with two stable mechanical equilibria. The transition between these mechanical equilibria is reflected by the occurrence of optical bistability, due to the the dependence of the resonance frequency on s. Yet, the direct observation of the optomechanically-induced optical bistability can easily be obscured in actual systems by other competing mechanisms including thermo-optic e↵ect due to two-photon absorption, free-carrier dispersion and the Kerr nonlinearity which otherwise have been actively pursued for realizing ultrafast, low-power optical switches and memory 19 . We designed our geometry to minimize these e↵ects by exciting a guided resonance which is delocalized throughout the PhC membrane. We estimate the total mode volume to be ⇡ 260( /n g ) 3 from simulation. Due to its large modal volume, the thermal and electronic nonlinearities (which scale inversely with the modal volume) are dramatically reduced. This is in contrast with many of the optomechanical structures being studied, which have modal volumes ⇡ ( /n g ) 3 and where thermal nonlinearities could be readily observed at even modest input powers. With the current membrane separation of the coupled PhC membrane, optomechanical detuning is larger than thermo-optic detuning that originates from linear absorption due to defects introduced during the fabrication processes, which is two orders of magnitude larger than the intrinsic material absorption of bulk silicon.
We solve for the the optical and mechanical equilibria in the presence of the thermo-optic e↵ect within the coupled-mode theory framework 20 . In particular, the stored optical energy in the system |a| 2 is given by where  is the full-width half-max linewidth of the optical resonance,  e is the ex-ternal coupling rate such that  e / represents the fraction of incident power coupled into the cavity, and P in is the power incident on the structure. Here the detuning of the laser excitation frequency ! l from the perturbed optical resonant frequency can be written as The third term in Eq. 2 is the thermo-optic detuning, with d!/dT = (d!/dn)(dn/dT ), n is the refractive index of silicon, d!/dn is obtained from simulations and approximately 2⇡ ⇥ 10 14 Hz and dn/dT is the thermo-optic coe cient of silicon equal to 2 ⇥ 10 4 K 1 . 21 The absorbed optical power and hence the temperature change of the system is given by where abs is the absorption coe cient of the system, C th is the heat capacity,  t is the thermal di↵usion rate. The fourth term in Eq. 2 is the optomechanical detuning.
In particular, the displacement of the membrane due to the respective photo-thermomechanical force and the repulsive gradient force is given by where K is the spring constant of the mechanical resonator and D is the thermalmechanical force coe cient in units of Newtons per Kelvin. 22 The above equations can be solved self-consistently to yield and hence the perturbed optical resonant To investigate the hysteresis and bistability in our devices, we employ a freespace coupling setup in a low vacuum condition described in Fig. 1(d). A low power (25 µW) wavelength sweep is shown in Fig. 2(a) (red curve), revealing a cavity resonance centered at 0 = 1581.55 nm. To account for interference fringes from parasitic reflections, we carefully fit both the optical resonance and the oscillating background (black line) to an expression which has the form where r d ( ) is the background reflectivity, and is the relative phase between the underlying background reflection and the optical cavity. Fitting parameters correspond to an optical cavity with  e = 0.3 and Q tot opt = ! 0 / = 3400. Using these parameters, we can model the reflectance of the system as a function of laser wavelength at multiple powers (P = 0.275 to 2.275 mW), shown in Fig.   3(a) and o↵set for clarity, and compare the results to our experimental observations, shown in Fig. 3(b). The experimental data were collected by sweeping the tunable laser output from short to long wavelength (red curve) and then back (blue curve) at a fixed tuning speed of 1 nm/s. We find excellent agreement between experiment and theory, which display an overall redshift of the mode and increasing hysteresis at higher powers. In particular, we directly compare the locations of the forward and backward bistable jumps, f and b respectively, in Fig. 3(c). The locations of these transitions were extracted from the data shown in Fig. 3  This can be seen in Fig. 3(d) as the thick red and blue sections of the curves, corresponding to periodic oscillations of the reflectance at harmonics of ⌦ m . Negative feedback between the membrane's motion and the photo-thermal-mechanical force occurs on the blue-detuned side of the resonance and works to damp the membrane oscillations 20 , allowing us to adequately treat the system as static at these detunings. While the optical spring e↵ect and dynamic back-action have been thoroughly investigated in our system 12 , this paper focuses on evaluating the static e↵ect of the repulsive optical and photo-thermo-mechanical forces. Qualitatively, we observe an overall redshift of the mode and increasing hysteresis at higher powers in Fig. 3(d).
When we compare the bistable jump locations in both the forward and backward sweeps to the values calculated in the same coupled-mode model, the forward sweep data display good agreement while the backward sweep data show discrepancy in the transition wavelengths. We attribute this discrepancy to the fact that our static model does not include the dynamic back-action 20 , which requires modeling over a wide range of time-scales spanning from the fundamental mechanical resonance frequency ⌦ m = 180 kHz to the optical frequency ! 0 = 189 THz and is beyond the scope of this paper. Nevertheless, we can heuristically understand the e↵ects of dynamic back-action as the cause of the discrepancy between our model and our experimental data due to detuning-dependent competition between mechanical amplification and cooling. The amplified mechanical oscillation corresponds to an oscillation of the optical resonance with an amplitude ! osc = g OM x osc (x osc ⇡ 3.3 nm; osc ⇡ 0.55 nm) which is greater than . Thus, when ! l is less than ! 0 0 , the optical resonance oscillates such that it spends time on both sides of ! l . In particular, when the resonance experiences maximum red-shift, l is greater than 0 0 , leading to motion damping at this point in the oscillation. Thus, during a single oscillation cycle, the competing processes of cooling and amplification are occurring at di↵erent points in time. During the backward wavelength sweep, we argue that the system will transition from the amplified state into the cooled state ( b ) before predicted by our static model, corresponding to the point at which cooling begins to dominate amplification over a single oscillatory cycle. Another caveat is that as the laser excitation is swept backward at the lowest sweeping speed such that the membrane transitions from an amplified state to a cooled state, the laser dwell time is shorter than the the time needed for the membrane to equilibrate. Hence the backward bistable transition is obscured by the residual ringing in the optical spectra. As we relate this observation to our goal of optical actuation of mechanical resonators, it is sensible to traverse from the cooled state to an amplified state (which is the forward-sweep case here), whereas the reverse direction presents di culties in evaluating the actuation range.
Finally, we neglect the Du ng nonlinearity in our mechanical model as the oscillation amplitude is still well within the linear regime for our structures: The amplitude is much less than the membrane thickness (185nm) and the compressive stress in the silicon device layer is alleviated by thin accordion structures as shown in Fig. 1(c).
In conclusion, we demonstrated actuation of a micron-scale membrane with a repulsive optical force using an extended guided resonance in a coupled silicon PhC membrane. The net red-shift displayed in the optical resonance of our doubly-bonded SOI platform is a result of an optomechanically induced red-shift, a thermo-optic red-shift, and a photo-thermo-mechanically induced blue-shift. Furthermore, simulations indicate that absorption in our system is dominated by surface defects and adsorbents, resulting in a linear absorption coe cient two orders of magnitude larger than that expected from bulk silicon. By minimizing these e↵ects through fabrication process and design modifications, we can further isolate and exploit the unique optomechanical properties of this platform. Since multi-photon nonlinearities do not occur until the excitation power exceeds ⇡ 1 W with the use of a delocalized optical mode, the extent of pulling of the PhC membrane can be many tens of nanometers.
Our silicon-based device provides a simple, non-intrusive solution to extending the actuation range of MEMS devices.