A simple reflective metalens based on reverse design for an ultra-high-efficiency free space wavelength splitter

We propose two kinds of high-efficiency free-space wave splitters in the mid-IR band using reverse design. The wavelength divider based on the abnormal reflection principle realizes a beam-splitting angle of 22.00° and 10.92° by controlling the phase distribution, and the reflection efficiency of both wavelengths exceeds 50%. The wavelength divider designed based on the concept of metalens simultaneously accomplishes the functions of focusing and beam splitting. It has a focal length of 100 μm and a relative focal position of 100 μm. Most importantly, the focusing efficiency for the two wavelengths reaches an impressive 48.59% and 72.51%, respectively.

3][24][25] The former has higher working efficiency due to no inherent heat loss, and it is easy to combine electric and dipole resonance in a single structure.However, the unit volume of this structure is large, and its height is equivalent to the working wavelength, making it challenging to integrate and prepare. In addtion, because the realization of this structure is mainly based on the Mie scattering effect and phase superposition effect, it is difficult to combine the structure with modulation materials, such as VO 2 , 26) graphene 27,28) and ITO, 29) to achieve active regulation.In contrast, metal nanostructures are much smaller than the working wavelength.Moreover, the local resonance is controlled by inducing the surface current, and so it can be perfectly combined with the modulation material.However, low overall efficiency is the biggest problem for these devices composed of metal nanostructure units.
The design process of optical devices based on metasurface can be divided into the following four steps: determine the basic shape and initial parameters of the unit according to experience, scan the electromagnetic response curve of a single structure under different structural parameters, solve the phase and amplitude curve required by the device according to the target function, select the unit structure corresponding to the target phase and arrange it on a 2D plane to achieve the phase profile.For the traditional metal resonant cell structure, its resonant effect will cause apparent phase and amplitude changes at a particular band.On the one hand, this makes it possible to obtain a phase modulation range of nearly 360°in the scanning step.However, on the other hand, it also makes the amplitude corresponding to different phases fluctuate significantly, which will further affect the overall working efficiency of the device, especially for multi-wavelength devices.
In order to solve the problem of low work efficiency, many related studies have been proposed.Xu et al. 30) proposed a 3D chirality-assisted metasurface concept relying on integrated magnetoelectric meta-atoms.The metasurface can independently control the phase and amplitude by controlling the mutually twisted three-layer integrated resonator, thus realizing a chiral-assisted high-efficiency metasurface.Zhu et al. 31) proposed a novel bilayer meta-atom design method based on a periodic unit cell, which can achieve all phase control and transmission efficiency up to 85%.Phan et al. 32) innovated in design methods, which can extend the high performance of topology optimization devices to the macro region in a computationally efficient way to achieve high efficiency and a large metasurface area.Zhang et al. 33) also investigated high-efficiency phase control Si metasurfaces based on aperiodic nanoarrays.
Nevertheless, these methods have certain shortcomings.On the one hand, extending the design freedom by adding layers will increase the difficulty of subsequent manufacturing and processing and the volume of the metasurface.In addition, most of the research still depends on searching the predetermined antenna element library, which is very timeconsuming and in many cases cannot find the element structure that meets the target requirements.On the other hand, topology optimization and aperiodic metasurface arrays are mainly focused on the design of dielectric metasurfaces, and the computational resources are enormous.This limits the application of functional metasurface in practical engineering.Therefore, inspired by the latest development of the computational inverse design method, 34) based on the strategy of inverse design we developed a high-efficiency metal metasurface element that can meet the requirements of any phase at two different wavelengths.The antenna structure mainly comprises 16 simple metal strips with the same width but different lengths, which can be more beneficial to the subsequent processing and manufacturing process and can save some optimization time.Furthermore, we designed two kinds of free space wavelength splitters to prove the value of the cell structure developed through reverse design in practical applications.Compared to the previous methods, the proposed design approach in this paper offers the advantage of achieving excellent wavelength demultiplexing functionality solely through the design of a single-layer antenna.This approach directly addresses the challenges associated with complex design schemes and stringent alignment requirements of multi-layer stacked metasurfaces.Moreover, the antenna structural design is simple and does not impose high demands on manufacturing precision, making it highly favorable for future fabrication processes.
The top view and cross-section of the designed foundation unit structure are shown in Fig. 1.It comprises, from top to bottom, a metal antenna structure, a dielectric layer and a metal substrate.The top antenna structure and metal substrate material is Au, and the thickness is set at 0.01 μm.Its characteristics can be expressed as follows: 35)  where the plasma frequency ω p = 1.37 × 10 6 rad/s and the collision frequency ω c = 4.07 × 10 13 rad/s.Al 2 O 3 is selected as the medium layer material, and its refractive index is set at 1.58.Al 2 O 3 is selected as the dielectric layer material, with the refractive index and thickness set at 1.58 and 0.5 μm, respectively.As an optimization object, the antenna structure is divided into 16 axisymmetric metal strips in the y-axis direction with a width of 0.178 μm.The length of the metal strip is l 1 ∼ l 8 from the center to each side, respectively.Moreover, to achieve the target amplitude and phase requirements as far as possible, we added a gap of g 1 ∼ g 8 in the middle of the metal strip to further increase the optimization variables.To ensure the correctness of the structural parameters, the length range of l i is set at 0 ∼ 0.30 μm, and the length range of g i is set at n i × l i , with n i set at 0 ∼ 1.In addition, the period p = 3.5 μm is determined through a combination of empirical values and scanning optimization of simple geometric structures.The purpose is to ensure that the unit structure obtained through inverse design can achieve any desired phase combination at the two target wavelengths.
The selection of this structure provides sufficient variables to obtain the corresponding structure for the target phase and amplitude requirements.On the other hand, the combination of simple structures is easier to manufacture compared to other complex shapes. 36,37)o achieve a high-efficiency photonic device based on a metasurface, it is necessary to ensure that each antenna unit has a target phase and can achieve high emissivity simultaneously.Therefore, we use the optimization algorithm to reverse design the antenna structural unit to achieve the above goals.Compared with the traditional method of scanning and fine-tuning the critical parameters of the design structure, the reverse design realized by the mathematical optimization algorithm can search more antenna structural units that can meet the requirements of the target, thus producing more excellent-performance metasurface devices.The widely used optimization algorithms mainly include the gradient descent algorithm, 38) genetic algorithm, 39) PSO algorithm 40) and binary search algorithm. 41)PSO is a robust stochastic evolutionary computation technique based on the movement and intelligence of swarms.At the same time, it also has the advantages of fast optimization speed and simple implementation, so it is used as the algorithm for reverse design in this paper.The whole optimization process is shown in Fig. 1(c).
(1) The structural parameters corresponding to 60 groups of different data are randomly generated within the specified range as the first-generation initialization structure.(2) The FOMs of each population are calculated by 3D-FDTD simulation.In the unit structure and subsequent overall metasurface simulation process, the light source is selected to be incident along the z-axis perpendicular to the plane of the metasurface unit, with polarization along the x-direction.A monitor is set behind the light source to detect the reflectivity and reflection phases.The boundary conditions and mesh accuracy are set at the periodic boundary and 10 nm, respectively.The construction of the FOM is crucial for the inverse design.After several trials, the FOM of the dual-frequency metasurface element is finally defined as, where λ represents the wavelength, P and R represent the reflection phase (in angular value) and reflectivity of the metasurface unit, respectively.P 1 and P 2 represent the target phase at 4.5 and 5 μm, respectively.According to Eq. ( 2), the performance of the metasurface unit is characterized by phase error and reflectivity.The weight coefficient η controls their relative influence on the final FOM.When the value of η increases, the difference between the final phase and the target phase is easier to reduce; when it decreases, the emissivity will be easier to improve.(3) The position and velocity of particles are updated according to the calculated FOM results, that is, the corresponding metal strip width and air gap width in the next-generation particle swarm.(4) Steps 2 and 3 are repeated until the optimal global position meets the minimum limit or reaches the set maximum number of iterations, and outputs each parameter corresponding to the optimal result.After the model and algorithm are built, the parameter of the metasurface element structure can be obtained by setting the target phase and weight factor in Eq. (2).
To verify the proposed model structure and reverse algorithm, we first optimize the single-wavelength metasurface unit to find a high reflectivity structural parameter that meets the phase of 90°at λ = 4.0 μm.In this process, we set the weight factor η to 0.5 to ensure that the phase and reflectivity meet the requirements of the target.The red curve in Fig. 1(d) shows the changing trend of the FOM in the optimization process of 100 iterations.As the number of iterations increases, the overall FOM gradually decreases until the optimal result (minimum value) is reached.It is worth noting that within the range of 80-100 iterations, the FOM value will remain the same, which also means that further increasing the number of iterations cannot continue to obtain better results.Figure 2(a) shows the final phase and reflectivity curve.Its phase has reached 90.64°, and the reflectivity is as high as 90.87%, fully meeting the requirements of the design target.
On this basis, we continue to carry out similar reverse designs and optimization for the dual-wavelength metasurface unit.For example, we set the priority of phase and reflectivity at the same level and set the target phase P 1 and P 2 at 0°and 90°, respectively.The maximum number of iterations is set at 100, and the minimum limit is set at 0.001.The blue curve in Fig. 1(c) shows the change curve of the FOM in the optimization process, and the FOM value set will gradually decrease to 10.57 with the continuous change of structural parameters.The final structural parameters are shown in Table I, and the phase and reflectivity curves are shown in Fig. 2(b).As expected, the phase reaches 0.01°and 90.61°at two wavelengths, respectively, and the reflectivity is more than 80.39%.
It can be seen from Fig. 2 that the output phase and reflectivity curves of the two structures obtained through reverse design are more gentle than those of traditional metal resonant structures.Furthermore, the maximum variation range at the mutation location is relatively small.In addition, the position of the target wavelength and the resonance position are ingeniously avoided so that high reflectivity can be obtained while meeting the target phase requirements.Therefore, the working principle of the unit structure obtained by reverse design must differ from that of the traditional metal resonant structure.
To analyze the proposed structure operating mechanism, we simulated its electric field distribution at 4.0 and 5.5 μm, respectively, as shown in Figs.2(c) and 2(d).As expected, irrespective of whether the distribution was at 4.0 and 5.5 μm, a large electric field like that of the traditional metal resonance structure did not gather on the whole structural surface.In contrast, only a tiny amount of electric field gathers at the edge of the metal strip, and this phenomenon becomes more slight when the incident wavelength is 5.5 μm.Then, we have a similar electric field analysis for other phase combination structures.Many simulation results show that the electric field distribution results of these unit structures are identical to those in Figs.2(c) and 2(d), and only a slight electric field resonance phenomenon occurs, only with different electric field gathering positions.Therefore, we judge that this slight electric field resonance phenomenon makes the designed unit structure meet the requirements of high reflectivity.At the same time, the different resonance positions endow the unit structure with the ability to realize any phase combination under dual wavelengths.The reverse design used in this paper is one of the best solutions to help us quickly find the structure that meets the above requirements.
After verifying the reverse algorithm of the two wavelengths, high reflectivity and arbitrary phase combination metasurface cells, we can customize the photonic devices, such as the free space wavelength divider.Currently, the realization methods of free space wavelength divider based on metasurface are mainly divided into anomalous beam deflection and metalens.
The metasurface based on abnormal beam deflection has been used in many critical applications, such as light detection and ranging.Its basic principle is to give the metasurface a constant phase gradient to obtain the beam deflection of the target angle.The required phase distribution to deflect the incident beam by θ is given by, 42) x k x x sin 2 sin , 3 0 0 where x is the relative position, k 0 is the wavenumber, λ 0 is the wavelength of the incident light, θ is the target incidence angle and x j ( ) is the phase of the desired position.With this principle, we can set the phase of the two wavelengths to the opposite gradient so that the two beams can be reflected according to their respective abnormal deflection angles to obtain an ideal free space wavelength divider.To realize the function of wavelength beam splitting, we first calculate the phase curve corresponding to the target splitting angle according to Eq. 3.Then, we introduce different phase curves into the reverse design and calculate the required metasurface units.Taking the phase gradients of +60°and −60°when λ = 4.0 μm and λ = 5.5 μm as an example, utilizing the aforementioned inverse design algorithm, we employed it to solve for the six metasurface unit structural parameters.The objective was to achieve phase distributions that simultaneously satisfy the conditions of 0°to 300°and 300°to 0°w ith a gradient of +60°and −60°when λ = 4.0 μm and λ = 5.5 μm, respectively.The reverse design results indicate that the phase errors of the metasurface units are all within 10°, and the reflectivity exceeds 67%.Finally, each metasurface element is arranged according to the specified phase gradient to form a new "metacell" for subsequent overall simulation.
We regard the metacell as a periodic structure for subsequent simulation in the overall simulation process.
The boundary conditions in the xand y-directions are set as periodic, and the boundary conditions in the z-direction are set as Perfectly Matched Layers, which is consistent with the element structural simulation.To prove that the designed structure can realize wavelength beam splitting at different angles, we simulated the metacells with phase gradients of ±60°and ±30°, respectively.The far-field electric field distribution results are shown in Figs.3(a), 3(b), 3(d) and 3(e).In the simulation process, when the incident light wavelength is 4.0 μm, its phase gradient is set at +60°and +30°, respectively.The corresponding theoretical calculation and simulation results of the beam deflection angle results are 10.98°, 5.47°and 11.02°, 5.45°, respectively.When the incident light wavelength is 5.5 μm, the phase gradient changes to a negative value, and the corresponding theoretical calculation and simulation results of the beam deflection angle are −15.18°,−7.52°and −15.26°, −7.53°, respectively.The maximum error between the theoretical results and the actual simulation results is only 0.08°, mainly because the unit structure obtained by reverse design can perfectly match the target phase gradient.The results also prove that the metacell composed of a reversely designed unit structure can perfectly realize the function of abnormal deflection at two wavelengths.This confirms that the method can meet the performance requirements of free space wave dividers by setting their respective phase gradients.
Subsequently, we conducted a more detailed simulation of these two devices to analyze their overall performance.When λ = 4.0 μm and λ = 5.5 μm, the reflectivity of these two metacells reaches 69.96%, 73.78% and 75.34%, 72.66%, respectively.In applications involving wavelength splitters, the deflection efficiency (or focusing efficiency) is another extremely important metric, which can generally be defined as the ratio of the power within the deflection angle (or focal  © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd position) range to the total emitted power: where P goal represents the optical power within the desired angle (or focal position) range, defined as the power confined within a circular region with a radius of three times the FWHM, and P total represents the total emitted optical power.Using these values, we calculated the deflection efficiency in the far field, which are 51.90%,51.18%, and 64.82%, 69.76%, respectively.This performance has surpassed most devices designed based on traditional metal resonant structures and is comparable to single-wavelength photonic devices based on dielectric metasurfaces.Furthermore, we also simulated and calculated the FWHM and extinction ratio of the two metacells.As shown in Figs.3(c) and 3(f), it is the section electric field distribution curve under the above wavelength and phase gradient.When the phase gradient is 60°, the FWHM of the two wavelengths reaches 2.69°and 1.89°, respectively, which can be reduced by increasing the number of cycles.The peak electric field strength of 5.5 μm is more significant, while the FWHM value of 4 μm is narrower, mainly due to the phase error and variable reflectivity corresponding to the two wavelengths.The same phenomenon occurs when the phase gradient is 30°.
Based on the simulation results, the extinction ratio of twophase gradients can be calculated to be 20.40,40.46 dB, and 23.06, 32.14 dB, respectively, which is sufficient to meet the practical application requirements of the wavelength divider.Furthermore, this method can be extended to the design of metasurfaces with different deflection angles.For example, by adjusting the phase gradient to 45°or 90°while keeping the other design steps unchanged, it is possible to achieve deflection angles of 19.54°and 39.73°, respectively.A metalens is a kind of photonic equipment that can simulate the lens focusing and imaging capabilities in the subwavelength thickness. 43,44)It can significantly reduce the physical size of optical equipment and has been applied to the Internet of Things system, optical sensors and achromatic lens arrays.The precise control of phase as a function of spatial position and wavelength is the key to focusing the light of different wavelengths on the target position.The phase distribution of the focusing metalens can generally be expressed as follows: 45) x where x and y are the spatial coordinates, λ is the incident wavelength and f is the lens focal distance.Using this formula and setting the focusing positions of the two wavelengths to different parameters, another kind of free space wavelength divider that can focus can be realized.Similar to the method shown in the previous section, we first set the focal length of the two wavelengths at 100 μm and set the focus positions x y , ( ) at 50 m, 0 m m m ( ) and 50 m, 0 m m m -( ) to obtain the corresponding phase distribution.In practical application, the phase distribution can be further adjusted according to the focal length and focus position of the target.The second step is to design the unit structure of each metalens according to the phase combination of the two wavelengths to obtain the best structural parameters that can meet the phase requirements.To achieve a high-performance wavelength divider, we set the area of the super lens at 300 × 300 μm 2 .Thus, we need to calculate the structural parameters of 86 × 86 metalens units.The theoretical calculated phase distribution of the central metalens unit (x = − 150 ∼ 150 μm, y = 0 μm) and the simulated phase distribution obtained from inverse design are illustrated in Figs.4(a) and 4(b), respectively.The phase error is so minimal that it is indistinguishable to observe any distinction between the two curves.Consequently, we have plotted the error curve in Fig. 4(c) to visualize the discrepancies.The simulation results show that the phase error of the metalens obtained by reverse design is less than 8°at both wavelengths.The reflectivity is more than 50%, as shown in Fig. 4(d), which fully applies to the high-efficiency ), respectively.This means that the focal lengths of the two wavelengths reach 100.14 and 99.74 μm, respectively, slightly different to the design value, mainly due to the thickness of the metalens itself.In addition, the distance between the two focal spots is 101.05 μm, and there is an error of 1.05 μm.The main reason is that some phase errors are inevitably introduced to ensure high reflectivity.Figure 5(c) shows the electric field intensity distribution curve at the focal position.The FWHM of these two spots reached 0.42 and 0.43 μm respectively, which reached the diffraction limit and is one of the advantages of metalens.Combined with the overall reflectivity curve in Fig. 5(d), we find that the focusing efficiency is 48.59% and 72.51%, respectively, when the incident wavelength is 4.0 and 5.5 μm.This focusing efficiency is basically at the same level as the dielectric metalens.Further calculation shows that its extinction ratio is 32.41 and 35.90 dB, respectively.Although the extinction ratio of the metalens is relatively low when the   032003-6 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd wavelength is 4.0 μm, it still meets the application requirements of the wavelength divider.
In summary, we propose a reverse design metasurface element structure based on a PSO algorithm.In addition, we theoretically design two different forms of dual-wavelength, high-efficiency free space beam splitters.First, the phase distribution of the metasurface is calculated based on the principle of abnormal reflection, and the corresponding unit structural parameters are optimized by reverse design to meet the requirements of phase and high reflectivity.The simulation results show that the designed device can perfectly realize the function of the beam splitter, and its beamsplitting angles are about 22.00°and 10.92°at different phase gradients.More importantly, the reflection efficiency of the device is more than 50%, which exceeds most of the metasurface device design based on resonant metal cells.Second, a wavelength divider is designed based on the concept of metalens.After reverse design optimization, the simulation results show that the device can achieve a beamsplitting function with a focal length of 100 μm and a phase focal length of 100 μm.After careful simulation and calculation, it is found that its focusing efficiency and extinction ratio reach 48.59%, 72.51%, and 32.41, 35.90dB, respectively, which fully meet the application requirements of the wavelength divider.This result provides an efficient approach to designing photonic devices similar to wavelength splitters.Furthermore, the method should be easily extendible to other operating wavelengths.

Fig. 1 .
Fig. 1.(a) Top view and (b) cross-section of the designed foundation unit structure.(c) Flow chart for reverse design of metasurface element structure based on particle swarm optimization (PSO) and finite-difference time domain (FDTD).(d) Figure-of-merit (FOM) change trend of a single-wavelength and dualwavelength metasurface unit structure in the reverse design process.

Fig. 2 .
Fig. 2. Reflection phase and reflectivity curves of a (a) single-and (b) double-wavelength metasurface structural unit.Electric field distribution diagram when the incident wavelength is (c) 4.0 μm and (d) 5.5 μm, respectively.

Fig. 3 . 5 ©
Fig. 3.When the phase gradient is ±60°, the far-field electric field distribution map with wavelengths (a) 4.0 μm and (b) 5.5 μm corresponding to the metasurface.(c) Section electric field distribution curve when the phase gradient is ±60°.When the phase gradient is ±30°, the far-field electric field distribution map with wavelengths (d) 4.0 μm and (e) 5.5 μm corresponding to the metasurface.(f) Section electric field distribution curve when the phase gradient is ±30°.

Fig. 4 .
Fig. 4. (a) Theoretical phase distribution and the simulated phase distribution obtained from inverse design at λ = 4.0 μm.(b) Theoretical phase distribution and the simulated phase distribution obtained from inverse design at λ = 5.5 μm.(c) Error between the simulated phase and the theoretical phase of the metalens.(d) Reflectivity curves corresponding to the designed structure.

Table I .
Structural parameters of metasurface elements obtained by the reverse design method.