# Second-Harmonic Generation of Blue Light in GaN Waveguides
## 1 Introduction
In this work, we show the feasibility of III-metal polar GaN waveguides for SHG using modal-dispersion phase matching (MDPM), which takes advantage of the fact that the waveguide modes of different orders have different effective refractive index dispersion relations [24; 25]. More specifically, we demonstrate the generation of light with \(\lambda\) = 450 nm in 1140 nm thick and 10 \(\upmu\)m wide GaN waveguides. Due to the dependence of the effective refractive index on the dimensions of the waveguide and the wavelength, the pump waveguide mode of order \(p\) at pump wavelength \(\lambda_{p}\) and the SHG waveguide mode of order \(q>p\) at \(\lambda_{\text{SHG}}=\lambda_{q}/2\) can be phase matched. The demonstration of SHG in AlGaN waveguides in the blue spectral region is understood as a step to extend the wavelengths of coherent light sources into the ultraviolet C spectral region via SHG. The final goal is to efficiently generate coherent light down to 200 nm through quasi-phase matching using periodically polled AlN waveguides [26]. The effective refractive index of the waveguide mode at a particular wavelength depends on the refractive indices of the waveguide and the substrate, and on the dimensions of the waveguide [27; 28]. To excite the largest nonlinear coefficient \(d_{33}\), the polarization of in-coupled and out-coupled light is set to transverse magnetic (TM), and therefore the extraordinary refractive indices of the GaN and sapphire substrate must be taken into account. We used the dispersion of the extraordinary index for III-metal-polar GaN waveguides as obtained by a prism coupling study and described by the Sellmeier dispersion formula [21]:  
\[n_{\text{e}}^{2}=1+\frac{A\lambda^{2}}{\lambda^{2}-B^{2}} \tag{1}\]  
where \(A\) = 4.3 and \(B\) = 195.4 nm. The extraordinary refractive index dispersion relationship for sapphire is given in the work of [PERSON] and [PERSON] [29]. In this study, an isotropic approximation [21; 30] and a planar waveguide approximation [27] were used for the calculation of effective refractive indices due to weak birefringence and the large width-to-thickness ratio of the waveguides used. The second-harmonic power in a planar waveguide, where linear losses at pump and second harmonic generation (SHG) wavelengths are neglected, is expressed as [24]:  
\[P_{2\omega}=\frac{32\omega^{2}[d_{33}]^{2}\epsilon_{2\omega}n_{\text{eff},\omega} ^{2}}{9\pi^{2}W\epsilon_{0}c_{0}^{3}\epsilon_{\omega}^{2}n_{\text{eff},2\omega} ^{3}}L^{2}P_{\omega}^{2}\Gamma h(\Delta\beta) \tag{2}\]  
where \(L\) is the interaction length, \(W\) is the waveguide thickness, \(P_{\omega}\) is the fundamental power at the waveguide front facet, \(n_{\text{eff},2\omega}\), and \(n_{\text{eff},\omega}\) are the effective extraordinary refractive indices of the waveguide modes at the fundamental and SHG frequencies, respectively. The function \(h\) (\(\Delta\beta\)) is given by:  
\[h(\Delta\beta)=\sin c^{2}\bigg{(}\frac{\Delta\beta L}{2}\bigg{)} \tag{3}\]  
This function represents a wave-vector mismatch term, where \(\Delta\beta\) describes the difference between the propagation constants of the pump and SH waves. The overlap integral \(\Gamma\)[24]:  
\[\Gamma_{p,q}(\lambda)=\frac{9\pi^{2}}{16W^{2}}\bigg{[}\int\Psi_{q,\lambda/2}( z)\Psi_{p,\lambda}^{2}(z)\text{d}z\bigg{]}^{2} \tag{4}\]  
is a dimensionless quantity normalized to 1 when the two interacting modes \(\psi_{p,\lambda(z)}\) and \(\psi_{q,\lambda/2(z)}\)--that are the transverse field distributions of the optical field amplitudes associated with the waveguide modes p and q at the fundamental and SH frequencies [24, 31]--perfectly overlap in the transverse direction. It limits the number of waveguide modes that lead to efficient nonlinear conversion.