Abstract

Metasurface plays a key role in various terahertz metadevices, while the designed terahertz metasurface still lacks flexibility and variety. On the other hand, inverse design has drawn plenty of attention due to its flexibility and robustness in the application of photonics. This provides an excellent opportunity for metasurface design as well as the development of multifunctional, high-performance terahertz devices. In this work, we demonstrate that, for the first time, a terahertz metasurface supported by the electromagnetically induced transparency (EIT) effect can be constructed by inverse design, which combines the particle swarm optimization algorithm with the finite-difference time-domain method. Incorporating germanium (Ge) film with inverse-designed metasurface, an ultrafast EIT modulation on the picosecond scale has been experimentally verified. The experimental results suggest a feasibility to build the terahertz EIT effect in the metasurface through an optimization algorithm of inverse design. Furthermore, this method can be further utilized to design multifunctional and high-performance terahertz devices, which is hard to accomplish in a traditional metamaterial structure. In a word, our method not only provides a novel way to design an ultrafast all-optical terahertz modulator based on artificial metamaterials but also shows the potential applications of inverse design on the terahertz devices.

© 2021 Chinese Laser Press

1. INTRODUCTION

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 [26]. These applications promote the development of terahertz modulators with ultrafast modulation speed, high modulation depth, and broad operation bandwidth [710]. 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 [1114]. 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 [2023]. 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 [3336], 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 [31,4244]. (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.

2. RESULT AND DISCUSSION

A. Inverse-Designed Metasurface and Its Functionality

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 [4951], 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]:

p¨(t)ωR2+ΓR1p˙(t)ωR+p(t)=f(t)κq(t)q¨(t)ωD2+ΓR2q˙(t)ωD+q(t)=κp(t),
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,
where χs(static) is the surface susceptibility, Dd(ω)=1(ω/ωD)2iΓR2(ω/ωD), and Dr(ω)=1(ω/ωR)2iΓR1(ω/ωR). Then, the complex transmittance of the normally incident electric field is given as
E˜(ω)=22+ζσs(ω),
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=|TsimTtar|. 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 [5961], 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.
 figure: Fig. 1.

Fig. 1. Structure chart of ultrafast all-optical terahertz modulation. (a) Schematic illustration of hybrid structure combining Ge film with inverse-designed metasurface at the pump light of 800 nm for terahertz modulation. (b) Processed inverse-designed metasurface structure without Ge film covering on the face. (c) Processed inverse-designed metasurface structure covering 200 nm thick Ge film on the face. The scale bar in the pictures is 50 μm.

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 figure: Fig. 2.

Fig. 2. Sketched scheme of inverse design of ultrafast terahertz modulator based on EIT effect utilizing the PSO algorithm and finite-difference time-domain method. (a) General design cycle of PSO algorithm. At this optimization procedure, we set the target function by using coupled harmonic oscillator model. The number of particles is set as 10, and the number of iterations is 100. (b) The fitness function tends to convergence with the increasing iterations. After 60 iterations, the fitness is convergent, and the result is what we want. (c) The transmittance of the terahertz wave at the last design is described in the red line, and the black line shows the target transmittance. All terahertz transmittance is obtained through finite-difference time-domain method.

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B. Light-Activated Germanium-Controlled Terahertz EIT Resonance Switching

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=TpeakTdip 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.

 figure: Fig. 3.

Fig. 3. Experimentally light-activated Ge-controlled terahertz EIT modulation for varying pump fluences and simulation results of EIT resonance modulation with different conductivities of Ge film. (a) Experimental modulation of terahertz EIT amplitudes for various pump fluences ranging from 0 to 2.2mJ/cm2. (b) The experimentally normalized resonance amplitude, defined by the difference between transmission at dip and transmission at peak, is changed following various pump fluences. (c) Simulated modulation of terahertz EIT amplitudes for different conductivities of Ge film ranging from 10 to 1000 S/m. (d) The simulated normalized resonance amplitude is changed following different conductivities.

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C. Ultrafast Dynamics of EIT Resonance Amplitude 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=TT0 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[tt0IRF/(2ln2)]2(A0+A1ett0τ),
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.
 figure: Fig. 4.

Fig. 4. Temporal evolution dynamics of ultrafast terahertz modulation behaviors under the pump fluence 2.2mJ/cm2. (a) Contour map of experimental terahertz transmission as a function of pump-probe time delay. (b) Top: Experimental terahertz transmission spectra at different pump-probe time delay from 5 to 0 ps. The state of EIT resonance switches from on to off. Bottom: Experimental terahertz transmission spectra at different pump probe time delay from 1 to 5 ps. The state of EIT resonance switches from off to on.

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D. Near-Field Distributions of Inverse-Designed Metasurface Combined with 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.

 figure: Fig. 5.

Fig. 5. Simulated near-field distributions as a function of Ge film conductivity ranging from 10 to 1000 S/m. E-field magnitudes are obtained at the position 1 μm above the metasurface.

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 figure: Fig. 6.

Fig. 6. (a) Schematic illustration of one-unit inverse-designed metasurface combined with 200 nm thick Ge film (the red part), including a cut wire and numerous crosses around it. The material of inverse-designed metasurface is gold with thickness of 200 nm (the golden part). The whole unit structure is mirror symmetric. The periods of one-unit metasurface are px=110μm in the x direction and py=180μm in the y direction. (b) Sketch of a half of one-unit metasurface. Each of cross consists of two orthogonal rectangles with length cy=5μm and width cx=3μm. The parameters of the cut wire in this sketch are w=10μm and L=63μm.

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 figure: Fig. 7.

Fig. 7. Experimentally measured group delay spectra of hybrid structure combining a reverse-designed metasurface with Ge film as a function of pump fluence ranging from 0 to 2.2mJ/cm2.

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 figure: Fig. 8.

Fig. 8. Negative differential transmission spectra of pure 200 nm thick Ge film are experimentally measured at the maximal amplitude of terahertz time-domain pulse signal as a function of optical-pump terahertz time delay. Experimental data points are fitted with equation in the main text and the delay time constants τ of pure Ge film at various pump fluences are extracted. The delay time constant is found to increase with the pump fluence increasing.

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 figure: Fig. 9.

Fig. 9. Modulations of the EIT resonance amplitude as a function of pump-probe time delay evaluated from the transmission spectra of Fig. 4 in the main text.

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3. CONCLUSION

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.

APPENDIX A

Sample Preparation: Using conventional microfabrication techniques, we fabricate the metasurface on commercially available 1 mm thick z-cut quartz substrate by patterning a metallic layer. A gold layer with a thickness of 200 nm was deposited by e-beam evaporation following the 10 nm thick chromium adhesion layer. Then the inverse-designed metasurface was fabricated using the standard photolithography technique with a preparational lithographic mask. After that, the sample was soaked in the potassium iodide solution to lift out the undesired gold film, and the remaining chromium adhesion layer was removed by soaking in acetone. The area of the finally prepared metasurface is 5mm×5mm, which is larger than the spot size of pump light (5mm in diameter) and terahertz wave (3mm in diameter). Subsequently, a 200 nm thick Ge film was thermally evaporated on the metasurface as a photoconductive semiconductor.

Terahertz Transmittance Measurement: We employ an optical pump terahertz probe setup from TuoTuo Technology (TTT-02-OPTP) to characterize the terahertz optical response of our inverse-designed metasurface cooperating with Ge film. The terahertz wave pulse is provided by nonlinear process when the femtosecond pulse beam is incident on 1 mm thick ⟨110⟩ ZnTe crystal and then focused onto the sample by two parabolic mirrors. Next, the terahertz information is modulated by our experimental samples and goes through two other parabolic mirrors then focused onto another 1 mm thick ⟨110⟩ ZnTe crystal for detection. A Ti:sapphire regenerative amplifier system is employed to generate a femtosecond pulse of 1 kHz repetition at a central wavelength of 800 nm for stimulating the samples as well as terahertz generation and detection. The terahertz time-domain waveform is recorded by the motion of the translational stage of the gated beam and then converted to frequency-domain spectra by Fourier transform. Thus, the frequency-dependent terahertz transmission amplitude is obtained by T(ω)=ES(ω)/ER(ω). The terahertz transmittance of different time delays is measured by controlling a delay stage on the path of the pump beam.

Electric Field and Transmittance Simulation: The inverse design and optical modulation of the hybrid structure combined with a metasurface and Ge film were simulated based on the finite-difference time-domain method. The period boundary condition is employed along the x direction, and the symmetric boundary condition is used in the y minimum in order to minimize the calculation time. The polarization of terahertz wave is set along the y direction. In the simulation, the metal gold of 200 nm thickness with conductivity σgold=4×107S/m was used to model metallic inverse-designed structure, and quartz with a refractive index of 1.98 was chosen as the transparent substrate. A 200 nm thick Ge film was set at the same layer of the metasurface with lower mesh order varying the conductivity from 10 to 1000 S/m. The near-field magnitude contribution was extracted from a field profile monitor placed 1 μm above the metasurface at the frequency of the EIT window.

Funding

National Natural Science Foundation of China (11802339, 11804387, 11805276, 11902358, 61801498, 61805282, 62075240); Scientific Researches Foundation of National University of Defense Technology (ZK18-01-03, ZK18-03-22, ZK18-03-36); Science Fund for Distinguished Young Scholars of Hunan Province (2020JJ2036).

Disclosures

The authors declare no conflicts of interest.

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8. T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nat. Photonics 10, 371–379 (2016). [CrossRef]  

9. X. C. Q. Yang, Q. Xu, C. Tian, Y. Xu, L. Cong, X. Zhang, Y. Li, C. Zhang, X. Zhang, J. Han, and W. Zhang, “Broadband terahertz rotator with an all-dielectric metasurface,” Photon. Res. 6, 1056–1061 (2018). [CrossRef]  

10. Z. Chen, X. Chen, L. Tao, K. Chen, M. Long, X. Liu, K. Yan, R. I. Stantchev, E. Pickwell-MacPherson, and J. B. Xu, “Graphene controlled Brewster angle device for ultra broadband terahertz modulation,” Nat. Commun. 9, 4909 (2018). [CrossRef]  

11. L. Cong, Y. K. Srivastava, H. Zhang, X. Zhang, J. Han, and R. Singh, “All-optical active THz metasurfaces for ultrafast polarization switching and dynamic beam splitting,” Light Sci. Appl. 7, 28 (2018). [CrossRef]  

12. C. C. Chang, Z. Zhao, D. Li, A. J. Taylor, S. Fan, and H. T. Chen, “Broadband linear-to-circular polarization conversion enabled by birefringent off-resonance reflective metasurfaces,” Phys. Rev. Lett. 123, 237401 (2019). [CrossRef]  

13. H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444, 597–600 (2006). [CrossRef]  

14. Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, Y. Zhang, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5, 041027 (2015). [CrossRef]  

15. H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008). [CrossRef]  

16. S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6, 1800570 (2018). [CrossRef]  

17. S. Xiao, T. Wang, T. Liu, X. Yan, Z. Li, and C. Xu, “Active modulation of electromagnetically induced transparency analogue in terahertz hybrid metal-graphene metamaterials,” Carbon 126, 271–278 (2018). [CrossRef]  

18. S. Han, L. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. G. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-dielectric active terahertz photonics driven by bound states in the continuum,” Adv. Mater. 31, 1901921 (2019). [CrossRef]  

19. L. Cong and R. Singh, “Symmetry-protected dual bound states in the continuum in metamaterials,” Adv. Opt. Mater. 7, 1900383 (2019). [CrossRef]  

20. M. Liu, Z. Tian, X. Zhang, J. Gu, C. Ouyang, J. Han, and W. Zhang, “Tailoring the plasmon-induced transparency resonances in terahertz metamaterials,” Opt. Express 25, 19844–19855 (2017). [CrossRef]  

21. X. Chen, S. Ghosh, Q. Xu, C. Ouyang, Y. Li, X. Zhang, Z. Tian, J. Gu, L. Liu, A. K. Azad, J. Han, and W. Zhang, “Active control of polarization-dependent near-field coupling in hybrid metasurfaces,” Appl. Phys. Lett. 113, 061111 (2018). [CrossRef]  

22. J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012). [CrossRef]  

23. T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photon. 5, 1800–1807 (2018). [CrossRef]  

24. Y. Hu, T. Jiang, J. Zhou, H. Hao, H. Sun, H. Ouyang, M. Tong, Y. Tang, H. Li, J. You, X. Zheng, Z. Xu, and X. Cheng, “Ultrafast terahertz transmission/group delay switching in photoactive WSe2-functionalized metaphotonic devices,” Nano Energy 68, 104280 (2020). [CrossRef]  

25. Y. K. Srivastava, A. Chaturvedi, M. Manjappa, A. Kumar, G. Dayal, C. Kloc, and R. Singh, “MoS2 for ultrafast all-optical switching and modulation of THz Fano metaphotonic devices,” Adv. Opt. Mater. 5, 1700762 (2017). [CrossRef]  

26. A. Kumar, A. Solanki, M. Manjappa, S. Ramesh, Y. K. Srivastava, P. Agarwal, T. C. Sum, and R. Singh, “Excitons in 2D perovskites for ultrafast terahertz photonic devices,” Sci. Adv. 6, eaax8821 (2020). [CrossRef]  

27. M. Manjappa, Y. K. Srivastava, A. Solanki, A. Kumar, T. C. Sum, and R. Singh, “Hybrid lead halide perovskites for ultrasensitive photoactive switching in terahertz metamaterial devices,” Adv. Mater. 29, 1605881 (2017). [CrossRef]  

28. Y. Hu, J. You, M. Tong, X. Zheng, Z. Xu, X. Cheng, and T. Jiang, “Pump-color selective control of ultrafast all-optical switching dynamics in metaphotonic devices,” Adv. Sci. 7, 2000799 (2020). [CrossRef]  

29. W. X. Lim, M. Manjappa, Y. K. Srivastava, L. Cong, A. Kumar, K. F. MacDonald, and R. Singh, “Ultrafast all-optical switching of germanium-based flexible metaphotonic devices,” Adv. Mater. 30, 1705331 (2018). [CrossRef]  

30. H. J. Joyce, J. L. Boland, C. L. Davies, S. A. Baig, and M. B. Johnston, “A review of the electrical properties of semiconductor nanowires: insights gained from terahertz conductivity spectroscopy,” Semicond. Sci. Technol. 31, 103003 (2016). [CrossRef]  

31. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011). [CrossRef]  

32. M. Minkov, I. A. D. Williamson, L. C. Andreani, D. Gerace, B. Lou, A. Y. Song, T. W. Hughes, and S. Fan, “inverse design of photonic crystals through automatic differentiation,” ACS Photon. 7, 1729–1741 (2020). [CrossRef]  

33. D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17, 3752–3757 (2017). [CrossRef]  

34. C. Sitawarin, W. Jin, Z. Lin, and A. W. Rodriguez, “Inverse-designed photonic fibers and metasurfaces for nonlinear frequency conversion [invited],” Photon. Res. 6, B82–B89 (2018). [CrossRef]  

35. W. Ma, F. Cheng, Y. Xu, Q. Wen, and Y. Liu, “Probabilistic representation and inverse design of metamaterials based on a deep generative model with semi-supervised learning strategy,” Adv. Mater. 31, 1901111 (2019). [CrossRef]  

36. Y. Chen, Y. Hu, J. Zhao, Y. Deng, Z. Wang, X. Cheng, D. Lei, Y. Deng, and H. Duan, “Topology optimization-based inverse design of plasmonic nanodimer with maximum near-field enhancement,” Adv. Funct. Mater. 30, 2000642 (2020). [CrossRef]  

37. A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015). [CrossRef]  

38. L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2018). [CrossRef]  

39. B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378–382 (2015). [CrossRef]  

40. S. Y. Y. Zhang, A. E.-J. Lim, G.-Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express 21, 1310–1316 (2013). [CrossRef]  

41. B. Shen, P. Wang, R. Polson, and R. Menon, “Ultra-high-efficiency metamaterial polarizer,” Optica 1, 356–360 (2014). [CrossRef]  

42. S. So, T. Badloe, J. Noh, J. Rho, and J. Bravo-Abad, “Deep learning enabled inverse design in nanophotonics,” Nanophotonics 9, 1041–1057 (2020). [CrossRef]  

43. S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018). [CrossRef]  

44. L. Su, D. Vercruysse, J. Skarda, N. V. Sapra, J. A. Petykiewicz, and J. Vučković, “Nanophotonic inverse design with SPINS: software architecture and practical considerations,” Appl. Phys. Rev. 7, 011407 (2020). [CrossRef]  

45. W. Ma, F. Cheng, and Y. Liu, “Deep-learning-enabled on-demand design of chiral metamaterials,” ACS Nano 12, 6326–6334 (2018). [CrossRef]  

46. M. D. Huntington, L. J. Lauhon, and T. W. Odom, “Subwavelength lattice optics by evolutionary design,” Nano Lett. 14, 7195–7200 (2014). [CrossRef]  

47. Z. Lin, X. Liang, M. Lončar, S. G. Johnson, and A. W. Rodriguez, “Cavity-enhanced second-harmonic generation via nonlinear-overlap optimization,” Optica 3, 233–238 (2016). [CrossRef]  

48. J. C. Mak, C. Sideris, J. Jeong, A. Hajimiri, and J. K. Poon, “Binary particle swarm optimized 2 x 2 power splitters in a standard foundry silicon photonic platform,” Opt. Lett. 41, 3868–3871 (2016). [CrossRef]  

49. T. Nakanishi, Y. Nakata, Y. Urade, and K. Okimura, “Broadband operation of active terahertz quarter-wave plate achieved with vanadium-dioxide-based metasurface switchable by current injection,” Appl. Phys. Lett. 117, 091102 (2020). [CrossRef]  

50. D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photon. J. 8, 5500308 (2016). [CrossRef]  

51. D. Wang, L. Zhang, Y. Gu, M. Q. Mehmood, Y. Gong, A. Srivastava, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Switchable ultrathin quarter-wave plate in terahertz using active phase-change metasurface,” Sci. Rep. 5, 15020 (2015). [CrossRef]  

52. M. Manjappa, A. Solanki, A. Kumar, T. C. Sum, and R. Singh, “Solution-processed lead iodide for ultrafast all-optical switching of terahertz photonic devices,” Adv. Mater. 31, 1901455 (2019). [CrossRef]  

53. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008). [CrossRef]  

54. P. Tassin, L. Zhang, R. Zhao, A. Jain, T. Koschny, and C. M. Soukoulis, “Electromagnetically induced transparency and absorption in metamaterials: the radiating two-oscillator model and its experimental confirmation,” Phys. Rev. Lett. 109, 187401 (2012). [CrossRef]  

55. R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings of the Sixth International Symposium on Micro Machine and Human Science (MHS’95) (1995), pp. 39–43.

56. D.-C. Wang, R. Tang, Z. Feng, S. Sun, S. Xiao, and W. Tan, “Symmetry-assisted spectral line shapes manipulation in dielectric double-Fano metasurfaces,” Adv. Opt. Mater. 9, 2001874 (2021). [CrossRef]  

57. D.-C. Wang, S. Sun, Z. Feng, W. Tan, and C.-W. Qiu, “Multipolar-interference-assisted terahertz waveplates via all-dielectric metamaterials,” Appl. Phys. Lett. 113, 201103 (2018). [CrossRef]  

58. T. W. Hughes, M. Minkov, I. A. D. Williamson, and S. Fan, “Adjoint method and inverse design for nonlinear nanophotonic devices,” ACS Photon. 5, 4781–4787 (2018). [CrossRef]  

59. Z. Tao, J. You, J. Zhang, X. Zheng, H. Liu, and T. Jiang, “Optical circular dichroism engineering in chiral metamaterials utilizing a deep learning network,” Opt. Lett. 45, 1403–1406 (2020). [CrossRef]  

60. Z. Tao, J. Zhang, J. You, H. Hao, H. Ouyang, Q. Yan, S. Du, Z. Zhao, Q. Yang, X. Zheng, and T. Jiang, “Exploiting deep learning network in optical chirality tuning and manipulation of diffractive chiral metamaterials,” Nanophotonics 9, 2945–2956 (2020). [CrossRef]  

61. S. Du, J. You, J. Zhang, Z. Tao, H. Hao, Y. Tang, X. Zheng, and T. Jiang, “Expedited circular dichroism prediction and engineering in two-dimensional diffractive chiral metamaterials leveraging a powerful model-agnostic data enhancement algorithm,” Nanophotonics 10, 20200570 (2020). [CrossRef]  

62. M. ElKabbash, A. R. Rashed, B. Kucukoz, Q. Nguyen, A. Karatay, G. Yaglioglu, E. Ozbay, H. Caglayan, and G. Strangi, “Ultrafast transient optical loss dynamics in exciton–plasmon nano-assemblies,” Nanoscale 9, 6558–6566 (2017). [CrossRef]  

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2021 (1)

D.-C. Wang, R. Tang, Z. Feng, S. Sun, S. Xiao, and W. Tan, “Symmetry-assisted spectral line shapes manipulation in dielectric double-Fano metasurfaces,” Adv. Opt. Mater. 9, 2001874 (2021).
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2020 (12)

Z. Tao, J. You, J. Zhang, X. Zheng, H. Liu, and T. Jiang, “Optical circular dichroism engineering in chiral metamaterials utilizing a deep learning network,” Opt. Lett. 45, 1403–1406 (2020).
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Z. Tao, J. Zhang, J. You, H. Hao, H. Ouyang, Q. Yan, S. Du, Z. Zhao, Q. Yang, X. Zheng, and T. Jiang, “Exploiting deep learning network in optical chirality tuning and manipulation of diffractive chiral metamaterials,” Nanophotonics 9, 2945–2956 (2020).
[Crossref]

S. Du, J. You, J. Zhang, Z. Tao, H. Hao, Y. Tang, X. Zheng, and T. Jiang, “Expedited circular dichroism prediction and engineering in two-dimensional diffractive chiral metamaterials leveraging a powerful model-agnostic data enhancement algorithm,” Nanophotonics 10, 20200570 (2020).
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Y. Chen, Y. Hu, J. Zhao, Y. Deng, Z. Wang, X. Cheng, D. Lei, Y. Deng, and H. Duan, “Topology optimization-based inverse design of plasmonic nanodimer with maximum near-field enhancement,” Adv. Funct. Mater. 30, 2000642 (2020).
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S. So, T. Badloe, J. Noh, J. Rho, and J. Bravo-Abad, “Deep learning enabled inverse design in nanophotonics,” Nanophotonics 9, 1041–1057 (2020).
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L. Su, D. Vercruysse, J. Skarda, N. V. Sapra, J. A. Petykiewicz, and J. Vučković, “Nanophotonic inverse design with SPINS: software architecture and practical considerations,” Appl. Phys. Rev. 7, 011407 (2020).
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T. Nakanishi, Y. Nakata, Y. Urade, and K. Okimura, “Broadband operation of active terahertz quarter-wave plate achieved with vanadium-dioxide-based metasurface switchable by current injection,” Appl. Phys. Lett. 117, 091102 (2020).
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X. Zang, W. Xu, M. Gu, B. Yao, L. Chen, Y. Peng, J. Xie, A. V. Balakin, A. P. Shkurinov, Y. Zhu, and S. Zhuang, “Polarization-insensitive metalens with extended focal depth and longitudinal high-tolerance imaging,” Adv. Opt. Mater. 8, 1901342 (2020).
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Y. Hu, T. Jiang, J. Zhou, H. Hao, H. Sun, H. Ouyang, M. Tong, Y. Tang, H. Li, J. You, X. Zheng, Z. Xu, and X. Cheng, “Ultrafast terahertz transmission/group delay switching in photoactive WSe2-functionalized metaphotonic devices,” Nano Energy 68, 104280 (2020).
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A. Kumar, A. Solanki, M. Manjappa, S. Ramesh, Y. K. Srivastava, P. Agarwal, T. C. Sum, and R. Singh, “Excitons in 2D perovskites for ultrafast terahertz photonic devices,” Sci. Adv. 6, eaax8821 (2020).
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Y. Hu, J. You, M. Tong, X. Zheng, Z. Xu, X. Cheng, and T. Jiang, “Pump-color selective control of ultrafast all-optical switching dynamics in metaphotonic devices,” Adv. Sci. 7, 2000799 (2020).
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M. Minkov, I. A. D. Williamson, L. C. Andreani, D. Gerace, B. Lou, A. Y. Song, T. W. Hughes, and S. Fan, “inverse design of photonic crystals through automatic differentiation,” ACS Photon. 7, 1729–1741 (2020).
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2019 (7)

S. Han, L. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. G. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-dielectric active terahertz photonics driven by bound states in the continuum,” Adv. Mater. 31, 1901921 (2019).
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L. Cong and R. Singh, “Symmetry-protected dual bound states in the continuum in metamaterials,” Adv. Opt. Mater. 7, 1900383 (2019).
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S. Zhong, “Progress in terahertz nondestructive testing: a review,” Front. Mech. Eng. 14, 273–281 (2019).
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Y. Hu, T. Jiang, J. Zhou, H. Hao, H. Sun, H. Ouyang, M. Tong, Y. Tang, H. Li, J. You, X. Zheng, Z. Xu, and X. Cheng, “Ultrafast terahertz frequency and phase tuning by all-optical molecularization of metasurfaces,” Adv. Opt. Mater. 7, 1970084 (2019).
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C. C. Chang, Z. Zhao, D. Li, A. J. Taylor, S. Fan, and H. T. Chen, “Broadband linear-to-circular polarization conversion enabled by birefringent off-resonance reflective metasurfaces,” Phys. Rev. Lett. 123, 237401 (2019).
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M. Manjappa, A. Solanki, A. Kumar, T. C. Sum, and R. Singh, “Solution-processed lead iodide for ultrafast all-optical switching of terahertz photonic devices,” Adv. Mater. 31, 1901455 (2019).
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W. Ma, F. Cheng, Y. Xu, Q. Wen, and Y. Liu, “Probabilistic representation and inverse design of metamaterials based on a deep generative model with semi-supervised learning strategy,” Adv. Mater. 31, 1901111 (2019).
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2018 (14)

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
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L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2018).
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W. Ma, F. Cheng, and Y. Liu, “Deep-learning-enabled on-demand design of chiral metamaterials,” ACS Nano 12, 6326–6334 (2018).
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D.-C. Wang, S. Sun, Z. Feng, W. Tan, and C.-W. Qiu, “Multipolar-interference-assisted terahertz waveplates via all-dielectric metamaterials,” Appl. Phys. Lett. 113, 201103 (2018).
[Crossref]

T. W. Hughes, M. Minkov, I. A. D. Williamson, and S. Fan, “Adjoint method and inverse design for nonlinear nanophotonic devices,” ACS Photon. 5, 4781–4787 (2018).
[Crossref]

X. C. Q. Yang, Q. Xu, C. Tian, Y. Xu, L. Cong, X. Zhang, Y. Li, C. Zhang, X. Zhang, J. Han, and W. Zhang, “Broadband terahertz rotator with an all-dielectric metasurface,” Photon. Res. 6, 1056–1061 (2018).
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Z. Chen, X. Chen, L. Tao, K. Chen, M. Long, X. Liu, K. Yan, R. I. Stantchev, E. Pickwell-MacPherson, and J. B. Xu, “Graphene controlled Brewster angle device for ultra broadband terahertz modulation,” Nat. Commun. 9, 4909 (2018).
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L. Cong, Y. K. Srivastava, H. Zhang, X. Zhang, J. Han, and R. Singh, “All-optical active THz metasurfaces for ultrafast polarization switching and dynamic beam splitting,” Light Sci. Appl. 7, 28 (2018).
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S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6, 1800570 (2018).
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S. Xiao, T. Wang, T. Liu, X. Yan, Z. Li, and C. Xu, “Active modulation of electromagnetically induced transparency analogue in terahertz hybrid metal-graphene metamaterials,” Carbon 126, 271–278 (2018).
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X. Chen, S. Ghosh, Q. Xu, C. Ouyang, Y. Li, X. Zhang, Z. Tian, J. Gu, L. Liu, A. K. Azad, J. Han, and W. Zhang, “Active control of polarization-dependent near-field coupling in hybrid metasurfaces,” Appl. Phys. Lett. 113, 061111 (2018).
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T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photon. 5, 1800–1807 (2018).
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C. Sitawarin, W. Jin, Z. Lin, and A. W. Rodriguez, “Inverse-designed photonic fibers and metasurfaces for nonlinear frequency conversion [invited],” Photon. Res. 6, B82–B89 (2018).
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W. X. Lim, M. Manjappa, Y. K. Srivastava, L. Cong, A. Kumar, K. F. MacDonald, and R. Singh, “Ultrafast all-optical switching of germanium-based flexible metaphotonic devices,” Adv. Mater. 30, 1705331 (2018).
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2017 (5)

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17, 3752–3757 (2017).
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M. Liu, Z. Tian, X. Zhang, J. Gu, C. Ouyang, J. Han, and W. Zhang, “Tailoring the plasmon-induced transparency resonances in terahertz metamaterials,” Opt. Express 25, 19844–19855 (2017).
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M. Manjappa, Y. K. Srivastava, A. Solanki, A. Kumar, T. C. Sum, and R. Singh, “Hybrid lead halide perovskites for ultrasensitive photoactive switching in terahertz metamaterial devices,” Adv. Mater. 29, 1605881 (2017).
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Y. K. Srivastava, A. Chaturvedi, M. Manjappa, A. Kumar, G. Dayal, C. Kloc, and R. Singh, “MoS2 for ultrafast all-optical switching and modulation of THz Fano metaphotonic devices,” Adv. Opt. Mater. 5, 1700762 (2017).
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M. ElKabbash, A. R. Rashed, B. Kucukoz, Q. Nguyen, A. Karatay, G. Yaglioglu, E. Ozbay, H. Caglayan, and G. Strangi, “Ultrafast transient optical loss dynamics in exciton–plasmon nano-assemblies,” Nanoscale 9, 6558–6566 (2017).
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2016 (5)

Z. Lin, X. Liang, M. Lončar, S. G. Johnson, and A. W. Rodriguez, “Cavity-enhanced second-harmonic generation via nonlinear-overlap optimization,” Optica 3, 233–238 (2016).
[Crossref]

J. C. Mak, C. Sideris, J. Jeong, A. Hajimiri, and J. K. Poon, “Binary particle swarm optimized 2 x 2 power splitters in a standard foundry silicon photonic platform,” Opt. Lett. 41, 3868–3871 (2016).
[Crossref]

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photon. J. 8, 5500308 (2016).
[Crossref]

H. J. Joyce, J. L. Boland, C. L. Davies, S. A. Baig, and M. B. Johnston, “A review of the electrical properties of semiconductor nanowires: insights gained from terahertz conductivity spectroscopy,” Semicond. Sci. Technol. 31, 103003 (2016).
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T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nat. Photonics 10, 371–379 (2016).
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2015 (4)

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, Y. Zhang, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5, 041027 (2015).
[Crossref]

D. Wang, L. Zhang, Y. Gu, M. Q. Mehmood, Y. Gong, A. Srivastava, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Switchable ultrathin quarter-wave plate in terahertz using active phase-change metasurface,” Sci. Rep. 5, 15020 (2015).
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B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378–382 (2015).
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A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
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2014 (2)

B. Shen, P. Wang, R. Polson, and R. Menon, “Ultra-high-efficiency metamaterial polarizer,” Optica 1, 356–360 (2014).
[Crossref]

M. D. Huntington, L. J. Lauhon, and T. W. Odom, “Subwavelength lattice optics by evolutionary design,” Nano Lett. 14, 7195–7200 (2014).
[Crossref]

2013 (2)

S. Y. Y. Zhang, A. E.-J. Lim, G.-Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express 21, 1310–1316 (2013).
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S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7, 977–981 (2013).
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2012 (3)

Y. Kawano, “Terahertz sensing and imaging based on nanostructured semiconductors and carbon materials,” Laser Photon. Rev. 6, 246–257 (2012).
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J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
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P. Tassin, L. Zhang, R. Zhao, A. Jain, T. Koschny, and C. M. Soukoulis, “Electromagnetically induced transparency and absorption in metamaterials: the radiating two-oscillator model and its experimental confirmation,” Phys. Rev. Lett. 109, 187401 (2012).
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2011 (2)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
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P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging–Modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
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2008 (2)

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).
[Crossref]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
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2007 (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
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2006 (1)

H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444, 597–600 (2006).
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Achanta, V. G.

S. Han, L. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. G. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-dielectric active terahertz photonics driven by bound states in the continuum,” Adv. Mater. 31, 1901921 (2019).
[Crossref]

Agarwal, P.

A. Kumar, A. Solanki, M. Manjappa, S. Ramesh, Y. K. Srivastava, P. Agarwal, T. C. Sum, and R. Singh, “Excitons in 2D perovskites for ultrafast terahertz photonic devices,” Sci. Adv. 6, eaax8821 (2020).
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Almond, N. W.

S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6, 1800570 (2018).
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Ambacher, O.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7, 977–981 (2013).
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An, Z.

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, Y. Zhang, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5, 041027 (2015).
[Crossref]

Andreani, L. C.

M. Minkov, I. A. D. Williamson, L. C. Andreani, D. Gerace, B. Lou, A. Y. Song, T. W. Hughes, and S. Fan, “inverse design of photonic crystals through automatic differentiation,” ACS Photon. 7, 1729–1741 (2020).
[Crossref]

Antes, J.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7, 977–981 (2013).
[Crossref]

Averitt, R. D.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).
[Crossref]

H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444, 597–600 (2006).
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Azad, A. K.

X. Chen, S. Ghosh, Q. Xu, C. Ouyang, Y. Li, X. Zhang, Z. Tian, J. Gu, L. Liu, A. K. Azad, J. Han, and W. Zhang, “Active control of polarization-dependent near-field coupling in hybrid metasurfaces,” Appl. Phys. Lett. 113, 061111 (2018).
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J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
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H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).
[Crossref]

Babinec, T. M.

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
[Crossref]

Badloe, T.

S. So, T. Badloe, J. Noh, J. Rho, and J. Bravo-Abad, “Deep learning enabled inverse design in nanophotonics,” Nanophotonics 9, 1041–1057 (2020).
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Baehr-Jones, T.

Baig, S. A.

H. J. Joyce, J. L. Boland, C. L. Davies, S. A. Baig, and M. B. Johnston, “A review of the electrical properties of semiconductor nanowires: insights gained from terahertz conductivity spectroscopy,” Semicond. Sci. Technol. 31, 103003 (2016).
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Balakin, A. V.

X. Zang, W. Xu, M. Gu, B. Yao, L. Chen, Y. Peng, J. Xie, A. V. Balakin, A. P. Shkurinov, Y. Zhu, and S. Zhuang, “Polarization-insensitive metalens with extended focal depth and longitudinal high-tolerance imaging,” Adv. Opt. Mater. 8, 1901342 (2020).
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Beere, H. E.

S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6, 1800570 (2018).
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Boes, F.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7, 977–981 (2013).
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Boland, J. L.

H. J. Joyce, J. L. Boland, C. L. Davies, S. A. Baig, and M. B. Johnston, “A review of the electrical properties of semiconductor nanowires: insights gained from terahertz conductivity spectroscopy,” Semicond. Sci. Technol. 31, 103003 (2016).
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Figures (9)

Fig. 1.
Fig. 1. Structure chart of ultrafast all-optical terahertz modulation. (a) Schematic illustration of hybrid structure combining Ge film with inverse-designed metasurface at the pump light of 800 nm for terahertz modulation. (b) Processed inverse-designed metasurface structure without Ge film covering on the face. (c) Processed inverse-designed metasurface structure covering 200 nm thick Ge film on the face. The scale bar in the pictures is 50 μm.
Fig. 2.
Fig. 2. Sketched scheme of inverse design of ultrafast terahertz modulator based on EIT effect utilizing the PSO algorithm and finite-difference time-domain method. (a) General design cycle of PSO algorithm. At this optimization procedure, we set the target function by using coupled harmonic oscillator model. The number of particles is set as 10, and the number of iterations is 100. (b) The fitness function tends to convergence with the increasing iterations. After 60 iterations, the fitness is convergent, and the result is what we want. (c) The transmittance of the terahertz wave at the last design is described in the red line, and the black line shows the target transmittance. All terahertz transmittance is obtained through finite-difference time-domain method.
Fig. 3.
Fig. 3. Experimentally light-activated Ge-controlled terahertz EIT modulation for varying pump fluences and simulation results of EIT resonance modulation with different conductivities of Ge film. (a) Experimental modulation of terahertz EIT amplitudes for various pump fluences ranging from 0 to 2.2mJ/cm2. (b) The experimentally normalized resonance amplitude, defined by the difference between transmission at dip and transmission at peak, is changed following various pump fluences. (c) Simulated modulation of terahertz EIT amplitudes for different conductivities of Ge film ranging from 10 to 1000 S/m. (d) The simulated normalized resonance amplitude is changed following different conductivities.
Fig. 4.
Fig. 4. Temporal evolution dynamics of ultrafast terahertz modulation behaviors under the pump fluence 2.2mJ/cm2. (a) Contour map of experimental terahertz transmission as a function of pump-probe time delay. (b) Top: Experimental terahertz transmission spectra at different pump-probe time delay from 5 to 0 ps. The state of EIT resonance switches from on to off. Bottom: Experimental terahertz transmission spectra at different pump probe time delay from 1 to 5 ps. The state of EIT resonance switches from off to on.
Fig. 5.
Fig. 5. Simulated near-field distributions as a function of Ge film conductivity ranging from 10 to 1000 S/m. E-field magnitudes are obtained at the position 1 μm above the metasurface.
Fig. 6.
Fig. 6. (a) Schematic illustration of one-unit inverse-designed metasurface combined with 200 nm thick Ge film (the red part), including a cut wire and numerous crosses around it. The material of inverse-designed metasurface is gold with thickness of 200 nm (the golden part). The whole unit structure is mirror symmetric. The periods of one-unit metasurface are px=110μm in the x direction and py=180μm in the y direction. (b) Sketch of a half of one-unit metasurface. Each of cross consists of two orthogonal rectangles with length cy=5μm and width cx=3μm. The parameters of the cut wire in this sketch are w=10μm and L=63μm.
Fig. 7.
Fig. 7. Experimentally measured group delay spectra of hybrid structure combining a reverse-designed metasurface with Ge film as a function of pump fluence ranging from 0 to 2.2mJ/cm2.
Fig. 8.
Fig. 8. Negative differential transmission spectra of pure 200 nm thick Ge film are experimentally measured at the maximal amplitude of terahertz time-domain pulse signal as a function of optical-pump terahertz time delay. Experimental data points are fitted with equation in the main text and the delay time constants τ of pure Ge film at various pump fluences are extracted. The delay time constant is found to increase with the pump fluence increasing.
Fig. 9.
Fig. 9. Modulations of the EIT resonance amplitude as a function of pump-probe time delay evaluated from the transmission spectra of Fig. 4 in the main text.

Equations (4)

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p¨(t)ωR2+ΓR1p˙(t)ωR+p(t)=f(t)κq(t)q¨(t)ωD2+ΓR2q˙(t)ωD+q(t)=κp(t),
σs(ω)=iωε0χs(static)Dd(ω)Dd(ω)Dr(ω)κ2,
E˜(ω)=22+ζσs(ω),
ΔTT(t)=e[tt0IRF/(2ln2)]2(A0+A1ett0τ),