# Second-Harmonic Generation of Blue Light in GaN Waveguides
## 3 Results and Discussion  
Figure 4 shows the second-harmonic response of GaN waveguides to excitation with different wavelengths ranging from 850 nm to 950 nm. The green curves show the spectral distribution of the excitation source and the blue curves show the corresponding SH response. A clear, relatively strong peak of the SHG signal was observed at a constant wavelength of 450 nm while scanning the pump wavelength spectrum. Maximum SHG response (blue curve) can be seen in Figure 4g,h with the central pump wavelengths of 895 nm and 900 nm. In order to explain the SHG peak at 450 nm the dispersions of the effective refractive indices for the pump and SHG waveguide modes were analyzed. Figure 3: Light propagation observed by CCD camera 1, shown in Figure 2b, mounted above the sample: (**a**) At in-coupling and (**b**) out-coupling positions. (**c**) Out-coupling recorded by CCD camera 2, shown in Figure 2b. White dashed lines represent waveguide positions where they are not easily seen. Note that all waveguides have a stripe width of 10 μm. Figure 2: (**a**) Top view of the experimental setup with the end-fire coupling system. Femtosecond pulses of vertical polarization are adequately attenuated with a neutral density (ND) filter and coupled into GaN waveguides with their polar axis vertical. The out-coupled signal is vertically polarized and is filtered with a shortpass filter and the SHG signal is analyzed with the spectrograph connected to the photon counting camera. (**b**) Side-view of the end-fire coupling system. Coupling lenses and a sample holder are mounted on _xyz_ translation stages, which enable precise positioning of the focal point of the laser beam onto the front facet of the waveguide. In Figure 5a the dispersion curves for a 1140 nm thick GaN waveguide on a sapphire substrate are shown. For the calculation, a planar waveguide approximation was used [27]. This was justified as the waveguides were 10 \(\upmu\)m wide. Extraordinary refractive indices for GaN in the measured pump- and SHG-wavelength ranges are given by Equation (1). As indicated by the circles in Figure 5, the MDPM condition is true for the interaction between the fundamental pump mode and the fourth SHG mode, and for the first pump mode and fifth SHG mode, both at a pump wavelength \(\lambda\) of approximately 900 nm:  
\[\text{TM}_{0}^{\lambda}+\text{TM}_{0}^{\lambda} \rightarrow\text{TM}_{4}^{\lambda/2} \tag{5}\] \[\text{TM}_{1}^{\lambda}+\text{TM}_{1}^{\lambda} \rightarrow\text{TM}_{5}^{\lambda/2} \tag{6}\]  
where \(\lambda\) is the pump wavelength. Both possible waveguide interactions are schematically shown in Figure 5b. Figure 4: SHG signal counts (blue) measured in 10 min from a 1 mm long GaN waveguide at different central wavelengths of the pump (green). Note the different wavelength scale for the pump and SHG. The fulfillment of the phase matching is only one of the necessary conditions for efficient SHG generation. In waveguide interactions, the overlap integral \(\Gamma\), detailed in Equation (4), must also be considered. Although phase-matching conditions for two combinations of waveguide modes at the pump wavelength of 900 nm are met, the overlap integrals reveal that the interaction from Equation (5) is dominant and thus responsible for the measured SHG peak. Taking only the overlap into account, the conversion efficiency for the MDPM interaction described by Equation (6) is 35 times lower than that described by Equation (5). Furthermore, it could not be measured separately in our waveguides. In Table 1 the overlap integrals are compared for other possible interactions indicated by black circles in Figure 5. The interaction:  
\[\text{TM}_{1}^{\lambda}+\text{TM}_{1}^{\lambda}\rightarrow\text{TM}_{6}^{ \lambda/2} \tag{7}\]  
Figure 5: (**a**) Dispersions of the effective refractive indices for the pump (dashed blue lines) and SHG (green lines) waveguide modes in a GaN waveguide with a thickness of 1140 nm.