1. Introduction

Since the proposal of generalized Snell’s law, metasurface has become a powerful technology to steer the light beam direction and achieve anomalous reflection and transmission properties [1]. By using basic light wave physical concepts such as geometric phase (also known as Pancharatnam-Berry phase) and resonance-based phase modulation [2–4], special Huygens metasurface [5,6] and Bayesian beam [7] etc. can be designed through careful selection of the meta-atom geometric parameters and material properties. Many interesting and important applications can be realized such as metalens, structural color and super-resolution imaging. Since the shape of the meta-atom in the light receiving surface plane is anisotropic, a phase response associated with the rotation angle can be generated by rotating the meta-atom along the central z axis. The resonance phase modulation is achieved by changing the size of the meta-atom structure, as the resonance frequency changes, the phase also changes at such specific frequency accordingly. As a result, beam control can be realized according to the design concept of gradient phase modulation [8].

Later, accurate phase control strategy based on digital coding concept has been proposed [9,10], where meta-atom units with two phase responses of 0 and π are defined as digital encoding ‘0’ and ‘1’, and extended to 2-bit array coding with 0, π/2, π, 3π/2 phase differences, which achieved beam control and low scattering in the terahertz and microwave bands with good codability [11,12]. It has been demonstrated in the literature that the 2-bit coded metasurface can be applied to microwave, terahertz and optical bands, and the tunable coding metasurface can actively manipulate the incident wave [13–15]. Recently, according to the coding metasurface concept, Chu et al. has proposed a 1-bit random-flip metasurfaces (RFM) based on reciprocity theorem and space flipping theory, using random-flip meta-atoms to achieve reciprocity-protected transmission and diffuse reflection near 697 nm in visible band and the microwave band near 11 GHz, respectively [16,17]. RFM refers to two atoms (head and tail) designed to be spatially flipped in a single layer, displaying a 180° reflection phase difference in a specific band. The flipped states of the two elements are reversed according to the ± z axis, similar to the spin up and spin down spin states of the electrons. By optimizing the design of phase array with initial random distribution, RFM can realize diffuse reflection phenomenon and distortion-free transmission simultaneously as the transmission phase response of the two meta-atoms are the same. This phenomenon shows great application potential for anti-glare glass and clear optical display. However, 1-bit RFM can only realize the 0 and π phases obtained by flipping meta-atom up and down, thus the optimization degree of freedom is limited.

On the other hand, 2-bit coding metasurface can realize four discrete phase distributions with higher degree of freedom of controlling phases of electromagnetic waves, and it has greater flexibility in controlling the coding sequence. In Ref. [9], Cui et al. designed a lateral 2-bit metasurface composed of square metal patch of four different sizes printed on a dielectric substrate that maps four gradient phases of the 0 - 2π interval, in which the radar scattering cross section (RCS) reduction is below -10 dB over a much wider frequency band from 7.5 to 15 GHz, thus effectively extending the range of low scattering frequency band. However, there does not exist a 2-bit coding strategy in the optical wavelength regime and it is not straightforward to apply the same lateral 2-bit coding strategy directly to the optical frequencies. Based on similar design concept of 1-bit RFM, here in this work, we propose a new strategy of double-layer random-flip meta-atom (DLRFM) to achieve 2-bit vertical metasurface coding, which results a wide and tunable near-infrared band coverage with excellent diffusive reflection and reciprocity protected transmission. Considering the basic idea of maintaining reciprocity theorem and space flipping theory, we adopt the superposition of two layers of random flipping structure to realize reciprocity protection and the space flipping respectively, thus to realize the same transmission phase and four reflection gradient phases (0, π/2, π, 3π/2) simultaneously. The efficiency of reflection diffusion is maximized when the phase difference is Δφ≈90°. Finally, the 2-bit DLRFM achieves the effective wavelength range of 719 nm - 955 nm, successfully extending the application range to the near infrared and can be transferred for longer wavelengths which demonstrates general and flexible design principles.

2. Meta-atom design methods

In our design, we use cylindrical metallic structures with two different radii embedded in the upper and lower dielectric layers (see in Fig. 1(a)) to form a double-layer random-flip metasurface. Gold nanorods are embedded into two thin layers of silica substrate separately. The period of a unit structure is set as P = P_x = P_y = 250 nm, the total thickness of the double-layer structure is H = 320 nm, and the thickness of the upper and lower layers is identical with H/2 = 160 nm, while the embedded gold nanorods have the same height of h = 50 nm. In order to reduce the shielding effect of the superstructure on the lower structure, and make the flip of the lower structure have the same effect on the reflection phase as the flip of the upper structure, the phase difference between the four constructed DLRFMs is equally dependent on the flip position of the upper and lower layers, we set the diameters of upper and lower nanorods as D1 = 100 nm and D2 = 160 nm, respectively. In addition, in order to avoid the random homogenization of reflection and transmission phases of deep subwavelength scale meta-atom, and to preserve the reflection and transmission phase characteristics of each meta-atom under periodic conditions, a 3 × 3 array is adopted to compose the meta-atom unit similar to previously reported method [17]. According to the different flip state of each layer structure, four kinds of DLRFM structures with four relative reflection gradient phases corresponding to 0, π/2, π, 3π/2 need to be designed. In one layer of DLRFM, the gold nanorods are named “head” or “tail” according to their position near the upper or lower surface of the structure, which are referred to as “h” and “t” for short. According to the combination of “h” and “t” states of the upper and lower layers of DLRFM structure, we can obtain four structures and name them respectively. For example, we call the combination of the upper layer “h” structure and the lower layer “t” structure of DLRFM “h-t”, and other corresponding relationships are shown in Fig. 1 (a). In order to realize the phase regulation encoding metasurface, we code the four structures successively as “0”, “1”, “2” and “3” according to the relative gradient reflection phase of the four structures, and the corresponding relationship is shown in Fig. 1(a).

 

Fig. 1. Meta-atom and their phase distribution. (a)”h-t”, “t-t”, “t-h”, “h-h” meta-atom structures; (b)The double layer random flipping principle, the vector ball composed of the red and blue hemispheres points in the direction of the head structure, i.e. red at top and blue at bottom represents the head structure, similarly, when the meta-atom is flipped over, blue at top and red at bottom represents the tail structure, where the vector arrow in the vector ball also indicates the direction of the head; (c, d) The reflection and transmission phase of meta-atom; (e, f) The reflection and transmission coefficients of meta-atom.

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According to the reciprocity theorem, when the single-layer random flip structure is flipped, that is, the normal of the structure is flipped in the direction of ± z to the inversed direction of ∓z, the transmission coefficient would be exactly the same (t'=t) while the reflection coefficient is quite different (r'≠r), as shown in the top two figures in Fig. 1(b). Based on this principle, we can extend it to a DLRFM stacked with two layers of RFM by derivation. Take the “h-h” structure encoded “3” and the “h-t” structure encoded “0” as an example, the transmission coefficient of the two elements after passing through the upper layers of the structure is the same (t_h’) under the same incident condition. However, when the incident light continues to pass through the lower layers of the structure, the lower layers of the “h-h” structure are flipped compared with the “h-t” structure. According to the above-mentioned reciprocity theorem, the transmission coefficient will not change after the structure is flipped, but the reflection coefficient will change, the transmission coefficients will remain the same (t_hh= t_ht), while the reflection coefficients will be different (r_hh≠r_ht). We can deduce the same relationship between other structures such as “h-t” and “t-t”, “t-h” and “t-t”, and “h-h” and “t-h”, as shown at the bottom of Fig. 1(b). Therefore, we believe the vertical 2-bit metasurface of DLRFM also has the same ability of 1-bit RFM to achieve reciprocal transmission and diffuse reflection, and provides a higher degree of freedom of coding optimization to achieve better diffuse reflection effect through DLRFM.

The structure of the meta-atoms composed of silica square lattice and gold nanorods has C4 symmetry. Therefore, the meta-atom is insensitive to the direction of linear polarization. In this regard, we use Lumerical FDTD simulation software to simulate the reflection and transmission phase distribution of the four meta-atoms between 400 nm - 1400 nm under periodic boundary conditions with TE wave incidence, as shown in Fig. 1(c) and (d). It can be found that the gradient phase difference of the four structures is close to 90° at the reflection phase of 762 nm, which maximizes the efficiency of diffusion, and the phase differences of the spectrum of 719 nm to 955 nm are also at the range of the effective phase difference interval of 2-bit encoding metasurface (45° < $\Delta \varphi $ < 135°).

In the transmission phase spectrum shown in Fig. 1(c), it is found that the transmission phases of the four double-layer random flipping state meta-atoms are relatively close, except that the “t-h” structure which we encode as “2”, has a larger transmission phase offset than the other three structures. We believe that the coupling absorption caused by the zero distance between the upper and lower structures leads to some phase offset in the near infrared band. However, in the far-field radiation pattern obtained by our simulation later, we can see that this offset has no influence on the overall transmission.

3. Array optimization

Here, we fix the size of the metasurface coding array as 8 × 8, and each meta-unit is a 3 × 3 repeating meta-atoms. We adopt the modified simulated annealing algorithm and the array factor as the fitness function to optimize the diffuse reflection effect of the whole array at the incident wavelength of 762 nm as described in the Supplement 1. The array coding mechanism is shown in the lower left corner illustration in Fig. 2. The fitness function used in our simulated annealing algorithm to judge the degree of optimization is far-field pattern array factor function and we realize the fast array far-field pattern calculation and speed up the optimization efficiency based on the two- dimensional Inverse Fast Fourier Transform function (IFFT) [18,19]. The array factor function is shown in Eq. (1):

 

Fig. 2. Schematic illustration of a double-layer RFM based on four coding meta-atoms.

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In the visible space, $\sin {\theta ^2} \le 1$, so in the uv plane, the visible region corresponds to the circle region, and the visible range of u and v is [-1, 1]. In the uv plane, define K × K sampling points (u_m, v_n). The K is the number of samples, and the sampling point value is k = 0, 1, 2, …, K-1. Since we adopt square matrix, the number of sampling points of row and column are the same, which is supposed to be:

And since the matrix factor function conforms to the two-dimensional IFFT, the matrix factor in Eq. (1) is transformed into:

In IFFT, the number of sampling points k = 0, 1, 2, …, K-1, that is, u = v = 0, …, $\frac{{K - 1}}{K}\frac{\lambda }{d}$, and perform Fast Fourier Transform shift (FFT-shift) operation and complement operation to shift the range of u to the visible range [-1, 1]. Whether the complement operation is performed depends on the size of λ/d; if it is 1/2, no complement is required; other values need periodic sample point to complement operation according to their size.

In addition to array far-field pattern calculation, numerical optimization is multi-round iteration of array correction with classical simulated annealing algorithm to avoid the influence of local minimum on the overall optimization process. We take the array factor of each array as the fitness function of the optimization algorithm, and finally minimize the value of $max\; ({AF({u,\; v} )} )$. In addition, the minimum and average values of each round of the iterative process of algorithm optimization are shown in Fig. S1(a) and (b), which finally converges to obtain the optimized array. The specific optimization flow chart is shown in Fig. S2 in the Supplement 1.

4. Result

In order to verify the diffuse reflection and reciprocity-protected transmission effect of DLRFM, we adopt CST studio suite simulation software to simulate the metasurface at multiple angles to obtain the far-field pattern. For comparison, we have simulated a metasurface with the same double layer array size filled with pure"1” elements and call such non-coded meta-atom pure tail-tail metasurface (PTTM), as shown in Fig. 4. There is always a powerful main lobe in the transmission direction, which proves that the structure of DLRFM has no influence on transmission. In the reflection direction, compared with the single powerful main lobe formed by PTTM, the radiation lobe obtained by DLRFM is more dispersed with much smaller energy, which proves that the diffuse reflection is achieved. In addition, the variation of the electric field intensity profile in the reflection direction and transmission direction of PTTM and DLRFM irradiated from the top down by the incident wave is shown visualizations respectively (see in Visualization 1 and Visualization 2). According to the non-uniform distribution of the reflection direction and the uniformity of the transmission direction of the electric field intensity in the profile, the comparison with PTTM can prove that DLRFM realized the modulation of diffuse reflection and distortion-free transmission. In addition, we have also simulated the far-field radiation spectrum of the 1-bit RFM [16,17] which operates at the same 762 nm incident wavelength. We have observed that DLRFM and RFM both show a complete and undistorted main lobe in the transmission direction, while the lobe formed by DLRFM appeared to be more uniform with smaller energy than those of RFM in the reflection direction, which proves that DLRFM has better diffuse reflection effect with the same transmission without distortion. We believe that the reason for this phenomenon is not only the beam splitting achieved by 2-bit coded phase modulation, but also caused by the Fabry-Perot resonator formed in the double-layer metal structure which realizes the interference elimination in the reflection channel, and leads to the weakening of the energy lobe in the reflection direction. We have found that in addition to the wavelength range of the previously confirmed effective phase difference (719 nm – 955 nm), diffuse reflection and distortion-free transmission effects still exist in other nearby ranges, such as at 700 nm wavelength. The complete far-field radiation patterns of DLRFM at 700 nm – 955 nm incident wavelength for comparison and verification with PTTM at the same incident wavelength can be seen in the Supplement 1 Fig. S3, proving that DLRFM has good diffuse reflection and reciprocity protective transmission effect at the above incident wavelengths. We speculate that the reason is that although some of the validity of the meta-atom phase difference is destroyed at other wavelengths near the effective phase difference band, however, these meta-atoms still fit the effective phase difference of the coding metasurface such as three-phase mapping [11,12], and thus still have diffuse reflection effect, greatly expanding the diffuse reflection bandwidth.

We have also simulated the total reflection/transmission and zero-order diffraction in reflection/transmission in Lumerical FDTD simulation software to quantify the diffuse reflection and distortion-free transmission effects of DLRFM. Due to the metasurface reflection phase modulation, when the incidence angle is ${\theta _i}$, in addition to the zero-order diffraction in the reflection direction $- {\theta _i}$, there are higher-order diffraction related to the period of the meta-atoms and wavelength, and the higher-order diffraction will produce the diffraction angle shift. Therefore, we make use of the characteristics of high-order diffraction to reduce the zero-order diffraction intensity in the reflection direction by phase modulation while maintaining the undistorted zero-order diffraction in the transmission direction. In Fig. 3 (a) and (b), we simulated the zero-order diffraction (${R_{{G_0}}}$, ${T_{{G_0}}}$) and total reflection (${R_{total}}$, ${T_{total}}$) of DLRFM at 650 to 1200 nm in the reflection direction and transmission direction respectively. It can be seen that ${R_{total}}$ is 0.583 and ${R_{{G_0}}}$ is 0.058 at the target wavelength of 762 nm. The zero-order diffraction in the reflection direction is modulated by only 10% of the total reflection, which proves the excellent diffuse reflection effect of DLRFM, and the ${T_{{G_0}}}$ in the transmission direction is maintained at 71.9% of the ${T_{total}}$, which can also show the transmission is basically undistorted. For more concrete proof, we also simulate the effect of single-layer RFM at 650 to 800 nm, and find that although the higher-order loss in the transmission direction is very small, the zero-order diffraction in the reflection direction accounts for 49.1% of the total reflection, so the excellent performance of diffuse reflection of DLRFM can be demonstrated and the zero-order diffraction efficiency of DLRFM transmission can be further improved. We believe that the proximity of double-layer metal materials will lead to a certain resonance scattering, resulting in zero-order diffraction offset of transmission. Subsequently, Au can be replaced by dielectric materials with better light transmission, or the structure shape can be adjusted to improve the zero-order diffraction efficiency of transmission.

 

Fig. 3. Total and zero-order diffraction in the reflected and transmitted directions of DLRFM at 650 to 1200 nm (a, b) and RFM at 650 to 800 nm (c, d), respectively.

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In addition, we have also verified the far-field radiation effect in the case of multi-angle incidence in the Fig. 4. Under the same incident light, we simulate the reflection and transmission of DLRFM under multi-angle incident light. In addition, in order to demonstrate the excellent effect of DLRFM, we have simulated PTTM under the same conditions as a comparison, and normalized the effect of PTTM based on Fig. S4 (see in Supplement 1). As shown in Fig. S4, as the incident angles varies from 0 to 30°, there are still clear diffuse reflection effect observed with non-distorted transmission phenomenon. It is proved that the corresponding effect of DLRFM will not be affected in the case of multi-angle incidence.

 

Fig. 4. The left side of (a-b) is the 2D far field pattern of PTTM and DLRFM at 0° incident angles at 762 nm; the right side of (a-b) is the 3D far field pattern at 0°, 10°, 20° and 30° incident angles at 762 nm incident light.

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We have also applied the same principle and designed a DLRFMs with larger size, and the parameter group is P = 600 nm, D1 = 150 nm, D2 = 240 nm, H = 500 nm, h = 50 nm. Wesimulate its reflection phase and far-field pattern at an incident angle of 0°, in Lumerical FDTD and CST studio suite, as shown in Fig. 5 (a) and (b) respectively. We can see the larger DLRFM still can achieve the phase difference which is close to 90°, and like the first DLRFM we designed, it can be seen that there are obvious diffuse reflection and distortion free transmission effects from the multi-beam of the far-field pattern reflection part and the only remaining main beam in the transmission direction. Figure 5 (c) and (d) show the total reflection and transmission and the zero-order reflection and transmission of large DLRFM. It can be found that at the target wavelength of 1450 nm, although the total reflectivity is very small, the zero-order diffraction in the reflection direction can also have a large reduction effect, accounting for only 12.5% of the total reflection, which proves the realization of diffuse reflection. Moreover, the zero-order diffraction in the transmission direction reaches 97.7%, which also proves that the transmitted main beam has no distortion. It demonstrates that the DLRFM with the larger size can realize the red shift of diffuse reflection and distort-free transmission effect in the near infrared band. Different structural specifications can be selected according to the requirements of operation wavelengths, proving the applicability of this structure in multiple bands. Moreover, it can be seen from Fig. 5 (c) and (d) that the transmission-reflection ratio of this large-size DLRFM is increased compared with the aforementioned small-size DLRFM. Therefore, it can be considered that the transmission-reflection ratio can be modulated by adjusting the geometric parameters of meta-atoms to achieve a wider range of applications.

 

Fig. 5. The modified large DLRFM structure. (a) Phase distribution diagram. (b) 3D far-field radiation pattern. Total and zero order diffraction in the direction of (c) reflection and (d) transmission.

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5. Conclusions and discussions

In conclusion, we have proposed a double-layer random-flip metasurface, combined with the reciprocity theorem of 2-bit coding and double-layer structure superposition, and optimized the array coding array by modified simulated annealing algorithm, which successfully realizes the effect of both diffuse reflection and reciprocally protected transmission in the near infrared bands with larger bandwidth. The diffuse effect is greatly optimized as the zero-order diffraction intensity in the reflection direction is significantly reduced. Moreover, the applicable wavelength range of the DLRFM can be redshifted by scaling the size of the meta-atoms and such proposed design of DLRFM can also be fabricated by layer-by-layer exposure of electron beam lithography which is well demonstrated in the literature. The overall transmission efficiency can be improved by using dielectric meta-atoms with low optical losses in the future. The advantages of better diffuse reflection control provide design solutions for anti-reflection optical component integrated in advanced optical and optoelectronic devices at near infrared bands.