1. Introduction

A conventional optical system typically suffers from the diffractive nature of light and cannot produce a spot with a size smaller than the Abbe diffraction limit of 0.5λ/NA, where λ is the wavelength of the illumination and NA is the numerical aperture. From the viewpoint of Fourier optics, the limited resolution is attributed to the cutoff of the spatial frequencies contributing to the plane of interest. Over the last few decades, several super-resolution methods have been demonstrated by exploiting and collecting non-propagating evanescent waves, which deliver higher spatial frequencies of an object into the imaging plane, such as the scanning near-field optical microscopy (SNOM) [1], superlens [2], and hyperlens [3]. However, near-field methods generally comprise of complex optical constructions and require precise operations. Alternative methods involve the fluorescence radiation manipulation [4–6]. A typical super-resolution microscope based on this technique is the well-known stimulated emission depletion microscope (STED) [4,7]. Although this method delivers high-resolution images for biological samples, it relies on dye labelling which poses limitations for certain types of samples, especially in non-invasive applications.

Recently, a far-field super-resolution technology based on the optical super-oscillatory phenomenon was proposed and applied in various optical systems, such as a confocal scanning microscope [8], a high density magnetic recording system [9] and a telescope [10]. This unique phenomenon occurs in the region where the band-limited function is able to oscillate much faster than its highest Fourier component [8–10]. Currently, super-oscillatory lenses (SOLs) contain complex structures to precisely manipulate the amplitude or phase of optical fields. Some examples include binary amplitude masks [8,9,11–13], phase-type metasurface masks [13,14], spatial light modulators (SLMs) [15,16], photonic quasicrystal lenses [17–21], among others. The delicate interference of transmitted or reflected light by these specially designed structures is essential to realize an optical super-oscillatory field. However, a slight disturbance of the light field can destroy the delicate interference, thereby resulting in a break of the super-oscillatory field, especially for the highly-compressed hotspot. Therefore, most previously reported SOLs operate in a narrowband wavelength regime and present a strong dependence on incident wavelengths [8–12,15–17,22]. Benefiting from the extraordinary capability of a metasurface to control the electromagnetic wave with thicknesses on the order of the wavelength [23–30], an ultra-broadband SOL was proposed to realize far-field sub-diffraction focusing [14]. The axial chromatic aberration, also observed in previously reported metasurface-based devices, could be removed through the combination of a metasurface-based filter and a commercial achromatic focusing lens [30]. However, bulky conventional lenses prevented the realization of a compact and light-weight optical system in a realistic application. Even though single-layer achromatic SOLs have been reported [13], their phase-type achromatic SOL was fabricated on a dielectric substrate. In this case, several incident wavelengths caused a change of the transmitted phase profile due to the inherent material dispersion. Therefore, the material dispersion must be carefully considered in the optimization process.

In this work, ASOMs are realized by employing metasurfaces composed of rotating high-aspect-ratio titanium dioxide (TiO₂) nanofins to precisely control the phase profiles of the transmitted light. In contrast to conventional optical elements exhibiting material dispersions, the proposed metasurface-based devices can provide dispersionless phase profiles for various incident wavelengths. Furthermore, simultaneous optimizations for three wavelengths can effectively remove the axial chromatic aberration appearing in previous metasurface-based devices [14,23–26]. In our demonstrative simulations, through the genetic algorithm (GA) optimization to control multi-objective and multi-constraint optimization models, ASOMs at three visible wavelengths of 473 nm, 532 nm and 632.8 nm are constructed to overcome the diffraction limit at the same observation plane. Optimized ASOMs are further demonstrated through the electromagnetic simulation and the numerical results agree well with the desired resolution. Spot size of central hotspots for one ASOM at the preset plane are 0.706, 0.722 and 0.750 times the diffraction limit. Flexible and arbitrary phase manipulations of the metasurface gives more freedom in designing a multi-functional SOL and the ultrathin single-layer structure guarantees a light-weight, low-cost and compact optical system.

2. Light response of unit cell and designs of ASOMs

ASOMs consist of a large number of high-aspect-ratio TiO₂ nanofins with various orientations. As illustrated in Fig. 1, nanofins with two vertical orientations possess a designed binary Pancharatnam-Berry (PB) phase profile when the left circularly polarized (LCP) light passes through the structure and converts to the right circularly polarized (RCP) light [23]. At three visible wavelengths of 473 nm, 532 nm and 632.8 nm, the generated phase profile is dispersionless but can form respective sub-diffraction foci at the same plane through simultaneous optimizations in the design.

 

Fig. 1 Schematic of a metasurface for achromatic sub-diffraction focusing. ASOM: achromatic super-oscillatory metasurface; R: red light; G: green light; B: blue light.

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In principle, the phase shift (Ф) of the cross-polarized light depends on the rotating angle (φ) of the nanofin satisfying the relationship: $\Phi \text{=}-2\text{φ}$ for LCP incidence [23]. To further demonstrate it numerically, a TiO₂ block has a fixed orientation (φ), 160 nm length, 50 nm width and 600 nm height on a glass substrate with 200 × 200 nm period, as shown in Fig. 2(a). The parameters are optimized through a commercial electromagnetic software (CST Microwave Studio) for a high conversion efficiency and can be fabricated by a previously reported method [31]. When LCP light normally illuminates from the bottom of the glass substrate at three wavelengths of 473 nm, 532 nm and 632.8 nm, transmitted phase and amplitude modulations of the cross-polarized light are calculated, as shown in Fig. 2(b). The phase shift changes from 0° to 360° with a nearly linear relationship with the orientation of the nanofin at the wavelength of 632.8 nm. Deviations between the theoretical and numerical results may be attributed to the different light interactions with respect to the orientation. As the PB phase is intrinsically independent of the operating frequency due to its dispersionless property, the cross-polarized phase profiles at wavelengths of 473 nm and 532 nm are parallel to the red line for phase in Fig. 2(b). The average amplitudes of the cross-polarization conversion are 0.86, 0.978 and 0.823 at wavelengths of 473 nm, 532 nm and 632.8 nm, and can be further improved through a reflective layer [28]. At each single wavelength, the change of the rotating angle slightly affects the transmitted amplitude, thus only the phase profile needs to be optimized in our designs while the amplitude profile is considered to be uniform.

 

Fig. 2 Light response of a high-aspect-ratio TiO₂ nanofin. (a) Schematic of the TiO₂ nanofin. The rectangular structure rotates an angle (φ) with respect to the x axis. (b) Phase and amplitude modulations of the cross-polarization light with various orientations at wavelengths of 473 nm, 532 nm and 632.8 nm. Average amplitudes of the LCP/RCP conversion are 0.86, 0.978 and 0.823, respectively.

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To precisely characterize the required achromatic super-oscillatory hotspots for various wavelengths, a multi-objective and multi-constraint optimization model is established, as presented in Eqs. (1) and (2). Objective functions contain three parts for various wavelengths to minimize the full-width-at-half-maximum (FWHM) of the central super-oscillatory hotspot; constraint conditions mainly contain two parts: the first three inequalities are used to control the dark regions between the main hotspots and the side-lobes for various wavelengths, and the last equality is for the phase-modulation constraint. The mathematical expressions are presented as follows:

The multi-objective and multi-constraint optimization model here is also known as the Pareto optimal model. For the multi-objective model, it is challenging to minimize three FWHMs minimal simultaneously due to conflicting objective functions, but there are a number of moderate optimal solutions, which are suitable for the model. The normal approaches to solve the Pareto optimal model involve controlling the weighted coefficient w of each objective function and converting the model to the single-objective model with the objective function expressed as:

3. Optical performance analysis of ASOMs

In the following examples, LCP beam illuminates the ASOMs and propagates in air at three wavelengths of 473 nm, 532 nm and 632.8 nm. Diameters of the ASOMs are 20 μm and the preset plane is located at a distance of 10 μm from the nanofin surface; the total number N equals to 50, i.e. each discrete unit is 200 nm in the Fig. 2(a). According to the aforementioned optimization procedure, after sufficient iterations to pursue the minimal value of the objective function, the transmitted binary phase profile of the ASOM₁ is achieved, as shown in Fig. 3(a). With the relationship between the phase modulation and the rotating angle in Fig. 2(b), the binary ASOM₁ can be constructed, where the orientations of 0° and 90° correspond to the phase modulations of 0 and π, respectively. Theoretical intensity profiles calculated by VAS at the preset plane are shown in Figs. 3(c-e), and the suppressed FWHMs are 250.5 nm, 277.6 nm and 326.3 nm at wavelengths of 473 nm, 532 nm and 632.8 nm. These overcome the diffraction limit of 334.5 nm, 376.2 nm and 447.5 nm for three wavelengths, with sub-diffraction ratios of 0.749, 0.738 and 0.729, respectively. Using the definition of the super-oscillation [35], the local wavevector ($k_{local}=\nabla \Psi$, phase gradient) is calculated from the optical field at the preset plane. Super-oscillatory regions are located at the intensity minima, where the local phase can oscillate rapidly and $k_{local}$are much higher than the highest Fourier component of $k_{\max }=2\pi /\lambda \ast NA$, as indicated by the shaded regions in Figs. 3(c-e).

 

Fig. 3 Optimized results through GA algorithm. (a) Transmission of the ASOM₁, containing binary phase modulations, either 0 or π. (b) ASOM₁ is 20 μm diameter; the zoom-in section shows the central region of ASOM₁ with only two orientations: 0° or 90°, corresponding to a phase change of π. (c-e) Distributions of the spot (solid lines) and local wavevector (dashed lines) across z = 10 μm at wavelengths of 473 nm, 532 nm and 632.8 nm.

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ASOM₁ is further validated numerically and a feasible method proposed in the previous reference [36] is used to reduce the computation time. Interactions between light and ASOM₁ are calculated in CST software, where the upper and lower boundary from the structure are 0 μm and 1 μm, respectively, and open boundary conditions are used along x, y and z directions. The total mesh cells in the CST model from the upper to lower boundary are about 4 x10⁷ to 5 x10⁷ to guarantee at least two mesh cells in the minimum feature size. The field distribution at a distance of 0.1 μm from the nanofin surface is extracted and then used to calculate other axial distributions by VAS theory. The axial intensity distributions on the propagating direction and the intensity profiles across the preset plane (z = 10 μm) are shown in Fig. 4. Performance of optimized ASOM₁ are presented in Table 1. FWHMs of the central spots are 0.763, 0.756 and 0.755 times the diffraction limit at the wavelengths of 473 nm, 532 nm and 632.8 nm, respectively. Deviations between the simulated and theoretical results may be attributed to the insufficient discrete sampling, since desired circular symmetric modulations can appear square-like, as shown in the inset image in Fig. 3(b). The increase of resolution comes at a price of losing the focusing intensity and energy focused into the central hotspots. However, the low levels of throughput efficiencies are acceptable compared to the ultralow transmission efficiencies of SNOM and complex near-field manipulations. It is also noted that our binary ASOM₁ will work well under a linear polarized light because the phase shift of two perpendicular orientations is also π for the transmitted cross-polarized light [30]. It implies a polarization independence of the binary ASOM₁.

 

Fig. 4 (a-c) Intensity distributions along the propagating direction and (d-f) Intensity profiles across z = 10 μm for ASOM₁ at wavelengths of 473 nm, 532 nm and 632.8 nm.

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Table 1. Performance of optimized ASOMs. The incident intensity is 1.

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It is evident that the intensities at the preset plane are lower than the peak/nearby intensities on the axial axis. This can be attributed to the fact that the energy concentration problem is not considered in the framework of the design. Hence, the energy concentration on the axial direction is included in the multi-objective and multi-constraint model. Intensities around the preset plane are set to be smaller than the central intensity at z = f. Thus three more constraints are added into Eq. (2) as follows:

 

Fig. 5 (a-c) Intensity distributions of ASOM₂ along the propagating direction, the shaded region represents the constrained region of the energy concentration. An artificial scale bar is used for a clear display near the foci. (d-f) Intensity profiles across z = 10 μm at the wavelengths of 473 nm, 532 nm and 632.8 nm.

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In the above examples, the nanofins only manipulate the transmitted phase profile with two values of 0 and π. In principle, an arbitrarily desired phase profile is possible according to the array of orientations of nanofins. In order to demonstrate the flexible phase manipulations of nanofins, ASOM₃ with 4-step phase in [0,2π] is designed for achromatic focusing, as shown in Fig. 6. Performance of optimized ASOM₃ are presented in Table 1. FWHMs of central spots at z = 10 μm are 0.722, 0.750 and 0.706 times the diffraction limit at wavelengths of 473 nm, 532nm and 632.8 nm. Compressed FWHMs and focusing intensities of ASOM₃ are better than binary ASOM₁. It means that increasing the number of phase levels between 0 and 2π would be beneficial towards improving the quality of the super-oscillatory hotspot [13]. Benefiting from flexible and arbitrary phase modulations of the metasurface, it is possible to design a high-performance component with more phase levels in [0, 2π] to balance conflicting factors of a super-oscillatory field.

 

Fig. 6 (a) Transmission of the ASOM₃, containing four phase modulations in [0, 2π]: 0, π/2, π and 3π/2. (b-d) Intensity profiles across z = 10 μm at wavelengths of 473 nm, 532 nm and 632.8 nm.

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4. Conclusion

An achromatic sub-diffraction method is proposed through utilizing metasurfaces composed of arrayed nanofins to simultaneously focus light at three visible wavelengths of 473 nm, 532 nm and 632.8 nm into the same axial plane with sub-diffraction resolution. Metasurface-based devices provide dispersionless phase profiles for various incident wavelengths and neglect the influence of the material dispersion in optimizations. Simultaneous optimizations for three wavelengths can effectively remove the axial chromatic aberration. Binary and multi-step phase-type ASOMs are optimized through multi-objective and multi-constraint models and further verified numerically. Simulated results are consistent with the desired expectations and FWHMs of central hotspots for one ASOM at the preset plane are 0.706, 0.722 and 0.750 times the diffraction limit. Additionally, the optimization models can be modified for practical applications, like real-time super-resolution imaging by adjusting the ratio of the intensities of the central spot and side-lobes, and high-quality needle focusing by adding a constraint condition for the intensity uniformity along the axial direction. We expect that the ultrathin achromatic metasurfaces can be helpful in bringing about a considerable reduction of the volume, weight and cost of an optical system by replacing bulky conventional doublet/triplet lenses.