Optical Amplification in Er-Doped LiNbO3-on-Insulator Photonic Wire Pumped at 1480 nm Wavelength

$1.5~\mu $ m optical amplification in a ridge-type Er-doped LiNbO3-on-insulator (Er:LNOI) photonic wire (PW) pumped at 1480 nm wavelength is studied theoretically. The study considers a three-level Er3+ model with inclusion of excited-state absorption and cooperative upconversion. Based on the model, steady-state rate equations and pump/signal power propagation equations have been established, and closed expression was derived for population density of each level involved. Quantitative relations of population inversion, signal gain or gain coefficient to propagation distance, pump power, scattering loss of pump/signal wave and Er3+ concentration are established. In comparison with Ti-diffused Er:LiNbO3 (Ti:Er:LN) waveguide, the Er:LNOI PW exhibits much more efficient gain performance because of ultra-compact mode field and hence ultra-high light intensity (photon density) in it. These include two orders of magnitude lower threshold pump power, tens of times lower saturation pump power and three times higher saturation gain. Due to the same reason, the $^{4}\text{I}_{11/2}$ population of Er3+ in the Er:LNOI PW is four times higher than that in the Ti:Er:LN waveguide. Consequently, the 1480 nm upconversion process of Er3+ in the Er:LNOI PW is much stronger than that in the Ti:Er:LN. It strengthens at higher pump power and definitely affects the $1.5~\mu \text{m}$ gain at higher pump level. Combined effects of ultra-high pump intensity, upconversion process and large non-radiative multi-phonon relaxation rate from $^{4}\text{I}_{11/2}$ to $^{4}\text{I}_{13/2}$ level result in some interesting features, including non-monotonic variation of gain coefficient with propagation distance, slight drop of gain in saturation region and gain increase with pump loss at higher pump power. These combined effects make it possible to realize $2.7~\mu \text{m}$ lasing and an output power of sub-mW is theoretically expected for a 1480 nm pump power of 100 mW. The study also shows that amplified spontaneous emission is significant in the Er:LNOI PW but affects little the signal gain performance.