2.1 Broadband Achromatic Deflector Design

2.1.1 Dispersion Conditions of Broadband Achromatic Refraction

As shown in Figure 2a, the plane wave is incident normally on a metasurface, which induces a phase gradient. A typical metasurface consists of a series of sub-wavelength structures called meta-atoms. As a result, the desired continuous phase distribution can only be realized approximately in a discretized fashion. According to Generalized Snell's refraction law,^[11] we write the phase induced by meta-atom i as function of frequency as ${\varphi }_i(f)$,

$${\varphi }_i(f)=-\frac{2\pi f}{c}\sin ({\theta }_f)\times (i-1)\mathrm{\Delta }x+{\varphi }_1$$(1)

where c is velocity of light, Δx is the size of meta-atoms, and θ_f is the refraction angle dependent on f.^[35] For an achromatic deflector, θ_f is a constant among the whole spectrum. Hence the dispersion of the phase induced by all meta-atoms has to depend linearly on the frequency. Meanwhile, for different meta-atoms, the phase curves have different slopes determined by the position of atoms. Our main aim is to design meta-atoms that have this property with the desired phase change as function of frequency.

2.2 Meta-Atom Design

In order to realize broadband achromatic refraction, we have to design a meta-atom with broadband linear phase response and high transmission. Our design consists of three parts. First, we propose a meta-atom A, which consists of three layers with metal (yellow) as shown in Figure 2b. There are two parallel metal gratings with both pairs being orthogonal to each other. The middle layer contains a circular ring with diagonal metal bar structure. This meta-atom can realize broadband linear phase response and high transmission. In addition, it can induce an arbitrary phase change varying from 0 to 2π.^{[63, 64]} Second, we design a meta-atom B which consists of a multi-layer split ring slot structure as shown in Figure 3a. This structure can also realize broadband linear phase response and high transmission. The difference with meta-atom A is that with meta-atom B arbitrary dispersion can be realized by changing the number of layers and the width of the slot. The physics mechanisms behind the phase delay of meta-atom A and meta-atom B are all based on the propagation phase principle.^{[65, 66]} We finally cascade the two kinds of metasurface structures to obtain the desired phase change and dispersion property.

As shown in Figure 2b–d, in the first structure three layers with a metal structure are alternated by two dielectric layers (blue). The metal is made of copper with conductivity σ = 5.8×10⁷ S m⁻¹ and thickness 0.036 mm. The substrate is made of FR-4 with relative permittivity of 4.3, a loss tangent of 0.025. To realize broadband highly efficient transmission and a phase change that depends linearly on the frequency, we choose the following initial values for the parameters shown in Figure 2b–d. The period p of meta-atom is 6 mm. The width l₁, the thickness h₀, and the length of metal gratings is 1.6, 0.036, and 6 mm, respectively. The distance l₂ between metal gratings is 1.4 mm. The radius r₀ and the width w₀ of circular ring is 2.8 and 0.3 mm, respectively. α and β are 80° and 45°, respectively. The thickness d₀ of substrate is 2 mm. The electromagnetic behavior of meta-atom A is characterized using CST Microwave Studio and the method can be seen in Section 4.

In order to understand this broadband highly efficient transmission structure, we utilize the TMM to explore the property of the meta-atom.^[67] The TMM is applicable for a multi-layer structure. As illustrated in Figure 2h, when there is a y-polarized incident wave from the left it is almost totally reflected, whereas when a x-polarized wave is incident from the left, a y-polarized wave with large amplitude is transmitted into the right medium because of the process of repeating reflection and transmission inside the meta-atom. The specific analytical process can be seen in Part A, Supporting Information.

Next, we study the phase controlling properties. As shown in Figure 2f, the transmission phase is determined by α and the rotation angle (β) of middle layer circular ring structure. For the frequency 11 GHz, when β is 45° and α changes from 22° to 76°, the phase change is from 0 to π. When β is −45° (the circular ring structure turns 90° clockwise as shown in the left inset in Figure 2f and α changes from 20° to 74°, the meta-atom can induce a phase change between π and 2π. At the same time, the transmission efficiency is always over 0.9. In addition, we show in Figure 2g the frequency dependence for several choices for β and α. The phase curves show a linear dependence between frequencies 9 and 13 GHz as required and they are parallel to each other. The amplitude curves show that transmission rate is always over 0.9 over the entire band. As a result, changing α and rotation direction of the diameter bar inside the circular ring gives phase coverage from 0 to 2π without affecting the dispersion relation.^[48]

In the second step of the design process, we consider multi layers metal sheets with split ring slots (yellow part) separated by air gaps (green part) as shown in Figure 3a. The thickness of the air gap is ≈$\frac{1}{4}\lambda$.^[68] The metal is aluminum with conductivity σ = 3.5×10⁷ S m^-1 and the thickness 0.2 mm. In order to support the metal sheet structures, the air gaps are filled with foam material with permittivity 1.15 and loss tangent 0.0057. The density is 110 kg m⁻³ and the compression strength is 2.9 MPa. Similarly, to obtain broadband high-efficiency transmission and a linear phase response, the parameters of meta-atom B are chosen as following. The period s of meta-atom B is 12 mm. The width w and the external radius r of the slot is 1.6 and 5.2 mm. $\gamma$is 170°. The thickness h_m of the metal sheet is 0.2 mm. The thickness g of the foam is 6 mm. We mainly focus on this structure responding to y-polarized waves. The transmission and reflection of a single-layer structure is shown in Figure 3c, which is for a y-polarized wave incident normally and propagating in the negative z-direction. The amplitude of t_yy is larger than 0.9 for frequencies from 9 to 13 GHz.

Here, we adopt equivalent circuit principle to explore the physical mechanism of the broadband high-efficiency transmission property.^[69-72] First, we observe in Figure 3d the surface current distributions on the metal for the frequency 10.8 GHz where |t_yy| is maximum. There are strong opposite resonant currents on both sides of the split ring slot, which are indicated by the pink arrows. In addition, strong currents also exist at the two gaps between the two semicircle slots indicated by the red circles in Figure 3b. We establish an equivalent circuit to analyze the electromagnetic characteristics of this structure, as shown in Figure 3e. The split ring slot is considered to be equivalent to a capacitance C; either of the two gaps indicated by the red circles is equivalent to be an inductor L₁ and a resister R₁ in series. These two equivalent circuits are parallel.

We simplify the equivalent circuit in Figure 3e to a LC parallel circuit, which is shown in Figure 3f. Thus, the single-layer split ring slot structure can be considered as a first-order band-pass filter, which could realize high-efficiency broadband transmission from 9 to 13 GHz. According to |t_yy| in Figure 3c, we can derive the values of R₀, L₀, and C₀ of the LC parallel circuit by utilizing circuit simulation software AWR Design Environment and find R₀, L₀, and C₀ is 5.13×10⁻⁵ Ω, 110.37 nH, and 0.08064 pF. The resonant frequency ω₀ derived from the circuit simulation software is 10.6 GHz. Then, we plot S₂₁ of equivalent circuit calculated by AWR as shown in Figure 3g. It coincides well with S₂₁ of physical model calculated by CST. The results above indicate the reliability of our equivalent circuit model theory. A multi-layer structure can be equivalent to a n^th-order band-pass filt, which is shown in Figure 3i.^[73] Furthermore, we study the relationship between the transmission bandwidth and the order of the filter, that is, the number of layers of the structure. The transmission property is shown in Figure 3h for different numbers of metal sheet layers with the same structural parameters. We find that the high-efficiency transmission bandwidth gradually narrows in agreement with theoretical predictions.^[73] However, the three-layer structure still can realize high-efficiency transmission from 9 to 12 GHz.

Next, we consider the phase response of the split ring slot structure. In Figure 4a the phase of t_yy is shown as function of frequency for several numbers of layers. A linear dependance of the phase on frequency is observed from 9.2 to 12.9 GHz, which we will call the linear region. When the number of layer increases, the slope of phase curves increases. We also consider the influence of the width w of the slot for the double-layer structure as a typical research. Figure 4b illustrates the variations in the amplitude and phase as function of frequency for several values of w. For smaller w the high-efficiency transmission band becomes narrower while the slope of the phase curves becomes larger. Single-layer and three-layer structures show similar properties. All phase curves show a linear dependance on frequency in the band from 9.3 to 12.6 GHz.

In order to increase the slope of the phase curves and reduce the thickness of this structure as much as possible, we propose a double split ring slot structure. The front view of the structure is shown in Figure 4c, where a second slot is added inside the original circular slot. The parameters of the added structure are w₁ = 1.8 mm and r₁ = 2.8 mm while the other parameters remain unchanged. We compare the transmission of the three-layer double split ring slot structure with the three-layer single split ring slot structure. The results are shown in Figure 4d. It is seen that by adding the slot, the slope of the phase curve increases while the bandwidth where the transmission is high remains. The linear region of the phase curves is from 9.5 to 12.3 GHz. Similarly, when we reduce the width w of the original slot, the slope of the phase curves increases while the bandwidth of high transmission gradually reduces. Thus, adding a slot is an effective way to widen the controlling range of dispersion property without increasing the thickness of the structure. In a word, we can carefully design meta-atom B to achieve dispersion relation of transmission as we desire.

In the third step of the design process, we cascade four meta-atom A with period of 6 mm placed 2 × 2 and one meta-atom B with period of 12 mm to form 4 types of meta-atom C with period of 12 mm. Meta-atom A and meta-atom B are separated by foam material (green part) with thickness of 6 mm. Every layer of the 4 structures is shown in Figure 4e. With these structures we can realize broadband high efficiency transmission with the slope of phase curves gradually increasing. At the same time, we can control the phase response and its slope independently by changing the circular ring with the metal diagonal bar structure and the width of split ring slot respectively. The cascaded meta-atom also has the cross-polarization conversion function.

2.3 High-Efficiency Broadband Transmission Achromatic Deflector Design and Experimental Verification

According to working frequency band of our meta-atoms, we design an achromatic refraction metasurface with $\frac{\pi }{3}$ phase difference between adjacent meta-atoms at 12 GHz. On the basis of theoretical Equation (1), it can be calculated that refraction angle θ_t is 20.3°. To have the angle of refraction constant for all frequencies in the band of interest, the difference of slope in phase curves induced upon transmission between adjacent meta-atoms should be $\frac{1}{36}\pi$ Rad GHz⁻¹.

To realize this phase relationship and dispersion property, we design a broadband achromatic refraction metasurface which contains 16 meta-atoms. All of the meta-atoms are composed of four types of structures as shown in Figure 4e. We design the structures by adjusting α and the rotation direction β of the circular ring with diagonal metal bar structure, and furthermore the outer slot radius r and width w of the metal sheet. Other parameters remain fixed as described mentioned above. We arrange 16 meta-atoms into 1D linear array, which is shown in Figure 4f with marked structure type of every meta-atom. Amplitude curves of them are shown in Figure S2, Supporting Information. Phase curves are shown in Figure 5a. The transmissivity of all meta-atoms is above 0.8 from 9.8 to 12.4 GHz. At the same time, all of phase curves show great linear characteristics from 9.6 to 12.5 GHz and the slope of curves increases evenly. In order to better assess the linearity of the phase response, we also give the spectral derivative of the phase profile in Figure S3, Supporting Information.^[74]

20 groups of 1D linear array are configured repeatedly along y-axis with a period of 12 mm and the whole area is 192×240 mm². We carry out simulation of the model in CST Microwave Studio. X-polarized plane wave is incident along z-axis. We take the direction of maximum radiation energy as the deflection angle and observe y-polarized wave from 9 to 13 GHz. The deflection angle curve is in Figure 5b. From 9.2 to 12.3 GHz, the deflection angle is always between 20.0° and 21°, which is consistent with theoretical value, 20.3°. The slight deviation may result from phase error of elements and finite size of model.

To furtherly demonstrate our design theory and simulation, we fabricate a sample and the size is the same as simulation model (16 × 20 cells with a total size of 192 × 240 mm²). Two layers of sample are shown in Figure 5c,d and all of layers can be seen in Figure S4, Supporting Information. Then, we test far field characteristics of our sample in microwave anechoic, as shown in Figure 5e. The method can be seen in Section 4. The measurement results of deflection angle are shown in Figure 5b. The deflection angle is between 20.0° and 21.0° from 9.3 to 12.3 GHz, which is in good agreement with simulation results. Inevitable fabrication errors and imperfections of the incoming wave fronts generated by the microwave horn lead to tiny difference.

In order to analyze the effect of achromatic refraction, we give the simulation and experiment normalized far-field patterns at 9.3, 10.8, and 12.3 GHz in Figure 5f–h. We can find that the main lobe of the patterns refracts and the simulation (experiment) refraction angle is 20.5° (21.0°), 20.5° (20.5°), and 20.0° (20.0°) at 9.3, 10.8, and 12.3 GHz, respectively. The edge diffraction component of co-polarization in the normal direction hardly has effect on the observation of deflection component because we merely observe cross-polarization component. This contributes to high polarization purity of our device. The measured 2D scattering power spectrums varying against angle and frequency of co-polarization and cross-polarization are shown in Figure 5i,j. In the operation band, the power distribution concentration angle of cross-polarization is almost consistent with theoretical deflection angle. Instead, outside of operation band, most co-polarization power is reflected around normal direction.

Next, we calculate working efficiency of our device. The efficiency is defined as the ratio between the power of the main beam regions and the total power of the incident wave. As shown in Figure 6a, the experiment efficiency is in consistent with simulation result. The fluctucation of experiment results comes from noise random error, measurement error, and sample fabrication error. The efficiency of experiment is over 70% from 9.7 to 12.3 GHz. It reaches peak (86.7%) at 11.3 GHz. The average efficiency is 77.5%. The lost power mainly results from reflection, edge diffraction and the loss of substrate.

At last, we test the near-field characteristics of the device. The test schematic diagram is shown in Figure 6b. The method can be seen in Section 4. Figure 6f–h give the y-polarization electric field distributions at 9.3, 10.8, and 12.3 GHz. When the wave is incident on the metasurface, the propagation direction of electric field obviously refracts. The flat transmission wave front proves the high efficiency of the device. The experimental results are in good agreement with simulation results.

We also give a comparison between chromatic and achromatic deflector. We design a chromatic deflector with meta-atom A and optimize six meta-atoms with $\frac{\pi }{3}$ phase interval between adjacent meta-atoms. The phase curves of all meta-atoms are shown in Figure 6i. We can see that phase curves are parallel to each other. In another word, the phase interval is always $\frac{\pi }{3}$ among the whole band. In order to ensure the fairness of comparison, the period and the size of chromatic deflector is the same as achromatic deflector. Then, we plot deflection angle varying against frequency of chromatic deflector by conducting simulation (finite-different-time-domain method) in Figure 6j. It is obvious that the deflection angle decreases with frequency increasing. The simulation is in good agreement with theory. On the contrary, for our achromatic deflector, the deflection angle remains stable as we desire. This comparison furtherly proves the correctness of our achromatic deflector theory and design.

2.4 Application for Achromatic Focusing Lens

2.4.1 Dispersion Conditions of the Broadband Achromatic Focusing Lens

Here, we take the design of the 1D achromatic lens as an example. The schematic diagram is shown in Figure 7b. Discontinuous phase distribution of lens is expressed as following formula,

$${\varphi }_i(f)={\varphi }_1+\frac{2\pi f}{c}\left(\sqrt{{[(i-1)\times \mathrm{\Delta }x]}^2+F^2}-F\right)$$(3)

where ${\varphi }_i(f)$ represents the phase of the i^th meta-atom varying against frequency f. Δx is the period of meta-atom. F is the focal length of the lens, which is a constant in the achromatic lens. The dispersion of phase induced by every meta-atom in the achromatic lens also depends linearly on frequency. The phase curves for different meta-atoms have different slopes determined by the location of atoms.