**3. Results**

The transverse section of the waveguides was designed for Type-I phase-matched SHG from the TE00 mode at ω (λ ≈ 1.6 μm) to the TM00 mode at 2ω (λ ≈ 800 nm): (a) the thickness of the Al0.19Ga0.81As film was chosen so as to ensure strong modal birefringence while keeping the interacting modes well-confined; (b) the wire/rib width was then adjusted in order to precisely set the phase-matching wavelength [33]. The electric field amplitude profiles of both modes are shown in Figure 3. It can be observed that the 40 nm thick membrane does not significantly affect the lateral confinement for both modes. Accordingly, phase-matching is obtained for an almost identical width (≈ 1 μm), and the SHG efficiency expected from numerical simulations (not shown) is also very similar for the two devices: η = 300% W<sup>−</sup>1mm−<sup>2</sup> (nanowire) and η = 401% W<sup>−</sup>1mm−<sup>2</sup> (nanorib).

**Figure 3.** TE00 amplitude at ω (Ey, left) and TM00 amplitude at 2ω (Ex, right) in the suspended nanowire (top) and rib waveguide (bottom).

Propagation losses at ω and 2ω were measured by acquiring Fabry–Perot transmission interference fringes in on-purpose processed 200 μm long waveguides terminated by flat ICP etched facets (which have higher reflectivity than the tapered counterparts). The combined loss–reflection coefficient R' = R exp(-αL) can be extracted by the contrast K of the transmission fringes as follows:

$$\mathbf{K} = \frac{\mathbf{T\_{max}} - \mathbf{T\_{min}}}{\mathbf{T\_{max}} + \mathbf{T\_{min}}} \tag{1}$$

$$\text{R}\prime = \frac{1-\sqrt{1-\text{K}^2}}{\text{K}}\tag{2}$$

where L is the length of the waveguide and Tmax and Tmin are the maximum and minimum transmission values, respectively. The modal reflectivity R was calculated at both ω and 2ω via 3D FDTD modeling, and the propagation loss coefficient was then found as:

$$\alpha = \frac{1}{L} \ln \left( \frac{\mathcal{R}}{\mathcal{R} \nu} \right) \tag{3}$$

The coupling efficiency κ of a waveguide terminated by inverted tapers, assumed to be equal at the input and at the output, was finally obtained by measuring its overall transmission and dividing it by the propagation loss exp(-αL):

$$\kappa = \sqrt{\frac{T}{e^{-\alpha L}}} \tag{4}$$

The results of the above linear characterization are summarized in Table 1. For both designs, for <sup>L</sup> <sup>=</sup> 1 mm, the propagation loss at <sup>ω</sup> is quite limited (e-<sup>α</sup><sup>L</sup> <sup>≈</sup> 70%), while at 2<sup>ω</sup> it turned out to be one order of magnitude higher, due to the proximity between photon energy at 2ω and the forbidden band (740 nm) and to stronger scattering at the waveguide sidewalls at shorter wavelength. As for the input/output coupling, we estimate that the efficiency at ω for the rib waveguides can reach the same level as in nanowires after further optimization of design and processing. The low coupling efficiency at 2ω is to be ascribed to the multimode nature of the waveguide at this wavelength.


**Table 1.** Measured linear optical features.

Figure 4 shows the SHG efficiency spectra acquired by injecting and tuning the TE polarized telecom-range laser into a 1 mm nanowire (black trace) and a 200 μm long nanorib (red trace) waveguide, collecting the outcoupled TM mode at 2ω. The internal efficiency η was calculated by normalizing the overall efficiency PSHG/Pin<sup>2</sup> to the coupling efficiency at ω and 2ω:

$$\eta = \frac{1}{\kappa\_{\text{av}}^2 \kappa\_{2\text{av}}} \frac{P\_{\text{SHG}}}{P\_{\text{in}}^2} \tag{5}$$

with peak values of 16% W−<sup>1</sup> (wire) and 3% W−<sup>1</sup> (rib). The normalized efficiency equations (ηnorm = η/L2) of the two devices are expected to be very similar; nevertheless, the ratio (ηwire/ηrib) does not scale as the square of the ratio of the lengths (Lwire/Lrib) 2 . This is due to propagation loss at

2ω, which limits the interaction length to << Lwire. By taking into account the effect of propagation loss on the efficiency η, we can calculate the normalized SHG efficiency ηnorm, defined as follows: [36]

$$\eta\_{\parallel} = \eta\_{\text{norm}} \text{L}^2 \exp[- (\alpha\_{\omega \nu} + \alpha\_{2\omega \nu}/2) \text{L}] \frac{\sin \text{h}^2 \left[ (\alpha\_{\omega \nu} - \alpha\_{2\omega \nu}/2) \frac{1}{2} \right]}{\left[ (\alpha\_{\omega \nu} - \alpha\_{2\omega \nu}/2) \frac{1}{2} \right]^2} \tag{6}$$

obtaining 128% W<sup>−</sup>1mm−<sup>2</sup> (wire) and 119% W<sup>−</sup>1mm−<sup>2</sup> (rib). The results are summarized in Table 2.

**Figure 4.** Nonlinear second-harmonic generation (SHG) efficiency spectra for the nanowire (black) and nanorib (red) waveguides.

**Table 2.** Measured nonlinear optical features.

