High-efficiency focusing metalens based on metagrating arrays

Jia Shi; Guanlong Wang; Longhuang Tang; Xiang Wang; Shaona Wang; Cuijuan Guo; Hua Bai; Pingjuan Niu; Jianquan Yao; Jidong Weng

doi:10.1364/PRJ.542798

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Abstract

The flexible and precise control of wavefronts of electromagnetic waves has always been a hot issue, and the emergence of metasurfaces has provided a platform to solve this problem, but their design and optimization remain challenging. Here, we demonstrate two design and optimization methods for metagrating-based metalenses based on the highest manipulation efficiency and highest diffraction efficiency. The metalens operating at 0.14 THz with numerical apertures of 0.434 is designed by these two methods for comparison. Then, the metalens is fabricated with photocuring 3D printing technology and an imaging system is built to characterize the distribution of focal spots. With the highest manipulation efficiency, the metalens shows a focal spot with the diameter of

0.93 λ

and depth of focus (DOF) of

22.7 λ

, and the manipulation and diffraction efficiencies reach 98.1% and 58.3%. With the highest diffraction efficiency, the metalens shows a focal spot with the diameter of

0.91 λ

and DOF of

24.6 λ

, and the manipulation and diffraction efficiencies reach 94.6% and 62.5%. The results show that the metalenses designed by both methods can perform a filamentous focal spot in the sub-wavelength scale with a long DOF; simultaneous high manipulation and diffraction efficiencies are obtained. A transmission imaging manner is used to verify the imaging capability of the metalenses, and the measurements are satisfactorily congruous with the anticipated results. The proposed methods can stably generate focal spots beyond the physical diffraction limit, which has a broad application in terahertz imaging, communications, etc.

1. INTRODUCTION

With the frequency ranging from 0.1 to 10 THz, the terahertz band is a special band that can provide huge available bandwidth. The terahertz wave between microwave and infrared light contains abundant physical and chemical information, so that most biochemical molecules can be identified by their absorption [1 –4]. Therefore, the terahertz wave shows wide-ranging and prospective applications in communications [5 –9], imaging [10 –13], radars [14,15], non-destructive detections [16,17], and biosensing [18,19]. Due to the wavelength and characteristics of a terahertz wave, many natural materials show a relatively weak response to it. As the weight and volume of conventional elements are no longer sufficient to meet the demands of miniaturization and lightweighting, metasurfaces, planar artificial media, have become an alternative to conventional optical elements [20,21]. Metalenses are an innovative metadevice in modern photonics and optics based on sub-wavelength elements [22]. Metalenses can control the polarization, phase, and amplitude of light by changing the shape, rotation, and height of each unit cell, and therefore metalenses have the advantage of being smaller, lighter, and easier to integrate [23 –28]. The door for the miniaturization of optical devices has been opened by metalenses. In recent years, there has been a great deal of development by manipulating the incident light in sub-wavelength scale to control diffraction and manipulate the phase [29 –31]. Some remarkable works have been focused on the broadband operation frequency or long depth of focus, which ensures the excellent focusing performance [32 –34]. Their flexibility and excellent performance have led to a wide range of applications, such as optical tweezers, generating orbital angular momentum waves, and achromatic focusing [35 –42]. Limited by inherent Ohmic loss [43,44], metal metalenses have suffered from low efficiency in terahertz bands. All-dielectric metalenses provide a new solution for the design of terahertz devices due to the high efficiency in manipulating the electromagnetic wavefront [45 –50].

In this paper, we report two design and optimization methods for metagrating-based metalenses at 0.14 THz based on the highest manipulation efficiency and highest diffraction efficiency. These two metalenses are based on tunable metagrating arrays by angular deflection according to the generalized Snell’s law [51]. Each group of unit cells in a metagrating is designed with different diffraction periods and optimized in parameters space for the highest efficiency. Then, two all-dielectric metalenses designed by these two methods are fabricated with UV-sensitive resin by using photocuring 3D printing. The focal spot distribution is measured and their imaging capability is verified. According to the experimental results, both designed metalenses obtain a filamentous focal spot in the sub-wavelength scale with a long depth of focus (DOF), which is beyond the physical diffraction limit. It provides general methods for the design and optimization of high-efficiency metalenses.

2. PHYSICAL MODELING AND SIMULATION

A. Principle of Metagrating

It has been of great significance to realize the arbitrary regulation of electromagnetic waves in different bands. The asymmetric scattering directivity metagrating can deflect the incident electromagnetic wave by transforming it to a specified diffraction order. According to Ref. [52], the grating equation $P (n_{1} \sin θ_{1} - n_{2} \sin θ_{2}) = m λ,$ (1)where $P$ is the grating periodicity along the diffraction direction; $n_{1}$ and $θ_{1}$ are the refractive index of incident media and incident angle, respectively. Similarly, $n_{2}$ and $θ_{2}$ are respectively the refractive index of transmission media and transmission angle, $m$ is the diffraction order, and $λ$ is the wavelength of incident electromagnetic wave. The conventional metagrating with symmetric scattering patterns distributes power to different diffraction orders, and the same energy distribution is obtained for both positive and negative diffraction orders [53]. Different from the symmetric scattering response, the metagrating with asymmetric scattering patterns can break this symmetry. The energy of the rest of the diffraction order is suppressed, so that most of the energy is deflected to the desired diffraction order [54,55]. In this work, we combine diffraction metagratings with asymmetric scattering patterns to achieve an angle-suppressed effect, which breaks this symmetry so that the energy is no longer equally distributed into positive and negative diffraction orders. The unit cells are designed with asymmetric structure formed by multiple different cylinders. The metagrating array is as shown in Fig. 1(a), and it is filled with these unit cells as shown in Fig. 1(b) compactly. With a normally incident electromagnetic wave in the metagrating, the transmitted energy can be redistributed so that more energy is directed into the $T_{- 1}$ diffraction order. Each unit cell contains multiple cylinders with different sizes; $h$ and $d_{n}$ are the height and diameter of the cylinders, $g$ is the gap between the adjacent cylinders in the unit cell, and $P_{x}$ is the diffraction period that controls the diffraction order and bending angle of the electromagnetic wave. With the diffraction period increased from 2.1 mm to 114.3 mm, the bending angle of the incident electromagnetic wave is controlled from 82.9° to 1.1°, as shown in Fig. 1(c).

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$(a) The model of electromagnetic wave normal incidence into a metagrating array. (b) The schematic diagram of an asymmetric unit cell. (c) The bending angles of T−1 order with different diffraction periods Px. (d) The top view and (e) the section view of the schematic diagram of a metalens.$

Figure 1.(a) The model of electromagnetic wave normal incidence into a metagrating array. (b) The schematic diagram of an asymmetric unit cell. (c) The bending angles of $T_{- 1}$ order with different diffraction periods $P_{x}$ . (d) The top view and (e) the section view of the schematic diagram of a metalens.

To improve the efficiency of the metagrating array and decrease energy loss, metagrating arrays are arranged as densely as possible, which requires that the non-diffracting period $P_{y}$ should be set as small as possible [56]. The cylinders in the unit cell are controlled by the parameters $d_{n}$ , $h$ , $g$ , and adjusting these parameters can lead to different efficiencies. In our work, we designed four asymmetric metagrating arrays with different diffraction periods. The metalenses include these four periodic circular metagrating arrays from inside to outside, which is designed to focus the incident electromagnetic wave beyond the physical diffraction limit. The schematic diagram of the metalens is as shown in Figs. 1(d) and 1(e).

B. Design and Optimization of Metalens Based on Asymmetric Scattering Metagrating

For demonstration, the metalenses operating at 0.14 THz are designed with the focal length of 80 mm and aperture of 80 mm. Four metagrating arrays with different diffraction periods are arranged in the metalens. The desired diffraction order $m$ for metagratings is designed as $- 1$ , which means most energy is concentrated on the $T_{- 1}$ order. Then, the unit cells are optimized according to the two methods, which are the highest manipulation efficiency and the highest diffraction efficiency. The incident terahertz wave transmitted through the metagrating is split into different diffraction orders; the diffraction efficiency is defined as the amount of power deflected into a particular diffraction order divided by the incident power [57,58]. The manipulation efficiency is defined as the amount of power deflected into a particular diffraction order divided by the total transmission power transmitted through the metagrating [56,57]. For the diffraction order of $T_{- 1}$ , the manipulation efficiency is $T_{- 1} / T_{total}$ .

First, the bending angles and diffraction periods for each metagrating array are obtained by Eq. (1). By the design of the diffraction period, the unit cells are designed to deflect the incident terahertz waves to 24.23°, 20.56°, 16.04°, and 7.48°, so that all the unit cells can bend the incident terahertz wave to the desired focal spot to achieve high-efficiency focusing. Accordingly, the diffraction periods are 5.22 mm, 6.10 mm, 7.76 mm, and 16.47 mm. Figure 2 shows the simulated electric field distributions for a metagrating array with diffraction periods. The metagrating with the diffraction period $P_{x 2}$ is further used to analyze the efficiency and operating frequency window. Figures 3(a) and 3(b) show the manipulation efficiency and diffraction efficiency in parameters space at different operating frequencies. The results show that the operating frequency window can cover 0.10–0.18 THz with high manipulation efficiency and diffraction efficiency (manipulation efficiency $> 50 %$ , diffraction efficiency $> 35 %$ ). The transmission spectra of different diffraction orders $T_{0}$ , $T_{+ 1}$ , $T_{- 1}$ and total transmittance $T_{total}$ are shown in Fig. 3(c). The result shows that the metagrating achieves a higher manipulation efficiency in the operating frequency window within 0.10–0.18 THz than in other wavebands.

$The schematic simulated electric field distributions of bending incident terahertz waves to different angles with different diffraction periods of 5.22 mm, 6.10 mm, 7.76 mm, and 16.47 mm. (a) Px1, (b) Px2, (c) Px3, and (d) Px4.$

Figure 2.The schematic simulated electric field distributions of bending incident terahertz waves to different angles with different diffraction periods of 5.22 mm, 6.10 mm, 7.76 mm, and 16.47 mm. (a) $P_{x 1}$ , (b) $P_{x 2}$ , (c) $P_{x 3}$ , and (d) $P_{x 4}$ .

$(a) The manipulation efficiency and (b) diffraction efficiency of T−1 order at different frequencies for the diffraction period Px2 with different parameters h and g. (c) The transmission spectra of different diffraction orders of diffraction period Px2. (d) The manipulation efficiency and (e) the diffraction efficiency of diffraction period Px2 with different geometrical parameters Δd, g, and h at 0.14 THz.$

Figure 3.(a) The manipulation efficiency and (b) diffraction efficiency of $T_{- 1}$ order at different frequencies for the diffraction period $P_{x 2}$ with different parameters $h$ and $g$ . (c) The transmission spectra of different diffraction orders of diffraction period $P_{x 2}$ . (d) The manipulation efficiency and (e) the diffraction efficiency of diffraction period $P_{x 2}$ with different geometrical parameters $Δ d$ , $g$ , and $h$ at 0.14 THz.

Then, two optimization methods are demonstrated based on highest manipulation efficiency and diffraction efficiency. The geometrical structural design is optimized for each unit cell in parameters space. In the design of each unit cell, the diameters of the cylinders are designed as an arithmetic progression, and the difference is expressed as $Δ d$ . Considering the space size constraint, in the parameters space of $P_{x 1} - P_{x 3}$ , $Δ d$ varies from 0.5 mm to 1 mm, $g$ varies from 0.1 mm to 1 mm, and $h$ varies from 2 mm to 3 mm. In the parameters space of $P_{x 4}$ , $Δ d$ varies from 0.1 mm to 0.3 mm, $g$ varies from 0.1 mm to 0.5 mm, and $h$ varies from 2 mm to 3 mm. By optimizing the parameter space, the energy is mainly concentrated into the designed order, and the highest manipulation and diffraction efficiencies can be obtained. The diffraction period $P_{x 2}$ is used as a demonstration to analyze the effect of different geometrical parameters on the manipulation and diffraction efficiencies. The wavefront manipulation efficiencies in the parameters space are as shown in Fig. 3(d) and the corresponding diffraction efficiencies are as shown in Fig. 3(e). The results show that the highest manipulation efficiency can be obtained when $Δ d$ is 0.5 mm, $h$ is 2.7 mm, and $g$ is 1 mm. The highest diffraction efficiency can be obtained when $Δ d$ is 0.5 mm, $h$ is 2.6 mm, and $g$ is 0.9 mm.

Derived from the process of $P_{x 2}$ , the optimized geometrical parameters of other diffraction periods can be obtained, and are listed in Table 1. The substrate thickness of the unit cell is set as 1 mm, and $n$ is the number of cylinders contained by each unit cell. $d_{1}$ is the diameter of the smallest cylinder contained in each unit cell, and $Δ d$ is the difference between cylinder diameters. Figure 4(a) demonstrates the manipulation and diffraction efficiencies of the $T_{- 1}$ order of the metalens optimized with the highest manipulation efficiency, and Fig. 4(b) demonstrates that of the metalens optimized with the highest diffraction efficiency. The results show that the optimized manipulation efficiencies of these four diffraction periods are all higher than 94.2%. The highest manipulation efficiency reaches 98.1%; accordingly, the diffraction efficiency is 48.4%. The optimized diffraction efficiencies are all higher than 51.7%. The highest diffraction efficiency reaches 62.5%; correspondingly, the manipulation efficiency is 94.6%.Table 1.

Design Parameters of Each Diffraction Period with Two Optimization Methods

	Unit Cell	$P_{x}$ (mm)	$n$	$d_{1}$ (mm)	$Δ d$ (mm)	$h$ (mm)	$g$ (mm)
Metalens with the highest manipulation efficiency	$P_{x 1}$	5.22	2	1	0.6	2.6	0.7
	$P_{x 2}$	6.10	2	1	0.5	2.7	1
	$P_{x 3}$	7.76	2	1	0.7	2.7	0.8
	$P_{x 4}$	16.47	8	0.5	0.2	2.8	0.3
Metalens with the highest diffraction efficiency	$P_{x 1}$	5.22	2	1	0.6	2.5	0.5
	$P_{x 2}$	6.10	2	1	0.5	2.6	0.9
	$P_{x 3}$	7.76	2	1	0.7	2.5	0.5
	$P_{x 4}$	16.47	8	0.5	0.2	2.9	0.2

$(a) The efficiencies of designed unit cells with the highest manipulation efficiency and (b) the highest diffraction efficiency optimization methods. (c) The simulated electric field distribution of the metalens with the highest manipulation efficiency and (d) the highest diffraction efficiency.$

Figure 4.(a) The efficiencies of designed unit cells with the highest manipulation efficiency and (b) the highest diffraction efficiency optimization methods. (c) The simulated electric field distribution of the metalens with the highest manipulation efficiency and (d) the highest diffraction efficiency.

With optimizing structural design of the unit cells, they are arranged and bent circumferentially to form two separate metalenses. The designed metalenses have a diameter of 77.2 mm and a focal length of 80 mm, and the numerical aperture (NA) is 0.434. The overall electric field distribution diagrams in the $x - z$ plane are simulated with a normally incident terahertz wave with a frequency of 0.14 THz as shown in Figs. 4(c) and 4(d). Figure 4(c) shows the intensity distributions on the focal plane demonstrated by the metalens optimized with the highest manipulation efficiency, and Fig. 4(d) shows that with the highest diffraction efficiency. Two filamentous focal spots can be distinctly observed, and the peak intensity is located at 89.5 mm and 87.3 mm, respectively. The results indicate that the metalens with the highest manipulation efficiency shows a focal spot with the diameter of 1.84 mm ( $0.86 λ$ ) and DOF of 25.1 mm ( $11.7 λ$ ), ranging from 77.5 mm to 102.6 mm. With the highest diffraction efficiency, the metalens shows a focal spot with the diameter of 1.76 mm ( $0.82 λ$ ) and DOF of 24.9 mm ( $11.6 λ$ ), ranging from 76.0 mm to 100.9 mm.

3. EXPERIMENTS AND RESULTS

$(a) The microscope images of fabricated metalenses with the metalens designed with the highest manipulation efficiency and (b) the highest diffraction efficiency. (c) The schematic diagram of the scanning transmission system for the characterization of the metalens. (d) The normalized intensity distributions of the focal spot measured by the knife-edge method in x−y plane of the metalens with the highest manipulation efficiency and (e) the metalens with the highest diffraction efficiency. (f) The experimental measurement normalized intensity distributions of the depth of focus along z-axis of the metalens with the highest manipulation efficiency and (g) the metalens with the highest diffraction efficiency.$

Figure 5.(a) The microscope images of fabricated metalenses with the metalens designed with the highest manipulation efficiency and (b) the highest diffraction efficiency. (c) The schematic diagram of the scanning transmission system for the characterization of the metalens. (d) The normalized intensity distributions of the focal spot measured by the knife-edge method in $x - y$ plane of the metalens with the highest manipulation efficiency and (e) the metalens with the highest diffraction efficiency. (f) The experimental measurement normalized intensity distributions of the depth of focus along $z$ -axis of the metalens with the highest manipulation efficiency and (g) the metalens with the highest diffraction efficiency.

To characterize the focusing performance of these two metalenses, an imaging system is built as shown in Fig. 5(c). The terahertz source generates terahertz waves of 0.14 THz normally incident onto a beam expansion group, which is composed with two flat-convex lenses. The beam expansion group can expand the terahertz waves to the same aperture as the metalens, so that the terahertz wave is allowed to be normally incident onto the metalens and focused on the detector. The focal spots along the $x$ -axis and $y$ -axis are measured individually using the knife-edge method. Figures 5(d) and 5(e) show the measured optimum size of the focal spots with the two metalenses, respectively. It demonstrates that the focal spot is measured as $1.95 mm \times 2.04 mm$ ( $0.93 λ$ ) when the highest manipulation efficiency is obtained, and $1.92 mm \times 1.98 mm$ ( $0.91 λ$ ) when the highest diffraction efficiency is obtained. The results show that the diameter of the focal spot obtained by both metalenses is less than 2.04 mm. According to the Rayleigh criterion $D = 0.61 λ / NA,$ (2)the theoretical diffraction limit of the metalenses with a numerical aperture of 0.434 operating at 0.14 THz is calculated to be 3.01 mm. The result shows that the physical diffraction limit can be stably broken through by both methods. The DOF is measured by scanning detection in the $z$ -direction. Figures 5(f) and 5(g) show the measured DOF and a filamentous focal spot is obtained. Figure 5(f) shows that most of the energy of the metalens with the highest manipulation efficiency is distributed from 68.5 mm to 117.1 mm, with a DOF of 48.6 mm ( $22.7 λ$ ). Figure 5(g) shows that the metalens with the highest diffraction efficiency distributes most of its energy from 62.5 mm to 115.1 mm with a DOF of 52.6 mm ( $24.6 λ$ ). The measured energy peaks are respectively located at 86.1 mm and 82.8 mm. The results demonstrate that the measured electric field distributions agree well with the theoretical design. The metalens optimized with the highest manipulation efficiency generates a diameter of the focal spot 2.3% ( $0.02 λ$ ) larger than that of the other metalens, and the DOF reduces by 7.6% ( $1.9 λ$ ). The focusing performance of the designed metalens has been compared with some published works as shown in Table 3. It has shown that it is challenging to design a focusing metalens with the sub-wavelength scale focal spot and long DOF, and simultaneously retain high efficiencies in the terahertz band. In this work, our strategy has provided a technical roadmap for the design, optimization, and fabrication of high-performance focusing metalenses in the terahertz band.Table 3.

Performance Comparison with Some Published Results

Reference	Materials	Diffraction Efficiency	Manipulation Efficiency	DOF	Focal Spot Diameter
[42]	$Si / {SiO}_{2}$	47.2%	34.9%	$7.89 λ$	$1.11 λ$
[59]	Si	-	35%	$23 λ$	$\sim 2.3 λ$
[60]	Si	-	75.3%	$4.8 λ$	$\sim 0.97 λ$
[61]	$Si / {SiO}_{2}$	-	43.1%	$1.23 λ$	$\sim 1 λ$
[62]	$Au / {SiO}_{2}$	-	61.62%	-	$\sim 1.73 λ$
[63]	Graphene/Si	-	-	$\sim 8.54 λ$	$1.72 λ$
[64]	Si	-	95%	$\sim 3.2 λ$	$1.48 λ$
[65]	Si	-	$< 90 %$	-	$1.15 λ$
[66]	Si	-	60%	$\sim 21.3 λ$	$1.01 λ$
Metalens 1 (this work)	Resin	58.3%	98.1%	$22.7 λ$	$0.93 λ$
Metalens 2 (this work)	Resin	62.5%	94.6%	$24.6 λ$	$0.91 λ$

To further verify the imaging ability of these two metalenses, the 1951 United States Air Force (USAF) resolution test chart (Edmund, Positive Target) is used to verify the imaging ability. A transmission terahertz imaging system experimental configuration is shown in Fig. 6(a). The USAF 1951 test chart is placed at the focal spot, and the transverse and longitudinal regions with line width and spacing of 2 mm in the USAF 1951 test chart are imaged separately as shown in Fig. 6(a). Figures 6(b)–6(e) show the recovered transmitted terahertz image in experiments demonstrated by these two metalenses, and Figs. 6(f) and 6(g) show the longitudinal and transverse normalized distribution signal along with the white dashed lines. The line width and spacing of 2 mm, which is less than the wavelength of 2.14 mm, can be clearly observed from the results. It confirms that the metalens optimized by these two methods has the ability of penetrable imaging beyond the physical diffraction limit.

$(a) The experimental configuration of transmission terahertz imaging system. (b), (c) The transmitted terahertz image of the marked area of USAF 1951 test chart demonstrated by the metalens with the highest manipulation efficiency, and with (d), (e) the highest diffraction efficiency. (f) Normalized distribution of intensity along the white dashed line of the longitudinal and (g) transverse signals with the highest manipulation and diffraction efficiencies in the transmitted image.$

Figure 6.(a) The experimental configuration of transmission terahertz imaging system. (b), (c) The transmitted terahertz image of the marked area of USAF 1951 test chart demonstrated by the metalens with the highest manipulation efficiency, and with (d), (e) the highest diffraction efficiency. (f) Normalized distribution of intensity along the white dashed line of the longitudinal and (g) transverse signals with the highest manipulation and diffraction efficiencies in the transmitted image.

4. CONCLUSION

In this work, two metagrating-based metalenses are designed based on dielectric material operating at 0.14 THz with the NA of 0.434. We have provided two general solutions for design and optimization of a metalens based on the highest manipulation efficiency and the highest diffraction efficiency. The metalenses designed by both methods can perform a filamentous focal spot in the sub-wavelength scale with a long DOF. With the optimization by the proposed methods, the highest manipulation efficiencies are higher than 94.2%. The highest diffraction efficiencies are higher than 51.7%. The metalens with the highest manipulation efficiency shows a focal spot with the diameter of $2.04 mm \times 1.95 mm$ ( $0.93 λ$ ) and DOF of $22.7 λ$ , and the manipulation and diffraction efficiencies reach 98.1% and 58.3%. With the highest diffraction efficiency, the metalens shows a focal spot with the diameter of $1.92 mm \times 1.98 mm$ ( $0.91 λ$ ) and DOF of $24.6 λ$ , and the manipulation and diffraction efficiencies reach 94.6% and 62.5%. The sub-wavelength scale focal spot is obtained beyond the physical diffraction limit with a long DOF. By the effective geometrical design of the meta-atom in the parameters space, the operation bandwidth and the DOF also can be customized for desired operating frequency. But it may bring the decreased diffraction and manipulation efficiencies and large loss. The NA can also be customized according to requirements. The metalenses can be fabricated quickly, efficiently, accurately, and cost-effectively by photocuring 3D printing technology. The imaging capability is verified with a USAF 1951 test chart, and the spacing and line width of 2 mm can be clearly observed in the recovered terahertz image. Both proposed methods can realize the design of focusing metalenses with excellent performance; they can both obtain a sub-wavelength scale focal spot with long DOF simultaneously with high manipulation and diffraction efficiencies. The choice of methods is determined by the demand of transmission loss and focusing performance for specific applications. If the lower transmission loss is desired, the metalens with the highest manipulation efficiency is better to be chosen. Although the existing design process requires a long time, we are endeavoring to integrate inverse design into this methodology to enhance the speed of the design and optimization. In summary, this work provides a technical roadmap for the design, optimization, and fabrication of high-performance focusing metalenses in the terahertz band, and holds great promise for extensive applications in terahertz imaging, communications, etc.

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