3.13.1 Optical Properties
The fiber Mach-Zender interferometer is used to calibrate the wavelength scans of the zipper cavity modes. For the zipper cavity mode (T E+,1) of the device studied in Fig. 2(c), Fig. 3, and Fig. 4, the taper-loaded optical Q-factor was measured to be QT = 2.8× 104 with a transmission contrast (fractional dropped power) of
∆T = 27.5%. The resonance wavelength is λ ∼ 1543 nm. The fit value (see below) of the component of optical loss attributed to absorption is Qa= 4.5×105.
3.13.2 Geometry
As discussed in the main text, the zipper cavity device under study had l = 36 µm, w = 650 nm, s = 120 nm, and t = 400 nm, as measured by calibrated SEM inspection. The etched air holes were measured to be 330 nm by 330 nm in area. The total number of air holes per beam is 55.
3.13.3 Silicon Nitride Material Properties
The material properties of silicon nitride were scoured from a number of sources and journal articles. Where possible we have used parameters most closely associate with LPCVD silicon nitride on h100i Si. The density of LPCVD silicon nitride is taken to be ρ = 3100 kg/m3, the Young’s modulus Y ∼ 290 GPa, the tensile stress S ∼0.75 GPa, the coefficient of thermal expansionηT E = 3.3×10−6 K−1, the thermal conducivity κth ∼20 W/m/K, and the specific heatcsh = 0.7 J/g/K.
3.13.4 Thermal Properties of the Zipper Cavity
Due to the air-filling-fraction of the etched holes in the zipper cavity nanobeams, the thermal conductivity of the patterned beams was taken as Γth = 75% of the
bulk value. A simple estimate for the thermal resistance of the zipper cavity is then given by Rth ∼ l/(8twΓthκth) ≈ 1.15 × 106 K/W, where the factor of 1/8 comes from the ability for heat to escape out either end of the nanobeams and in either direction. The physical mass of the zipper cavity, taking into account the etched holes, is approximately m = 43 picograms. The heat capacity of the zipper cavity is then roughly ch = 3×10−11 J/K. From the heat capacity and the thermal resistance, the thermal decay rate is estimated to be γth = 1/Rthch ∼ 2.9 ×104 s−1. Finite-element-method simulations of thermal properties of the zipper cavity yield an effective thermal resistance of Rth = 1.09×106 K/W and a thermal decay rate of γth = 1/Rthch ∼ 5.26×104 s−1 for temperature at the center of the zipper cavity, in reasonable correspondence to the estimated values. The thermal tuning rate (dominated by the thermo-optic effect) for the device under study was measured to be δλc/δT ∼0.0149 nm/K using a thermo-electrically heated stage and a thermo-couple placed a few millimeters from the sample.
3.13.5 Optomechanical Properties of the Zipper Cavity
The bare mechanical resonance frequency of the h1d zipper cavity is measured to be ΩM ∼ 8 MHz, in good corresondence with the FEM-simulated value when S = 0.75 GPa of tensile stress is introduced into the silicon nitride film. The measured mechanicalQ-factor is approximatelyQM ∼50 for the differential mode, and roughly QM ∼ 150 for the common mode of motion of the nanobeams. This difference is attributed to the squeeze-film-like damping [76] of the differential motion due to gas
“squeezed” in between the beams. It should be mentioned that all measurements in the work described here were done in a nitrogen purged box (i.e., not under vacuum).
The FEM-simulated optomechanical coupling length, based upon SEM images of the device under test, is LOM = 2.09 µm. The inferred optomechanical coupling length value, based upon the peak measured optical spring effect for various optical input powers, is LOM = 1.575 µm. Although good correspondence is found between simulated and measured LOM, the measured value of LOM = 1.575 µm is used to fit
the remaining cavity parameters as described below.
3.13.6 Wavelength-Scan Fitting
The steady-state equations of motion of the zipper cavity, as given by eqs. (1.126- 1.128), are numerically solved with variable parameter Qa (all other parameters are fixed to values given above). The resulting wavelength dependent transmission curve is then fit to the measured (low-pass filtered) curve for a variety of optical input powers in order to determine the fit value of Qa. Optical input power is calibrated as described in the Methods section using a calibrated power meter and measuring the system response for optical power sent in both directions down the fiber taper.
From the fit transmission curve (Fig. 4(a)), the laser-cavity detuning (Fig. 4(b)) at each point within the intensity image of Fig. 4(c) can be determined. Calculating the optomechanical damping versus laser-cavity detuning, with and without the thermo- optic correction, is then used to model the expected RF power in the mechanical resonance line using eqs. (1.175-1.178).
Chapter 4
Design and Simulation Principles of Photonic and Phononic Crystal Optomechanical Resonators:
Optomechanical Crystals
This chapter is reproduced and adapted from Ref. [4].
4.1 Summary
Periodically structured materials can sustain both optical and mechanical excitations which are tailored by the geometry. Here we analyze the properties of dispersively cou- pled planar photonic and phononic crystals: optomechanical crystals. In particular, the properties of co-resonant optical and mechanical cavities in quasi-1D (patterned nanobeam) and quasi-2D (patterned membrane) geometries are studied. It is shown that the mechanical Q and optomechanical coupling in these structures can vary by many orders of magnitude with modest changes in geometry. An intuitive picture is developed based upon a perturbation theory for shifting material boundaries that allows the optomechanical properties to be designed and optimized. Several designs are presented with mechanical frequency ∼ 1-10 GHz, optical Q-factor Qo > 107, motional masses meff ≈ 100 femtograms, optomechanical coupling length LOM < 5 µm, and a radiation-limited mechanical Q-factor Qm >107.