## 1 Introduction  
Ranging from the frequency conversion to photon pair emission, the second-order nonlinear phenomena are fundamental tools in integrated photonics. The choice of the material to generate this nonlinear conversion depends on the specific application. An application which is of our interest is the quadratic comb generation [1]. Optical frequency combs (OFCs) are light sources whose spectra contain thousands of equally spaced laser lines and have revolutionized precise optical frequency measurements. The self-referencing allowed by octave spanning OFCs was even awarded with the Nobel Prize in Physics in 2005. In 2015, optical frequency comb generation has been demonstrated in silicon microring resonators by using four wave mixing and self-phase modulation [2]. Due to these nonlinear processes, the frequency comb generation was limited in its frequency span. To extend this span beyond one octave, the use of both second-order and third-order nonlinearities was shown in MgO:LiNbO\({}_{3}\) microcavities [3]. In fact, second-order nonlinearities can directly produce simultaneous octave-distant OFCs with intracavity phase modulation. Via cascaded second-order processes, they enlarge the generated frequency spectrum in these \"quadratic comb\" sources [4; 5; 6; 7]. Critical in this approach is the use of the second-order nonlinearities. Therefore, to use a material to achieve a quadratic comb source, it is mandatory that the same material shows second-harmonic generation (SHG), which in turn can be used as a tool to scout for the best material for quadratic OFCs by measuring the second-order nonlinear dielectric susceptibility \(\chi^{(2)}\) of materials. A large \(\chi^{(2)}\) does not suffice to make the material suitable for quadratic comb applications in integrated photonics. For instance, thanks to its ferroelectric properties, lithium niobate can achieve a high \(\chi^{(2)}\) but monolithic integration is difficult [8; 9]. Efficient SHG was reported in III-V semiconductors, such as AlN [10] and GaN [11]. Indeed, GaAs, InP, and their alloys have large \(\chi^{(2)}\) values and can be integrated in small photonic chips, even though their mass manufacturing is still challenging [12; 13; 14]. With respect to integration, silicon or silicon-based materials are the preferred platform [15]. However, silicon is a centro-symmetric material and does not show any second-order nonlinearity in the dipole approximation [16]. However, by strain [17; 18], by alloying, as in silicon nitride [19], and by applying a controlled DC electric field [20; 21], SHG has been reported. Even though its origin is still debated--a strain-induced \(\chi^{(2)}\)[22] or an effective \(\chi^{(2)}\) caused by a dressed third-order nonlinear dielectric susceptibility \(\chi^{(3)}\)[18]--this enables the potential of silicon for quadratic OFCs. It is therefore the aim of this paper to discuss the design of a microring resonator that shows SHG for a range of parameter variations which is representative of the typical fabrication tolerance in integrated photonics. We want to propose a method which yields high generation efficiency despite the uncontrolled variations in the microring resonator geometry. The SHG in the microring resonator is allowed by the quasi-phase matching due to the typical 4-bar symmetry in a ring made of a zinc-blend or a diamond lattice material (details in Appendix B). The use of this symmetry has been already reported and discussed in many papers, such as [23]. As a model system we used silicon leveraging the reports on strained silicon or electric-field-induced SHG without entering into the debate on its origin. We simply assumed that a \(\chi^{(2)}_{2 xy}\) element was present in the studied system. We studied various microresonator geometries, allowing SHG to find the best compromise between performances and experimental requirements. We optimized the geometries while also taking into account the random fabrication defects. This paper is organized as follows. First, we present the method and the general results.