Metasurface-assisted phase-matching-free second harmonic generation in lithium niobate waveguides

The phase-matching condition is a key aspect in nonlinear wavelength conversion processes, which requires the momenta of the photons involved in the processes to be conserved. Conventionally, nonlinear phase matching is achieved using either birefringent or periodically poled nonlinear crystals, which requires careful dispersion engineering and is usually narrowband. In recent years, metasurfaces consisting of densely packed arrays of optical antennas have been demonstrated to provide an effective optical momentum to bend light in arbitrary ways. Here, we demonstrate that gradient metasurface structures consisting of phased array antennas are able to circumvent the phase-matching requirement in on-chip nonlinear wavelength conversion. We experimentally demonstrate phase-matching-free second harmonic generation over many coherent lengths in thin film lithium niobate waveguides patterned with the gradient metasurfaces. Efficient second harmonic generation in the metasurface-based devices is observed over a wide range of pump wavelengths (λ = 1580–1650 nm).


Supplementary Note 2 | Phased-antenna design and simulation techniques
In Lumerical FDTD simulations, the LiNbO3 waveguide is orientated along the z direction, and the cross section of the waveguide is in the x-y plane. The optical refractive indices of LiNbO3 are obtained from the Palik database 1 and are anisotropic with no (ordinary index of refraction) in the y and z directions and ne (extraordinary index of refraction) in the x direction. ne used in FDTD simulations is shown in Supplementary Figure 2c. The diagonal terms of the second-order nonlinear susceptibility tensor of LiNbO3 are set to be χ (2) xxx = 66 pm/V, χ (2) zzz = 6 pm/V and χ (2) yyy = 0 pm/V. The optical refractive indices of a-Si are obtained from ellipsometric measurements and are shown in Supplementary Figure 2b). The optical refractive indices of SiO2 are also from the Palik database. The amplitude of incident fundamental waveguide mode is set to be 10 9 V/m.
The phase response of a-Si nano-antennas is obtained also using FDTD simulations. The width and height of the nano-antennas are kept constant (75 nm). The centre-to-centre distance between adjacent antennas is kept to be dz = 140 nm. These numbers are chosen according to our fabrication capabilities. In our FDTD simulations, a nano-antenna is placed on a LiNbO3 substrate and in FDTD simulations. The extinction coefficient is assumed to be zero.

Supplementary Note 3 | Mode power loss through the antenna region
One major advantage of using dielectric a-Si nano-antennas is that the insertion loss is greatly reduced compared to plasmonic nano-antennas. We conducted simulations to study the propagation loss of five different modes (i.e., TE00, TE20, TE40, TM30 and TM40) as a result of their interaction with a single phased antenna array at three different wavelengths: λ = 750 nm, 775 nm and 800 nm (Supplementary Figure 3). This propagation loss is due to undesired optical scattering by the antennas into air or the substrate, as well as power absorption by the antennas. The simulation results show that for the TE00 mode, the power loss is 7%, 12% and 14% of the input power at λ = 750 nm, 775 nm and 800 nm, respectively. Similarly, the power loss is 10%, 12%, and 9% for the TE20 mode, and 11%, 11% and 9% for the TE40 mode at the three wavelengths, respectively. As for the TM modes, the propagation loss is much smaller. For example, the power loss is 1.7%, 1.6% and 1.6% for the TM30 mode, and 2.1%, 2.0% and 2.5% for the TM50 mode at those three wavelengths, respectively. Overall, the insertion losses are small in our all-dielectric devices.

Supplementary Note 4 | Power evolution after phased antenna arrays
To investigate how the generated SH signal evolves after the phased antenna arrays, we conducted full-wave simulations to monitor the SHG power at λ = 775 nm up to a propagation distance of 100 m for three different devices, which contain one, three and five phased antenna arrays, respectively. The simulation results show that the SHG power monotonically increases in the antenna-array-covered waveguide section (red curves in Supplementary Figure 4) and then oscillates with a small amplitude in the following bare waveguide (black curves in Supplementary   Figure 4).
As stated in the main text, in the antenna-array-covered section, the fundamental mode at λ ~ 775 nm unidirectionally couples into higher-order modes. As a result, optical power is continuously transferred from the pump at λ = 1550 nm to the SH signal, leading to a monotonic increase of the SHG power. After the antenna-array-covered section, there is no further mode conversion at the SH wavelength. Each converted higher-order mode at λ ~ 775 nm interacts with the fundamental mode at λ = 1550 nm weakly and thus optical power is coupled back and forth between the two modes in a period way determined by their phase mismatch. This effect could lead to a periodic variation of the SHG power. In addition, different higher-order modes at λ ~ 775 nm propagate with different phase velocities and their interference could also cause variations of the SHG power along the waveguide. In summary, the SHG power is accumulated monotonically in the antennaarray-covered section and maintains its intensity (although with small oscillations) after that section.

Supplementary Note 5 | Robustness against the waveguide parameters
We conducted full-wave simulations to study the effects of waveguide geometries, such as waveguide width and etching thickness, on device performance (Supplementary Figure 5). The simulation results show that the performance is tolerant to the waveguide geometry changes. The change to the SHG power in a device with three sets of phased antenna arrays is small when the lengths of the nano-rod antennas deviate from their designed values by ± 10% (a), when the antenna arrays are offset from the centre of the waveguide up to 100 nm (b), when the width of the LiNbO3 waveguide deviates from its designed values by ± 100 nm (c), and when the waveguide etching depth deviates from its designed value by ± 50 nm (d). For example, Supplementary Figure   5c compares the SHG enhancement for waveguides of different widths (i.e., 2600 nm as used in fabricated devices, 2500 nm and 2700 nm as controls) patterned with three phased antenna arrays.
The SHG enhancement at the end of the antenna arrays for these three waveguides exhibits a variation of ± 6%. Supplementary Figure 5d shows that the performance is robust against variations of the waveguide etching depth. For three different etching depths of 300 nm (as used in fabricated devices), 250 nm and 350 nm of thin-film LiNbO3, the SHG enhancement factors exhibit a variation of ± 1.2%.