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Abstract
Generating multiple local oscillator (LO) beams by beam splitters is a crucial aspect of large heterodyne array receivers operating at terahertz (THz) frequencies, with over 100 pixels. Metasurfaces have received considerable attention due to their unique and flexible wavefront modulation capabilities. Nevertheless, the design of beam-splitting metasurfaces faces significant challenges in increasing the number of diffraction beams, improving power efficiency, and achieving greater homogeneity. A SA-GS-based design model for beam-splitting metasurfaces is proposed to achieve multi-beam with high power efficiency and homogeneity. As a proof of concept, we have designed and optimized a 16-beam splitting metasurface from 0.82 THz to 1.6 THz. The objective is to develop large-pixel THz multi-beam heterodyne array receivers and optical systems. The number of beams is also extended to 100-, 144-, 225-, and 289-beam configurations, with power efficiencies of 93.55%, 93.92%, 96.01%, and 96.18% at 0.85 THz, respectively. Moreover, the main beams exhibit excellent homogeneity. This model can be employed in the design of multi-beam metasurfaces with variable deflection angles and intensity ratios. Finally, the multi-beam splitting metasurface is fabricated, and the experimental measurement agrees with the simulation. This work presents an effective approach for the inverse design of beam splitters or holographic imaging devices.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The heterodyne array receiver includes a mixer and a local oscillator (LO) source, which is approaching quantum-noise limited sensitivity. To improve the imaging and observing efficiency, large-pixel heterodyne array receivers are necessary. The challenge to develop next-generation large arrays now is to generate a large LO array with high power efficiency and good homogeneity among individual beams. For generating multiple terahertz (THz) LO beams, two common approaches have been reported: solid-state waveguide multiple beams generator based on frequency multipliers driven by a common microwave source, and reflective phase gratings for generating multiple beams by diffracting a single THz source. Considering the challenges of providing suitable THz sources and the complexity of synchronizing the frequencies or phases of multiple individual sources, the phase grating approach is more beneficial and practical. Nowadays, there are many reports about LO power allocation techniques based on Fourier phase grating, and the working frequency has been extended from 345 GHz to 4.7 THz [1,2]. In addition, the number of beams has been increased from 2 × 2 beams to 10 × 10 beams [3,4]. These Fourier phase grating beam splitters have been successfully used for beam splitting of THz signals, where the diffracted beam direction can be freely regulated and the beam conversion efficiency is high. However, the size of the two-dimensional microstructure of the grating decreases with increasing operating frequency. The complexity of high-precision micromachining directly affects the performance of diffracted beams. At the same time, due to the limitation of role dispersion, it is difficult to further improve its working bandwidth.
Metasurface has a planar structure preparation process that is simple to process and facilitates modular integration [5–9]. It is obtained by periodically arranging cells of specific shapes and has electromagnetic modulation capabilities that natural materials do not possess. Similar to the working mechanism of phase grating, by rationally designing the wavefront phase of the metasurface, diverse functional devices can be obtained. In recent years, several works on beam-splitting metasurfaces have been reported, with the working frequencies extending from GHz to THz and visible region. The beam-splitting metasurfaces have been widely used in the design of polarization beam splitters [10–13], tunable beam splitters [14–19], multi-function beam splitters [20,21], and so on. The power efficiency, which is an important evaluation criterion for beam splitters, can also reach more than 90% [22–24]. In addition, the advantages of metasurfaces in increasing bandwidth have been demonstrated due to the high degree of freedom of the unit cells design, achieving higher than 50% of the working bandwidth [9,25–30]. However, existing metasurface-based devices still have significant deficiencies in the performance characterization of beam-splitting number and power homogeneity. To improve the performance of multi-beam THz heterodyne array receivers, it is necessary to further enhance the number of beams, power efficiency, power homogeneity, and working bandwidth. Therefore, a generalized model is needed to design beam-splitting metasurfaces quickly and accurately to generate more beams.
The far-field scattering pattern of the metasurface is mainly controlled by the wavefront phase. In previous designs of beam-splitting metasurfaces, phase distributions were often obtained using trial-and-error models. Obviously, this is not desirable in realizing more beams. In recent years, machine learning has shown advantages in solving phase distributions and is widely used in holographic imaging metasurface design [31–33]. The most commonly used model is the Gerchberg-Saxton (GS) algorithm, which has the advantage of achieving a predetermined complex imaging distribution. Based on the GS algorithm, metasurfaces have been designed to realize various far-field diffraction patterns. However, the randomness of the initial phase of the GS algorithm makes the image resolution low. Global optimization algorithms, such as Simulated Annealing (SA) algorithm and Genetic Algorithm (GA), etc., have been applied to the design of metasurfaces to further improve the design efficiency and accuracy. Therefore, by combining machine learning, it is expected to realize an efficient design model for beam-splitting metasurfaces.
The metasurface design is dominated by multi-focal lenses, digital coding, and the number of splitting beams is usually less than 8 [34,35]. The number of beams is only enlarged by a square factor through phase addition operations. The power difference between the beams is usually higher than 3% due to the error introduced by phase addition [28]. To improve the targeting efficiency, phase computation in the 2D direction is required. In this study, a design model for SA-GS-based multi-beam splitting THz metasurfaces is proposed. Through the combination of the SA and GS algorithms, arbitrary multi-beam splitting metasurfaces with more splitting beams, higher power efficiency and homogeneity can be quickly realized. More than 10 × 10 beams are simulated with over 93% power efficiency and maximum 0.351% power difference at 0.85 THz. The widest operating frequency of the multi-beam splitting metasurface reaches 64.46% from 0.82 THz to 1.6 THz, which can be applied in the LO of large array receivers. In addition, a metasurface with a splitting ratio of 1:0.52 and variable beam deflection angle is also realized. The experimental measurement results show the correctness of the model. This generic model is highly scalable and can be easily applied to achieve a more function of diffraction beams in the far-field.
2. Design model based on SA-GS algorithm
Amplitude and phase are key to the design of the metasurface function. For metasurface with a high reflection or transmission coefficient, the wavefront phase distribution determines the intensity distribution in the far-field. The GS algorithm is an efficient phase-solving model to retrieve the phase distribution (the metasurface) of the hologram plane (the reflected far-field plane). In our design, the field at the image plane can be treated as the Fourier transformation of the field at the hologram plane. Since the GS algorithm starts with a random initial phase, the results of the phase distribution are unstable, affecting the power efficiency and scattering main beams homogeneity of the beam-splitting metasurface. Therefore, it is difficult for general GS algorithms to realize metasurfaces with high efficiency and homogeneity beams while the number of beams increases.
In our model, the GS algorithm is combined with the SA algorithm to optimally obtain the optimal phase of the metasurface, as indicated in Fig. 1. The first step requires setting up the target far-field intensity distribution, consisting of an M × N matrix, representing the number of metasurface cells and far-field scattering plane pixels. For the multi-beam splitting metasurface design, the main beams can be set to different intensities such as A0, A1, …, An, …, Am, and other diffracted beams are set to 0. For the metasurface with homogeneity beam intensity, the main beams’ target intensities are set to 1. The second step is to retrieve the metasurface phase distribution based on the target far-field intensity. The GS algorithm uses the defined intensity at the hologram plane and the image plane to alternate between the two-dimensional Fourier transform and the inverse Fourier transformation at each iteration to improve the phase estimation. For the initial iteration, a random phase is applied to all diffracted beams. To improve the computational robustness and reduce the error, the SA algorithm is introduced to optimize the initial phase of the GS algorithm. The phase distribution obtained after 200 iterations of the GS algorithm is used as the initial value of each SA algorithm iteration. The initial temperature and the cooling rate are crucial parameters in the SA algorithm [36]. After optimization, the initial temperature is set to 360 °C and the cooling rate is 0.993. In each cooling, a new temperature and phase distribution is generated as
where, i is the current iteration. T is the temperature. c is the cooling rate. P is the phase distribution. Rand (M × N) is the random generation of an M × N matrix with values between 0 and 1. Therefore, as the number of iterations increases, the amount of phase variation decreases to find an optimized phase distribution. In the computational process, the error function is defined as the difference between the target intensity and the Fourier transform of the holographic plane, and the error is accepted with the Metropolis criterion. The error function is expressed as where, Itheory and Itraget are the holographic plane of theory and target, respectively. A smaller error means that the far-field plane intensity is close to the target and the phase recovery results are better. As the temperature decreases, the acceptance probability gradually increases and the results stabilize. The accepted new phase will be calculated as the initial phase of the GS algorithm. In our case, the above SA algorithm operation is repeated 1120 times and finishes the iterative process. As the temperature continues to decrease, the probability that the model will accept the error increases. After completing all the iterations, the optimal phase is obtained. The final step is to design the metasurface unit cells and arrange them according to the optimal phase. When electromagnetic waves are incident on the metasurface, the reflected multi-beam with predetermined intensity ratios and angles can be obtained.As a verification, the multi-beam metasurface phase distribution based on the above process is computed in terms of n × n = 4 × 4 beams. The metasurface and holographic plane sizes are taken to be M × N = 52 × 52, implying that the space is divided into 52 × 52 points. The intensity of the 16 pixels at the center of the target intensities are set to A = 1 and other intensities are set to 0. The main beams are spaced 1 pixel apart. Therefore, when M = N, the adjacent beam angle is 360° / M. And, the beams’ coverage angle is about n × (n-1) × 360°/M. Therefore, the theoretical angle between adjacent beams is calculated with 360° / 52 = 6.92°. The variation of the beam angle is achieved by changing the values of M and N. As shown in Fig. 2(a), the error stabilizes around 1.667 after 620 iterations, which is due to the acceptance probability becomes higher. This demonstrates that the robustness of the SA-GS-based model obtains the optimal phase. It takes only 1 minute to calculate a metasurface phase distribution. The optimal phase is shown in Fig. 2(b) and contains 52 × 52 cells. By Fourier transform, the theoretical far-field intensity is shown in Fig. 2(c). As a comparison, we also use a general GS algorithm to compute the 16-beam splitting metasurface phase distribution. The obtained theoretical far-field intensity result is shown in Fig. 2(d). It is worth noting that the corresponding error value reaches 8.836, which is higher than the model proposed in this paper. It can be seen that the phase calculation model for SA-GS proposed in this study has a better solution. Since there are acceptance probabilities in the SA algorithm, the result may be different for each calculation, but the effect is the same. Moreover, the main beams have a higher power efficiency and better homogeneity, and there is a significant reduction in the power of the other scattered beams.
Fig. 2. Calculation processing and theoretical far-field intensity of the 4 × 4 beams metasurface phase distribution. (a) The error during iteration. (b) The optimal phase distribution by the SA-GS algorithm. (c) The theoretical far-field intensity distribution of the SA-GS algorithm by Fourier transform. (d) The theoretical far-field intensity distribution of the general GS algorithm by Fourier transform.
3. Broadband THz multi-beam splitting metasurface
The key to a broadband working metasurface is the cell design. In this paper, a reflective metasurface is used as a simulation, as shown in Fig. 3(a), where the metasurface cell consists of metal-medium-metal. The metal is aluminum (Al), which has low loss in the THz band, and the dielectric is a flexible polyimide (PI) material. The PI has a dielectric constant of 3.1 and a loss angle tangent of 0.05. The top layer of the cell is a square patch that is rotated to achieve the Pancharatnam–Berry (PB) phase (±2θ) when incident with a left circularly polarized (LCP) wave or a right circularly polarized (RCP) wave. Ideally, the cells should act as a half-wave plate to convert the incident LCP/RCP beam into orthogonal output polarization. To achieve broadband modulation, the period of the metasurface cell is typically in the 1/2- to 1/4-wavelength range. The size of the square patch affects the operating bandwidth and polarization conversion efficiency. The optimized cell period p is 100 m, the PI dielectric thickness h is 35 m, the patch length l is 85 m and the width w is 15 m, and the metal thickness is 0.2 m.
Fig. 3. Design of the unit cells. (a) The reflection unit cell. (b) The reflection phase of 6 cells. (c) The cross-polarization (LCP-RCP) and co-polarization (LCP-LCP) amplitude of 6 cells respectively.
To simplify the calculation and verification process, the phase will be quantized to order 6. The cell is computed with periodic boundary conditions in the x and y directions and open boundary in the z direction for phase and reflectivity. The incident wave is LCP wave. As shown in Fig. 3(b), a 60° phase difference is generated when the rotation angles θ are -60°, -30°, 0°, 30°, 60°, and 90°. The cross-polarization (LCP-RCP) and co-polarization (LCP-LCP) amplitudes of the unit cells are shown in Fig. 3(c), where the curves coincide when θ is 0 and 90 degrees, and similarly for other θ values. The cross-polarization amplitude is above 0.8 and the co-polarization amplitude is below 0.2 from 0.82 THz to 1.6 THz. This shows that high reflectivity can be achieved using these cells and a wide bandwidth. The multi-beam splitting metasurface can be obtained by arranging the cell according to the optimal phase calculated by the SA-GS model in this paper.
The phase of Fig. 2(b) is quantized into Fig. 4(a) so that it contains 6 cells with a phase difference of 60°, comprising a total of 52 × 52 cells. The numerical simulations are performed using full-wave simulation software and open (add space) boundaries, and the far-field scattering pattern of the beam-splitting metasurface is shown in Fig. 4(b) at 0.85 THz. The simulation results are shown for the absolute value of the electric field in the direction of the electromagnetic wave reflection. When incident with LCP/RCP wave, 4 × 4 main beams can be obtained in the far-field space with very low sidelobe power. As an important parameter for analyzing the performance of beam-splitting metasurfaces, we define the power efficiency as
here, Pmain is the power of the main beams and Pfull is the total power over the full hemispherical plane. The numerical calculation shows that the power efficiency is 90.77% at 0.85 THz. The maximum difference between the main beams is 0.9 dB with good homogeneity. The adjacent beam angle is 6.9°, which is close to the theoretical predictions. The errors between the theoretical results and the simulation mainly appear in the meshing accuracy and artificial angle segment during the simulation. Through simulation verification, as shown in Fig. 4(c) and (d), the power efficiency is higher than 80%, and the maximum difference of the main beams is within 2 dB from 0.82 THz to 1.6 THz, which corresponds to a working bandwidth of 64.46%. The validation results demonstrate the feasibility of the multi-beam splitting metasurface design model based on the SA-GS model. The designed 16-beam splitting metasurface can be applied in large heterodyne array receivers to obtain high power efficiency of LO distribution.Fig. 4. The phase distribution and numerical far-field scattering pattern of 16-beam splitting metasurface after 6-order quantization, and its power efficiency and maximum difference over a wide bandwidth range. (a) The phase distribution after 6-order quantization. (b) The numerical far-field scattering pattern of 16-beam splitting metasurface at 0.85 THz. (c) The power efficiency from 0.7 THz to 1.7 THz. (d) The maximum difference between main beams from 0.7 THz to 1.7 THz.
The generation of multiple LO beams is a crucial aspect of large heterodyne array receivers (over 100 pixels). More beams are conducive to improved detection efficiency and range. When the number of splitting beams is larger, the power allocated to each main beam is smaller. As a result, the allowable error in designing the metasurface is much smaller, making it more difficult to design. In the proposed model, by expanding the values of M and N, the number of main beams can be further increased. We designed 10 × 10 beams, 12 × 12 beams, 15 × 15 beams, and 17 × 17 beams metasurfaces. Due to the high robustness of the model mentioned above, good power efficiency and homogeneity are maintained for more beams as well.
The optimal phase distributions in Fig. 5(a), (c), (e) and (g) are obtained after the SA-GS model and phase 6-order quantization, and the numbers of 10 × 10, 12 × 12, 15 × 15, and 17 × 17 beams splitters cells are M × N = 74 × 74, 88 × 88, 100 × 100 and 110 × 110, respectively. As shown in Fig. 5(b), (d), (f), and (h), the far-field scattering pattern is obtained. The numerical results are shown in Table 1 using full-wave simulation at 0.85 THz. The power efficiencies are 93.55%, 93.92%, 96.01%, and 96.18%, respectively, indicating that most of the power is allocated by the main beams. The maximum power differences between the main beams are 0.112%, 0.178%, 0.271%, and 0.351% with good homogeneity, respectively. According to the theoretical calculations, the adjacent beam angle of the 10 × 10, 12 × 12, 15 × 15, and 17 × 17 beams are 360° / 74 = 4.8°, 360° / 88 = 4.09°, 360° / 100 = 3.6°, and 360° / 110 = 3.27°, respectively. The simulated results are consistent with the theoretical calculations in Table 1. Four multi-beam splitting metasurfaces are summarized to demonstrate the advantages of the proposed design approach in improving power efficiency and homogeneity. This model can also be further extended to the design of metasurfaces with more beams. Moreover, by changing the values of M and N, as well as the positions and A values of the pixel points, it is possible to theoretically realize a multi-beam with arbitrary angles and intensity ratios.
Fig. 5. The phase distributions after 6-order quantization of 100 (a), 144 (c), 225 (e), and 289 (g) multi-beam splitting metasurfaces, respectively. The far-field scattering patterns of 100 (b), 144 (d), 225 (f), and 289 (h) multi-beam splitting metasurfaces at 0.85 THz, respectively.
Table 1. The performance of 10 × 10 beams, 12 × 12 beams, 15 × 15 beams, and 17 × 17 beams splitting metasurface at 0.85 THz
4. Variable splitting ratio metasurface
The beam-splitting metasurface with a variable splitting ratio can be assembled into the optical systems and thus assists in the realization of the convenient optical path design. The model proposed in this paper can be used to design the metasurface with a variable splitting ratio. The target far-field scattering intensity is defined as beams with different positions and values, as shown in the inset in Fig. 6(a). The ‘H’ and ‘L’ are used to denote two sets of beams with an intensity ratio of 1:0.5, located in the upper left and lower right of the space, respectively. In the target matrix, both M and N are 50 pixels. According to the above proposed SA-GS-based design model, the error is 3.4 after 1120 iterations in Fig. 6(a). And the optimal phase distribution can be obtained, which is shown in Fig. 6(b) after 6-order quantization. By Fourier transform, the theoretical far-field intensity demonstrates a low sidelobe and the average intensity ratio of two beam sets is 1:0.5, as shown in Fig. 6(c). The far-field scattering pattern shown in Fig. 6(d) is obtained by full-wave simulation of the metasurface. The intensity ratio of ‘H’ and ‘L’ is 1:0.52, which agrees well with the target intensity distribution. This small difference is due to phase quantization in the simulation. Therefore, the model proposed in this paper can also be used in more optical systems such as holographic imaging.
Fig. 6. Design and verification of variable intensity ratio and position beam metasurfaces. (a) The error iteration obtained by the SA-GS model (insert: the target far-field scattering intensity). (b) The phase distribution after optimization and 6-order quantization. (c) The theoretical results by the Fourier transform calculation. (d) The simulated far-field scattering pattern at 0.85 THz.
5. Fabrication and measurement of metasurface
The metasurface is comprised of a metal-dielectric-metal three-layer structure with a simple preparation process as illustrated in Fig. 7(a). A metallic aluminum layer is deposited on a silicon (Si) substrate to serve as a reflective backing. A 35 m PI is formed by spin-coating several times on the Al. The top metal patch is obtained through a standard photolithography process. We fabricate a 16-beam splitting metasurface with a + 1 m error in the metal patches, as shown in Fig. 7(b). For measurement, the signals are incident at an angle and are received in relative space. In the measurement, a signal source with a frequency of 0.85 THz is used, with an output power of approximately 50 µW. Since the output beam is a focused Gaussian beam, the output beam needs to be lens-coupled to form a beam with a width of about 15 mm before entering the metasurface. The reflected beam is then quantified by a pyroelectric detector, which scans in coordinates via a PC-controlled rotating platform. To determine the angular resolution of the system, a pinhole with a diameter of 2 mm is placed in front of the detector. The distance between the detector and the metasurface is set to 150 mm. The range of the scanning area is adjusted to effectively include the 16 diffracted beams. The measurement system is shown in Fig. 7(c).
Fig. 7. Fabrication and measurement of the 16-beam splitting metasurface. (a) The fabrication process. (b) The 16-beam splitting metasurface. (c) The measurement system.
When the beam is incident at 20°, the reflected beam is deflected accordingly as shown in Fig. 8. The intensity of the edge beams is reduced by the angle of incidence. The far-field scattering pattern is obtained by full-wave simulation, with a maximum difference of 2.2 dB between the main beams. And the 16 beams are tested using the measurement system and have a maximum difference of 2.9 dB, which showed good homogeneity. It can be seen that the designed metasurface permits a large angle of incidence and maintains high beam-splitting performance. Furthermore, the beams’ coverage angle is about 44°, close to the theoretical 48.44°. The difference between measurement and simulation comes from errors in the fabrication of the metasurface and the measurement system.
Fig. 8. The simulated and measured results at 20° oblique incidence. (a) The simulated results. (b) The measured results.
6. Conclusion
In conclusion, we propose an alternative model for multi-beam splitting metasurfaces based on the SA-GS algorithm to realize beams with homogenized or variable intensity ratios. The 16, 100, 144, 225, and 289 beams splitting metasurfaces have been designed according to the intended target far-field intensity. With the incident LCP/RCP wave, the predefined number of beams can be achieved in the range of 0.82-1.6 THz with 64.46% working bandwidth. This model has a power efficiency of more than 93% and good homogeneity in the design of metasurfaces with over 100 beams. By varying the number of pixel points in the target matrix, it is possible to realize a multi-beam with an arbitrary angle. In addition, this model can also be used to design metasurfaces with variable deflection angles and splitting ratios. We design a metasurface with a splitting ratio of 1:0.52 and the results are consistent with the target. Finally, the reliability of the model is validated through theoretical, computational, and experimental means. These multi-beam splitting metasurfaces can be applied in large-pixel (over 100 pixels) heterodyne array receivers. This study presents an exemplary model for the design of beam-splitting metasurfaces and offers a promising solution for electromagnetic modulation and diverse functional devices.
Funding
National Natural Science Foundation of China (62001235, 12273012).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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