# A Micro-Processor-Based Feedback Stabilization Scheme for High-Q, Non-Linear Silicon Resonators
## 1 Introduction  
Due to strong optical mode confinement and significant non-linearities, silicon photonics is an attractive platform for non-linear optics [1]. A wealth of useful non-linear modes of operation have been demonstrated on this platform, including: Kerr non-linearity for Four-wave mixing (FWM) [2; 3; 4], super-continuum generation [5] and photon pair generation [6]; and thermal refractive index non-linearity for optical bistability and all-optical signal processing functions [7]. Enhancement of the non-linear effects in waveguide devices has been achieved through both material [8] and device design. In many cases, micro-resonator devices are used to enhance the non-linear interaction for low power, compact device designs [3; 9; 10; 11; 12]. The increased photon lifetime in the cavity and resonant enhancement of the field in a small mode volume allows for improved light-matter interaction, but additionally increases detuning effects on the resonator due to thermal refractive index variation of the resonant wavelength. In non-linear applications it is desirable to operate the system at the point of maximum resonant enhancement, which in turn requires the alignment of the resonator wavelength and input laser signal. Typically, resonator-laser alignment is achieved at a given injection power by tuning the laser from the blue spectral side of the resonant line into the cavity. Gradually, the laser wavelength is increased, increasing the power in the cavity and therefore the thermally induced refractive index of the resonant line shift. The thermal refractive index increase shifts the resonant line to longer wavelengths as the laser wavelength approaches the cold cavity peak. Finally, at some detuning to the red side of the cold cavity peak, the thermally-induced refractive index shift is no longer sufficient to align the cavity resonance with the laser line and the system reverts to the cold cavity state. Figure 1 shows a typical measured transmission spectrum for a silicon waveguide resonator where the injected power is sufficient to induce a thermal refractive index effects. The device is a \(\pi\)-phase shifted Bragg grating, the details of which will be discussed in Section 2.1. In addition to the thermal refractive index detuning induced by optical absorption, devices are also subject to variations in the environmental conditions, with silicon exhibiting a relatively high thermal coefficient of refractive index, \(1.8\times 10^{-4}\)\(\mathrm{K^{-1}}\)[13]. For example, the resonant wavelength of a Bragg grating device is given by \(\lambda_{B}=2n_{eff}\Lambda_{0}\), where \(n_{eff}\) is the modal effective index and \(\Lambda_{0}\) is the grating period. So a change of 0.1 K on a grating with a period of 318 nm and an effective index of 2.445, can result in a resonance shift of over 10 pm. This is on the order of the linewidth of a high-Q cavity resonance, and therefore can significantly affect the operation of the device given a fixed wavelength laser injection, essentially detuning the device resonant wavelength from the laser signal. Furthermore, a shift of the laser wavelength to the red with respect to the cavity working point can essentially'switch off' the non-linear device operation as the resonant enhancement reverts to the cold cavity case. This produces discontinuities in any experimental setup using the device in this mode and is to be avoided. In a temperature controlled laboratory setting with devices mounted on Peltier control stages device stability is often only in the order of minutes due to the picometre tuning range that is tolerable between the laser wavelength and cavity wavelength. There have been a number of demonstrations of cavity stabilization in silicon photonic devices using both electrical [14] and thermal [15] elements on-chip. These schemes have been primarily designed for tracking of the cavity line in linear optical applications such as data communications. Since the cavity lineshape is symmetric in these regimes, active control schemes using dithering are suitable.