Open Access
Add to BibSonomy
Abstract
Metalens with extended depth of focus (EDOF) can extend the mapping area of the image, which leads to novel applications in imaging and microscopy. Since there are still some disadvantages for existing EDOF metalenses based on forward design, such as asymmetric point spread function (PSF) and non-uniformly distributed focal spot, which impair the quality of images, we propose a double-process genetic algorithm (DPGA) optimization to inversely design the EDOF metalens for addressing these drawbacks. By separately adopting different mutation operators in successive two genetic algorithm (GA) processes, DPGA exhibits significant advantages in searching for the ideal solution in the whole parameter space. Here, the 1D and 2D EDOF metalenses operating at 980 nm are separately designed via this method, and both of them exhibit significant depth of focus (DOF) improvement to that of conventional focusing. Furthermore, a uniformly distributed focal spot can be maintained well, which can guarantee stable imaging quality along the longitudinal direction. The proposed EDOF metalenses have considerable potential applications in biological microscopy and imaging, and the scheme of DPGA can be promoted to the inverse design of other nanophotonics devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Depth of focus (DOF) is a crucial technical parameter of the lens, which determine the mapping area of the image plane. Through a lens with extended depth of focus (EDOF), a bright image of the object can be captured within a larger distance along the optical axis than that of a conventional focusing lens, which can significantly alleviate the rigorous requirement of refocusing for imaging systems. For instance, a refocus-free camera can be achieved by integrating with an EDOF lens. In the past decades, EDOF lenses have been demonstrated by using wavefront coding [1], then providing novel applications in imaging [2]. However, the bulky volume of traditional lenses brings tremendous difficulty in fabrication and integration. Recently, metasurfaces have attracted lots of attention due to their unprecedented abilities on controlling the wavefront properties of electromagnetic waves [3–7], such as amplitude [8–10], phase [11–14], polarization [15,16] and etc, by spatially arranging the two-dimension (2D) array of meta-atoms. Such unique characteristics of metasurfaces enable various applications, involving vortex beam generators [17,18], absorbers [19,20], and metalenses [21–25]. Among them, the metalens is regarded as a powerful alternative to the traditional optical lens. Compared to traditional bulky lenses, metalenses have significant superiority in practical applications due to their subwavelength thickness and lower fabrication difficulty. The growing demand for lowering the volume and complexity of imaging system sufficiently promote the research on subwavelength diffractive optical elements (DOEs) with EDOF property, including metalenses and multi-level lenses (MDLs). And experimental demonstrations of EDOF metalenses [26] and MDLs [27,28] exhibit novel applications, such as varifocal imaging [29,30], depth sensing [31], achromatic computational imaging [32,33], and light-sheet fluorescence microscopy (LSFM) [34].
However, the commonly used methods of designing EDOF metalenses have several disadvantages. For example, one kind of EDOF metalenses refers to radial modulation (RM) and angular modulation (AM) [26,35,36], such as light-sword optical elements [37] and axicon lenses [38] which depend on continuously variable phase profiles, resulting in enormous difficulty in the fabrication. The cubic metalens generates asymmetric point spread functions (PSFs), leading to blurry images [29,39]. The SQUBIC and log-asphere metalenses converge incident light into discretely distributed focal points to achieve EDOF [32,34], yet cannot hold a uniformly distributed focal spot and harm the axial resolutions. In general, all the above-mentioned cases lead to a trade-off between the DOF improvement and uniformity of the focal spot. Besides, these metalenses are all designed by a specific phase profile, which provides an analytic solution for a desired function, and such a method is called forward design. The forward design method relies on prior knowledge of the desired optical response and thus is unsuitable for non-intuitive problems. Furthermore, it is hard for manual searching to find an ideal structure when there is a large number of design parameters. Since it is still challenging to design an EDOF metalens with large DOF improvement while maintaining a uniformly distributed DOF, inverse design methods become great substitutes to deal with it. In particular, recent research [30] reported a 1.8 mm inverse-designed MDL reaches 1195 mm of extreme DOF, which provides an astonishing result about the inverse design of achieving a non-intuitive solution, and such a large DOF can enable some remarkable applications in 3D imaging.
Benefiting from the development of computer science and artificial intelligence (AI), inverse design methods have shown significant advantages in handling problems with huge data and a high degree of freedom [40,41]. In the inverse design, the desired function can be mathematically depicted as a figure of merit (FOM), which is then optimized by some intelligent optimization algorithms, such as evolutionary optimization, adjoint methods [42,43], and deep learning (DL) [44]. As one of the evolutionary optimization methods, genetic algorithm (GA) has been widely employed in scientific research and industrial circles during the past decades. Unlike the adjoint method such as topology optimization [43], the gradient-free property of GA makes it powerful for searching the global optimum in problems without analytic solutions. Evolving the design parameters by mimicking the natural selection of creatures, non-intuitive solutions with desired responses can be ultimately found by GA. As the GA-optimized metalens has shown significant improvement in the focusing efficiency than that of the conventional cylindrical metalens [45], and the GA-generated light-sheet [46] (cylindrical metalens) with lower signal-to-noise ratio has brought superiority over other EDOF optical patterns in the application of LSFM [34], many reports about the inverse design of metasurface based on GA [47–49] sufficiently prove that GA still has considerable potential applications in designing metasurfaces with remarkable performances.
By encoding the structure of the metalens into a gene sequence and representing the desired function as a fitness function, an inverse design strategy based on GA for designing EDOF metalens is provided in this paper, and the ideal metalens can be obtained through the iterative evolution of the GA process. Furtherly, since we find it difficult to search an ideal structure of the metalens through a single GA process due to the high degree of design freedom of metalens, a double-process genetic algorithm (DPGA) optimization is proposed, which is conducted by combining two independent GA processes with different mutation operators. Here, we first design a 1D cylindrical EDOF metalens with a 100 µm aperture size through the DPGA optimization and expand it to a 50 µm 2D EDOF metalens design by utilizing the rotational symmetry of 2D metalens. To characterize the optical performances of our inverse-designed EDOF metalenses, a comparative study of our inverse-designed metalenses and conventional hyperbolic metalenses is employed by full-wave simulation. The simulation results show that the DOF of inverse-designed 1D (2D) EDOF metalens reaches 76.4 µm (83.5 µm), which is about 17.3 (5.39) times higher than that of hyperbolic metalens. In addition, a uniformly distributed focal spot can be generated by both the 1D and 2D EDOF metalenses, while the point-like PSFs are maintained well by 2D EDOF metalens within the DOF range, which is at significant advantages than the existing EDOF metalenses made by specific phase profiles, such as cubic, SQUBIC and etc. The near-infrared EDOF metalenses designed in this paper enable promising applications in biological microscopy and imaging. Furthermore, DPGA provides a special strategy of making full use of the performances of different optimization operators to address the difficulties in finding the optimum result in a huge parameter space, which can be further promoted to the inverse design of other nanophotonics devices.
2. Design methods
2.1 Metalens design
The schematic diagrams of the metalenses and corresponding meta-atoms in this paper are shown in Fig. 1. Both the 1D and 2D meta-atoms are made by Silicon (Si) nano-antenna for the top layer and grass (SiO2) substrate for the bottom layer. In the 1D metalens, the meta-atom acts like a grating ridge, which is composed of a nano-stripe and the lattice equi-spaced from the square substrate along the x-direction, and the cross-section of the 1D meta-atom is represented in Fig. 1(a). As for the 2D meta-atom, a cylindrical nanopillar is arranged on the square lattice, which is shown in Fig. 1(d). The height of the nano-stripes (nanopillars) and the lattice period of the substrate are fixed as constant H = 600 nm and P = 450 nm, respectively. The wavefront modulation of the transmitted light for both the 1D and 2D meta-atoms is achieved by varying the dimensions of the top layer. As the width (radius) of a nano-stripe (nanopillar) is changed, an additional phase shift is introduced to the transmitted light. Complete structures of the 1D and 2D metalenses are represented in Fig. 1(b) and 1(e), respectively, which consist of arrays of the meta-atoms. Since the memory requirement and computing time of simulation immediately rise with the increasing size of the metalens, as a consideration for the balance of computing resources and practical application, the aperture size of the 1D (2D) metalens is set as 100 µm (50 µm). After all the fundamental parameters of the metalenses are determined, we perform a parameter sweeping via finite difference time domain (FDTD) method (Lumerical Inc.) for the meta-atoms to ensure that enough phase shift can be offered by meta-atoms. The relationships between the dimension and the phase shift of the 1D and 2D meta-atoms are plotted in Fig. 1(c) and 1(f), respectively. The illumination source is set as the 980 nm x-polarized plane wave. As can be seen, the phase shift covers 0-2π when the width (radius) of nano-stripe (nanopillar) is modified from 100 nm (50 nm) to 250 nm (150 nm), and the transmittances are all above 85% within the parameters range. After the parameter sweeping, a phase library of the meta-atom has been established.
Fig. 1. The structure diagrams and results of parameter sweeping for the 1D and 2D metalenses. The structures of the (a)1D and (d) 2D meta-atoms, both the 1D and 2D meta-atoms are composed of silicon (Si) and grass (SiO2); The complete structures of the (b) 1D and (e) 2D metalenses; (c) Simulated phase shift and transmission as a function of the width for a nano-stripe; (f) Simulated phase shift and transmission as a function of the radius for a nanopillar.
In the forward design method, we can spatially vary the dimensions of meta-atoms on the metalens to meet a specific phase profile, then realize target function, such as conventional focusing which is achieved by commonly used hyperbolic phase profile represented in Eq. (1):
2.2 Workflow of genetic algorithm
Since the conventional focusing has a limited DOF due to the diffractive limitation and there are no specific solutions to address the above-mentioned problems for the forward-designed EDOF metalenses. We consider an inverse design strategy based on GA to design the EDOF metalens. The design details of the GA process are discussed in the following content.
The flow chart of GA process in this paper is represented in Fig. 2(a). The whole process can be specified into 5 steps: (a) Population initialization, (b) Fitness computation, (c) Selection, (d) Crossover and (e) Mutation.
Fig. 2. The diagrams of implementation details for the GA in this paper. (a) The workflow of the GA process; the implementation details of the (b) crossover, (c) single-point mutation, and (d) multi-point mutation, the di,j means the j-th gene point of the i-th individual and the d’i,j means a new generated parameter; (e) The flowchart of double-process GA optimization, which is applied by adopting different mutation operators in the successive two GA processes.
As the first step of GA, it is important to encode the design parameters into a certain form suitable for GA implementation. In our problem, we treat the distribution of meta-atoms on a metalens as the gene sequence of an individual in the GA process, then the gene sequence for both the 1D and 2D metalenses can be set as a vector $\boldsymbol{x}$ which is represented in Eq. (2):
Here, ${d_i}$ denotes the width (radius) of the i-th nano-stripe (nanopillar) along one side of the 1D (2D) metalens and it ranges from 100 nm (50 nm) to 250 nm (150 nm). In other words, assuming that the dimension of the vector $\textrm{x}$ is n, there are total 2n-1 nano-stripes (nanopillars) along the x-axis. The complete structures of 1D and 2D metalenses can be constructed by symmetric flip and rotation operations, respectively. After the meta-atoms are encoded as the gene sequence, a population containing N of individuals is generated according to a specific initialization method. We emphasize that we adopt random initialization for the population of 1D metalens, but preset all the individuals in the population as hyperbolic metalenses with 100 µm focal length for the 2D metalens.
The quality of the individual in the population is determined by the fitness function during the GA process, which leads the evolution direction of the population. In order to make the optical field uniformly distributed within the target DOF range, the fitness function is set as the following Eq. (3):
In the selection step, the population is updated by random selection, and the selected rate of each individual in the population is calculated by the following Eq. (4):
In Eq. (4), $pr{o_i}$ means the possibility of being selected for the i-th individual, $fitnes{s_i}$ is the fitness value of the i-th individual which is calculated by Eq. (3), and N is the population size. Since excellent individuals in the population are easy to be lost during the iteration because of the random selection, an elite number k is set to avoid this problem. The best k individuals in the original population remain in the updated population and then the updated population will be expanded to the original population size by random selection. This operation can ensure that the population will not be led to a more terrible direction by random selection, which makes good help on the convergence of optimization.
Figure 2(b) illustrate the implementation details of the crossover step. In this step, two parents will be randomly chosen from the elite individuals, and then the gene sequences of the two parents are separately divided into four segments based on a randomly generated cross-point. The offspring will be created by exchanging the gene segments from the two parents. The crossover operator enables the excellent gene segments to be maintained for the next generation by passing it to the offspring. In other words, the partial dimension distributions of the parent metalenses will be exchanged by the crossover operator then leading to new combinations of the dimension distribution. The crossover operator makes great effects on keeping the diversity of the individuals.
The final step is the mutation, just like the gene mutation of creatures, the parameter at the mutation point will be randomly changed by a new one within the parameters range. This operator occupies a notable part of the complete GA process, which makes it possible to find better solutions when the population has been confined into a locally optimum region within the whole parameter space. In this paper, two different types of mutation operators are considered. They are called single-point mutation and multi-point mutation, respectively. As shown in Fig. 2(c) and 2(d), one but multiple mutation points will be randomly generated for single-point and multi-point mutation, respectively.
The complete GA process consists of the above-mentioned 5 steps. In our design, the steps (b-e) are repeated and then stopped until a 1D or 2D EDOF metalens with an ideal focal spot is achieved during the iteration.
2.3 Implementation of double-process optimization
Assuming that one individual has n parameters, the difficulty of searching the global optimum within the whole parameter space directly rises with the increase of n. Although sizes of the 1D and 2D metalenses have been limited to 100 µm and 50 µm, respectively, there are still high dimensions of the design parameters, resulting in a parameter space that is too large to explore the whole of it for GA. Besides, we cannot set a large population size which is enough to fulfill the whole parameter space because of the finite computational resources and times. Therefore, it is necessary to find a solution to alleviate the difficulty of searching global optimum in the whole data space. As mentioned before, we have proposed two mutation operators which are called single-point mutation and multi-point mutation, respectively. In the GA process with multi-point mutation, significant improvements will suddenly occur to the population while the population will suffer severe oscillations since the individuals have been widely changed. In contrast, the population develops slowly but steadily in the GA process with single-point mutation. As a result, we propose a scheme of combining two independent GA processes with multi-point and single-point mutation, respectively, which is mentioned as the DPGA. Figure 2(e) plots the workflow of the DPGA, we start the optimization via the first GA process with multi-point mutation, then the population can reach a local region that is nearer to the global optimum. Subsequently, the converged result of the first GA process is treated as the initial population for the second GA process with single-point mutation. The second process can further improve the population and an optimum result can be finally obtained at the end of the second GA process. It is worth noting that we confirm that the population has converged when the fitness value had no significant improvement for a long time. As can be seen in the following section, DPGA performs well in achieving our goals.
3. Simulation results and discussion
3.1 1D EDOF metalens
As mentioned before, we start the DPGA optimization via the GA process with multi-point mutation. The field distribution variations of the 1D metalens during the first GA process are plotted in Fig. 3. As shown in Fig. 3(a), there is no focused light in the farfield of the metalens due to the random initialization. Figure 3(b) and 3(c) plot the electric intensity distribution in the x-z plane of a typical individual from the population of 100th iteration and the 350th iteration, respectively. Comparing Fig. 3(a-c), a gradually emerged focal spot can be clearly observed in the farfield of metalenses, which means that the ideal metalenses are correctly screened by the GA evolution. When the number of iteration runs to 350, a focal spot seems to be maintained in the target DOF range. Furtherly, to demonstrate the different performances of the first and the second GA processes, the variations of the fitness value during the first and second GA processes are plotted in Fig. 3(d) and 3(e), respectively. It can be observed that the fitness value suffers sudden jumps during the first GA process. In the first GA process, the population travels over the parameter space with a large step due to the multi-point mutation, and the remained elite individuals ensure that the population will not evolve towards a more terrible direction. On the contrary, the single-point mutation provides a smaller search step, and the population can be steadily optimized in the current region, so the fitness value of the second GA process varies continuously. These operations make great effects on searching the global optimum in the whole parameter space.
Fig. 3. The field distribution variations of the 1D metalenses during the GA process with multi-point mutation, including the normalized electric intensity distribution in the x-z plane of a typical individual from the (a) 0th (initial population), (b) 100th, and (c) 350th population; the normalized fitness value as a function of the iterative number in the (d) first and (e) second GA processes, respectively.
As mentioned before, we get the final result at the end of the second GA process. To verify the DOF improvement, we make a comparative study on the inverse-designed metalens and hyperbolic metalens. The hyperbolic metalens has the same aperture (100 µm) and working wavelength (980 nm) as the inverse-designed one, and the focal length is set as 100 µm. The optical responses of 1D hyperbolic and inverse-designed EDOF metalenses are simulated and the corresponding simulation results are plotted in Fig. 4. As shown in Fig. 4(a), a significantly larger focal spot can be directly observed in the farfield of the 1D inverse-designed EDOF metalens, which makes a powerful proof on the DOF improvement. Figure 4(b) shows that the focusing characteristic of the EDOF metalens is distinctly different from that of the hyperbolic metalens, as the power has been spread uniformly over a large range. To better characterize the performance of our EDOF metalenses, we identify the DOF as the range along the optical axis, where the electric intensity is larger than 80% of the maximum [43]. Then we can get that the actual DOF of the 1D inverse-designed EDOF metalens has reached 76.4 µm, ranging from z = 74.8 µm to z = 151.2 µm, which is 17.3 times to that of the 1D hyperbolic one (DOF = 4.4 µm). Figure 4(c) plots the width distributions of meta-atoms of the EDOF and hyperbolic metalenses, the width distribution of the EDOF metalens has been completely changed through the inverse design but exhibits a gradient-varied distribution which is similar to the hyperbolic metalens. The intensity distribution along the x direction at z = 100 µm of the EDOF metalens shown in Fig. 4(d) perform nearly to the hyperbolic metalens, indicating the light has been effectively converged into the main lobe though the side lobes have been inevitably improved, and the corresponding full width at half maximum (FHWM) of EDOF metalens is 3.22 µm.
Fig. 4. The simulation results for the 1D hyperbolic metalens (right) and inverse-designed EDOF metalens (left), respectively. The normalized electric intensity distribution (a) in the x-z plane and (b) along the z-axis; (c) the width distribution of the meta-atoms; (d) the normalized electric intensity distribution along the x-direction at z = 100 µm.
The detailed simulation results of the 1D inverse-designed EDOF metalens are plotted in Fig. 5. Figure 5(a) represents the local and global views of the electric intensity distribution in the x-z plane, and the local view only focuses on the target DOF region (from z = 75 µm to 150 µm). As shown in the left of Fig. 5(a), a significantly extended focal spot is maintained well in the whole target DOF range, which strongly proves the effects of our DPGA optimization. Figures 5(b-d) separately show the electric intensity distributions at three different positions in the actual DOF range (start point, maximum intensity point, and end point). All of them exhibit a focusing state and most of the power is converged into the main lobe. Besides, the FWHMs and the focusing efficiencies within the DOF range are calculated. The focusing efficiency is defined as the ratio of the power within a circular area with a radius of three times the FHWM in the focal plane to the incident power. The minimum FWHM is approximately 1.46 µm, and the maximum and average focusing efficiencies reach about 31.8% and 30%, respectively.
Fig. 5. The detailed simulation results of the 1D EDOF metalens, including (a) local view (left) and global view (right) of the normalized electric intensity distribution in the x-z plane; the radial intensity distribution along the x-direction at z = (b) 74.8 µm, (c) 145 µm, and (d) 151.2 µm, respectively.
Off-axis aberrations of the 1D EDOF metalens are also tested. The electric intensity distributions in the x-z plane with different incident angles are simulated and plotted in Fig. 6(a-c), respectively. As shown in Fig. 6(a-c), when the light illuminates with 5°-15° of incident angles, the uniformly distributed focal spots, which are similar to the normal incidence, can still be maintained well. These results indicate that our 1D EDOF metalens can undergo a measure of deviations of incident angles.
Fig. 6. The normalized electric intensity distribution in the x-z plane of the 1D EDOF metalens with different incident angles of (a)5°, (b)10°, and (c)15°, respectively.
3.2 2D EDOF metalens
Then we analyze the effects of DPGA optimization in the inverse design of 2D EDOF metalens. The simulation results are plotted in Fig. 7. As mentioned before, we start the optimization for 2D metalens by presetting the initial population as hyperbolic metalenses with a 100 µm focal length. As the aperture of metalens reduces from 100 µm to 50 µm and the numerical aperture (NA) correspondingly reduces from 0.44 to 0.24, we can observe a larger DOF of 2D hyperbolic metalens in Fig. 7(a) compared with 1D metalens because of the inverse relationship between the NA and the DOF for hyperbolic metalens. Besides, a larger focal spot can also be observed in the electric field distributions of the 2D EDOF metalens from Fig. 7(a) and 7(b). The actual DOF of the 2D EDOF metalens is 83.5 µm covering from z = 59.2 µm to z = 142.7 µm, and corresponding NA ranges from 0.39 to 0.17. Nearly 5.39 times improvement of DOF to the hyperbolic one (DOF = 15.5 µm, covering from z = 91.4 µm - 106.9 µm) effectively demonstrate the feasibility of DPGA for the 2D situation. Although the actual focal point has changed via optimization, the significantly extended DOF range still covers the original focal point.
Fig. 7. The normalized electric intensity distributions (a) in the x-z plane and (b) along the z-axis for the 2D inverse-designed EDOF metalens (left column) and hyperbolic metalens (right column), respectively.
To demonstrate the DOF improvement and make further analysis on the focusing characteristics of our 2D inverse-designed EDOF metalens, the field distributions in the cross-section are simulated and corresponding results are summarized in Fig. 8. The phase profile of the 2D EDOF metalens is calculated by the interpolation, and a gradient-varied phase profile can be correspondingly observed in Fig. 8(a). Figure 8(b) provides a more clear view of the intensity distribution in the x-z plane. Figure 8(c) shows the electric intensity distributions in the x-y plane at six positions from z = 60 µm to z = 160 µm with an interval of 20 µm, and the corresponding electric intensities along the x-direction and y-direction are plotted by blue lines and red lines in Fig. 8(d), respectively. All the electric field distributions exhibit rotationally symmetric point-like profiles, and the corresponding radial intensity distributions fit as Gaussian-like curves, indicating that the focusing state is achieved within the target DOF range after optimization. The corresponding FWHMs are 1.18 µm, 1.80 µm, 3.55 µm, 3.33 µm, 3.50 µm and 3.66 µm, respectively, which can greatly improve the imaging quality. Furtherly. The focusing efficiencies within the DOF range are calculated, and the maximum efficiency reaches 32.8%, which is an acceptable result for the practical application.
Fig. 8. The detailed simulation results of the 2D EDOF metalens. (a) The phase profile (mod by 2$\pi $.) of the 2D EDOF metalens; (b) The local view of the normalized electric intensity distribution in the x-z plane. (c) The normalized electric intensity distributions in the x-y plane and (d) corresponding radial distributions along the x-direction (blue lines) and y-direction (red lines) at z = 60 µm, 80 µm, 100 µm, 120 µm, 140 µm, and 160 µm, respectively.
Finally, to make an intuitive comparison of our work to others, we summarize the crucial technical parameters of the inverse-designed EDOF metalenses reported in recent years and list them in Table 1. It is worth to be noted that most of the former research is concerned more about the visible spectrum due to its wide applications in real life, thus the selected material has a lower refractive index than Si. As can be seen in Table.1, our 1D metalens has outperformed on the DOF improvement and focusing efficiency to Ref. [42] with a nearly NA, and the 2D EDOF metalens in our research exhibits significantly larger DOF improvement than that in Ref. [42], meanwhile comparative to that in Ref. [33], though, with lower NA. Reference [44] reports a data-driven methodology for fast predicting the near-field responses of the scatters on the metalens, then significantly decreasing the simulation time of the whole lens, and the results are especially great. References [33] and [34] make experimental demonstrations of the inverse-designed EDOF metalens with a 1 mm aperture size, which put high demands on the fabrication technology and computational resources. Normally, there is an inverse relationship between the DOF and NA [29,30], a metalens with lower NA is easier to get a larger DOF. Just as the results are shown in our work and Ref. [44], the 2D EDOF metalens in Ref. [44] get a lower NA range and higher DOF (0.25 - 0.07, 250 µm) than that (0.39 - 0.17, 83.5 µm) in our work. But through comprehensively comparing all the results in the Table 1, we can observe that the relationship is not absolute in the free inverse design of EDOF meta-optics The main reason of resulting in such differences is that the different FOMs are carried out in our research, different FOMs lead the optimization move towards respective directions, then we will finally get unique results, respectively. In addition, it is worth to be noted that the aperture does also perform an important part in extending DOF, since more freedoms can be offered by a metalens with a larger aperture size. In summary, there are lots of influences to design an EDOF metalens, and all of them should be considered when we desire an EDOF metalens via the inverse design method. Furthermore, as the FOM (fitness function) occupies the most important part of the algorithm, we believe that the results in our paper can be further improved by optimizing it.
Table 1. Summarized parameters about inverse-designed EDOF metalens in the recent reports
4. Conclusion
In this paper, 1D and 2D EDOF metalenses operating at 980 nm are separately designed by the proposed DPGA optimization. We make a comparative study on the inverse-designed EDOF metalenses and hyperbolic metalenses, and the results show that the DOF of the inverse-designed 1D (2D) EDOF metalens with a 100 µm (50 µm) aperture size reaches 76.4 µm (83.5 µm), which is about 17.3 (5.39) times higher than that of hyperbolic metalens. Furthermore, our inverse-designed EDOF metalenses present a uniformly distributed focal spot in a large distance, respectively, meanwhile maintaining symmetric PSFs which is at a significant advantage over some existing forward-designed EDOF metalenses, such as SQUBIC, cubic and etc. The above results sufficiently prove the great effects of our DPGA in achieving our goal. The inverse-designed EDOF metalenses in this paper have potential applications in biological imaging and microscopy at near-infrared spectrum. Besides, the DPGA can provide well inspiration for the inverse design of nanophotonics.
Funding
National Natural Science Foundation of China (12272407, 60907003, 61805278, 62275269, 62275271); National Key Research and Development Program of China (2022YFF0706005); China Postdoctoral Science Foundation (2018M633704); Foundation of NUDT (JC13-02-13, ZK17-03-01); Natural Science Foundation of Hunan Province (13JJ3001); China Guangdong Guangxi Joint Science Key Foundation (2021GXNSFDA076001); Guangxi Major Projects of Science and Technology (grant No.2020AA21077007).
Acknowledgments
S.X designed the metalenses and wrote the scripts of the inverse design and manuscript; F.Z, D.W, and J.W helped with the writing and preparation of the manuscript and data processing; Y.W helped with the simulation of metalenses and data collections; X.H, H.C, Z.Z, and Y.Y contributed the literature researches; Z.Z, Z.Z, and J.Y contributed the literature researches, funding and preparation of the manuscript.
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.
References
1. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef]
2. Z. Zalevsky, “Extended depth of focus imaging: a review,” J. Photonics Energy 1, 018001 (2010). [CrossRef]
3. Y. Xie, W. Wang, H. Chen, A. Konneker, B.-I. Popa, and S. A. Cummer, “Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface,” Nat. Commun. 5(1), 1–5 (2014). [CrossRef]
4. N. M. Estakhri and A. Alu, “Wave-front transformation with gradient metasurfaces,” Phys. Rev. X 6(4), 041008 (2016). [CrossRef]
5. S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics 7(6), 1041–1068 (2018). [CrossRef]
6. K. E. Chong, L. Wang, I. Staude, A. R. James, J. Dominguez, S. Liu, G. S. Subramania, M. Decker, D. N. Neshev, and I. Brener, “Efficient polarization-insensitive complex wavefront control using Huygens’ metasurfaces based on dielectric resonant meta-atoms,” ACS Photonics 3(4), 514–519 (2016). [CrossRef]
7. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). [CrossRef]
8. L. Liu, X. Zhang, M. Kenney, X. Su, N. Xu, C. Ouyang, Y. Shi, J. Han, W. Zhang, and S. Zhang, “Broadband metasurfaces with simultaneous control of phase and amplitude,” Adv. Mater. 26(29), 5031–5036 (2014). [CrossRef]
9. J. Zhang, X. Wei, I. D. Rukhlenko, H.-T. Chen, and W. Zhu, “Electrically tunable metasurface with independent frequency and amplitude modulations,” ACS Photonics 7(1), 265–271 (2019). [CrossRef]
10. L. Bao, Q. Ma, G. D. Bai, H. B. Jing, R. Y. Wu, X. Fu, C. Yang, J. Wu, and T. J. Cui, “Design of digital coding metasurfaces with independent controls of phase and amplitude responses,” Appl. Phys. Lett. 113(6), 063502 (2018). [CrossRef]
11. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]
12. S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, M.-K. Chen, H. Y. Kuo, B. H. Chen, Y. H. Chen, T.-T. Huang, and J.-H. Wang, “A broadband achromatic metalens in the visible,” Nat. Nanotechnol. 13(3), 227–232 (2018). [CrossRef]
13. Y. F. Yu, A. Y. Zhu, R. Paniagua-Domínguez, Y. H. Fu, B. Luk’yanchuk, and A. I. Kuznetsov, “High-transmission dielectric metasurface with 2π phase control at visible wavelengths,” Laser Photonics Rev. 9(4), 412–418 (2015). [CrossRef]
14. X. Zang, Y. Zhu, C. Mao, W. Xu, H. Ding, J. Xie, Q. Cheng, L. Chen, Y. Peng, and Q. Hu, “Manipulating terahertz plasmonic vortex based on geometric and dynamic phase,” Adv. Opt. Mater. 7(3), 1801328 (2019). [CrossRef]
15. J. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017). [CrossRef]
16. Y. Hu, X. Wang, X. Luo, X. Ou, L. Li, Y. Chen, P. Yang, S. Wang, and H. Duan, “All-dielectric metasurfaces for polarization manipulation: principles and emerging applications,” Nanophotonics 9(12), 3755–3780 (2020). [CrossRef]
17. K. E. Chong, I. Staude, A. James, J. Dominguez, S. Liu, S. Campione, G. S. Subramania, T. S. Luk, M. Decker, and D. N. Neshev, “Polarization-independent silicon metadevices for efficient optical wavefront control,” Nano Lett. 15(8), 5369–5374 (2015). [CrossRef]
18. S. Tang, X. Li, W. Pan, J. Zhou, T. Jiang, and F. Ding, “High-efficiency broadband vortex beam generator based on transmissive metasurface,” Opt. Express 27(4), 4281–4291 (2019). [CrossRef]
19. X. Liu, K. Fan, I. V. Shadrivov, and W. J. Padilla, “Experimental realization of a terahertz all-dielectric metasurface absorber,” Opt. Express 25(1), 191–201 (2017). [CrossRef]
20. X. Jiang, H. Yuan, D. Chen, Z. Zhang, T. Du, H. Ma, and J. Yang, “Metasurface based on inverse design for maximizing solar spectral absorption,” Adv. Opt. Mater. 9(19), 2100575 (2021). [CrossRef]
21. A. Arbabi, Y. Horie, A. J. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat. Commun. 6(1), 1–6 (2015). [CrossRef]
22. W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13(3), 220–226 (2018). [CrossRef]
23. X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C.-W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun. 3(1), 1–6 (2012). [CrossRef]
24. E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9(1), 1–9 (2018). [CrossRef]
25. Y. Wang, M. Peng, W. Cheng, Z. Peng, H. Cheng, X. Ren, S. Zang, Y. Shuai, H. Liu, and J. Wu, “Manipulation force analysis of nanoparticles with ultra-high numerical aperture metalens,” Opt. Express 30(16), 28479–28491 (2022). [CrossRef]
26. X. Zang, W. Xu, M. Gu, B. Yao, L. Chen, Y. Peng, J. Xie, A. V. Balakin, A. P. Shkurinov, and Y. Zhu, “Polarization-insensitive metalens with extended focal depth and longitudinal high-tolerance imaging,” Adv. Opt. Mater. 8(2), 1901342 (2020). [CrossRef]
27. Y. Wu, Y. Rivenson, Y. Zhang, Z. Wei, H. Günaydin, X. Lin, and A. Ozcan, “Extended depth-of-field in holographic imaging using deep-learning-based autofocusing and phase recovery,” Optica 5(6), 704–710 (2018). [CrossRef]
28. F. Zhou, F. Zhou, Y. Chen, J. Hua, W. Qiao, and L. Chen, “Vector light field display based on an intertwined flat lens with large depth of focus,” Optica 9(3), 288–294 (2022). [CrossRef]
29. J. E. Whitehead, A. Zhan, S. Colburn, L. Huang, and A. Majumdar, “Fast extended depth of focus meta-optics for varifocal functionality,” Photonics Res. 10(3), 828–833 (2022). [CrossRef]
30. S. Banerji, M. Meem, A. Majumder, B. Sensale-Rodriguez, and R. Menon, “Extreme-depth-of-focus imaging with a flat lens,” Optica 7(3), 214–217 (2020). [CrossRef]
31. S. Colburn and A. Majumdar, “Metasurface generation of paired accelerating and rotating optical beams for passive ranging and scene reconstruction,” ACS Photonics 7(6), 1529–1536 (2020). [CrossRef]
32. L. Huang, J. Whitehead, S. Colburn, and A. Majumdar, “Design and analysis of extended depth of focus metalenses for achromatic computational imaging,” Photonics Res. 8(10), 1613–1623 (2020). [CrossRef]
33. R. Bayati, S. Pestourie, Z. Colburn, S. G. Lin, A. Johnson, and Majumdar, “Inverse designed extended depth of focus meta-optics for broadband imaging in the visible,” Nanophotonics 11(11), 2531–2540 (2022). [CrossRef]
34. Y. Fan, M. K. Chen, M. Qiu, R. J. Lin, Y. Xu, J. Wen, T. Tang, X. Liu, W. Jin, and D. P. Tsai, “Experimental Demonstration of Genetic Algorithm Based Metalens Design for Generating Side-Lobe-Suppressed, Large Depth-of-Focus Light Sheet,” Laser Photonics Rev. 16, 2100425 (2022). [CrossRef]
35. Z. Zhang, Q. Yang, M. Gong, M. Chen, and Z. Long, “Metasurface lens with angular modulation for extended depth of focus imaging,” Opt. Lett. 45(3), 611–614 (2020). [CrossRef]
36. G. Mikuła, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express 15(15), 9184–9193 (2007). [CrossRef]
37. Z. Zhang, D. Wen, C. Zhang, M. Chen, W. Wang, S. Chen, and X. Chen, “Multifunctional light sword metasurface lens,” ACS Photonics 5(5), 1794–1799 (2018). [CrossRef]
38. C. Chen, Y. Wang, M. Jiang, J. Wang, J. Guan, B. Zhang, L. Wang, J. Lin, and P. Jin, “Parallel polarization illumination with a multifocal axicon metalens for improved polarization imaging,” Nano Lett. 20(7), 5428–5434 (2020). [CrossRef]
39. S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for full-color computational imaging,” Sci. Adv. 4(2), eaar2114 (2018). [CrossRef]
40. S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018). [CrossRef]
41. P. R. Wiecha, A. Arbouet, C. Girard, and O. L. Muskens, “Deep learning in nano-photonics: inverse design and beyond,” Photonics Res. 9(5), B182–B200 (2021). [CrossRef]
42. E. Bayati, R. Pestourie, S. Colburn, Z. Lin, S. G. Johnson, and A. Majumdar, “Inverse designed metalenses with extended depth of focus,” ACS Photonics 7(4), 873–878 (2020). [CrossRef]
43. Y. Zheng, M. Xu, M. Pu, F. Zhang, D. Sang, Y. Guo, X. Li, X. Ma, and X. Luo, “Designing high-efficiency extended depth-of-focus metalens via topology-shape optimization,” Nanophotonics 11(12), 1 (2022). [CrossRef]
44. M. V. Zhelyeznyakov, S. Brunton, and A. Majumdar, “Deep learning to accelerate scatterer-to-field mapping for inverse design of dielectric metasurfaces,” ACS Photonics 8(2), 481–488 (2021). [CrossRef]
45. H. Cai, S. Srinivasan, D. A. Czaplewski, A. B. Martinson, D. J. Gosztola, L. Stan, T. Loeffler, S. K. Sankaranarayanan, and D. López, “Inverse design of metasurfaces with non-local interactions,” npj Comput. Mater. 6(1), 1–8 (2020). [CrossRef]
46. Y. Fan, Y. Xu, M. Qiu, W. Jin, L. Zhang, E. Y. Lam, D. P. Tsai, and D. Lei, “Phase-controlled metasurface design via optimized genetic algorithm,” Nanophotonics 9(12), 3931–3939 (2020). [CrossRef]
47. S. Jafar-Zanjani, S. Inampudi, and H. Mosallaei, “Adaptive genetic algorithm for optical metasurfaces design,” Sci. Rep. 8(1), 1–16 (2018). [CrossRef]
48. Z. Jin, S. Mei, S. Chen, Y. Li, C. Zhang, Y. He, X. Yu, C. Yu, J. K. Yang, and B. Luk’yanchuk, “Complex inverse design of meta-optics by segmented hierarchical evolutionary algorithm,” ACS Nano 13(1), 821–829 (2019). [CrossRef]
49. M. M. Elsawy, S. Lanteri, R. Duvigneau, J. A. Fan, and P. Genevet, “Numerical optimization methods for metasurfaces,” Laser Photonics Rev. 14(10), 1900445 (2020). [CrossRef]
