Vectorial engineering of second-harmonic generation in silicon-based waveguides integrated with 2D materials

Abstract

Integrating 2D materials onto on-chip photonic devices holds significant potential for nonlinear frequency conversion across various applications. The lack of inversion symmetry in monolayers of transition metal dichalcogenides (TMDs), e.g., MoS$_2$, is particularly attractive for enabling nonlinear phenomena based on $χ^{(2)}$ in silicon-based photonic devices incorporated with these materials, which has been previously demonstrated. However, reports have largely overlooked the need to consider, in the nonlinear modal interaction, both the tensorial nature of the TMD's second-order susceptibility and the full vectorial nature of the electromagnetic fields. In this work, we investigate second-harmonic generation (SHG) in silicon nitride (SiN) waveguides integrated with a monolayer of MoS$_2$. We experimentally observed an enhancement in SHG in MoS$_2$-loaded waveguides compared to those without the monolayer. Notably, this enhancement occurred even when the dominant electric field component of the pump and/or signal mode was orthogonal to the TMD plane, highlighting co- and cross-polarized SHG interactions. This phenomenon cannot be predicted by the traditionally used scalar models. By taking into account the full vectorial and tensorial natures of the problem, we then designed a waveguide in which a TE pump mode is phase-matched to a TM second-harmonic mode. With a single 110-$μ$m-long MoS$_2$ flake, we experimentally achieved $14\times$ frequency conversion enhancement relative to the non-phase-matched case and $220\times$ enhancement relative to free-space (normal-incidence) excitation. Our work, thus, introduces fundamental guidelines for the design and optimization of nonlinear silicon-photonic devices based on 2D-material hybrid integration. These guidelines are material independent and may lead to significant further conversion efficiency enhancement.

Figures (5)

• Figure 1: MoS$_2$-loaded waveguide: models, design and 2D material transfer: (a-b) Schematics of SHG in the MoS$_2$-loaded waveguides as described by the (a) the scalar model and (b) the full vectorial model. Since the scalar model only considers the main electric field component of the modes and the 2D material does not interact with out-of-plane electric fields, according to this model, there is no SHG for vertical-polarization pumping. In contrast, the vectorial model predicts SHG at horizontal and vertical polarizations for the vertical-polarized pump. (c) Schematics of the SiN WG integrated with a MoS$_2$ monolayer. The WG is cladded in a SiO$_2$ substrate and has cross-section dimensions of 1 $\mu$m $\times$ 0.8 $\mu$m (width $\times$ height). There is a 100 nm spacing between the WG top surface and the MoS$_2$ flake. Also shown is the MoS$_2$ ($x'$, $z'$) and the WG ($x$, $z$) reference frames, highlighting the angle $\theta$ between them. (d) Optical micrograph of the 79 $\mu$m-long MoS$_2$ transferred onto the SiN WG.
• Figure 2: Experimental results of SHG in the MoS$_2$-loaded waveguides: (a) Main components of the experimental setup. The laser is an Erbium-doped fiber laser @ 1560 nm, with 150 fs of time duration and 89 MHz of repetition rate, delivering 750 W of input (off-chip) peak power. $\lambda/2$ stands for half-wave plate. The inset shows a 3D view of the coupling region, with the input and output objective lenses and the chip. See Methods for further details. (b) Schematics of the input and output polarization states. H and V are defined with respect to the chip plane. (c-f) Normalized SH-intensity collected at the chip output for WGs with MoS$_2$ (blue) and without it (red) at different polarization configurations, as indicated in the subplot legends. The light blue and light red curves represent low-pass filtered curves. To compare the signals with and without MoS$_2$ in each subplot, the curves are normalized by the square of the pump power collected at the chip output, eliminating the influence of insertion losses in the chip. For visual clarity, within each subplot, the curves are further normalized to the maximum SH intensity of the MoS$_2$-loaded waveguide.
• Figure 3: Collective contribution of signal modes for the conversion efficiency: (a-b) Overall SH-conversion efficiency as a function of the interaction length in the MoS$_2$-loaded WG for (a) TM$_{0}$ and (b) TE$_{0}$ pumping at 1550 nm, taking into account $\theta = 12^{\circ}$. The insets show the field profile of the pump mode in each case. (c-d) Overall SH-conversion efficiency as a function of the angle between the WG axis and MoS$_2$ armchair for (c) TM$_{0}$ and (d) TE$_{0}$ pumping at 1550 nm, taking into account $L$ = 79 $\mu$m. The simulations took into account the 16 signal modes. The green (orange) curve displays the conversion efficiency for the H (V) signal polarization with respect to the chip. The dashed gray lines indicate the interaction length of the MoS$_2$-loaded WG ($L$ = 79 $\mu$m) and the angle between the WG axis and MoS$_2$ armchair ($\theta = 12^{\circ}$).
• Figure 4: Simulation results of SHG in the MoS$_2$-loaded WGs for $\theta = 12^{\circ}$ and L = 79 $\mu$m: (a,b) Conversion efficiency ($\eta$), nonlinear coefficient ($\gamma$) and phase mismatch factor ($\Delta \beta$) for the 16 signal modes at 775 nm with (a) TM$_{0}$ and (b) TE$_{0}$ as the pump field. In both cases, the insets show the field profiles of the pump mode. The green (purple) color denotes TE (TM) modes. The color grading follows the $\eta$ and $\gamma$ scale (the higher the value, the darker the color), and the $\Delta \beta$ scale (the lower the $\Delta \beta$, the darker the color). (c) Normalized mode profile ($|\vec{e}|$) and $x$- and $z$-components of the $e$-field for the pump modes at 1550 nm and the signal mode with the highest $\eta$ (TM$_2$). The opacity attributed to the field profiles is defined by the ratio between the maxima of the relevant electric field component ($|\vec{e_x}|$ or $|\vec{e_z}|$) and that of the mode profile ($|\vec{e}|$). (d,e) Real (solid) and imaginary (dashed) parts of the electric field products between the pump and signal modes that contribute to the overlap integral along the MoS$_2$ domain for the (d) TM$_{0}$$\rightarrow$ TM$_{2}$ and the (e) TE$_{0}$$\rightarrow$ TM$_{2}$ SHG interactions. The color coded \ref{['eq:angleDependence_main']} is indicated in the legend above the plots for reference of the field products. The electric field components that yield the highest field products - indicated in red and purple, respectively, in (d) and (e) are indicated by squares of the same colors in (c).
• Figure 5: Phase matching analysis in the MoS$_2$-loaded WGs: (a) Effective index as a function of WG width for the pump TE$_0$ (green) mode at 1550 nm and signal modes at 775 nm (gray). The green star at 1.22 $\mu$m corresponds to phase matching for the TE$_{0}$$\rightarrow$ TM$_{2}$ interaction. (b) Mode profiles (norm of the electric field) for the phase-matched modes at the width of 1.22 $\mu$m. (c) SH intensity normalized by the pump output power squared divided by the effective mode area for horizontal pump polarization and vertical SH polarization, in the 1.2 $\mu$m-width WG with MoS$_2$ (blue), a 1.2 $\mu$m-width WG without MoS$_2$ (red), and a 1.0 $\mu$m-wide WG without MoS$_2$ (green). (d) Comparison between SH intensity, normalized by the pump output power squared divided by the effective mode area, as measured for normal incidence excitation of an MoS$_2$ flake (black) and for the 1.2 $\mu$m-wide WG (blue), with the pump at 1560 nm.