Inverse design of sub-diffraction focusing metalens by adjoint-based topology optimization

Breaking the diffraction limit to realize imaging at the nanoscale is challenging in scientific research. Traditional sub-diffraction focusing metalens is obtained by arranging artificially selected unit cells, of which the design process is passive and complex. This paper brings up an inverse design idea of planar sub-diffraction focusing metalens based on super-oscillatory theory to solve these problems, starting from a desired focusing performance. The sub-diffraction focusing metalens is then obtained by iterative topology optimization with different initial structures. We introduce the adjoint-based topology inverse optimization into the structural design of sub-diffraction focusing metalens, which provides another way to design a sub-diffraction metalens for far-field unmarked super-resolution imaging. Based on this idea, we achieve a sub-diffraction focusing characterized by a focal radius of 0.75 times the Rayleigh diffraction limit, optimizing from a diffraction-limited focusing metalens. Moreover, focal radii between 0.63 and 0.73 times the Rayleigh diffraction limit are achieved by optimizing 11 sets of random initial metasurface structures with even no focusing performance. The results indicate that our method is independent of the initial structure distribution, which can be extended to the inverse design of other functional metasurfaces in imaging, lithography, and other fields.


Introduction
Resolution is the core criterion of the microscopic imaging system.It is the perpetual pursuit of scientists to break the diffraction limit and realize super-resolution imaging.Various super-resolution imaging techniques [1][2][3][4][5] have been proposed since the Abbe diffraction limit came out, including the near-field technologies based on evanescent field manipulation and the far-field technologies based on propagation field regulation.
The microscopic technologies based on near-field manipulation, such as optical near-field scanning micro-imaging [6] and superlens constructed with the silver film [7][8][9][10], are not constrained by the diffraction limit, but their application scopes are limited by the requirement of near-field operation.Super-resolution fluorescence micro-imaging technologies, such as stimulated emission depletion microscopy [11,12] and stochastic optical reconstruction microscopy [13,14], do not require the evanescent field but depend on the nonlinear response of fluorescent molecules, so they are mainly used in biological and medical fields.Planar lens based on diffractive optical theory provides a new way to achieve far-field unmarked super-resolution imaging [15][16][17].The imaging process requires neither an evanescent field nor a marked sample, which greatly expands the application fields of super-resolution imaging technologies.The sub-diffraction focusing metalens (planar lens with metasurface), has been extensively studied recently due to the superior ability of metasurface in manipulating the amplitude, phase, and polarization state of the light [18][19][20][21][22][23].Exploring the far-field unmarked sub-diffraction focusing and imaging technology has important value in various fields such as micro-nano processing [24], super-resolution imaging [15,16], high precision measurement [25], high-density storage [26], and telescopic system [27].
As a typical far-field unmarked approach, optical super-oscillatory [28][29][30] effectively breaks the diffraction limit.The essence of the super-oscillatory phenomenon is that the band-limited function of the optical field oscillates locally faster than its highest Fourier component, which can be achieved by exploiting specially designed metasurfaces to precisely modulate the interference of the transmitted optical fields.In this way, a focal spot beyond the diffraction limit in the far field is obtained.The commonly used methods to design the metasurface include forward design and inverse design.Herein, the forward design is start from the structure of metalens and the inverse design is start from the far-field focal spot.One of the forward design methods is optimization-free algorithms [31] proposed by Huang kun et al.It has clear physical correspondence but some deficiencies, such as too much prior knowledge, few optimization variables, and parameter precision may not be appropriate for all sub-diffraction problems.Another method relies on optimization algorithms such as particle swarm optimization (PSO) and genetic algorithm, but it is often accompanied by a unit-cell-based method [19,24,32], which is passive and time-consuming.Similarly, there are two ways in inverse design.The first is to get the focal plane electric distribution by using multiple prolate spheroidal wave functions to construct a sub-diffraction focal spot.Then the focal plane electric field is calculated by the vector diffraction theory with unknown transmittance of the metasurface.The transmittance can be further determined by equaling the calculated and the constructed focal plane electric fields [33].The second use the inverse diffraction of the constructed focal plane electric field to get the near-field electric field distribution of the metalens, whose amplitude and phase distributions can be met by arranging certain meta-atoms [34].The design flows of the two ways are clear but for the former, the phase and amplitude of the metasurface are modulated continuously and it is difficult to manufacture accurately, while it is challenging for the latter to find the meta-atoms that simultaneously satisfy the requirements of amplitude and phase.The above-mentioned methods can achieve the desired sub-diffraction focusing, but there are still obstacles in the way to inversely design metalens from the focal spot to the structure, including complicated design processes and inconvenient manufacturing.
To this end, this paper starts from the desired focusing performance of the metalens, exploits the adjoint-based topology optimization method, and automatically optimizes with different initial structures to obtain a sub-diffraction focusing metalens, which successfully realizes the transformation of metalens from artificial design to intelligent design.When the initial structure is a diffraction-limited focusing metalens, we achieve the desired sub-diffraction focusing by iterative optimization, where the radius of the focal spot is 0.75 times the Rayleigh diffraction limit.Further, we design 11 sets of pseudo-random initial metasurface structures, and most of them are not even focused, but they all achieve the desired sub-diffraction focusing after the topology optimization.As a result, the radii of the focal spots are between 0.63 and 0.73 times the Rayleigh diffraction limit.The proposed method can simplify the design process and does not depend on the initial structure distribution, which is expected to be applied in imaging, lithography, and other fields.

Target parameters and evaluation criteria
To our knowledge, the size of the focal spot is divided into three parts by the Rayleigh criterion (0.61λ/NA) and the super-oscillatory criterion (0.38λ/NA) [31], and the one below the Rayleigh criterion belongs to the sub-diffraction focal spot.The super-oscillatory focal spot can be obtained by setting the super-resolution factor below 0.62 and there is no theoretical limit to the size of a super-oscillation focal spot.The desired super-resolution effects can be achieved by a specially designed super-oscillation phase.The designed focal spot in this paper is between the two criteria, which can not only ensure proper focal spot size but also maintain low side lobe intensity.Table 1 shows the target parameters of designed sub-diffraction focusing metalens.The metalens is composed of TiO 2 nanopillars (n = 2.58) and a SiO 2 substrate (n = 1.47).We plan to design a super-resolution focusing metalens that can realize 0.7 times diffraction limit even under arbitrarily polarized illumination with a maximum side lobe suppression ratio of less than 0.2.
There are many important indicators to evaluate the focusing performance of a lens, such as full width at half-maximum (FWHM), the radius of focal spot (R), maximum side lobe suppression ratio (M), and focusing efficiency (η), etc. Herein, the focusing efficiency is defined as the ratio between the intensity of the selected area in the focal plane (three times the FWHM [35,36]) and the incident intensity, which is expressed as η = I focal r<3•FWHM /I in .The maximum side lobe suppression ratio is defined as the ratio of the maximum side lobe intensity in the point spread function (PSF) to the peak intensity of the central focal spot.

Inverse design by adjoint-based topology optimization
In this paper, the polarization-insensitive sub-diffraction focusing metalens based on the transmission phase is inversely designed through adjoint-based topology optimization.Polarization insensitivity is achieved by employing rotationally symmetric square nanopillars and forcing the metalens to be symmetric about the x = y axis.The inverse design process is as follows.As shown in figure 1, this paper starts from the ideal near-field electric field distribution of the sub-diffraction focusing metalens, where the near-field of the metalens refers to the position half wavelength away from the upper surface of the metalens.Herein, the ideal phase distribution of the near-field electric field of metalens adopts the phase obtained by PSO, which is the sum of the traditional focusing phase and super-oscillation phase.Then employs adjoint-based topology optimization to promote the iterative near-field electric field close to the ideal one, so as to obtain the in-plane structural distribution of the sub-diffraction focusing metalens.Finally, simulations are conducted by finite-difference time-domain (FDTD) solutions and the far-field optical field modulation performance of designed metalens is verified by the plane wave expansion method (PWEM) [37].
The vector diffraction theory is adopted to calculate the ideal near-field electric field distribution of planar metalens.Assume there are no reflections and no loss for the planar metalens, the ideal near-field electric field distribution of metalens with a y-polarized incident plane wave (E in = ŷE 0 ) is expressed as [38,39]: where θ = arctan( √ x 2 + y 2 /f ) is the angle between the pixel (x, y) and the +z axis in the cylindrical coordinates, φ = arctan(y/x) is the angle between the pixel (x, y) and the +x axis in the cylindrical coordinates, 1/ √ cos θ is the amplitude factor, and φ(r) is the ideal phase characteristic imparted by the focusing metalens.
Topology optimization is one of the widely used gradient-based approaches that can be combined with other algorithms, such as the adjoint method, boundary optimization algorithm, and machine learning.The gradient optimization methods based on the design of adjoint source can be applied in the optimization of other metasurface functional devices, such as beam deflectors [40,41], ultrahigh numerical aperture metalens [39], and high-efficiency extended depth-of-focus metalens [42].In large-scale electromagnetic simulations, the adjoint method can significantly reduce the number of simulations for calculating the influence of structural change on merit function [43].Combined with adjoint-based topology optimization algorithm and electromagnetic simulation software, we realize the intelligent inverse design of sub-diffraction focusing metalens.Figure 2 shows the implementation principle of the adjoint method in topology optimization.In the forward simulation, the beam is incident from the substrate of metalens and converges to the focal plane through the metalens.The output near-field electric field E(x) is the electric field motivated by a normally incident electric field and the incident light is y-polarized.In the adjoint simulation, the incident light is the backward-propagating adjoint source that depends on the ideal near-field electric field of metalens, and it is illuminated from the near-field region of the metalens.The figure of merit (FOM) of optimization is defined as the inner product of the output field E(x) and ideal field [39] f where E d is the ideal electric field distribution in the near-field region of metalens, and * represents the complex conjugate operation.The adjoint source in adjoint simulation is the derivative concerning the electric field When the in-plane structure of the metalens changes, the optimization gradient can be calculated by FOM with respect to the variation of dielectric constant (see S1, supporting information for details) where E fwd represents the electric field in forward simulation, E adj represents the electric field in adjoint simulation and x ′ represents the structure design region.Topology optimization starts from the initial structure and gets the local optimal through iterative updating.In each generation, the structure will be blurred, filtered and binarized to achieve greater freedom of optimization.The binarized parameters will increase with the increase of generations.Therefore, each pixel of the design area is represented by a value that varies continuously from 0 to 1 in the optimization process and is going to be binary eventually with 0 (air) or 1 (dielectric).The pixel sizes in the x and y directions are 5 nm.The condition for iteration termination is that the sum of the changeable value of the five consecutive FOM is less than 0.01 or the number of iterations reaches the preset maximum number of iterations of 100.In the FDTD simulation models, the convergence condition is set as 1 × 10 −4 .The metalens is illuminated by normally incident y-polarized plane waves and a circular perfect electrical conductor aperture is placed between the source and the metalens to limit the area of injection.The total simulation domain is 21 µm × 21 µm × 1 µm and the symmetric boundary conditions are used in both x and y directions to reduce simulation time but perfectly matched layers in the z-direction.It takes about 4 min in every iteration of topology optimization including the forward simulation and the adjoint simulation.The electric field distribution of the far-field focal plane is calculated by PWEM and all the simulations and calculations are accomplished on our workstation with Intel Xeon Gold 6256 CPU, 3.60 GHz, and 512 GB of RAM.

Results and analysis
In order to demonstrate the capability of the proposed method for inverse design, two kinds of initial metasurface structures are provided.One is a diffraction-limited focusing metasurface with a ring-like structure obtained by arranging unit cells according to the focusing phase.The other is a random metasurface structure without any focusing ability, where the dimension of each unit cell is random distribution.Figure 3 shows the evolution results of metalens from diffraction-limited focusing to sub-diffraction focusing.Figures 3(a) and (d) are the diagrams of a quarter representative structure of metalens before and after optimization, respectively.The positions of the inner rings of the optimized structure change obviously compared with the initial structure.Note some unit cells may be missing or sticking after binarization as it is a topological deformation for topology optimization.In order to facilitate fabrication, the design of the manufacturability constraint is needed [44].Figures 3(b) and (e) demonstrate that the optimized focal spot has been reduced and there are some weak side lobes around the main lobe.The emergence of the weak side lobes arises from that the designed sub-diffraction focal spot has not reached the condition of super-oscillation.Despite the focal spot size can be further compressed under the super-oscillation condition, the side lobe intensity will be increasing sharply, which will affect the field of view and focusing efficiency.Therefore, we only design a sub-diffraction focal spot under a comprehensive consideration.Compared with figures 3(c) and (f) indicates that the sub-diffraction focusing based on super-oscillatory phenomenon results in a needle-liked optical field with a longer focal depth.At this point, the location of the most intense light is not in the focal plane, because the nearly perfect destructive interference re-adjusts the distribution of the optical field at the preset focal plane.Meanwhile, the optical intensity is much weaker.In order to better illustrate the compression capacity of the focal spot by this method, we draw the one-dimensional radial PSF intensity distributions along the x direction before and after optimization.As depicted in figure 3(g), the radius of the focal spot is decreased from 0.5822 µm to 0.4523 µm, which reaches 0.75 times the Rayleigh diffraction limit.Next, we analyze the uniformity of the focal spot in the x and y directions.Figure 3(h) shows that the FWHM and the radius slowly decrease from the diffraction limit values to around 0.7 times the diffraction limit values as the number of iterations increases.At the end of the iterations, FWHM x = 0.4050 µm, FWHM y = 0.4301 µm and the deviation is 25 nm; R x = 0.4523 µm, R y = 0.4773 µm and the deviation is 25 nm.The deviations are due to the asymmetric focal spot originating from the increasing intensities of the Ex and Ez components of the far-field focal plane electric field of metalens [45], and they will become much more obvious for metalens with larger NA (see S2, supporting information for details).With regard to the focusing efficiency (figure 3(i)), the values in the x and y directions are similar, with a difference of about 0.98%.The performance of low focusing efficiency is due to the compression of the main focal spot by optimized sub-diffraction focusing metalens, which is essentially a process of modulating the energy distribution of the optical field in the focal plane.The decrease in intensity of mainlobe results in a decrease in focusing efficiency.So far, the desired sub-diffraction focusing is obtained from a preset diffraction-limited structure.
In inverse designs, one can gain the desired goal via optimization regardless of the initial structure.In the above optimization, the initial structure already has focusing capability, while in the following we exchange it with metasurfaces without any focusing ability to confirm the generality of the proposed method.Hence, the optimization is carried on with a random initial metasurface structure and the optimized results indicate that the realization of the sub-diffraction focusing seems to be independent of the initial structure distribution.Figure 4(a) displays a quarter of the random initial structure that has no ring-like distribution.The resulting far-field distributions have no obvious focusing phenomenon, as shown in figures 4(b) and (c).Although the field intensities are not uniform, they are symmetric on account of the good symmetry of the device (along the x-, y-, and x = y axes).As anticipated, from figures 4(d)-(f), the optimized metasurface structure achieves a focusing effect and a needle-liked optical field distribution.Next, the corresponding one-dimensional radial PSF normalized intensity distributions along the x direction before and after optimization are illustrated in figure 4(g) for quantitative analysis.It is obvious that optimization achieves the perfect transformation of the beam from disorder to focusing and dramatically enhances the optical intensity.Moreover, after optimization, the radius of the focal spot is 0.3973 µm, which reaches 0.66 times the Rayleigh diffraction limit.Figures 4(h) and (i) demonstrate that the optimized metalens has realized the desired sub-diffraction focusing.It is worth noting that there are some data anomalies before the 5th iteration, because these iterations have no significant focusing behavior to be able to calculate the FWHM and thus the focusing efficiency, so we artificially set a large FWHM of 10 µm until it could be calculated from the PSF curve.Similarly, at the end of the iterations, the values of FWHM (radius/focusing efficiency) along the x and y directions are different, where the deviation is 33 nm (30 nm/1.1%).
To further verify the universality of the topology optimization, we regenerate other 10 groups of pseudo-random metasurface structures to optimize and summarize the far-field optical field indicators of the optimized metalens (see S3, supporting information for details).The results show that the radii of the focal spots are between 0.63 and 0.73 times the Rayleigh diffraction limit, the FWHMs are between 0.72 and 0.81 times the Rayleigh diffraction limit, and the values of the maximum side lobe suppression ratio M are basically below 0.2.The above data indicates the good robustness of the topology optimization method for designing the sub-diffraction focusing metalens.
In order to compare with the traditional design method, a metalens is designed by PSO and unit-cell-based method under the same focusing requirements (see S4, supporting information for details).For convenience, it is abbreviated as PSO method in the following.Firstly, the radial phase jump points are optimized by PSO.Then the artificially selected unit cells are arranged according to the optimized phase profile.As shown in figure 5(a), the designed metalens structure is a symmetric ring-like distribution.Figures 5(b)-(d) show the corresponding far-field optical field modulation performance.From figures 3 and 4, the far-field optical field modulation performances of metalens obtained by topology optimization seem to be consistent with that obtained by PSO method.Further, the focusing performance indicators comparison of these methods is investigated in table 2. Among them, x/y represents the values along the x-or y-axis.The calculation of error is expressed as Error = (X value − X theoretical ) /X theoretical .The focusing performance indicators of sub-diffraction focusing metalens obtained by topology optimization are basically consistent with theoretical values and that obtained by PSO method, which demonstrates the feasibility of the proposed method.It can be seen that the metalens optimized from a random initial metasurface structure can achieve the same focusing effect as the PSO method.Importantly, the process to achieve the structural distribution of metalens is automatic for topology optimization, while the PSO method is passive.In addition, compared with theoretical values, the errors of the FWHM and R of metalens optimized from the diffraction-limited focusing metalens are comparatively obvious.It comes from the fact that the optimized metalens is locally optimal rather than globally optimal, meaning that the indicators could be further improved.Positively, the results of topology optimization make some indicators better.For instance, the maximum side lobe suppression ratio M is greatly reduced while other indicators have suffered, which also proves that the performance indicators of the focal spot are mutually restricted [46].
In Table 3, we also summarize the performance comparison in previous literatures.We use the same parameters as the references, where D is the diameter of metalens.The meanings of other indexes in the table remain the same as above.The first two rows of G are the ratios of the radius of the focal spot to the diffraction limit and the last two rows of G are the ratios of the FWHM to the diffraction limit.Under the same parameters, the values of η and G calculated by our method are basically consistent with the reference values.So the proposed adjoint-based topology method is effective in designing a sub-diffraction metalens.It is initiative optimization and can obtain a similar super-resolution effect compared with traditional PSO combined with the unit-cell-based method.
The FWHM, radius of the focal spot, and focusing efficiency are theoretically polarization-insensitive due to the high degree of structural symmetry.Figures 6(a)-(c) are polarization insensitivity verifications of three indicators of FWHM, the radius of the focal spot, and focusing efficiency from left to right, respectively.The major axis of the asymmetric focal spot also changes from the x-axis to the y-axis when the polarization state of incident light changes from the x-axis to the y-axis.Meanwhile, no matter the metalens obtained from the PSO method, the diffraction-limited focusing metasurface structure or the random initial metasurface structure, the numerical distributions of the three indicators in the direction of the long axis and the short axis both satisfy polarization insensitivity.Although the values are fluctuating with the change of polarization angles, they almost keep constant.For the curves with the same color, the indicators along the x and y directions are different, which illustrates the focal spot is asymmetrical.Compared with the theoretical values, the FWHMs of three metalenses vary within a range of less than 15 nm, 55 nm, and 25 nm, respectively.The radii of the focal spot vary within a range of less than 25 nm, 45 nm, and 25 nm, respectively.As a result, we have realized the desired sub-diffraction focusing optimized from random initial metasurface structures, which can meet the focusing requirement even under arbitrarily polarized illumination.
Figure 7 shows the simulated imaging performances of the designed metalenses.The patterns whose feature sizes are smaller than the diffraction limit are selected to better illustrate the super-resolution ability of the designed sub-diffraction metalens.The imaging results are obtained by the convolution of the target patterns and the PSFs of these metalenses.The two rows of figure 7 are the simulated imaging results of double holes which are symmetrical about the center of the plane with a center distance of 430 nm (0.71 times the diffraction limit), and a letter 'E' , respectively.Obviously, the two targets both cannot be distinguished by diffraction-limited metalens, while they can be clearly observed by the three sub-diffraction metalenses, which can strongly prove the practicality of the sub-diffraction metalens designed by topology optimization.

Figure 2 .
Figure 2. The technical realization of the adjoint method in the topology optimization process.The white plane is the near-field region of the metalens.Here, the metalens is only a schematic and does not correspond to the focal spot image.

Figure 3 .
Figure 3.The results of topology optimization from the diffraction-limited focusing metasurface structure.(a) Diagram of a quarter representative structure of the diffraction-limited focusing metalens.(b) The intensity distribution of focal spot.(c) The optical field distribution along the propagation direction.The white dotted line represents the focal plane.(d) Diagram of a quarter representative structure of the optimized sub-diffraction focusing metalens.(e) The intensity distribution of focal spot.(f) The optical field distribution along the propagation direction.The white dotted line represents the focal plane.(g) The red and blue curves represent the one-dimensional radial PSF normalized intensity distributions before and after optimization, respectively.(h) The iterative curves of FWHM and radius along x and y directions.Among them, the solid lines represent radius and the dotted lines represent FWHM.The red, yellow, and blue curves represent the values of x direction, y direction, and theoretical, respectively.(i) The evolution of focusing efficiencies during the optimization process along x and y directions.

Figure 4 .
Figure 4.The results of topology optimization from the random initial metasurface structure.(a) Diagram of a quarter representative structure of metalens with random distribution.(b) The intensity distribution of focal spot.(c) The optical field distribution along the propagation direction.The white dotted line represents the focal plane.(d) Diagram of a quarter representative structure of optimized sub-diffraction focusing metalens.(e) The intensity distribution of focal spot.(f) The optical field distribution along the propagation direction.The white dotted line represents the focal plane.(g) The red and blue curves represent the one-dimensional radial PSF normalized intensity distributions before and after optimization, respectively.(h) The iterative curves of FWHM and radius along x and y directions.Among them, the solid lines represent radius and the dotted lines represent FWHM.The red, yellow, and blue curves represent the values of x direction, y direction, and theoretical, respectively.(i) The evolution of focusing efficiencies during the optimization process along x and y directions.

Figure 5 .
Figure 5.The light field modulation performance of sub-diffraction focusing metalens obtained by PSO and unit-cell-based method.(a) Schematic diagram of a quarter representative structure of the metalens.(b) Focal spot intensity distribution.(c) Light field distribution along the propagation direction.The white dotted line represents the focal plane.(d) The red and blue curves represent one-dimensional radial PSF normalized intensity distributions along the x and y directions, respectively.

Figure 6 .
Figure 6.(a) FWHM, (b) radius of the focal spot and (c) focusing efficiency of the metalens at different incident polarization angles.Among them, the solid lines represent the data along the x direction and the dotted lines represent the data along the y direction.The red, yellow, and blue curves show the results of the PSO method, optimization from the diffraction-limited focusing metalens structure, and optimization from the random initial metasurface structure.The black lines represent theoretical values.

Figure 7 .
Figure 7.The simulated imaging performances for the double holes and the letter 'E' .The white lines are the intensity distributions at the position of the green dotted lines.

Table 1 .
The target parameters of the sub-diffraction focusing metalens.
Figure 1.Schematic diagram of the inverse design process of sub-diffraction focusing metalens.

Table 2 .
Comparison of the focusing performance indicators for different design methods.

Table 3 .
Performance comparisons of metalens in different literatures.