The metasurface is a structured array developed in recent years to adjust the wavefront of electromagnetic waves, which has attracted extensive attention in the field of micro-nanophotonics [1,2]. Such superstructured surfaces mainly control the electromagnetic field through the accumulation of phase mutations in single or multilayer scatterers [3]. A metasurface is a two-dimensional metamaterial whose thickness is much smaller than the wavelength of the incident light [1] and can adjust and control the basic characteristics of the incident light such as phase and amplitude [4]. The fabrication of three-dimensional metamaterial optical devices requires delicate fabrication techniques, while two-dimensional metasurfaces contain only one or a few layers of planar nanostructure arrays, thus greatly reducing the loss of light transmission [5,6]. Generally speaking, researchers can realize metasurface functional devices such as biological detection [7–9], holographic imaging [10], vortex beam generation [11], and stealth [12] through resonance phase, waveguide transmission phase, and Pancharatnam–Berry phase [13].

Electromagnetic wave phase control is one of the most widely used research directions on metasurfaces [1,2]. The metasurface is composed of many meta-atoms, so we can control the phase of each meta-atom through the phase function [3], so as to design the properties of the metasurface we want. In recent years, dielectric nanostructures based on low loss [14] and high refractive index [15–17] have aroused great interest among researchers, especially the performance of semiconductor metasurfaces [18]. Dielectric nanostructures overcome the problem of loss in metallic metasurfaces, and they possess both electric and magnetic dipoles upon excitation [19]. Cui et al. [20] proposed a seven-layer (from top to bottom: quartz, complementary split ring resonator, polyimide, liquid crystal, polyimide, metal grinding, and silicon substrate) structure, using reflective-like [21] grating metasurfaces for frequency-steered beam steering. Based on the theoretical prediction of the grating principle [22], when the frequency of the vertically incident terahertz wave varies between 0.87 and 1.02 THz, the metasurface grating can steer the deflected beam from 59.5° to 47.3°. To generate arbitrary optical vortices [12], Wang et al. designed and demonstrated a chip-scale device based on an all-dielectric [23] two-dimensional polarization-independent metasurface [24,25] grating [26]. Based on the principles of multi-beam interference and nanofabrication techniques, the designed devices allow high efficiency [27–29] while also being able to generate on-chip multi-channel beams with different angular momentums [30–32]. In order to improve the diffraction efficiency, Pottier et al. [33] fabricated a planar concave integrated optical diffraction grating using a grating distributed Bragg reflector based on a dielectric material [34]. They combined the Bragg conditions that produce high reflection [35] with the grating diffraction conditions that provide wavelength diffusion, constructed a single-order diffraction grating using elliptical sections [36,37], and finally made the diffraction efficiency of the single-order grating reach 99%. In addition, all-dielectric metalenses [38] have aroused great interest among researchers due to their advantages of low loss [39], high efficiency [40], and low cost [4], especially all-dielectric metalenses based on silicon materials [41]. Metalenses [42] can be combined with large-scale semiconductor fabrication techniques for large-scale industrial production [43].

Most current grating metasurface designs are based on two or more different materials [44,45] to manipulate electromagnetic waves to achieve different functions [46]. Most metasurface photonic devices are designed and fabricated on thicker substrate materials [47–51]. The presence of the base structure leads to a decrease in device efficiency, especially for transmissive metasurface devices [52,53]. In this paper, we propose a substrate-free all-dielectric silicon grating structure. At the frequency of 0.1 THz, the subwavelength coding grating unit can achieve 2π phase mutation and has high transmission efficiency. Based on the classical grating equation, a new coded hypergrating is obtained by adding and subtracting two different sequences of coded hypergratings. Through the four-bit operation, the maximum deflection angle of the grating is 48.59°. We encode this baseless grating, and each coding unit can be moved and arranged randomly, which further improves the design flexibility of traditional grating. It is worth noting that, based on the traditional grating equation, we innovatively propose the addition principle of the coded grating to achieve the multifunctional operation of the traditional grating. In addition to the verification of the basic coded grating beam regulation, we also regulate the metasurface beam deflection of the cascaded silicon grating and realize the superposition of the two coded gratings, which provides a new idea for the design of multifunctional devices. Both simulation and experiment prove the feasibility of substrate-free silicon grating structure for beam regulation.

Metagratings combine the resonant scattering properties of the basic unit structure of metasurfaces [54–56] with the intrinsic diffractive properties of gratings [57]. According to the phase transition characteristics of the metasurface [58], we can encode each different unit structure of the metasurface [59]. For example, we can specify a unit structure with a phase difference of 90°, and its digital codes are “0,” “1,” “2,” and “3”. For the basic encoding grating of this gradient sequence, its diffraction characteristics conform to the classical grating equation. The traditional gratings we are familiar with are based on the grating equation. The diffraction level and efficiency are optimized and regulated, their design parameters are relatively single, and the functions realized are also relatively traditional. The coding grating addition principle proposed by us can realize the flexible regulation of the unit structure of traditional grating and realize the integration of functions. The coding principle of traditional applications in the process of metasurface design is mainly based on the principle of the phased array antenna, and the scattering characteristics of the coding metasurface are regulated by the Fourier transform characteristics of the near-field and far-field. The principle of coding raster addition proposed here is based on the extension of the classical raster equation, and the two raster equations are coded and added. The coding raster addition principle proposed by us is obviously different from the traditional metasurface coding principle in terms of form and basic physical principles.

The diffraction grating can control the propagation direction of the beam, and the diffraction direction is determined by the grating equation, sin(θk′)−sin(θi′)=k′λΛ.(1)

Among them, λ is the wavelength of the incident wave, Λ is the grating period, k′ is the diffraction order, θi′ is the incident angle, and θk′ is the diffraction angle. When the incident light is vertically incident, the first-order diffraction grating equation simplifies to sin(θk′)=λΛ.(2)

Assuming that there are n basic coded super-grating sequences S0,S1,S2,…,Sn, then the grating periods are Λ0,Λ1,Λ2,…,Λn, respectively. The first-order diffraction equations of each grating sequence can be expressed as $$\begin{gathered} sin(θ0)=λΛ0,sin(θ1)=λΛ1, \\ sin(θ2)=λΛ2,…,sin(θn)=λΛn. \end{gathered}$$(3)

Among them, θ0,θ1,θ2,…,θn represent the first-order reflection and diffraction angles of each sequence, respectively. Adding or subtracting these grating sequences in Eq. (3), the new super grating sequence Sm can be obtained as S0|Λ0±S1|Λ1±S2|Λ2±…±Sn|Λn=Sm.(4)

Each grating sequence of the first-order diffraction equation in Eq. (3) can also be subtracted, and the first-order scattering angle of the new grating sequence Sm can be expressed as sin(θm)=sin(θ0)±sin(θ1)±sin(θ2)±…±sin(θn)=λΛ0±λΛ1±λΛ2±…±λΛn.(5)

If we add and subtract the two grating sequences of S0 and S1, Eq. (5) can be simplified, and the corresponding scattering angle can be simplified as sin(θm)=sin(θ0)±sin(θ1)=λΛ0±λΛ1.(6)

We designed a transmission-type unit particle using silicon material to achieve 2π phase coverage and high transmittance. Figure 1 shows an encoded metaparticle composed of silicon (the dielectric constant is 11.9, the magnetic permeability is 1, and the losses are negligible) with a unit particle period length P of 1500 μm and a height h of 700 μm.

By changing the width w of the unit particle, the transmission phase of the unit cell can cover the range of 2π. After parameter scanning optimization, four unit structures with a phase difference of 90° can be obtained. When the y-polarized light is perpendicular to the supergrating along the z axis direction, the phase and amplitude characteristics of the cell structure are shown in Fig. 2. At 0.1 THz, the phases of the four units digitally coded as “0,” “1,” 2,” and “3” are −196.96°, −106.94°, −17.43°, and 72.47°, respectively. The transmission coefficients are all greater than 0.82. The geometric parameters of the grating strip width w are 951 μm, 496 μm, 231 μm, and 11 μm, respectively.

First, the basic sequence as shown in Fig. 3 is designed, encoding is carried out along the x axis direction, and the same encoded particles are used in the y axis direction. The basic sequence codes are S1=0123…, S2=00112233…, S3=000111222333…, all of which are composed of 40×30 coded particles. From the above coding composition, it can be seen that the coding periods of the basic sequences S1, S2, and S3 are Γ1=4 and P=6000μm; Γ2=8 and P=12,000μm; and Γ3=12 and P=18,000μm, respectively. According to the formula θ=arcsin(λ/Γ) of generalized Snell’s law [60], the anomalous transmission angles corresponding to the three basic sequences are calculated as θ1=30°, θ2=14.47°, and θ3=9.59°.

The encoding metasurfaces of the above-mentioned transmissive basic sequences S1, S2, and S3 are shown in Fig. 3. In the simulation, the terahertz wave electric field is vertically incident on the encoding metasurface along the y axis direction [61,62], and the boundary conditions in the x, y, and z axis directions are all set to open space [open (add space)].

Figure 3 shows the arrangement of the basic sequence metasurfaces, which correspond to the transmission basic sequences S1, S2, and S3 from top to bottom. According to the simulation calculation results, it can be obtained that θ1=30°, θ2=14°, θ3=10° (the transmission angle is the included angle with the positive semiaxis of the z axis). These numerical results have little error with the theoretically calculated data. The obvious deflection effect can be seen in both the far-field and the near-field, thus verifying that the designed basic sequence can achieve abnormal transmission.

Based on the far-field transmission angles of the three coded metasurface basic sequences obtained above, in order to realize the continuous control of the transmission angle, the coding design of the non-basic periodic sequence can be carried out according to the principle of coded grating addition. We know that, in the two-bit coding design, a new coding sequence can be obtained by performing four-bit operations on two known basic sequences [10].

The four-digit arithmetic rule is as follows: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 2 = 0, 2 + 3 = 1, 3 + 3 = 2, etc. The same rules of calculation apply to subtraction. For the convenience of description, a one-to-one correspondence is made between the coding sequence and the transmission angle. Through the above operations, the new sequence is S4=01302312…, S5=30011223…, S6=012012…, S7=301123… (see Figs. 4–6). According to the four-bit operation of the above-mentioned new sequence and Eq. (6), there are θ4=arcsin(λ4P+λ8P),(7)θ5=arcsin(λ4P−λ8P),(8)θ6=arcsin(λ4P+λ12P),(9)θ7=arcsin(λ4P−λ12P).(10)

Given that λ=3000μm and P=1500μm, the theoretical values of the transmission angles of S4, S5, S6, and S7 sequences can be calculated, respectively, according to the above four formulas: θ4=48.59°, θ5=14.47°, θ6=41.81°, θ7=19.47°.

The newly obtained S4, S5, S6, and S7 coded supergratings are shown in Figs. 4–6. The simulation calculates that the transmission angles of the four non-basic sequences are θ4=48°, θ5=14°, θ6=42°, θ7=19°. It can be seen that the theoretical and simulation results are similar, which verifies the feasibility of the coded grating addition principle to continuously control the abnormal transmission angle.

On the basis of the above addition principle, we can perform four-bit operations on the non-basic sequence again to obtain a new coding sequence. For example, the new sequence is S8=S6−S7=333111… (as shown in Fig. 7). It can be seen from the new sequence that the four-bit operation belongs to one-bit encoding, and the encoding period Γ=6P. Although the encoding period Γ of the S8 sequence is an integer multiple of P, the new sequence does not contain “0” and “2” unit elements, so the new sequence cannot be used as a basic sequence for four-bit operations. According to Eq. (6), it can be known that the transmission angle formula of the S8 sequence is θ8=arcsin(λ3P−λ6P).(11)

Through theoretical calculation, the anomalous transmission angle of S8 sequence is 19.47°. It can be seen from Fig. 7(b) that the S8 sequence can obtain two mainlobes that are symmetrical about the z axis, and the mainlobes are equivalent to the energy being evenly distributed and emitted from different directions when passing through the super-coded grating. The reason why two mainlobes can be generated can be seen from the coding sequence. In the simulation, when the light source is set as a plane wave, the metasurfaces 333111… and 111333… are the same metasurface. The above-mentioned two phase gradients are just opposite, which in turn causes the mainlobe to be exactly symmetrical about the z axis.

The above are multiple sets of non-basic periodic sequences obtained by four-bit operations using two-bit coded numbers. The simulation and numerical calculations are in good agreement, indicating that the coded grating addition operation can continuously control the abnormal transmission angle. Obtaining aperiodic encoded sequences by four-bit operations is also applicable to one-bit encoded numbers. We calculate the basic sequences S1′=0011… and S2′=00001111… according to Eq. (6) to obtain a new coding sequence S3′=11000011…, and the coding sequence is shown in Fig. 8.

The coding periods of coding sequences S1′ and S2′ are Γ1′=4P=6000μm and Γ2′=8P=12,000μm, respectively. According to the generalized Snell’s law formula θ=arcsin(λ/Γ), the anomalous transmission angles θ1′=30° and θ2′=14.47° corresponding to the two basic sequences are, respectively, calculated. The simulation results can be seen from Fig. 8. The S1′ sequence generates a symmetrical double beam with a transmission angle of 30°, and the S2′ sequence generates a symmetrical double beam with a transmission angle of 15°. S1′ sequence and S2′ sequence are coded and subtracted to obtain a new sequence S3′ with a coding period of 333,111…. According to Eq. (6), it can be calculated that the theoretical transmission angle of the S3′ sequence is 14.47°, while the simulation result is 15°. The theoretical calculation and the simulation have a good consistency.

In addition to using the coded grating addition principle [59] to control the deflection angle of scattered beams, we also verified the cascaded metasurfaces to control the deflection of multiple beams, as shown in Fig. 9. The basic sequences S1=0123… and S3=000111222333… are physically superimposed in space to form a cascaded coded super grating S4′. The cascaded grating can simultaneously generate two beams on a plane with deflection angles of 9° and 30°, respectively. These results verify that the substrate-free silicon grating structure [63] is also suitable for beam deflection with cascaded metasurfaces.

Two coded gratings are cascaded to realize the superposition of functions, which provides a new idea for the design of multifunctional devices. By superimposing different coded gratings, the performance of the system can be flexibly adjusted and optimized. For the aperiodic arrangement of the coded grating, it is inevitable to carry out a lot of operations when using the coding principle, and the coded grating is cascaded by the addition principle, which can reduce the cumbersome calculation and improve the design efficiency of multifunctional devices.

In order to verify the designed array structure experimentally, we fabricated two simple substrate-free supergrating samples P1 and P2, and conducted data tests. The periods of the two unit samples are both 500 μm. For sample P1, Γ1=4 and P=2000μm; for sample P2, Γ2=8 and P=4000μm. In order to meet the requirements of the experiment, we selected double-throw high-resistance silicon with a crystal orientation of 110 [64,65] and a thickness of 300 μm for experimental verification. The experimental process is shown in Fig. 10.

We utilize classical photolithography techniques [66,67] to fabricate substrate-free coded grating samples. First, a layer of SiO2 is oxidized on the entire surface of the silicon wafer, as shown in Fig. 10(b). The oxidized silicon wafer was cut according to the size of the mask [68], and the silicon wafer used in the experiment was thoroughly cleaned (cleaned with acetone and water) and then dried on a dryer for 10 min. We put the cleaned silicon wafer on the glue leveling table, coat a layer of tackifier and photoresist [69] on the silicon wafer in turn [as shown in Fig. 10(c)], and dry it for 5 min after coating and 1.5 min. As shown in Fig. 10(d), we align the silicon wafer with the mask and place it under a mercury lamp for exposure for 11 s. After the exposure, the silicon wafer is placed in the developer [70] (0.9 g NaOH + 100 mL H2O) for 8 s [Fig. 10(e)], and then the silicon wafer is quickly rinsed under flowing plasma water. As shown in Fig. 10(f), after the development, the silicon wafer is dried for 1.5 min, and then the silicon wafer is placed in a cleaning solution (HF:NH4F:H2O=3:6:10) for 10–12 min to wash off the bottom SiO2. We use the method of wet etching [71] to etch in the etching solution (20mLC4HNO+6mLC3H8O) for 4.5–5 h [wet etching is shown in Fig. 10(g); we put the silicon wafer in the etching solution, and the upper and lower layers will be etched at the same time] and get the experimental sample with SiO2 on the top layer [72]. Finally, the designed experimental structure can be obtained by washing off the SiO2 on the top layer. Since the etching time is too long, the SiO2 on the surface of the structure will be destroyed, resulting in uneven adhesion of SiO2 on the surface [73]. For the accuracy of the experiment, all the SiO2 on the silicon surface was washed off first, and then a layer of SiO2 was re-oxidized, so that a uniform layer of SiO2 was attached to the entire surface of the silicon wafer as shown in Fig. 10(i). After the oxidation is completed, a layer of photoresist [69] is evenly coated on the top layer of the grating bars [as shown in Fig. 10(j)] to prevent damage to the top silicon structure during the subsequent wet etching process. Finally, the bottom SiO2 and Si are, respectively, etched away to obtain the designed substrate-free silicon grating.

As shown in Fig. 11, through the above experimental process, we produced two coded samples P1 and P2, both of which consisted of 40×30 unit particles. Due to the limitation of experimental conditions and the inhomogeneity of wet etching, the thicknesses of the final grating samples P1 and P2 were 135 μm and 124 μm, respectively. The bar widths were 292 μm and 129 μm, respectively.

We built a tunable continuous terahertz wave system [74], and the test platform is shown in Fig. 12. First, we fix the experimental sample on the support frame and adjust the height of the support frame so that the incident light, the sample, and the receiving end are on the same horizontal plane. Based on the principle of photon mixing [75–77], two independent semiconductor lasers are used as the irradiation light source of the butterfly photoconductive antenna [78]. The incident light frequency ranges from 0.05 THz to 1 THz, and the computer is connected to the receiving end to display the test data. We set the parameters of the software before the experimental test, and we set the lowest frequency point to 0.05 THz and the highest frequency point to 0.5 THz. We tested the transmittance of samples P1 and P2 in the range of 0.05 THz to 0.4 THz, and the test results are shown in Fig. 13. It can be clearly seen from the figure that the transmittance of sample P1 at 0.1 THz is about 70%, the transmittance of sample P2 is about 69%, and the transmittance of cascaded metasurface is about 52%. The reasons for the deviation between the experimental data and the simulated data may be as follows: on the one hand, there may be preparation errors in the process of metasurface preparation, especially in the process of wet corrosion, when the width of the prepared grating strips and simulation errors will occur. In addition, because the wet corrosion method is to immerse the sample in the corrosive liquid for corrosion, a large number of bubbles will be generated during the corrosion process, and the sample will float on the surface of the corrosive liquid, resulting in different corrosion rates on the upper and lower surfaces, and eventually the thickness of the prepared sample is inconsistent. In addition, the substrate-free silicon grating structure is prepared, and the side of the grating strip may be corroded during the wet corrosion process, resulting in changes in the actual width of the grating strip and resulting in errors. On the other hand, because the designed structure is a baseless grating metasurface, the step meter cannot accurately measure the width and depth of the small grating gap during the test process, resulting in a gap between the actual measurement results and the theoretical results. Although there are errors in the test and simulation data, the overall trend of the transmission efficiency curve is consistent within the range of 0.05–0.4 THz. The highly efficient transmission properties of P1 and P2 can also be indirectly verified at 0.1 THz.

Laboratory tests use TeraSense’s terahertz source as well as terahertz imaging cameras. The terahertz camera bandwidth range is 0.05 THz to 0.7 THz, the high-speed image acquisition rate can reach up to 50 frames per second, and the number of pixels is 32×32. The experimental platform is shown in Fig. 14(a) and consists of an antenna speaker, lens, terahertz camera, display, etc. A light source with a frequency of 0.1 THz is emitted through an antenna horn, and the outgoing beam diverges outward in a scattered manner. After passing through the first lens, the beam is converted into a parallel beam, and then the second lens is used to converge the parallel light. The incident beam is concentrated on the metasurface, and after passing through the metasurface, it is received by a terahertz camera. The application is designed by LabView to receive and process data on the computer side. The experimental test results are shown in Fig. 14(b), and we can see that there is a gap between the simulation effect of the prepared coding array at 0.1 THz and the experimental results. In region (ii) of Figure 14, we select four points with obvious scattering effects, and by comparing the two-dimensional far-field scattering patterns of samples P1, P2, and cascaded metasurfaces, we find that the superposition of the cascade metasurface function is in line, thereby indirectly verifying the feasibility of cascade metasurface manipulation beam.

We propose a coding supergrating structure, and the coding units are all made of substrate-free silicon material. The parameters of the unit structure are optimized so that the unit particles have a complete 2π transmission phase coverage under the condition of 0.1 THz and co-polarization, and there is a phase difference of π/2 between each unit particle. We design multiple sets of simple basic sequences and solve the theoretical far-field scattering beam elevation angle through the generalized Snell theorem. For the far-field scattered beam angles of other non-basic sequences, the corresponding transmission angle can be obtained by redetermining the encoding period Γ through the four-bit operation of the basic sequence. We obtained coded supergratings with high transmittance and large deflection angle. We verified the control of the cascaded metasurface on the deflection angle of the incident light, and the theoretical values obtained are basically consistent with the simulation results. We demonstrate the feasibility of the encoded grating addition principle to continuously control the far-field scattering angle on a substrate-free all-dielectric encoded supergrating.