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Abstract
Conventional achromatic optical systems are matured to achieve effective chromatic aberration correction and diffraction-limited resolution by the multiple bulky lenses. The emergence of the super-oscillation phenomenon provides an effective method for non-invasive far-field super-resolution imaging. Nevertheless, most super-oscillatory lenses are significantly restricted by the chromatic aberration due to the reliance on delicate interference; on the other hand, most achromatic lenses cannot break the diffraction limit. In this article, a single-layer broadband achromatic metasurface comprising sub-wavelength anisotropic nanostructures has been proposed to achieve sub-diffraction focusing with a focal length of f=60 µm and a diameter of 20 µm in the visible ranging from 400 nm to 700 nm, which are capable of generating sub-diffraction focal spots under the left-handed circularly polarized incident light with arbitrary wavelength in the working bandwidth at the same focal plane. This method may find promising potentials in various applications such as super-resolution color imaging, light field cameras, and machine vision.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Conventional optical lenses are restricted by the diffraction-limited resolution and chromatic aberration. On the one hand, conventional optical systems suffered from the diffraction limit due to the cutoff of spatial frequency components, which can be defined as the Abbe diffraction limit 0.51λ/NA, where the NA is the numerical aperture of the optical system and the λ is the illumination wavelength [1]. Breaking the diffraction limit and realizing super-resolution imaging are of great significance to the advancement in the fields of physics, biology, material science, medicine, life science, and so forth. Past decades have witnessed tremendous efforts to break the diffraction limit, including exploiting the short-wavelength effect of surface plasmonic polaritons experimentally demonstrated by extraordinary Young’s interference [2–4]. However, these super-resolution methods are inevitably limited by complex near-field manipulations [5] or fluorescent dye labels embedded in samples [6]. Recently, an optical far-field super-resolution method based on super-oscillatory lens (SOL) has been proposed and applied to the unlabeled imaging field. Super-oscillation is a phenomenon where the band-limited functions oscillate much faster than the highest Fourier components [7], which originates from delicate interference of digital meta-surface-wave [8,9]. SOLs have been successfully applied to sub-diffraction focusing [10–16], super-resolution microscopy imaging [17–20], super-resolution telescope [21,22], supercritical microscopy imaging [23,24], optical hollow needles [25–27], optical trapping [28], high-density disk storage [29], and so on. However, most of them can only respond to a single wavelength or a narrow bandwidth range because the super-oscillatory light field is fragile albeit the slight changes.
On the other hand, chromatic aberration control is of great significance to the optical system. Conventional achromatic imaging systems with multiple bulky lenses lead to the increasing complexity of the equipment inevitably. Therefore, metasurfaces as flat optical components attracted adequate attentions with the miniaturization and high integration of the optical system. Benefiting from the precise control of metasurfaces at the subwavelength scale [30], many impressed phenomena and fantastic devices have been proposed continuously, including catenary optics [31,32], holography [33,34], vectorial beam [35–38], broadband spin Hall effect [39], and so on. As a typical application, metasurfaces comprising sub-wavelength spaced nanostructures have been proposed to eliminate the chromatic aberration over the continuous wavelength region through dispersion engineering. Recently, many impressive achromatic metasurfaces have been proposed to achieve broadband chromatic aberration control in the visible [40–45], near-infrared [46,47], mid-infrared [48,49], and terahertz region [50]. The methods for wavefront modulations of these fantastic achromatic multifarious devices are different, mainly including resonance phase, joint control of geometric phase and transmission phase, transmission phase, and so forth. Nevertheless, these devices cannot break the diffraction limit without exception, which restricts the applications to a certain extent [51,52].
Although SOLs have been successfully applied to achieve super-resolution imaging, the chromatic aberration still greatly limited the working bandwidth of the SOLs. In 2015, an ultrabroadband super-oscillatory lens composed of plasmonic metasurfaces was proposed to achieve broadband sub-diffraction focusing [12]. However, this method did not eliminate axial chromatic aberration. Subsequently, several feasible solutions had been proposed to solve the dispersion problem of SOLs. In 2017, a multi-wavelength achromatic super-oscillatory lens was proposed to achieve achromatic sub-diffraction focusing at 405 nm, 532 nm, and 633 nm [14]. Similarly, multi-wavelength achromatic super-oscillatory metasurfaces were proposed to realize achromatic sub-diffraction focusing in the visible. However, these methods can only achieve achromatic focusing at discrete wavelengths [15]. Meantime, a white-light super-oscillatory imaging system based on the metasurface filter was proposed to achieve broadband achromatic super-resolution imaging in the visible, but this method increased the complexity of the optical system inevitably [22]. Recently, a multi-wavelength achromatic super-oscillatory lens has been achieved within the bandwidth of 492 nm-592 nm [16]. However, this method cannot completely cover most of the visible light region.
Here, for the first time, a broadband achromatic super-oscillatory metasurface (BASOM) has been demonstrated to achieve achromatic sub-diffraction focusing under the left-handed circularly polarized (LCP) incident light in the visible. A lossless material with high refractive index, titanium dioxide (TiO2), was employed to achieve waveguide-like resonant mode within the working bandwidth 400-700 nm. The single-layer metasurface merging dynamic phase and geometric phase is designed by the optimized dispersion engineering technology. The simulation results of the proposed devices are consistent with the theoretical results, which further confirmed that the proposed method can be implemented to achieve broadband achromatic sub-diffraction focusing. This method is expected to further accelerate the development of full-color super-resolution imaging.
2. Design methods of BASOMs
The BASOM is composed of plentiful high-aspect-ratio TiO2 nanopillars with different sizes and different rotation directions deposed on a SiO2 glass substrate. Theoretically, BASOM can converge light at an arbitrary wavelength within the bandwidth to the same focal plane. As illustrated in Fig. 1, the proposed BASOM achieve achromatic sub-diffraction focusing through joint control of the geometric phase and the transmission phase. The geometric phase, also called Pancharatnam–Berry (PB) phase, originates from photonic spin-orbit interactions and is only related to the rotation angle of the anisotropic structure. However, the transmission phase (TP) is generated by the optical path difference during the transmission process of the electromagnetic wave, which is determined by the intrinsic properties of the nanostructures, including material and geometric parameters.
2.1 Design of the unit cell
In this article, CST Microwave Studio software was adopted to calculate the amplitude and phase response of dielectric nanopillars. In this study, as shown in Fig. 2(a), the anisotropic TiO2 nanopillars are placed on the hexagonal SiO2 substrate to increase the symmetry and reduce the optical coupling between the adjacent unit cells, the different geometric phases originate from the rotation angle(φ) of the unit cell, and the different transmission phases are generated by the nanopillars with different length(l) and width (w). The width and length were swept from 50 nm to 180 nm in steps of 5 nm. In Fig. 2(b), we selected a representative unit cell (l=150 nm, w=95 nm) and presented the amplitude and phase response curve of this unit cell. Here, the relevant parameters of TiO2 material are measured in the laboratory by spectroscopic ellipsometer which are the same as the TiO2 material used in the literature [51], and the refractive index of SiO2 is set as 1.46 for the convenience of calculation because the refractive index of SiO2 changes little in the visible. The optical field modulation of TiO2 nanopillars can be recognized as a truncated waveguide supporting for the Fabry-Perot resonance. As seen in Fig. 2(c), the waveguide-like cavity resonances show that the light field is strongly confined inside the dielectric nanopillars. Therefore, the optical coupling in the adjacent nanopillars is negligible because the high refractive index of the TiO2 material in the visible compared to the surrounding environment [41]. Moreover, as illustrated in Fig. 2(d), there are the phase responses of the unit cells with different sizes at several discrete wavelengths within the bandwidth. These phases folded in 0-2π can provide more options of nanostructures for the optimization of BASOM. The reason for the jagged boundary in the phase data space along the northeast direction in Fig. 2(d) is that the unit cells basically have no response of the PB phase at this time due to the isotropic structure. To prevent these structures from affecting the optimization results, these unit cells with relatively small cross-polarization amplitudes have been eliminated in advance. There will be more phase compensations by introducing more resonance units in a unit cell or increasing the height of the nanopillars appropriately. In this way, the BASOM can be designed as a wider bandwidth and can get a relatively more perfect phase matching. It is worth mentioning that this device has been demonstrated to be processed according to previous reports [53].
Fig. 2. The unit cell for a broadband achromatic sub-diffraction focusing metalens in the visible light region. (a) Schematic diagram of structure. The length(l), width(w), and the rotation angle(φ) of the nanopillars are changeable on the metasurface. The height(h) of all TiO2 nanopillars is fixed at 800 nm, standing on a SiO2 hexagonal substrate with side length p of 120 nm. (b) Phase response curve(blue) and amplitude response curve(red) of the unit cell (l=150 nm, w=95 nm) in the visible. (c) Normalized magnetic energy for the unit cell, the dotted line represents the boundary of the unit cell. (d) Phase data space of different length and width unit structures at different wavelengths.
2.2 Design of the BASOMs
According to the previous work [12], to achieve broadband super-oscillatory focusing, we design a SOL based on an ideal focusing phase which can be expressed as:
where λ represents the wavelength of the incident light and ${r} = \sqrt {{x^2} + {y^2}}$ represents the distance from arbitrary pixel (x, y) to the center (0, 0) of the designed metasurface. The preset focal length f of the BASOM devices designed in this article are all fixed as 60 µm. Furthermore, according to the Fresnel diffraction integral, the point spread function (PSF) of the system could be approximately expressed as [21]:In our design, the ideal phase of sub-diffraction achromatic focusing can be depicted as:
Fig. 3. Phase matching diagram. Optimized phase (dotted lines) and desired phase (solid lines) at (a) λ = 424.52 nm (b) λ = 444.66 nm (c) λ = 466.80 nm (d) λ = 491.26 nm (e) λ = 518.42 nm (f) λ = 548.77 nm (g) λ = 582.89 nm (h) λ = 621.53 nm (i) λ = 665.66 nm.
3. Numerical simulation and discussions
Here, the CST simulation software is used to calculate the optical field modulation effect of the designed devices under LCP incidence. According to the method proposed in the literature [34], we extracted the electric field distribution at a distance of 1µm from the exit surface of the metasurface, and then calculated the optical field distribution of the device along the propagation direction through the vector angular spectrum (VAS) theory to reduce the calculation time effectively. Meanwhile, the co-polarized part of the transmitted light had been filtered to eliminate the influence of the co-polarized light in the diffraction. Typically, BASOM1 is designed to produce a sub-diffraction focal spot about 0.8 times of the diffraction limit with the ratio M=0.2. To verify the achromatic super-oscillatory focusing effect of the device more intuitively, we selected several arbitrary wavelengths in the visible and gave the corresponding light field modulation effects. As shown in Fig. 4(a) and Fig. 4(b), it is evident that BASOM1 can produce a preset focusing effect within the bandwidth range. Moreover, BASOM1 produces a needle-like light field with an ultra-long depth of focus (DOF) within the working bandwidth. Theoretically, the phase of BASOM1 is composed of the binary super-oscillatory phase and the ideal focusing phase. On the one hand, the ideal focusing phase can be regarded as a quadratic phase under the paraxial approximation condition, and the long DOF originates from the superimposed slightly shifted focal spots produced by different discrete parts of the quadratic lens according to the metasurface-assisted law of refraction [54]. Meantime, the corresponding DOF will generally be longer when the NA is relatively small. On the other hand, optical super-oscillation originates from the superimposed multiple spatial frequencies, and the phenomenon of local rapid oscillating light field is formed under the modulation of the constructive interference or destructive interference. Therefore, the super-oscillatory light field tends to appear as long DOF or multifocal along the propagation direction, as described in the previous Refs. [25]. Although the achromatic behaviors are difficult to accurately judge from the needle-like light field, the achromatic performance can be demonstrated by observing the light field response of the device along the propagation direction, and it is not difficult to distinguish the achromatic performance in the region with short DOF located at 20-30 µm along the axial direction. Subsequently, we extracted the PSF of the optical system to calculate the generated sub-diffraction focal spot size accurately. According to the PSF curves calculated at the focal plane position, the simulated results shows the FWHMs of super-oscillatory focal spots are 1075 nm, 1135 nm, 1240 nm, 1332 nm, 1443 nm, 1522 nm, 1620 nm, and 1730 nm at 400 nm, 450 nm, 490 nm, 530 nm, 570 nm, 610 nm, 650 nm, and 690 nm, which are equal to 0.866, 0.813, 0.816, 0.810, 0.816, 0.804, 0.803, and 0.808 times of the Abbe diffraction limit, respectively. Besides, the ratios M on the entire focal plane at these discrete wavelengths are all lower than 0.22. The reason for the tiny differences at 400 nm is that some of the selected structures with larger duty ratios will produce a resonance phase. As shown in Fig. 2, the selected unit cell could produce a resonance within 400-430 nm, the guided mode resonance will have a negative impact on the broadband performance of the optimized device. At this time, the discrete optimization algorithm may have a relatively large difference between the adjacent wavelengths.
Fig. 4. The optical performance of BASOM1. (a) Light field distributions along the propagation direction (b) Intensity distributions at preset focal plane position z=60 µm (c) PSF of BASOM1 at the preset focal plane position.
To further demonstrate that this method can achieve a smaller sub-diffraction focal spot focusing, we designed BASOM2 to prove the versatility of the proposed method in this paper. At this time, we used the linear programming algorithm to calculate the radial π-phase-jump position as r1=0.154R, r2=0.315R, r3=0.481R, r4=0.725R, leading to 0.66 times of the Abbe diffraction limit. At this time, the M value of the designed metasurface on the entire focal plane is fixed as 0.3. Similarly, the CST simulation software and VAS theory are used to calculate the optical performances of BASOM2. As shown in Fig. 5, the light field distribution diagrams of BASOM2 in the propagation direction and focal plane position indicate that the device has achieved broadband achromatic focusing in 400-700 nm. According to the extracted PSF curves, BASOM2 can produce a smaller sub-diffraction focal spot at 60 µm in the broadband range. The focal spots generated at 400 nm, 450 nm, 490 nm, 530 nm, 570 nm, 610 nm, 650 nm, and 690 nm are equal to 0.75, 0.72, 0.70, 0.68, 0.69, 0.66, 0.67, 0.67, and 0.67 times of the Abbe diffraction limit.
Fig. 5. The optical performance of BASOM2. (a) Light field distributions along the propagation direction (b) Intensity distributions at preset focal plane position z=60 µm (c) PSF of BASOM2 at the preset focal plane position.
To quantify the optical performance of proposed BASOMs from 400 nm to 700 nm more clearly, several important properties of the devices are depicted in Fig. 6. In terms of the actual sub-diffraction focusing, there are many meaningful parameters, including FWHM, FOV, M value, focusing efficiency, Strehl ratio (SR), and so forth. Here, the focusing efficiency is defined as the ratio of the energy in the desired region (three times the corresponding FWHM) at the focal plane to the total energy of the incident light, and the SR is defined as the ratio of the central intensity of the super-oscillatory spot to the ideal Airy spot. According to the focusing performance calculated by the VAS theory, the average SR of BASOM1 and BASOM2 within the entire working bandwidth are 0.163 and 0.047, respectively. Although the SR and focusing efficiency within the working bandwidth of the device at the focal plane are generally very low due to the characteristics of the super-oscillation phenomenon and the polarization conversion efficiency, the intensity of the focal spots can be increased by increasing the energy of the incident light field simply. As shown in Fig. 6(a), the focal spots produced by BASOM1 are all about 0.8 times the diffraction limit, and the ratios(M) are stable at about 0.2. However, the super-oscillatory focal spots generated by BASOM2 are more likely to be fragile. Although the super-resolution effect produced by BASOM2 is basically stable around 0.66 times the diffraction limit, the intensity of the sidelobes has exceeded the preset M value (0.3) in most cases. The differences between simulation results and theoretical results can be summarized as the following reasons primarily. Firstly, the amplitudes produced by different unit structures are not uniform while the amplitude responses are assumed to be flat in the optimization. Although the matching degree of the amplitudes has a greater influence on the achromatic performance, these uneven amplitude distributions will cause some incompressible sidelobes and distinct differences compared to the ideal cases. If the super-resolution effect is further improved, the super-oscillation light field will become more fragile because the permitted error is smaller at this time. Secondly, when we used CST simulation software to calculate the full-wave simulation, the changes of phase modulations caused by the coupling effects in adjacent nanopillars are subsistent while they are ignored in the optimization. Thirdly, though the differences between the ideal phase and the optimized phase are negligible by the continuous iteration in the PSO algorithm, there are only a few discrete wavelengths selected in the algorithm. Therefore, we cannot guarantee the perfect phase matching at arbitrary wavelengths within the working bandwidth, although the phase response curves have little resonances which can minimize the errors in the broadband range. Finally, the phase shift caused by discrete sampling will also lead to a phase modulation error, and the discrete precision error is inevitable because the unit cells of the metasurface cannot be infinite, and the influence of this error on the light field modulation effect will become larger with the further compression of the super-oscillatory focal spot.
Fig. 6. FWHM, M value, and efficiency of the BASOMs. The value of FWHM in the figure represents the ratio of the FWHM of focal spot to the Abbe diffraction limit.
If the focal spot is further reduced, many characteristics cannot be satisfied at the same time, e.g., FOV, M, the central intensity of the focal spot, and so on. Although it has been proven that the confocal scanning method can be used to solve the difficulties caused by huge sidelobes in practical applications [19], the actual manipulations will be more complicated. Furthermore, the sidelobes can be compressed through optimization algorithms, which can be directly used in the imaging system as proved in the previous reports [55]. However, the super-oscillatory light field will become more fragile as the further compression to the width of mainlobes, leading to greater difficulties in the actual processing, because some minuscule errors may destroy the sub-diffraction focal spot. In general, it is imperative to improve the degree of phase matching and the uniformity of the amplitude distribution in this case. The relatively perfect phase matching can be achieved by introducing more resonators in the unit cell or increasing the height of the nanopillars directly, and the relatively uniform amplitude distribution can be realized by the symmetric polarization-insensitive unit cell. Moreover, this tougher optimization design process can be accomplished through topology optimization methods which have been widely emerged in the optical field in recent years [56,57]. The metalens optimized through topology optimization algorithm is designed as no intrinsic periodicity to maximize the degree of freedom at the selected plane. Theoretically, the phase modulations of the light field can also be regarded as the propagation phase to a certain extent. In this way, the topology optimization can be employed to achieve the BASOM with a large NA which requires more phase compensation. Finally, the proposed method in this study can also be employed to realize flexible light field modulations through polarization-multiplexed metasurface, and is no longer limited to the discrete working wavelengths in the previous work multi-wavelength achromatic polarization-multiplexed super-oscillatory metasurface, which can further improve the flexibility of the device.
4. Conclusion
In this article, a broadband achromatic metasurface has been proposed to achieve sub-diffraction focusing within the visible region 400-700 nm. The broadband achromatic sub-diffraction focusing has been realized through the joint control of the PB phase and transmission phase. The proposed BASOMs are optimized through the PSO algorithm, and the optical performances of designed metasurfaces are verified by CST simulation software. The simulation results calculated by the VAS theory indicate that the designed devices can achieve sub-diffraction achromatic focusing at the preset focal length, which are consistent with the theoretical results. We believe that the proposed method is expected to have great potential in the fields of super-resolution color imaging, computer vision, light field cameras, and so on.
Funding
National Natural Science Foundation of China (61675207, 61675208, 61822511); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019371).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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