Terahertz is an electromagnetic wave falling in the far-infrared band with the frequency ranging from 0.1 to 10 THz at the interface of electronics and photonics [1]. Terahertz technology has caught many attentions for its pregnant applications including imaging, communication, and non-destructive testing [2–6]. These applications promote the development of terahertz modulators with ultrafast modulation speed, high modulation depth, and broad operation bandwidth [7–10]. However, the interaction between terahertz waves and traditional materials is so weak that it is hard to adjust terahertz information, restricting applications of terahertz devices. Fortunately, periodically artificial metamaterials, which can confine electromagnetic fields in subwavelength volume, provide an opportunity to realize controllable terahertz modulation in amplitude, phase, and polarization [11–14]. To achieve efficient terahertz modulation, sensitive physical processes in the terahertz regime are required for active tunning, such as inductive-capacitive (LC) coupling [13,15], the electromagnetically induced transparency (EIT) effect [16,17], and bound state in the continuum [18,19]. As a super-sensitive physical phenomenon, the EIT effect, produced by the interaction between the bright and dark modes in metamaterials, is generally used for ultrafast and high-depth terahertz modulation due to its susceptiveness and high resonance amplitude [20–23]. The usual metamaterial supported EIT effect is designed by arrays of metal split ring resonators (SRRs). Cooperating SRR with two-dimensional materials such as transition metal dichalcogenides (TMDCs) [24,25], perovskite [26,27], and thin-film semiconductor (Ge, Si) [28,29], ultrafast terahertz switching is realized by external pump excitation. However, the performance of a traditionally designed structure is likely to be highly dependent on the characteristics of the terahertz devices. And the broad applicability of past structure is doubtful, as the greater demands on functionality [30]. Besides, demand on a novel terahertz device needs numerous time-consuming attempts. Thus, it is significant to improve the design approach for the development of terahertz devices.

The inverse design method based on optimization algorithms has been used for the optimal structure parameters and a large number of applications in the field of nanophononics including photonic crystal [31,32], metasurface [33–36], demultiplexer [37,38], and silicon optical waveguide [39,40], arousing further interest in the structure design of terahertz metamaterials. One of the most important advantages of inverse-designed structures is to achieve high performance beyond traditional structures [37,41]. By means of inverse design optimization algorithms, it is even available to obtain better performance than the traditional devices [41]. Compared to conventional design for photonic devices depending on intuition-based strategies, inverse design is welcome for several key reasons [31,42–44]. (1) Inverse design can automate to search for the optimal objective with the computer-assisted design process, which greatly reduces the artificial attempts. (2) To achieve considerable success in existing metamaterial structures, the schemes of inverse design can ignore the underlying physics, with simplicity and universality. (3) Not limited to periodic artificial structure, inverse design algorithms can design complex, aperiodic structures with novel functionality and high performance, which enriches the designing structures. The inverse design algorithms contain artificial neural network algorithm [45], genetic algorithm [46], topology optimization [47], direct-binary-search algorithm [41], and particle swarm optimization (PSO) [40,48]. Compared to other algorithms, the advantages of PSO are fewer parameters to adjust and the simple principle for ease of implementation. This algorithm has large adaptability not relying on the problem information with faster convergence and low requirement of computer performance, which is suitable for our actual design. According to practical applications on terahertz modulation, the algorithms of inverse design offer more opportunities for the development of terahertz devices.

Herein, we propose a novel concept of building an ultrafast all-optical terahertz modulator utilizing inverse design, incorporating the PSO algorithm with the finite-difference time-domain method. In this work, a metasurface with a Fano-type transmission spectrum supported by the EIT effect was designed and fabricated to achieve high-depth and ultrafast modulation by covering Ge film. The sensitivity characteristic and near-field contributions confirm the existence of the EIT effect in our inverse-designed metasurface. The ultrafast terahertz EIT resonance modulation is experimentally demonstrated through optical pump terahertz probe technology. The experimental data is consistent with the simulated results, indicating that it is robust for our scheme of inverse design to manufacture a feasible terahertz modulator. Finally, an ultrafast modulation speed on the picosecond scale is obtained in our hybrid structure combing inverse-designed metasurface with Ge film. Overall, we present an inverse design method for ultrafast terahertz modulation for the first time to our knowledge, which brings opportunities for novel terahertz devices.

To realize an ultrafast terahertz modulator, hybrid structure cooperating metasurfaces with semiconductors have been proposed in this work. Compared with approaches to electrical, thermal active control combined with phase material such as vanadium dioxide [49–51], all-optical terahertz modulation with active mediums can support faster modulation speed in the nanosecond or picosecond scale. A host of works reveal that the speed of modulation is mainly determined by the ultrafast dynamics based on the relaxation of semiconductor free carriers [24,26,29,52]. Here germanium (Ge) with carrier lifetime in the picosecond scale attracts us for ultrafast terahertz modulation [29]. As shown in Fig. 1(a), we utilize germanium (Ge) film combined with inverse-designed metasurface to control terahertz waves through exciting photon-generated carriers by pump light. The initial conductivity of Ge is set as 10 S/m without pump excitation. When the pump light arrives at the surface of semiconductor, the concentration of carriers is changed; at the meantime the conductivity changes, which provides an avenue to regulate terahertz wave information such as amplitude, phase, polarization. On the one hand, the short carrier relaxation time gives rise to ultrafast response; on the other hand, sensitive physical phenomena should be discovered for feasible and easy terahertz adjustment. Fortunately, the high sensitivity of the EIT effect to change in the resonant local electromagnetic field favors its utilization in switchable modulation. In a typical EIT metasurface unit cell, a pair of two coupled Lorentzian oscillators is utilized to describe the interaction between the bright and dark modes. To describe the EIT effect of the hybrid structure, a simple coupled harmonic oscillator model can be employed [53]: $$\begin{gathered} p¨(t)ωR2+ΓR1p˙(t)ωR+p(t)=f(t)−κq(t) \\ q¨(t)ωD2+ΓR2q˙(t)ωD+q(t)=−κp(t), \end{gathered}$$(1)where the subscripts R and D refer to bright and dark modes, respectively, p and q denote the amplitudes of two coupled modes, and Γ is the damping rates. κ is the coupling strength between the radiative mode at resonance frequency ωR and dark mode at resonance frequency ωD. f(t) is the driven force to manifest the coupling between the bright mode of the metasurface and external incident electromagnetic field. Then the surface conductivity σs(ω) of the EIT structure can be obtained by utilizing abovementioned parameters and formula: σs(ω)=−iωε0χs(static)Dd(ω)Dd(ω)Dr(ω)−κ2,(2)where χs(static) is the surface susceptibility, Dd(ω)=1−(ω/ωD)2−iΓR2(ω/ωD), and Dr(ω)=1−(ω/ωR)2−iΓR1(ω/ωR). Then, the complex transmittance of the normally incident electric field is given as E˜(ω)=22+ζσs(ω),(3)where ζ is the wave impedance of the external wave. More details and derivations can be found in Ref. [54]. In general, one can utilize this mode to learn the EIT effect and calculate its damping and coupling strength for physical analysis. Nevertheless, we use this model to design a structure with a certain EIT window which is accorded with our target, namely, inverse design can be of assistance for terahertz modulation devices. The method of this inverse design is an algorithm of PSO combined with finite-difference time-domain simulation. We use the standard “GBEST” particle swarm algorithm to achieve our target [55]. To better understand the process of this inverse design, we flowchart the details in the Fig. 2(a). Our proposed inverse-designed periodic metasurface consists of a cut wire and numerous crosses around the single cut wire in one unit. The periods of one-unit metasurface are px=110μm in the x direction and py=180μm in the y direction. The thickness of the metasurface is 0.2 μm. The polarization of incident terahertz wave is along the y direction. More details about geometric parameters of the structure are shown in Fig. 6 in Appendix A. To reduce the memory and time consumption of simulations, we set symmetrical boundary condition in the y minima and periodic boundary condition in the x direction. Unlike the square pixel elements in the inverse-designed waveguide [48], the reason why we replace it with a cross element is to consider the practical processing. Each pixel element in this metasurface can be selectively filled by Au or air, corresponding to the logical state “0” or “1,” respectively. In the terahertz regime, the conductivity of Au set by 4×107S/m is applicable, and the substrate is set as quartz with refractive index nsub=1.98. Turning now to the procedure of PSO, first we set a target with a certain transmittance Ttar shown in the red dash line in Fig. 2(a), the simulation result of the initial metasurface design is given by Tsim, and we define the fitness function as fit=|Tsim−Ttar|. The transmittance data are discrete at different frequencies. Then we simulate plentifully various metasurface structures in the iteration process until it is converged for searching for the optimal result (the minimum of the fitness function). These results of simulations in each iteration are called particle swarm in the PSO algorithm, and each of them is named a particle. Ten particles are set for one iterative loop. The averaged fitness of all simulation particles is described in Fig. 2(b), and after around 60 iterations, the results tend to converge. The last designed one-unit metasurface structure is shown in Fig. 6 (Appendix A), and the simulation result of the final design, which is well fitted to the target, is described in Fig. 2(c). Note that the final transmittance curve does not perfectly match the target. It is likely that the inverse design is in the local optimum but close to the global optimum. In our inverse design, the bright mode is mainly supported by a cut wire in one unit, and the dark mode can be selectively excited by altering the position and number of the crosses. The picture of really processed products is shown in Fig. 1(b), and the hybrid structure combined with the metasurface and covered Ge film is shown in Fig. 1(c). The advantage of inverse-designed EIT effect in the terahertz regime is that none of the specific physical explanations are necessary. In addition, the inverse-designed method has universal applications including but not limited to terahertz amplitude, phase, and polarization control. The composition of inverse-designed metasurface allows not only metal but also all dielectric for intriguing terahertz control such as EIT control and polarization conversion [56,57]. Compared to other algorithm models such as gradient-based methods [43,58] and artificial neural networks [59–61], the PSO can provide the ability to jump out of local optimum and has no need for lots of original data and a trained model. Thus, this design scheme has significant potential for terahertz modulators.

To further study the EIT effect caused by our designed metasurface, pump light at wavelength 800 nm (an optical pulse of energy 1.55 eV, which is higher than the energy bandgap of the Ge film) was use to excite photoinduced carrier concentrations of Ge and change its conductivity to modulate the shape of the EIT spectrum as shown in Fig. 3. Due to the high sensitivity to the electromagnetic environment, EIT resonance can switch on or off at different pumping energy fluences. Figure 3(a) displays the experimental transmission spectral dispersion of a hybrid structure combined with Ge film and a metasurface at various pumping energy fluences. Optical excitation of Ge film results in interband transition of electrons from the valence band to the conduction band. The injected free carriers give rise to the changes of local electromagnetic environment and lead to different coupled strength between the bright mode and dark mode. In other words, the free carriers of photoexcited Ge film alter the strength of terahertz electric field confined in our inverse-designed metasurface structure. This change generates a strong modulation of EIT resonance amplitude. With the optical pump energy fluence increasing from 0 to 2.2mJ/cm2, the EIT resonance (defined by a transmittance peak at 0.76 THz and dip at 0.67 THz) is increasingly suppressed as the concentration of photoexcited free carriers increases. The transmission resonance amplitude defined by δT=Tpeak−Tdip achieves maximum with no pump light and decreases with growing pump energy fluence. The normalized resonance amplitude extracted from transmittance spectra is shown in Fig. 3(b), reaching over 95% modulation depth in our experiment. All results show the trend toward smoothness with the resonance being almost completely faded away at pump energy over 2mJ/cm2. On the other hand, our simulation results described in Figs. 3(c) and 3(d) show the same trend as the experimental process. By changing the conductivity of Ge film covered over the inverse-designed metasurface, the local electromagnetic environment alters and contributes to modulation of terahertz wave information including amplitude and phase. In this work, we define the state of “switching on” at the maximal resonance amplitude and the state of “switching off” at the case of over 90% modulation of normalized resonance amplitude. In the simulation, we set the corresponding conductivities for references only to illustrate the tendency of EIT modulation [29]. With the conductivity of Ge film ranging from σ=10S/m (without pump) to σ=1000S/m (with high pump power), the EIT resonance amplitude falls to zeros gradually. It is clear that the EIT resonance is eliminated with the conductivity of Ge up to σ=600S/m, meaning in the switched-off state. Note that the resonance frequency in our experiment is different from the simulation. (The transmission peak of EIT is 0.79 THz in our simulation, and the dip is 0.57 THz.) The depth of EIT resonance also has distinction. Mainly, we attribute these differences of transmittance spectra between experiment and simulation (the structure we simulated is from inverse design) to the fabrication errors and the material characters. Another significant physical phenomenon based on EIT is the slow-light effect that is described in Fig. 7 (Appendix A). Without loss of generality, we quantitatively characterize the slow-light performance in our inverse-designed hybrid metasurface structure combined with Ge film as the group delay Δtg=−dω/dt, where ω=2πf is the terahertz angle frequency. As we can see, the group delay influenced by the slow-light phenomenon of the EIT effect reaches 1.65 ps at a certain frequency 0.76 THz (the peak of transmittance spectra of the EIT window) without pump light. Subsequently, the slow-light effect degrades together with the increasing pump fluence, presenting as the group delay at the peak of the transparency window trending to 0 ps. Both optical excited modulation of transmittance amplitude and the slow-light effect reveal, in a way, that EIT effect exists in our inverse-designed metasurface structure, offering a high sensitivity for carrying out THz modulation.

The ongoing request is to investigate the dynamically tunable terahertz EIT response of a hybrid structure integrating an inverse-designed metasurface with Ge film. By taking advantage of optical pump terahertz probe (OPTP) technology, we explore the possibility of the ultrafast switch dynamics for terahertz EIT resonance. As just mentioned before, the modulation behavior can be influenced by optical pump excitation, together with Ge carrier dynamics changing in the time domain. Therefore, by moving the pump-probe delay stage, we employ the time-resolved optical pump measurements to illustrate the ultrafast modulation processes at a selectable optical pump fluence of 2.2mJ/cm2, where the largest modulation depth can come true. The time delay given in Fig. 4 represents the relative time delay in the arrival of the pump light and the terahertz probe at the sample face. The time-domain terahertz spectra are experimentally measured by a terahertz time-domain spectrum (TDS) system for each relative time delay, and then the dynamics of terahertz modulation can be exhibited. First, we characterize the ultrafast relaxation time of the pure Ge film with femtosecond pulse laser pumping, which is shown in Fig. 8 (Appendix A). Depending on different carrier concentrations in time scale, the absolute transmission change defined by ΔT=T−T0 was measured by OPTP, where T is the transmission of a terahertz pulse through the sample at a real time, and T0 is that of the sample in absence of pump fluence. To improve the signal-to-noise ratio, we set the time delay of TDS at the maximum amplitude of terahertz pulse in the time domain to explore the terahertz transmittance. The negative differential transmission of pure Ge film −ΔT/T0 exhibits a characteristic curve performed by an exponential delay. The mono-exponential delay profiles convolved with the instrument response function can be used to fit the experimentally measured results, given by [62]−ΔTT(t)=e−[t−t0IRF/(2ln2)]2⊗(A0+A1et−t0τ),(4)where IRF is the full width at half-maximum of the instrument response function of the pump pulse, A0 is a constant referring to the amplitude without decay, A1 is the amplitude determining the weight of the exponential function with the delay time constant τ, and t0 corresponds to time zero of the exponential fit. The delay time constant τ is found to increase as a function of pump fluence in the supporting information, demonstrating an ultrafast relaxation process. It is in agreement with previous results [29]. Subsequently, we investigate the ultrafast modulation of the hybrid structure combining Ge film with an inverse-designed metasurface, as shown in Fig. 4. Due to the slow-light effect, which increases the optical path length, caused by the 200 nm thick metasurface, the modulation time was prolonged [29]. For simplicity, we set the time delay of 0 ps as the case where pump light arrives at the sample and the terahertz pulse reaches the maximal amplitude at the same time. The carrier’s concentration of Ge film rises rapidly with the pump light stimulation; in the meantime, conductivity increases and weakens the interaction between dark modes and bright modes, and then EIT resonance is regulated in the ultrafast scale. By altering the relative time delay stage, the terahertz transmittance spectra of the time domain are described in Fig. 4(a) by contour map, demonstrating the switching process of the EIT effect. For instance, at a time delay of −5ps, where the terahertz maximal amplitude arrives 5 ps before the arrival of the pump beam, the EIT phenomenon is switching on. This is because no photoexcited carriers of Ge film are generated, indicating that the local electric field modes of the metasurface remain unchanged and no EIT modulation is coming. While the EIT effect is in the case of switching off at a time delay of 0 ps due to the large density of the photoexcited carriers, coming into huge terahertz EIT modulation. It is worth noting that the modulation of EIT starts after the time delay of −5ps because of the arriving but not maximal pulse of terahertz. Following the time delay of OPTP, the EIT phenomenon is gradually eliminated and then recovers. According to the transmission spectra of the terahertz wave, the resonance amplitude decreases and recovers afterward, with the delay time going by. More clearly, we extract the terahertz transmittance curves in Fig. 4(b). For each delay time, the terahertz pulse is scanned through the metasurface and substrate after or before the femtosecond laser pulse. From the delay time of −5 to 0 ps, the EIT resonance switches from on to off, and after that the EIT resonance switches from off to on between the time delay of 1 to 5 ps, as indicated by the black arrows. The EIT resonance amplitude as a function of pump-probe delay is described in Fig. 9 in Appendix A. The whole process realizes an ultrafast all-optical terahertz EIT resonance switching in the picosecond scale (<15ps). In all, we experimentally perform an ultrafast EIT modulation in our hybrid structure of inverse-designed metasurface utilizing ultrafast carrier dynamics of Ge film.

In order to illuminate the physical mechanism of terahertz EIT modulation of the inverse-designed metasurface with Ge film, we perform the simulations of near-field distributions as a function of the conductivity of Ge film. To clarify the interaction between dark mode and bright mode, a field profile monitor monitoring the near field at 0.67 THz (the dip of transmittance spectra of EIT resonance) is placed 1 μm above the metasurface. It is visible that the near-field radiation of the metasurface entirety gradually decreases as shown in Figs. 5(a)–5(d), with increasing conductivity of Ge film from 10 to 1000 S/m. More details discovered in the near-field profiles show that the electric field distribution of cut wire nearly keeps unchanged with the increased conductivity of Ge film and performs a dipole resonance. Then the dipole resonance can reradiate the terahertz wave and play a role in bright mode in the EIT effect. In contrast to the cut wire in the metasurface, the near-field strength of the nearby cross group is higher, and weakens with growing conductivity of the Ge film. We consider these modes dark modes due to the strong confinement of electric field in the crosses. The increasing conductivity of Ge film reduces the strength of dark modes and meanwhile weakens the interaction between bright mode and dark mode, allowing for gradually declining EIT resonance amplitude. The tendency of EIT modulation in our experiment through various pump fluences is in accord with the transformation of near-field strength in simulation with different conductivities of Ge film.

In summary, using a PSO algorithm, we have performed an inverse-designed metasurface supporting the EIT effect. Through setting a goal fitness calculated by a coupled harmonic oscillator model, we search for the terahertz transmittance displaying a desired EIT window, and then we manufacture this inverse-designed structure so as to experimentally find the EIT effect, which confirmed a feasible scheme to design ultrasensitive terahertz physics phenomena. Utilizing its sensitive character, we further verify this EIT effect by covering Ge film in the structure and optical excitation realizes modulation of EIT resonance amplitude as well as slow-light switching. Furthermore, under the pump pulse excitation, dynamical modulation of this Ge film hybrid metasurface exhibits ultrafast switching characteristic in the picosecond time scale. Also, the near-field profiles from simulation confirm the interaction between bright mode and dark mode at various conductivity of Ge film, agreeing well with the modulation of terahertz EIT resonance amplitude at different pump energy densities. Compared to split ring resonators and other conventional metamaterial structures, the inverse-designed structure in the terahertz regime provides a more flexible method to design multifunctional terahertz devices. In the future, the optimization algorithm of inverse design could be extended to incorporate interesting physics and modulators such as multiple EIT resonance modulation and broadband polarization converters. Our results suggest that this method is useful for terahertz modulation and paves the way for advanced terahertz multifunction devices with high performance.