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High harmonic generation allows one to extend the frequency of laser to a much broader regime and to study the electron dynamics of matters. However, severely limited by the vague high-order process in natural material and the unfriendly state of the commonly applied gas and plasma media, the ambitious goal of custom-design high harmonics remains exceptionally challenging. Here, we demonstrate that high harmonics can be artificially designed and tailored based on a metamaterial route. With the localized reconstruction of magnetic field in a metamaterial, the nonlinear Thomson scattering, a ubiquitous electromagnetic process which people used to believe that it only occurs with the relativistic velocity, can be stimulated in a nonrelativistic limit, which drives anharmonic oscillation of free electrons and generates high harmonics. An explicit physical model and the numerical simulations perfectly demonstrate the artificial generation and tailoring of the high harmonics. This novel mechanism is entirely dominated by the artificial structure instead of the natural nonlinear compositions. It not only provides unprecedented design freedom to the high harmonic generation but breaks the rigorous prerequisite of the relativistic velocity of the nonlinear Thomson scattering process, which offers fascinating possibilities to the development of new light source and ultrafast optics, and opens up exciting opportunities for the advanced understanding of electrodynamics in condensed matters.

Thomson scattering, one of the most fundamental mechanisms in electrodynamics, is the elastic photon-electron scattering under moderate intensity electromagnetic (EM) radiation [

Metamaterial may be a promising concept for realizing the artificial HHG, since its unprecedented degree of design freedom has been extensively demonstrated in linear optics with a variety of extraordinary properties [

In this work, we propose an entirely artificial mechanism for generation of high harmonics based on a nonrelativistic nonlinear Thomson scattering process in a solid-state metamaterial. By locally redistributing the magnetic field in a metamaterial, the magnetic force becomes comparable to the electric one with a nonrelativistic velocity of free electrons in solids, which generates high harmonics without involving any external nonlinear materials. Numerical simulations perfectly verify the artificial generation of high harmonics, and the geometric impacts on it are studied as well to demonstrate the ultrahigh design freedom.

Conventionally, the relativistic velocity of the free electron plays a prerequisite role in the nonlinear Thomson scattering, which guarantees the evident magnetic force. In the nonrelativistic regime, however, the essential magnetic force is neglected in a majority of materials because the drift velocity is far slower than the speed of light, and the intrinsic magnetic field from the EM wave is negligibly weak. Meanwhile, the localized magnetic field redistribution of the metamaterial at resonance has primarily been acknowledged [

_{1}=15.6 _{1}=2.5 _{2}=3 _{2}=8.6 _{B}

With an

As the magnetic force is dominated by the localized fields, which arise from the resonance between the metamaterial and the incident EM fields, their amplitudes can be described as

To explicitly describe the high-order behavior of the metamaterial, we studied its equivalent susceptibility at different orders. Since the process occurs in a solid structure, the simple motion formula of the conventional Thomson scattering based on the plasma electron is not applicable, and the collisions should be considered. We thus modified the classical Drude model by substituting the magnetic force (see (_{avg} and_{avg} are the average values of the_{p} is the plasma frequency of the material forming the cut-wire resonator (mathematical details are described in Supplementary Materials). It can be found that the enhancement coefficients are the key factors determining the high-order nonlinear susceptibility, which are solely defined by the artificial geometry of the metamaterial. Therefore, by elaborately varying the structure, the high-order nonlinearity generated from the metamaterial can be designed and controlled at will.

Several characteristics of the artificial HHG can be predicted from (

High conductivity of the SRR would improve the intensity of the HHG, because its characteristic of low loss increases the current density and provides stronger localized magnetic field (Figure _{e0}. Therefore, by further analyzing (_{e0})^{N -1} can be obtained. The physical essence is easy to understand that the high mobility leads to fast drift velocity, strong magnetic force, and intense HHG. For the cut-wire resonator, high conductivity usually means high density of free electrons, which would not only induce much more frequent collision and dramatically decrease the mobility but shield the local magnetic and electric field. Both are negative factors and may weaken the HHG.

To verify the artificial mechanism of HHG, we optimized the metamaterial shown in Figure _{2}. Guided by the theory, the SRR is comprised of a 500 nm thick gold, and a 500 nm thick n-doped GaAs film is modeled as the cut-wire resonator due to its high mobility. The details of the materials and simulation settings are described in the Materials and Methods. All compositions are treated as linear materials in the simulation, and they are chosen solely due to their linear properties, such as mobility and conductivity. The metamaterial could be composed of any material that satisfies the fundamental requirements of the basic principle. It should also be noted that, despite the illustration of the mechanism in THz regime, this theory of artificial HHG can be easily extended to other frequencies by simply scaling the geometry of the metamaterial.

We first simulated the linear response of the metamaterial at the frequency domain to study its resonant behavior. Figure

With a Gaussian pulsed plane wave at 2.0 THz incident from the top, the temporal response of the metamaterial was then simulated. The peak amplitude of the incident electric field was set as 1×10^{7} V/m, corresponding to the power density of 1.3×10^{7} W/cm^{2}, which is orders of magnitude lower than the general requirement of the conventional nonlinear Thomson scattering (~10^{18} W/cm^{2}) [^{4} V/m, and the peak value at

We also simulated the metamaterials containing only SRR or cut-wire resonator for comparison, and the same Gaussian plane wave at 2.0 THz illuminated from the top. Figure

To further illustrate the parity of the harmonic order can be distinguished by the polarization state, we defined a parameter_{c}_{y}_{x}_{y}_{x}_{c}_{c}_{c}_{c}

_{c}

We probed the

As specified in the theory, the proposed metamaterial radiates longitudinally in both directions, and the reflected spectra are thereby examined as shown in Figure ^{4} V/m, and that of the

To confirm the artificial designability and tailoring of the high harmonics generated from the metamaterial, we manipulated the geometry of the unit cell and studied the influence on the amplitude of the HHG. For clarity and better contrast, only the dominant electric field components of the transmitted harmonics have been studied, namely,

The other geometric characteristic we studied is the location of the cut-wire resonator. By tailoring the distance between cut wire and bottom of the SRR from 0 (attached) to 7

Inspired by these two examples, it can be conveniently predicted that other geometric constants would also have an effective impact on the high-order nonlinear response, including the width (_{1}) and length (_{1}) of the SRR, and the lattice constant of the metamaterial (

All these simulated results profoundly demonstrate the proposed artificial mechanism of HHG without involving any sophisticated mesoscopic and quantum mechanisms in materials or external nonlinear insertions. The HHG essentially arises from the fundamental nonlinear Thomson scattering in a nonrelativistic limit, which is dominated by the artificial metamaterial structure instead of the natural compositions, making the method available to almost all the conducting materials, such as doped silicon and graphene, not limited to the GaAs (details are described in Figure

The HHG from the metamaterial contains both even and odd orders, and the parity can be distinguished by the polarization state, which is in sharp contrast to the majority of natural isotropic media, including the commonly illustrated semiconductor crystal and rare gas, which can only generate odd order harmonics because of the symmetry inversion [

Since the proposed artificial mechanism provides an approach of achieving the nonlinear Thomson scattering in solid-state materials with no request of relativistic velocity, the demand of the intense illumination of the light would be substantially relaxed. To quantitatively verify it, we compare the amplitude of the normalized vector potential in the definition form of _{0} is the electron mass. In the metamaterial, the _{0} of the free electrons in the cut-wire resonator can be theoretically calculated as 0.75, which is 1.6×10^{3} times stronger than that of the plasma electrons, corresponding to 2.6×10^{6} times higher power density of the incident laser (calculation details are described in Supplementary Materials). This significant enhancement overcomes the longstanding obstacle of the near-light speed in the study of the nonlinear Thomson scattering and makes its occurrence in low velocity and in condensed phase material possible. This novel mechanism may become a generic tool to advance the understanding of the fundamental electrodynamics of the condensed matters. Given the fact that the proposed design is only a proof-of-concept, we believe, by optimizing the structure of the metamaterial, the nonlinear scattering would be further improved.

More importantly, by simply engineering the geometry of the metamaterials, the artificially generated high harmonics can be precisely designed and exactly tailored with ultrahigh degree of freedom. Although the metamaterial is numerically demonstrated in THz regime, due to the universal existence of the Lorentz force, this theory of artificial high harmonics can be easily extended to wide wavelength ranging from microwave to infrared and visible light by properly scaling the metamaterial (details are described in Figures

These fascinating features of the proposed mechanism would open groundbreaking possibilities and tons of novel phenomena to the high-order optical nonlinearity, metamaterial, ultrafast physics, and electrodynamics of matters. For instance, the active modulation and feedback tuning of the high harmonics, and the flat self-focusing lens and holography for high harmonics, and compact attosecond laser might all become achievable.

We theoretically demonstrated an artificial mechanism for high harmonic generation based on a nonrelativistic Thomson scattering in a metamaterial. As adequately described by an explicit physical model, the locally reconstructed magnetic field in the metamaterial stimulates the nonlinear Thomson scattering in solid with a nonrelativistic velocity, which drives the free electrons to oscillate in an anharmonic way and further generates the high harmonics. A series of numerical simulations perfectly support the proposed theory with the artificial generation of high harmonics, and the geometric variation of the metamaterial leads to an apparent control and tailoring. This purely artificial mechanism provides extraordinary degree of design freedom to high harmonic generation with a metamaterial-based approach and allows the occurrence of nonlinear Thomson scattering in a nonrelativistic limit, which would open myriad application possibilities and novel potentials to high-order nonlinearity and advanced understanding of the electron dynamics in condensed matters.

In THz regime, the substrate was SiO_{2} with the permittivity of 4.2+0.026^{7} S/m and mobility of 29.5 cm^{2}/V•s [^{17} cm^{−3}, corresponding to the dc conductivity of 3.1×10^{4} S/m and the dc mobility of 3800 cm^{2}/Vs with the damping frequency of 2_{0} and_{e0} are the dc conductivity and mobility, respectively, and

The metamaterial was simulated in both frequency and time domains. In both domains, single unit cell was simulated with the periodic boundary condition in

In frequency domain, the S-parameters of the metamaterial were simulated ranging from 1.4 THz to 2.6 THz with the step of 0.01 THz. Therefore, the reflection (

In time domain, the_{0} is the wavenumber at the frequency_{0} and^{7} V/m,^{12} rad/s,_{0}=7 ps, and

The authors declare no competing financial interests.

This work was supported by the Basic Science Center Project of NSFC (no. 51788104), as well as National Natural Science Foundation of China (nos. 51532004 and 11704216), and the China Postdoctoral Science Foundation (nos. 2015M580096 and 2017T100074).

Figure S1: the influence of conductivity of SRR on the metamaterial. Figure S2: transmission spectra in the time domain of the metamaterial. Figure S3: transmitted high-harmonic spectra of the metamaterial in