Large second harmonic generation enhancement in Si3N4 waveguides by all-optically induced quasi-phase-matching

Efficient second harmonic generation in integrated platforms is usually achieved by resonant structures, intermodal phase-matching or quasi-phase matching by periodically poling ferroelectric waveguides. However, in all these structures, it is impossible to reconfigure the phase-matching condition in an all-optical way. Here, we demonstrate that a Watt-level laser causes a periodic modification of the second-order susceptibility in a silicon nitride waveguide, allowing for quasi-phase-matching between the pump and second harmonic modes for arbitrary wavelengths inside the erbium band. The grating is long-term inscribed, and leads to a second harmonic generation enhancement of more than 30 dB. We estimate a χ (2) on the order of 0.3 pm/V, with a maximum conversion efficiency of 0.05% W−1. We explain the observed phenomenon with the coherent photogalvanic effect model, which correctly agrees with the retrieved experimental parameters.

a dependence given by the equation below 1 .
ph ∝ * 1 | p | 2 | sh | ( 2 | p | 4 − | sh | 2 ) exp ( ( sh − 2 p )) + c. c. (1) where is the local electric field of the pump or second harmonic (SH) (denoted by the subscript p or sh, respectively), is the phase of the corresponding fields (same subscript), 1 and 2 are coefficients of multi-photon absorption, and c.c. indicates the complex conjugate. The exponential term denotes the coherent nature of the process, and leads to spatially periodic charge separation. Charges therefore migrate in a preferential direction and ph gets associated to a space-charge DC electric field DC , whose periodicity allows for quasi phase matching the second harmonic generation (SHG). The relation between the two is given by the photoconductivity , proportional to the total density of carriers  Figure 1). Moreover, some variation in the energy traps has also to be considered when moving from neutral to positively charged donor (negatively charged acceptor) defects 4 . Considering these ranges, a two-step process cannot be excluded. In this case, an electron from the donor level is promoted to the acceptor one by one SH photon or by two pump photons. Subsequently, a first order CPE process involving the interference between the two pump photons and the one SH photon absorptions would take place giving the correct coherent phase term for quasi-phase matching (QPM): sh − 2 p .
Moreover, SiO 2 -SiN structures can localize electrons and holes for multi-year lifetimes 7 . Since these Si-Si defects also entail localized states ~1.4 eV below the conduction band, they can potentially trap free carriers as well. It would result in a long-lived and thermalization-immune localization of the photo-carriers, in agreement with the observed grating persistence.
In our experiments, we also observed the adverse effect of green light from third harmonic generation (THG) on the SH enhancement dynamics, manifesting itself for instance by oscillations in the growth curves in waveguide (i). The erasure of  (2) gratings by visible radiation was reported in fibres as well 8 . If the electrons are promoted from defects to the conduction band via single-photon absorption (without any interference process), there is no preferential ejection direction on average. They are thus subject to the built-in electric field only and entail a drift current d opposite to the photogalvanic one.
In our case, green light has energy of 2.4 eV, sufficient to excite the carriers trapped in defect states near the conduction band and lead to their recombination, and to the grating erasure. Moreover, THG, as well as SHG, is a coherent process. Thus, in the presence of strong THG, the CPE can involve a coherent interference between the absorption of one pump photon and one third harmonic (TH) photon. However, this process entails a grating period inversely proportional to | th − |, being th the propagation constant of the TH field, counteracting the growth of the grating with the right periodicity for QPM of the SHG.

Supplementary Note 2: Evaluation of  (2) after grating inscription
The overlap integral S between the pump and SH mode, appearing in Equation 1 of the main text, has an analytical formula derived from the coupled mode theory 9 and given by the equation below.
In this formula, the integration is carried out over the entire waveguide cross section. The electric fields are normalized to the power flux ( z ) across the waveguide cross section given by the mode solver: The local value of the space charge field DC has an orientation and a magnitude that depends upon the phase difference between the pump and SH. The (2) grating, which originates from this field via the third-order susceptibility tensor ( (2) = 3 (3) DC ), also follows this rule. In fact, this (2) dependence holds whatever is the microscopic model behind the phenomenon, as verified in the main text by measuring the QPM peaks. Yet, in our waveguides, the SH propagates on a dipolar or tripolar mode (see Supplementary Figure 2). The electric field in each lobe is in quadrature with the field in the adjacent lobe(s). On the contrary, the pump field in the fundamental mode has a constant phase over the waveguide cross section. The space charge field DC must therefore have opposite signs in each lobe region. Consequent, (2) also switches sign from one lobe to the other, as does the SH electric field phasor. The overlap integral can thus be rewritten as We performed the evaluation of (2) and of grating period by computing the waveguide dispersion and mode profile with a finite-element solver (COMSOL Multiphysics). In all cases, the SH is assumed to propagate on the high order mode that yields the smallest index difference (i.e. which yields the best natural phase matching) and the best mode overlap with the pump. QPM then occurs via CPE