Tunable unidirectional nonlinear emission from transition-metal-dichalcogenide metasurfaces

Nonlinear light sources are central to a myriad of applications, driving a quest for their miniaturisation down to the nanoscale. In this quest, nonlinear metasurfaces hold a great promise, as they enhance nonlinear effects through their resonant photonic environment and high refractive index, such as in high-index dielectric metasurfaces. However, despite the sub-diffractive operation of dielectric metasurfaces at the fundamental wave, this condition is not fulfilled for the nonlinearly generated harmonic waves, thereby all nonlinear metasurfaces to date emit multiple diffractive beams. Here, we demonstrate the enhanced single-beam second- and third-harmonic generation in a metasurface of crystalline transition-metal-dichalcogenide material, offering the highest refractive index. We show that the interplay between the resonances of the metasurface allows for tuning of the unidirectional second-harmonic radiation in forward or backward direction, not possible in any bulk nonlinear crystal. Our results open new opportunities for metasurface-based nonlinear light-sources, including nonlinear mirrors and entangled-photon generation.

In this work, we presented linear and nonlinear response of transition-metal-dichalcogenide (TMD) MoS2 metasurface. Specifically, exploiting the high index of MoS2, we observed sub-diffractive second harmonic (SH) and third harmonic (TH) response. Intriguingly, the presented MoS2 metasurface possess the capability to switch the SH emissions in forward and backward directions.

Simulated Linear Response
We employed scanning electron microscopy to extract the parameters of the fabricated metasurfaces and name them as A, B and C, according to their periodicity and diameter of the truncated cone. The periodicity and the diameter increase as we move from A toward C. To the perform the numerical simulations, the values of refractive index and extinction coefficient, for bulk MoS2, are extracted from the work of Beal and Hughes 1 , as shown in Supplementary  Figure 1.The simulated transmittance spectrum of MoS2 metasurfaces A, B, and C is presented in Supplementary  Figure 1b. Two clear transmittance dips in the SH spectrum (700-800 nm) correspond to the resonant excitation of electric dipole (ED) and magnetic dipole (MD) in each metasurface. However, two weak resonant dips, around 700-715 nm, in the linear spectrum of metasurface B and C corresponds to magnetic quadrupole (MQ). The resonance of MQ mode is weak because of absorption coefficient of MoS2 around 700nm. The simulated transmittance spectra find qualitative agreement with the experimental results. The small discrepancy likely originates from the uncertainty of the extracted parameters, shape of the cone, and Fabry-Perot resonances present between the layered TMDs 2 . Moreover, in Supplementary Figure 1b, along with other high order Mie-resonances below 500 nm, one can observer resonance corresponding to the first order diffraction, which we call the wood's anomaly (WA). Figure 1. Simulated linear spectrum. a Refractive index and extinction coefficient of Bulk MoS2. b Simulated transmission spectrum of three different metasurfaces (A, B and C) having same height (around 150nm) but different radii, 100 nm, 110 nm and 120 nm, respectively. For clear understanding spectra for three different metasurfaces are vertically displaced by = 1. Resonances around 500 nm are Wood's anomalies associated with the onset of the first diffraction order. Figure 2. Measured transmission spectra in the IR spectrum a metasurface A. b metasurface B. c metasurface C.

Supplementary
To avoid emissions of multiple diffraction orders with SH and TH emissions, we wanted our metasurfaces to be transparent in the infrared (IR) spectrum, which is the spectrum of our tunable femtosecond laser pump. All of the fabricated metasurfaces A, B, and C are transparent in the IR spectrum, which can be observed from the measured transmittance response as shown in the Supplementary Figure 2a THG intensity was obtained by Thorlabs CMOS camera with 10 ms integration time, as shown in Supplementary  Figure 3a. Then a 532 nm laser was used to calibrate the camera intensity to power, which was shown in Supplementary Figure 3b. After getting the ratio of camera counts to actual power from the calibration laser, we can convert the THG counts in camera to power. Therefore, by dividing the power of 1550 nm pump laser, the THG conversion efficiency is 1.01×10 -9 at average power of 68 mW. The peak power of the laser is = = 4.3 kW, where = 80 MHz is the repetition rate of the laser, = 200 fs is the pulse duration. The pump beam diameter is measured to be 5.48 µm (full width at half maximum of 4.6 µm), resulting in pump peak intensity of 18.2 GW/cm 2 . As SHG intensity is around 30 times weaker than THG at 1550 nm pump from the spectrum in Supplementary Figure  3c, the SHG efficiency is estimated to be around 3.4×10 -11 with the same pump power. From the wavelength dependence Figure 5a of the main text, SHG efficiency can be increased by 10 times when tuning the pump wavelength to 1400 nm, reaching maximum SHG efficiency of 3.4×10 -10 .

Simulated Third Harmonic Response
We perform full wave nonlinear simulations to calculate the wavelength dependent THG efficiency of the metasurfaces A and B, as shown in the Supplementary Figure

Simulated Second Harmonic Response
The intensity and the position of SHG peak strongly depends upon the modal overlap between the SH current source and induced mode eigenfields. Hence, the exact knowledge of the location of the second order nonlinear current source in the meta-atom is very important. Therefore, to find out the location, we performed full wave nonlinear simulations and calculated the SH conversion efficiency for four different scenarios: origin of the second order nonlinearity is at  Supplementary Figure 6a, then the coupling of the SH light with MQ and MD modes is quite optimal, showing strong SH peaks. However, weak SH peak around 1500 nm is due to the poor overlap of SH source with ED eigenfields. The same physics lies behind the different intensity of SH peaks at 1420 nm, 1500 nm and 1750 nm, in other two scenarios.

SHG Efficiency of Vertical Cylinder and Asymmetric Conical Metamolecule
We employed quasinormal modes (QNMs) theory to calculate the SHG efficiency for three different shapes. The dimensions are presented in Supplementary Table1. All other dimensions like height (150 nm) and pitch (300 nm) is same. The calculated SHG efficiency, with QNMs theory, of truncated cone meta-atom (our fabricated meta-atoms) is presented in the main text. Whilst, the SHG efficiency of asymmetric cone and vertical cylinder is presented in the Supplementary Figure

Simulated Polarization Resolved Second Harmonic Response
We calculated the polarization dependent SH response of the presented MoS2 metasurface in the forward direction (shown in the main text) and backward direction, as illustrated in Supplementary Figure 8. Polarization dependent SHG is function of two angles, (angle between armchair direction and lab frame axis) and (angle between pump polarization and lab frame). In the simulations, we fixed the = and simulated SH efficiency as function of pump wavelength and pump polarization, as depicted in Supplementary Figure 8a. The mapping shows three peaks around 1420 nm, 1560 nm and 1750 nm. These peaks are associated with resonant excitation of the three modes MQ, ED and MD respectively. In Supplementary Figure 8b, the polar plots titled , and represents the extracted SH conversion efficiency at three photonic modes i.e. MQ, ED and MD, as function of pump polarization when = . Whilst, the * is the calculated SH conversion efficiency in the vicinity of ED mode as indicated by ED* in Supplementary Figure 8a. In case of magnetic modes, the coupling of the SH light with MQ and MD is quite optimal. Therefore, only one mode radiates the SH light and we receive the six SH peaks of equal amplitude. However, because of the interference of the magnetic modes at ED and ED*, we receive four stronger and two weaker SH peaks and two stronger and four weaker SH peaks, respectively.
Additionally, the polarization resolved SH response for asymmetric cone in the forward and backward directions is depicted in Supplementary Figure 9a and Supplementary Figure 9b respectively. Whilst, Supplementary Figure 9c and Supplementary Figure 9d illustrate the polarization dependent SHG efficiency of the cylindrical metamolecule in the forward and backward direction respectively. In case of cylinder the coupling of the SH light is quite optimal.

Effect of Shape and Symmetry of Metamolecule on SHG
The shape and symmetry of metamolecule strongly affects the shape of polarization resolved SHG. For fully asymmetric cone, Supplementary Figure 9a  It can be observed, in either direction, the response of the MD mode around 1600 nm, is quite strong in comparison to the MQ and ED modes. However, the contribution of MQ and ED modes is also visible in either direction around 1400 nm or 1500 nm respectively. For vertical cylinder meta-atoms, Supplementary Figure 9c

Second Harmonic Generation Directionality
The capability of the MoS2 metsurface to switch the SH emissions in forward and backward directions influenced by the following factors, (i) interference of the induced and electric and magnetic modes, (ii) intrinsic property of MoS2, and (iii) location of the second order nonlinear tensor. We extracted the SHG directionality by taking the ratio of SH emissions in the forward and backward directions and plotted against the pump wavelength and its polarization for different shapes, (i) truncated cone and vertical cylinder meta-atoms (shown in the main text) and (ii) fully asymmetric cone, as shown in the Supplementary Figure 10. The switching of SHG directionality can be observed around 1480 nm, as function of pump polarization. Along with modes interference, this feature strongly depends upon the shape and symmetry of the meta-atoms as well. Among three different shapes i.e truncated cone, fully asymmetric cone, and vertical cylinder, the metasurface consists of vertical cylinder meta-atoms shows best performance and acts as virtual mirror for SH light.