Generation of spatiotemporally tailored terahertz wavepackets by nonlinear metasurfaces

The past two decades have witnessed an ever-growing number of emerging applications that utilize terahertz (THz) waves, ranging from advanced biomedical imaging, through novel security applications, fast wireless communications, and new abilities to study and control matter in all of its phases. The development and deployment of these emerging technologies is however held back, due to a substantial lack of simple methods for efficient generation, detection and manipulation of THz waves. Recently it was shown that uniform nonlinear metasurfaces can efficiently generate broadband single-cycle THz pulses. Here we show that judicious engineering of the single-emitters that comprise the metasurface, enables to obtain unprecedented control of the spatiotemporal properties of the emitted THz wavepackets. We specifically demonstrate generation of propagating spatiotemporal quadrupole and few-cycles THz pulses with engineered angular dispersion. Our results place nonlinear metasurfaces as a new promising tool for generating application-tailored THz fields with controlled spatial and temporal characteristics.


Supplementary Information
Generation of spatiotemporally tailored THz wavepackets by nonlinear metasurfaces

Flying doughnuts
Maxwell's equations have many well-known solutions that have separable temporal and spatial structures. In addition, there is a group of solutions that draw a lot of attention, in which the spatial and temporal components are inseparable 1-4 . One of the most known wavepackets among this group are the so-called "Flying doughnuts"(FD) 2,5,6 . These are single-cycle pulses with a doughnut-like electromagnetic profile, which move at the speed of light. FD are also known for having a field component along the propagation direction when focused, which can be used for electron acceleration and intriguing light-matter interaction in the nanometric scale. The electromagnetic field of FD was derived before 1,2 and was shown to be: 3 (1) In this work we present a single-cycle 10 "flying" field, which also carries a longitudinal polarization components when focused. Moreover, it has the form that is similar to the ypolarization component of an FD. Supplementary Figure 2 shows the resemblance between the profile of the 10 wavepacket profile as simulated with accordance to the NLMS presented in Another way to achieve this will be by a more complex design of the NLMS, which will include manipulation of the locally emitted polarization state, e.g., by circularly symmetric rotation of the SRRs along the metasurface.

Supplementary Note 3
Nonlinear Raman-Nath diffraction in THz generation For a simple, one-dimensional nonlinear periodic structure along the x-axis with a lattice constant of Λ, the energy conservation condition in optical rectification process is: where 1 and 2 are the the wave-vectors of the broadband NIR beam. THz is the emitted THz wave-vector and is an odd number marking the diffraction order, as also depicted in Supplementary Figure 3. For NLMPC with 0.5 duty cycle only odd Fourier components construct the spatial structure of the nonlinear response tensor, therefore the emitted THz wave is directed only to the odd diffraction orders.
The momentum conservation condition along the x-axis requires: where is the diffraction angle of the ℎ order and in is the NIR incidence angle relative to the modulation direction.

Calculation of number of cycles in a diffracted pulse
The profile of the THz electric field on the NLMPC plane can be described by: where Π( ) is the unit-box function, given by the finite size of the NLMPC, is the NLMPC length, is a square wave function and Λ is the period of the NLMPC.
The spatial frequency distribution is given by its Fourier transform: Each term denotes a diffraction order. While an infinite grating is described by ( − The spatial bandwidth of the diffraction pattern of a certain temporal frequency, can be used to calculate the temporal bandwidth at a certain diffraction angle: With as the Raman-Nath diffraction angle: With as the emitted wave vector length -= 2 , and consequently = 2 .
The bandwidth of the pulse at a specific emission angle is therefore: In addition, the number of cycles in a diffracted pulse ( c ) is proportionate to: Therefore c is propotional to the number of periods, p = Λ . For the case described in the main text, it is shown that for Λ = , i.e. p = 1 the NLMS emits a single cyclec = 1.
Consequently we get that c = p . Supplementary

Supplementary Note 5
Carrier-envelope shaping of the few-cycle pulse The use of nonlinear metasurfaces as THz generation platform allows to control the emitted wave by manipulation of the local nonlinear response tensor across the metasurface. In this work we demonstrated it mainly by binary modification of the phase, however, it is possible to control additional nonlinear emission properties. Specifically, the amplitude of the local THz emission can be controlled, either by local geometrical modification of the single building block, or, more simply, by adjustment of the single-emitters concentration. The local distance between the single emitters can be easily tuned, as long as nearfield effects and collective effects are avoided.
The constant amplitude of the nonlinear response tensor along the NLMPC, is followed by constant carrier envelope amplitude along the few-cycle pulse. Similarly, a modulated amplitude of the NLMPC is translated to carrier-envelope modulation.