Photo-Excited Silicon-Based Spatial Terahertz Modulators 

: The increasing development of terahertz (THz) technology has led to various potential applications in THz imaging, spectroscopy and communications. These devices capable of actively manipulating the amplitude, phase and frequency of THz waves are thus gaining numerous interests. All-optical silicon-based spatial terahertz modulators (STMs), as a simple, cost-effective, and reconfigurable technique, are standing the focus of research. Beginning with a fundamental concept of THz radiation, this paper systematically summarized the modulation mechanism and theoretical model for this kind of STM, reviewed the recent advancements in THz functional devices implemented by this optical method and yet, discussed the performance-improved measures with an emphasis on the reflection reduction. Despite that, there has been considerable progress in realizing high-performance STMs, and novel design is urgent to realize higher modulation rate and more functionality.


Introduction
Terahertz (THz) radiation situated between microwave and infrared regions of electromagnetic spectrum has become increasingly attractive for its extraordinary properties. Without a standard definition, THz band is commonly referred as the range of 30 μm to 3 mm in wavelength and 0.1 THz to 10 THz in frequency [1]. This special frequency spectrum motivates THz technology preferably meritorious in numerous sectors spanning from industry, defense, medical to scientific research [2,3]. Representatives are THz spectroscopy and sensing, THz imaging and near-field microscopy, and THz radar and communications [4,5]. With the significant advancement of THz source and detection techniques, physical properties of materials, for instance, carrier density calculation, molecular identification and internal structure analysis have been investigated employing THz spectroscopy and sensing [6,7]. THz image and near-field microscopy is mainly used for non-destructive inspection, medical diagnose as well as security detection. As for THz radar and communications, it is one of the most promising applications especially in the fields of interplanetary communications, short-range atmosphere communications [8] and wireless LAN communications. Such widespread deployment of THz technology applications has pushed THz research into the center stage. However, its full exploitation remains technically challenging due to the typical natural materials' poor response to THz radiation. The imperious demand for active devices which could manipulate THz radiation efficiently is driving THz modulation a hotspot in THz research. As a fundamental functionality, THz modulation is highly requested to satisfy the demand for fast, high-efficiency, low cost, easy fabrication and integration indeed. Although numerous efforts have been reported with electrical gating or thermal controlling scheme [9,10], optically controlled THz modulators are increasingly important and in demand, attributing to the strict conditions of material and working ambient in thermal-controlled modulators and the relatively high requirements of materials and processing in electric-controlled ones [9,11].
Generally, semiconductor materials are employed to fabricate THz optical modulators. When optical illumination is radiated to a semiconductor, free carriers could be generated once its energy (  ) is larger than that of bandgap (E g ) of semiconductor, i.e., g E   . The photo-induced carriers (electrons or holes) density (N) is proportionally related to the fluence (energy per unit area) of incident photons and commonly described by Drude model, where ()  is the complex conductivity and ε 0 , τ, m and e are the permittivity of free space, scattering rate, electron mass and electron charge, respectively. ω p represents the plasma frequency, , a characteristic frequency below which the material acts "metallic" while above which it is transparent to optical radiation. This phenomenon indicates that semiconductors are suitable for all-optical modulation of THz waves: A high-reflectance region is temporarily formed onto the semiconductor when the laser is incident and thus the co-projected THz radiation on this area is modulated.
Regular semiconductor materials, such as silicon (Si), gallium arsenide (GaAs) and germanium (Ge), have been resoundingly put to use in optical-controlled THz waves modulators [12][13][14][15][16][17][18][19]. Si-based modulators are the priority of implementing THz spatial modulation and have been most investigated. In this paper, we mainly concentrate on the researches of THz spatial modulation by optically excited electron plasma in photo-active Si substrate. Starting with the generating mechanism of photo-carriers, THz wave manipulation by laser illuminating Si wafer is theoretically analyzed in Section 2. Section 3 introduces such modulation scheme based experimental demonstration on functional elements. Corresponding performance enhancement measures are exhibited in Section 4. Finally the outlook is concluded in Section 5. 3

Modulation Mechanism by Photo-Exciting Si
Si, existing in the fourth group in periodic table and possessing a wide bandgap (1.12 eV), is one of the best candidates to manage THz radiation. Owing to its indirect bandgap characteristic, laser with a wavelength less than 1107 nm can be capable to produce photo-carriers, efficiently controlling free carriers density. Consequently pronounced changes of complex refractive index and hence the optical properties of Si may occur, which makes the transmissivity and reflectivity in the THz range vary. As early as 1983, Salzmann et al. utilized 337 nm pulsed laser to irradiate high-resistivity Si wafer and realized subnanosecond level of amplitude switching of 119 μm radiation in both transmission and reflection [20]. Fekete et al. theoretically studied and experimentally demonstrated the validity of light-controllable THz reflectivity of high-resistivity Si wafers under both 405 nm and 810 nm illumination [21]. Whereafter, three different wavelengths of 1064 nm, 532 nm and 355 nm were demonstrated in 214 μm radiation modulation, the transmitted waves intensity decreased while that of reflected increased when laser-on [22]. The observed change of THz waves transmissivity and reflectivity going through a pulse laser excited high-resistivity Si wafer is caused by the temporal and spatial perturbation of free carriers density. The dynamic behavior of free carriers density yields the dynamics of the complex dielectric constant in the Si wafer. The THz optical properties is thus regulated. Such modulation scheme could be theoretically characterized by using Drude's theory and concluded in three steps: free carriers density formulation, complex refractive index formulation and transmission/reflection formulation.

A. Free carriers density formulation
When an effective illumination (λ≤1107 nm) is interacted with Si, the dynamic behavior of free carriers include generation, recombination and diffusion, which could be described by the continuity equation [23][24][25]: in which x denotes the spatial coordinate parallel to the Si surface while z refers to the direction of the propagating THz beam, that is, perpendicular to Si. The carriers generation rate G(x,z,t) and recombination rate R(x,z,t) are both time-and space-dependent. Since the absorption depth d is much smaller than the dimension of the homogeneously irradiated Si area, it is justified to make a one-dimensional approximation for eq. (2). Now therefore the carriers generation rate G(x,z,t) can be written as which signifies that the free electron-hole carriers are generated at a depth z through band-to-band absorption by a single pulse of intensity I(t) and photon energy  incident on silicon at an angle with a reflectivity R and an absorption coefficient  .
As for the recombination, there mainly exist three components consisting of Auger recombination, radiative recombination and that via lattice imperfections and impurities which are characterized by 3  , 2  and 1  , respectively. The recombination term R(x,z,t) is thus given as   32 where 1  and 2  are constant for a specific Si sample, while   3 ( , ), N z t T  is a function of the induced carriers density ( , ) N z t and the carrier-lattice temperature T [26].
The J  term delineates the coupled diffusion process of a photo-generated electron-hole plasma and the associated lattice-carrier temperature and J is termed as the electron-hole current density, written as Assuming a surface recombination occurred on the top surface of Si, the carriers density ( , ) N z t satisfies the boundary condition: where h is the Si sample thickness, N i is the intrinsic equilibrium carriers density for Si, and S denotes the surface recombination rate depending on the surface condition.
Because the material parameters such as J and 3  are lattice-temperature-dependent, heat generation should be considered which is mainly induced by two processes. One is in the laser absorption process, where the excess energy is lost by the electron-hole pairs instantaneously and assumed to be . The other occurs via the recombination process where the where C is the specific heat and D L is the lattice thermal diffusivity, which meets the boundary condition shown in eq. (8) for the reason that little heat is lost from the surface on a nanosecond time scale while the heat is produced at the surface once 0 in which K T is the thermal conductivity and D L =K T /C [28]. For a given laser pulse, for example the sine-square-shaped laser used in [23], the carriers density is thus numerically solved.

B. Complex refractive index formulation
According to Fresnel formula, the transmission of normally incident THz waves through the photo-excited Si wafer is determined by the complex refractive index with 0  denoting the permittivity of vacuum.

C. Transmission/reflection formulation
Obviously, the complex refractive index of Si wafer under illumination is varied with z. The transmission/reflection of THz waves through such substrate can be calculated by Fresnel transfer matrix, which treats the inhomogeneous medium as a multilayer structure composed of m homogenous thin parallel layers with an identical thickness of z  . The relevant complex refractive index j n and characteristic matrix j M for the jth layer are as eqs. (14) and (15) 7 The reflection and transmission coefficients of r and t are then derived as Accordingly, the reflectivity and transmittivity are given by In brief, the effective photoexcitation on Si leads to a rapid change of free carriers density as well as refractive index. The resultant refractive index variation contributes to the pronounced changes of THz transmission and reflection. The above theory not only provides a guideline to design or optimize a modulation system beforehand for desirable modulation performance including transmission, modulation depth and lateral resolution [25], but also can be used to analyze carriers dynamics in photo-excited Si through the measured THz time-domain spectroscopy (THZ-TDS) [29].

Functional Implementation in THz Manipulation
Although pulsed illumination was applied in the beginning, it is not required but alternative on account of longer carrier lifetime of Si which could allow high-concentration excess carries for THz radiation modulation under low-power continuous wave (CW) excitation, making such optically-manipulated scheme more cost-effective and pow-efficient [31]. Examples are experimental demonstration in Ref. [32] and theoretical simulation in Ref. [25]. Alius and Dodel applied 1060 nm CW laser to high-resistivity Si wafer, realizing a maximum intensity modulation of 70%. Both of phase-and frequency-modulation were achieved when two such illumination sources were in use [32]. Kannegulla et al. directly deduced the analytical express for carriers density when a CW laser is applied, and also theoretically discussed the physical parameters' influence on the THz modulation, including the substrate thickness, THz frequency, optical illumination width and wavelength [25]. This kind of scheme could realize high-performance (e.g., high modulation efficiency, large modulation depth, fast modulation speed) THz modulation in a simple, cost-effective manner. Not merely does such optical modulation show unique superiority encompassing structural simplicity, economic and flexible fabrication, and wide-range frequency modulation over these metamaterials-based counterparts. But also, various extraordinary manipulation of THz radiation, such as amplitude, frequency, phase, and polarization has been imple- 8 mented when cooperated with different photopatterned illumination, so did THz beam steering, focusing, filtering, imaging, encoding information on THz waves, etc.
Early to 1996, Brand examined the diffraction by different photo-induced plasma patterns, for the first time, involving several varieties of slits and Fresnel zone plates [33], where, however, most of the beam proceeded in the straight-through direction, only a small fraction was diffracted to the sides. Then, he found most of the reflected radiation would be diffracted into the adjacent interference maxima when a pre-designed cutout mask, which consisted of periodic clear and opaque regions with equal width, was aligned between the optical illumination and Si wafer [34]. This mask projected a grating pattern and thus generated a diffraction grating onto Si wafer. The incident THz radiation was interacted with photo-induced grating and diffracted following Fraunhofer diffraction principle. Thus the direction of reflected beam could be tuned by varying the period of the projected pattern. Wire gratings, as a textbook example, have been widely investigated.
Then derivative grating patterns were adopted by Okada et al. for frequency tuning [35,36]. Combining with a liquid-crystal spatial light modulator (SLM), one-dimensional corrugations (wire-grating patterns), periodically constituted by a broader transparent region and a narrower photoinduced wire, were formed at the surface of a silicon prism by a prepatterned 800 nm pulse [35]. They confirmed that the photopatterned structure possessed metallic properties and a characteristic minimum at 0.53 THz was observed due to the excitation of surface waves. A line grating patterned at the same surface with uniform fringes also come to an identical consequence [36]. A sharp minimum at lower frequency was observed and the frequency of the characteristic sharp minimum red-shifted with increased period of grating patterns, as shown in Figure 1. A same trend was also obtained for corresponding 2-dimensional square array structure, and each square occupied half a period, once the period kept unchanged. Such uniform periodical rectangular patterns enable real-time, wide-range frequency modulation of THz radiation just by tuning the photogenerated structural periodicity without any microfabrication process. LG. (b) Dependence of spectral minimum frequency on structural periodicity in photo-designed THz devices. [36] 9 In [37], Cheng and Liu systematically studied several representative types of transmitted THz waves modulations under various photo-excited patterns. These patterns were generated and programmatically reconfigured by a digital light processing (DLP) projector, by which the patterns could be modified and reconfigured optically in a high speed. The experimental setup and modulation types are depicted in Figure 2. With the gray scale varying from bottom to top, the photo-excitation intensity increased and the transmitted THz power reduced, THz intensity modulation was implemented for the illumination patterns in the left column. As for the wire-grid patterns, the reconfigurable THz quasi-optical component shown in Figure 2 (a) acted as a photo-induced polarizer. The polarization angle was tuned by generating rotated images using DLP. Such THz waveform polarizer with subcycle switch-on time was also demonstrated in [38], in which the wire grating was produced by using a shadow mask to shape transverse beam profile of irradiating laser. In addition, chiral patterned photoexcitation, gammadion patterns for instance presented in Figure 3, was also employed to control THz polarization [39]. The resultant polarization rotation was sensitive to the patterns' handedness, with its angle increased with the excitation density and spectra shifted to higher frequencies as the periodicities decreased. These two patterns provide alternative avenues to rotate THz polarization, suggesting that we can design the spectra of the polarization rotation.  Fig. 3 Schematic of the experimental setup by using 2-dimensional reflection type phase modulator to produce chiral patterned photocarrier distribution. [39] Except for using a prepatterned radiation, the interference of two titled plane waves is also a valid approach to produce grating pattern on the Si wafer. Rizza et al. theoretically investigated the THz dielectric response through a Si slab with such interference pattern [40]. The numerically evaluated results, indicated the interference-induced optical response, could be tailored by varying the photo-generated grating and it ranged from birefringent to hyperbolic to anisotropic negative dielectric while free of microfabrication. But unfortunately, so far there are no reports on the experimental achievements.
In [36], Okada et al. verified the feasibility of frequency modulation in reflection by the virtual grating structure formed on Si wafer. And comparable conclusion was made by Busch et al. that an increase in line densities, i.e. a decrease in grating period, shifted the peak of penetrated THz signal towards higher frequencies [41]. They also presented a new finding that such technique allowed for an optically controlled beam steering meanwhile. But the tunability of steering is limited compared with those carried out by photo-induced Fresnel zone plate (PI-FZP) [42][43][44]. Under the wave-front spatial encoding of PI-FZP, both THz beam steering and forming were executed simultaneously. To demonstrate the joint functionalities of beam steering and forming, representative PI-FZP patterns were mapped as shown in Figure 4 (a). The relevant 2-dimensional THz beam profiles in Figure 4 (b)-(e) were measured by these situation sequentially that without PI-FZP, with a PI-FZP of 70 nm focal length and 0°, with a PI-FZP of 50 nm focal length and 0° and with a PI-FZP of 50 mm focal length and 6° steering along 135° direction. Evidently, the use of programmable PI-FZP allowed for progressively more focused beams and also offset from the center as expected but no need for any circuit fabrication or patterning on Si wafer as well as modifying the physical arrangement of the experimental setup. Such dynamically adjustable THz beam steering and forming make it more practical and in demand in adaptive THz wireless communications. Another strategy of THz wave-front encoding is the utilization of spatial THz modulator (STM) to produce special THz beam based on the computer generated holography [45], whose prototype is exhibited in Figure 5. The required pattern of photo-generated carriers was produced via using 11 a modulated control beam by a conventional SLM to illuminate the surface of a Si wafer. Thus the governed transmission of the THz beam by the density of carriers led to independent modulation of transparency at different positions of the Si wafer. Projected a THz computer generated hologram on the Si wafer, a THz vortex beam with a special phase and amplitude distribution was generated. Such route to encode information in a wave front has paved a new avenue for THz imaging, THz information processing and THz communications. Except for these functional elements listed above, tunable THz filters were also designed by an optically gating Si wafer combined with two blazed gratings [46]. As shown in Figure 6 (a), THz radiation diffracting from the blazed grating was focused depending on its angle of incidence at different spatial positions, resulting in spectrally resolved, spatially separated focus points. The light induced absorption in this illuminated spot led to spectral notches. For example, frequency-filtering at 420 GHz and 840 GHz is shown in Figure 6 (b), where two accessible spectral regions are corresponding to the first and second orders of grating diffraction. In combination with such spatially separated spectrum at the intermediate focus, arbitrary frequencies of THz radiation could be blocked with a customized diffraction grating on the basis of the required filter parameters.

Performance-Enhanced Photo-Excited Si-Based THz Spatial Modulators
Since this kind of photo-excited Si-based THz spatial modulators has such numerous potential applications, they have been investigated intensively. However devices to control the THz radiation are still rather scarce and generally inefficient. The photoconductivity of standard Si samples tends to saturate, resulting in limited efficiency of most Si-based photomodulators. This necessitates high-power laser sources to achieve significant modulation, which limits the applications of THz systems outside of research laboratories. As such, in recent years, a myriad of coatings such as organic overlayers [47][48][49], phase-transition materials [50], 2D materials [51][52][53] and surface-passivation films [54] have been deposited to boost photomodulation by enhancing the light matter interaction. Despite significant improvements have been realized in THz modulation efficiency, the inherent bottlenecks of Si-based photoconductive materials themselves are still being encountered, for example, the high reflection for both the excitation laser and THz radiation at the air-Si interface. Due to the abrupt transition in the refractive index between the surface of air and bare-Si, the resultant reflectivity is as high as 35%~55% across a broad wavelength range of 400~1000 nm and ~40% for THz waves. Such high reflectivity not only cripples the modulation efficiency, but also introduces disturbance or noise into the quasi-optical THz system, which greatly limits the achievable tenability and versatility of Si-based STMs. To overcome such deficiencies, our group proposed two different all-optical spatial STMs based on nano-or micro-structuring the Si surface [55,56].
In our first design, silicon nanotip (SiNT) array was utilized as antireflection layers for THz wave for the first time [55]. By adopting a silver-assisted chemical etch, numerous tapered nan- 13 owires were fabricated on the Si wafer. These nanowires have a diameter ranging from 200 nm and 350 nm. And more importantly they are slightly bundled, approximately perpendicular to the surface, which leaves parts of Si substrate expose to air. The typical SEM image and the schematic of this SiNT-based STM and are shown in Figure 7 (a) and (c), respectively. With such a nano-textured surface, significantly increased transmittance as high as 90% of THz wave was observed and this improvement is strongly dependent with length of nanowires, as shown in Figure 7 (b). Consequently a low-loss and spectrally broadband optical-driven THz modulator was implemented. Moreover, this SiNT array also avails to light harvesting, contributing to a 2~3 times larger modulation efficiency compared with these bare-Si ones. However this performance-enhanced SiNT-based STM requires a rigorous and subtle balance between the tip shape, wire length and filling ratio but it can only realize a limited improved modulation depth (MD) of 75%. To find a more practicable scheme, an easier anisotropic wet chemical etching technique was adopted [56]. By utilizing the anisotropic response of Si to KOH, pyramid-resembled microstructures were produced on the Si surface, as the SEM image shows in Figure 8 (a). These pyramids are all in a diverse size roughly ranging from 2 μm to 10 μm, randomly distributed, closely compacted and even partially embedded with each other. This special structure lead to a 1.6-times expanded active area for spatial THz modulation as well as 1.4~2-times enhanced light harvesting. Although no THz transmission enhancement was observed, this micropyramid array (MPA) based STM achieved a remarkable increase in modula- 14 tion efficiency, as large as 93.8% under 638 nm laser with a low power density of 1 W/cm 2 . Furthermore, comparing with bare-Si, this Si-MPA based STM exhibited a reversal relation to laser wavelength, where the modulation efficiency increased with the decrease of illumination wavelength. This different wavelength dependence renders the Si-MPA a superior candidate to constructing a fast THz imager with digital light processing projector (DLP). As a proof of concept, precise THz beam mapping and fast real-time imaging with 225 pixels were demonstrated [56]. Very recently, we further realized an enhanced photomodulation efficiency under low illumination density by microtexturing and passivating the Si surface, where as high as 91% MD was obtained at only 0.53 W/cm 2 photodoping [57]. It is demonstrated that this enhancement in efficiency is mainly contributed by the effect of passivation. When a surface is well passivated, the surface recombination of photo-induced carriers is decreased, carrier lifetime is lengthened, and the carrier density in Si is increased, thereby the efficiency is improved. This result also gives a more general explanation to the enhancement by cooperating with various heterogeneous films [47][48][49][50][51][52][53], all of which are possibly ascribed to surface passivation including chemical passivation and field-effect passivation. Moreover, the periodically aligned pyramids formed by microstructuring resembled as a mesa array. This array could significantly suppress the lateral diffusion induced by longer diffusion and break the trade-off between resolution and MD.
All of these designs demonstrate that nano-or micro-textured surfaces of Si wafers benefit to decrease the reflection of THz radiation and enhance the light harvesting. But to be unsatisfied, none of the proposed schemes can switch on/off THz radiation and simultaneously increase the transmission of THz radiation. Seeking a new design to achieve these two goals in a simple way will be one of focuses for the future research.

Outlook
Even though considerable progress has been realized in high-performance STMs, they mainly concentrate on enhancing the modulation efficiency. While modulation speed, as another critical factor in THz imaging and communications systems, is merely investigated for all-optical Si-based STMs. In such optical pumping configurations, the concentration of photo-induced carriers increases with increasing pump power up until saturation, where the carrier recombination rate becomes the determining factor for modulation speed. However, the modulation depth mainly depends on the accumulation of carriers while the modulation speed is determined by the carrier lifetime. Thus there must be a trade-off. Designing a new high-speed and high efficient structure for THz modulation is in an urgent need for actual scenarios. Moreover, the current state-of-the-art THz functional devices are mainly on the basis of bare Si substrate. Its inherent deficiencies include high reflection, limited efficiency and speed render, and its versatility and multifunctionality remain underexploited. Further research with performance-enhanced STMs is also of significance, especially cooperated with other advanced technologies such as coding metasurface.

Funding
Science Challenge Project (TZ2018003); National Natural Science Foundation of China (61831012); International Science and Technology Cooperation Programme (2015DFR50870); Sichuan Province Science and Technology Support Program (2021JDTD0026).