## 2 Conditions for SHG  
### Method  
As a model material, we study SHG in a Si microring (Figure 1). As proposed in [18] and due to good knowledge of the material parameters, we consider a silicon waveguide with a thin cladding of Si\({}_{3}\)N\({}_{4}\) of thickness \(e_{SiN}=140\) nm. The waveguide width in the microring is \(w\), its thickness is \(e\), and the internal microring radius is \(R_{in}\). The light propagating in such a microring can be described by the resonant modes of the electromagnetic field. Each mode is labeled by three numbers: the planar number, \(q\), giving the number of the electric field extrema along \(z\); the radial number, \(p\), giving the number of extrema along \(r\); and the azimuthal number, \(m\), corresponding to the number of wavelengths on one round-trip. These modes exist in two different polarizations: either the electric field is along \(z\) (\(Ez\)) and the mode is called TM, or the magnetic field is along \(z\) (\(Hz\)) and the mode is called TE [24]. The fundamental field (pump of wavelength \(\lambda_{f}\)) is coupled to the microring by a straight bus waveguide which is also used to collect the second harmonic (SH) field (wavelength \(\lambda_{SH}\)).  
The SHG occurs if the energy and the angular momentum are conserved. The first conservation requirement is straightforward and reads as \(\lambda_{f}=2\times\lambda_{SH}\). The second one depends on which modes are coupled through the susceptibility tensor. Without loosing generality and to simplify the discussion, we assume a homogeneous second-order nonlinear tensor with a single \(\chi^{(2)}_{zxy}\
eq 0\). In practice, this value of \(\chi^{(2)}\) can be set either by the proper engineering of the strain [25, 26, 27] or by adding properly designed p-i-n junctions across the waveguide in the ring [20]. With the assumption \(\chi^{(2)}_{zxy}\
eq 0\), a \(Hz\) polarized fundamental mode generates a \(Ez\) polarized SH mode. The conservation of the angular momentum reads, therefore, \(m_{SH}=2\times m_{f}+2\), where \(m_{SH}\) is the SH azimuthal number and \(m_{f}\) is the pump azimuthal number (see Appendix B). This situation is similar to the quasi-phase matching condition which exploits the 4-bar symmetry [23, 28].  
We determine the geometries and wavelengths meeting these two requirements by finite element method simulations using the COMSOL Multiphysics software [29]. In addition, in this work we want to propose a design that can be realized and tested in our laboratory. Therefore, after a preliminary computation, we limit the study to the case \(q_{f}=q_{SH}=1\), \(p_{f}=1\), and \(p_{SH}=3\) to keep \(\lambda_{f}<2.6\)\(\upmu\)m because of the available laser source in our laboratory. Moreover, the shorter wavelength limit \(\lambda_{f}>2.2\)\(\upmu\)m is imposed to avoid two-photon absorption (TPA) in silicon. TPA is a process that will compete with SHG, affecting in a drastic way the conversion efficiency. For a given thickness, width, and fundamental azimuthal number, the program performs iterations modifying the microring radius until the SHG conditions are satisfied. The refractive index dispersions used in the modeling for the different materials are taken from experimental ellipsometry measurements and are given in Appendix A.