## 1 Introduction  
The capacity of the current optical communication systems operating at the mainstream wavelengths, i.e., around 1310 nm and 1550 nm is approaching a shortage due to the explosive growth in information being transmitted. In order to exploit new frequency resources for telecommunications, transmitting signals at mid-infrared (MIR) wavelengths has been put on the agenda [1; 2; 3; 4; 5; 6], among which the 2 \(\upmu\)m wavelength band (1900 to 2100 nm) attracts special attention as signals in this band can be significantly amplified by thulium-doped fiber amplifiers (TDFAs) [7] and glass fiber in this band is still transparent [8]. Indeed, besides the amplifier, many other functional devices necessary for the 2-\(\upmu\)m-band optical communication system have been also commercially available or laboratory developed including the laser [9; 10], modulator [11], low-loss hollow-core fiber [12], wavelength-division multiplexer [13; 14], detector [15; 16], etc. Meanwhile, nonlinear devices like the wavelength converter are also indispensable and, in order to leverage the manufacturing scalability for such devices, it is preferred to use the on-chip nonlinear waveguides with the fabrication processes compatible with the mature silicon-based complementary metal-oxide-semiconductor (CMOS) technology. As to the CMOS-compatible nonlinear waveguides, silicon (Si) waveguides and stoichiometric silicon nitride (Si\({}_{3}\)N\({}_{4}\)) waveguides are currently the workhorses at the telecommunication wavelengths [17; 18; 19; 20; 21]. However, Si still suffers from the two-photon absorption (TPA) below the wavelength 2200 nm [22], despite that linear Si waveguides can possess a low loss in the 2 \(\upmu\)m band [23] and the Si nonlinear waveguides have been used for parametric conversion around 2 \(\upmu\)m [24]. Indeed, key nonlinear devices like the parametric amplifier [25] or the frequency comb generator [26] have been demonstrated using the Si nonlinear waveguides in the MIR wavelengths but just over the wavelength 2200 nm. For Si\({}_{3}\)N\({}_{4}\), its low linear refractive index (RI) inevitably gives a large footprint of any Si\({}_{3}\)N\({}_{4}\)-based photonic device and the low nonlinear RI usually produces a poor energy efficiency of devices based on the Si\({}_{3}\)N\({}_{4}\) nonlinear waveguides. Moreover, in spite of demonstrations of high-performance Si\({}_{3}\)N\({}_{4}\) waveguides fabricated by using some advanced processes [27], strain issues are still present in Si\({}_{3}\)N\({}_{4}\) film, especially in the thick Si\({}_{3}\)N\({}_{4}\) film, which are yet necessary for achieving compact devices at long wavelengths like 2 \(\upmu\)m and for waveguides with proper dispersion. Besides, titanium dioxide (TiO\({}_{2}\)) waveguides [4] and silicon germanium (SiGe) waveguides [28] were also demonstrated to transmit optical signals at 2 \(\upmu\)m, but, up to now, no reports can be found for them being used for nonlinear processes in this band. Recently, silicon-rich nitride (SRN) has emerged as a promising candidate for the on-chip nonlinear optics [29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39] since SRN possesses larger linear and nonlinear RIs from a Si excess compared to the stoichiometric Si\({}_{3}\)N\({}_{4}\) and, more importantly, the SRN film can have a substantially reduced stress [40]. While the SRN waveguides have been applied for many nonlinear processes like the octave-spanning supercontinuum generation (SCG) [31] and the frequency comb generation (FCG) [32], they were almost operating at the near infrared wavelengths. A SRN waveguide was very recently used to transmit optical signals at 2 \(\upmu\)m and the wavelength conversion was also demonstrated with a pulse pump in the 2 \(\upmu\)m band, but the waveguide was quite thin (300 nm) and the dispersion was not optimized [39].