1. Introduction

In comparison to other sub-diffraction imaging methods that exploit near-field evanescent waves [1] or manipulate fluorescence radiation [2], the optical super-oscillatory imaging method not only avoids the challenge of complex near-field operation or pre-labeling with special dyes, but also can deliver spatial frequencies higher than the cutoff frequency of optical system in the far-field [3]. Theoretically, the super-oscillatory lens (SOL) based on the super-oscillatory phenomenon works by controlling the amplitude [4,5] or phase [6,7] of light precisely to generate the destructive interference for a desired sub-diffraction hotspot. It has been applied in the fields of super-resolution imaging [4–7], heat-assisted magnetic recording [8], and optical metrology [9].

Generally, SOLs are optimized based on a reverse design strategy. First, the customized features of the sub-diffraction spot are designed, such as compressed spot size, field of view (FOV), and maximum intensity of sidelobes. Then, the geometric parameters of SOLs are optimized by diffracting the light field in the focal plane backward to the plane of the SOL structure [10]. Recently, the working wavelength was considered in the optimization to construct achromatic or broadband SOLs [11–16]. Additionally, the working FOV was extended to 4° with a SOL specially optimized for oblique incidence of light [17], however, it is still quite challenging to design a SOL operating at a large incident angle due to the great challenge of aberration correction, especially for the single-layer SOLs [18–20]. More recently, inspired by traditional wide-field imaging systems, such as the typical landscape lens [21], metasurface doublet was proposed to achieve diffraction-limited foci with a large incident angle [22,23]. The double-layer structure design and the flexible phase modulation of metasurface [24,25] provide more degrees of freedom to substantially correct optical aberrations, especially for coma and field curvature. By using the catenary metasurface [26,27], phase modulation with high efficiency was achieved and imaging for extreme-angle was demonstrated [28]. However, a sub-diffraction focusing with a large incident angle has yet to be reported.

Here, a super-oscillatory metasurface doublet (SMD) is proposed to realize sub-diffraction focusing with a large incident angle (up to 25°). Our SMD contains two metasurfaces: the first one mainly modulates specific spatial frequencies of light under different incident angles and the second metasurface can focus the light within the designate field of view into the focal plane to realize the sub-diffraction focusing. Rectangular nanopillars with different orientations and high polarization conversion efficiency are used to provide the required geometric phase modulations. The constructed SMD is further verified by electromagnetic simulation and the sub-diffraction focus is formed exactly within the designed incident angle of up to 25°.

2. Design of the super-oscillatory metasurface doublet

The design strategy of SMD includes two steps: the first step is to achieve the diffraction-limited focusing with a large incident angle by metasurface doublet as proposed in the Refs. [22,23]; the second one is to optimize the super-oscillatory phase for sub-diffraction focusing and superimpose it onto the front surface of metasurface doublet. To correct the aberration under a large incident angle, our metasurface doublet contains two layers of nanostructures on two sides of a common K9 glass substrate. The front and back layers work as a concave lens and a convex lens, respectively. Instead of the parabolic phase profile in the traditional focusing phase, these two metasurfaces were designed with even aspherical phase profile to correct aberration more effectively, which is expressed as:

A commercial optical design software (Zemax OpticStudio) was used to optimize the coefficients a_i by minimizing the focal spot size with distinct incident angles. The optimization procedure worked as follows: firstly, the systemic parameters, including the diameter of the entrance pupil, the focal length, and the working wavelength (for the proof-of-concept example, they were set to 20 µm, 20 µm, and 632.8 nm, respectively), were configured as required; secondly, the effective focal length and working f-number were set as constraints in the merit function. The back focal length, the coefficients of the even aspherical phase profiles, and substrate thickness were added as optimization parameters in turn to minimize the focal spot size with different incident angles. Theoretically, it is beneficial to improve the focusing performance with more even aspherical coefficients. However, the focusing performance has little dependence on these coefficients when the focus is nearly diffraction-limited. To achieve a better balance between the focusing performance and optimization efficiency, the number of even aspherical coefficients was chosen to be 5. Finally, the substrate thickness and back focal length were optimized to be 30 µm and 21.5 µm. The optimized coefficients are presented in Table 1.

Table 1. Optimized coefficients of two metasurfaces.

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Essentially, the optimized doublet can be simplified as an equivalent singlet working perfectly for different incident angles, as illustrated in Figs. 1(a) and 1(b). To achieve the sub-diffraction behavior, a pure binary phase (i.e., super-oscillatory phase) is superimposed on the singlet, which provides an ideal focusing phase function. The additional super-oscillatory phase ${\varphi _{\textrm{SO}}}$ should be set in the entrance pupil plane, i.e. the front surface of our doublet. Thus, the intensity distribution in the focal plane I can be obtained through the scalar angular spectrum theory [29] as follows.

 

Fig. 1. (a) Layout of the metasurface doublet. (b) Schematic of the super-oscillatory singlet working for a large incident angle. (c) Phase profiles of the front metasurface with or without the additional super-oscillatory phase. (d) The phase profile of the rear metasurface.

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To obtain a specific sub-diffraction pattern, a single-objective optimization model with multi-constraint is established as follows:

The phase profile of the front metasurface in SMD was finally obtained by superimposing the front surface’s phase of metasurface doublet ${\varphi _{\textrm{front}}}$ with the super-oscillatory phase ${\varphi _{\textrm{SO}}}$, as shown in Fig. 1(c). Clearly, the phase of the front metasurface has an opposite sign to the rear one, and there are several phase abrupt changes induced by the optimized super-oscillatory phase. As a result, the rays can be slightly diverged and specially modulated by the front metasurface and then converged by the rear metasurface so that the designated super-oscillatory field is constructed correctly in the focal plane.

To realize the optimized phase profiles, the rectangular nanopillars of titanium dioxide (TiO₂, the refraction index is 2.872 at 632.8 nm) with varying orientations based on the principle of geometric phase (i.e., Pancharatnam-Berry phase) [31] were used as the unit cells in SMD. Theoretically, for a geometric phase design, the modulated phase of emitting right-handed circularly polarized (RCP) light should be two times of the orientation with the left-handed circularly polarized (LCP) incidence [32]. The CST microwave studio software was employed to calculate the light response of the nanopillars and optimize the geometric parameters of the unit cell for a high polarization conversion efficiency. The transmitted amplitude is approximately unchanged as the length (L), width (W), height (H), and period (P) are 250 nm, 80 nm, 600 nm, and 250 nm, respectively, as shown in Figs. 2(a) and 2(b). The modulated phase follows a 2-fold linear relationship with the change in orientation, which proves the theory of geometric phase and good performance of our unit cell design.

 

Fig. 2. (a) Schematic of the TiO₂ rectangular nanopillar. The nanopillar’s dimensions are L = 200 nm, W = 80 nm, H = 600 nm, and P = 250 nm. (b) Simulated amplitude and phase modulations of transmitted RCP with LCP incidence for varied orientations. The dashed black line represents the theoretical phase.

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3. Focusing performance of the super-oscillatory metasurface doublet

The performance of our SMD was characterized by Zemax. Two surface types, Binary 2 and Grid Phase, were used to simulate the even aspherical phase and the super-oscillatory phase. For comparison, a super-oscillatory metasurface singlet (SMS) with the same optical parameters as SMD was also simulated. Their intensity distributions in the focal plane with different incident angles of up to 25° are plotted in Fig. 3(a). All the mainlobes of the SMD are well suppressed as required and are nearly identical, while the SMS manifests some distinct aberrations even with a small incident angle. We can find that the noticeable coma aberration in the focal plane of the SMS with an incident angle of larger than 15°. Additionally, the FWHMs and Strehl ratios with different incident angles are calculated in Figs. 3(c) and 3(d), which are all approximately equal to the design value in our proposed SMD. Slight deviations can be mainly attributed to the residual astigmatism in our design. However, the FWHMs of the SMS increase significantly with a larger incident angle, and the Strehl ratios indicate the efficiency of SMS decreases as the incident angle.

 

Fig. 3. (a) Simulated intensity distributions in the focal plane of the super-oscillatory metasurface doublet (upper row) and super-oscillatory metasurface singlet (bottom row) with different incident angles of up to 25° by Zemax. Scale bar, 2 µm. FWHMs and Strehl ratios are compared in (b) and (c), respectively. The dashed black line represents the FWHM of the diffraction-limited focus.

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The focusing performance of our constructed SMD was further verified with CST. The incident light was set to LCP at a wavelength of 632.8 nm with different incident angles of 0°, 10°, 20°, and 25° along the x-axis. The following strategy was adopted to accelerate computation: first, CST was used to calculate the interactions between the nanostructures and light, and the field distribution was extracted at a distance of 0.8 µm from the rear surface of the SMD; then, the scalar angular spectrum method was used to compute the light distribution at a specific plane of interest.

In Fig. 4(a), the intensity distributions along the propagating direction were calculated for different planes, and the focal plane was set to z = 21.5 µm. It can be found that the sub-diffraction spots are exactly formed for all incident angles. In Fig. 4(b), it is shown that the asymmetry of the sub-diffraction spot along the x-axis becomes more pronounced as the incident angle increases. As presented in Fig. 4(c), the FWHMs in the x-direction are 0.68, 0.7, 0.74, and 0.82 times of the diffraction limit for each of the incident angles, while the FWHMs in the y-direction are all nearly unchanged since the incident angle was set along the x-axis in CST. The difference between the theoretical and numerical results is mainly caused by the insufficient discrete sampling, since phase profiles change quicker in the edge area of the metasurfaces than the central area, as seen in Figs. 1(c) and 1(d). Though the nanopillar has a subwavelength size of 250 nm, only five nanopillars can be arranged for each 2π-period in the outmost area of the rear metasurface. Nonetheless, it is obvious that the simulated results support our design within the whole designed incident angle, which verifies the feasibility of our proposed SMD. The transverse focal displacements for SMD at distinct incident angles are 0 µm, 3.04 µm, 6.17 µm, and 7.64 µm, respectively, as illustrated in Fig. 4(d), which are consistent with the theoretical values of 0 µm, 3.47 µm, 6.84 µm, and 8.45 µm. The focusing efficiencies, defined by the ratio of the energy within the sub-diffraction spot and the entire incident energy, were calculated to be 9.24%, 9.57%, 9.63%, and 8.58%, respectively. They stay constant with different incident angles, which shows the good performance of our method. The low focusing efficiency is mainly ascribed to the energy re-distribution and it is a typical feature of super-oscillatory behavior, i.e., a compressed spot size is usually accompanied by the inevitable loss of the focal energy and the appearance of side-lobes.

 

Fig. 4. (a) Intensity distributions along the propagating direction. (b) Focal intensity distributions. Scale bar, 1 µm. (c) FWHMs in x and y directions for distinct incident angles of 0°, 10°, 20°, and 25°. The dashed black line represents the FWHM of the diffraction-limited focus. (d) Transverse displacements of foci.

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4. Discussion

Our design is obtained under the assumption that the designed metasurface can modulate the light field perfectly as required, which means that the change of incident angle can hardly affect the phase and amplitude modulation of metasurface. However, there are several influencing factors in our method, which may degrade the performance of the SMD and introduce some small deviations between simulation and design. In our analysis, the deviations in our results may arise from three aspects: firstly, a deviated phase profile caused by discrete sampling. In our design, we consider the size of each unit cell to be infinitesimal for the convenience of optimization, while it is obviously impossible to be realized by metasurfaces. This inevitable discrete arrangement of the unit cell leads to some deviated phase modulation; secondly, inhomogeneous amplitude modulation. The proposed SMD is assumed to have only phase modulation with the same amplitude profile. However, due to the inherent shadowing effect and coupling phenomenon in the metasurface [19], the amplitude modulation of the SMD varies with the position of the orientated nanopillar and incident angle, contributing to a nonuniform amplitude profile; thirdly, scattering light induced by the residual co-polarized components. Since the metasurfaces used in the SMD are based on the geometric phase, there exist two polarization conversions. The residual co-polarized components cannot be eliminated completely, which incurs some scattering light in the focal plane of the SMD.

The light field can only be simulated in the order of micrometer in the visible region with CST due to the limited computational memory of the computer. Though the results simulated with Zemax are not as accurate as that with CST, we can still zoom in on our design with Zemax to the order of millimeter to further demonstrate the advantage of our method. First, we scaled the designed doublet by a factor of 25 in Zemax, which means the new doublet has an entrance pupil diameter of 0.5 mm and a focal length of 0.5 mm (we note that metasurface doublets with similar size have been fabricated and demonstrated experimentally in previous reports [22,23]); Second, another super-oscillatory phase constrained by an FWHM of 0.65 times diffraction limit and an M of 0.3 was optimized and superimposed on the front surface. The normalized π-phase-jump positions were 0.1, 0.15, 0.2 0.3, 0.45, 0.55, and 0.7, respectively. For comparison, an SMS with the same optical configuration was simulated. The spots of the SMD show nearly no difference within the designed FOV and are quite symmetric as illustrated in Fig. 5(a). The FWHMs of the SMD plotted in Fig. 5(b) are approximately 0.66 times of the diffraction limit, which supports our design strategy. By contrast, the FWHMs of the SMS are only close to the design value when the incident angle is less than 0.5°. It can be found that the SMD largely outperforms the SMS, since the uncorrected aberration is more evident with a larger size of the device. Additionally, the simulation results suggest that our proposed SMD can still achieve sub-diffraction focusing even with an incident angle of larger than 25°. However, compared to the theoretical result, the sub-diffraction spot would be widened and become asymmetrical as a result of the uncorrected optical aberration induced by the large incident angle. This also indicates the fragility of the super-oscillatory field.

 

Fig. 5. (a) Simulated intensity distributions in the focal plane of the super-oscillatory metasurface doublet (upper row) and super-oscillatory metasurface singlet (bottom row) with different incident angles by Zemax. Scale bar, 2 µm. FWHMs in y direction are plotted in (b) and (c), respectively.

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5. Conclusion

In conclusion, a super-oscillatory metasurface doublet comprised of TiO₂ nanopillars is proposed to operate at a wavelength of 632.8 nm with a f-number of 1, a focal length of 20 µm, and 50° field of view. In our proof-of-concept demonstration, the super-oscillatory metasurface doublet can not only achieve sub-diffraction focusing but also work accurately with a quite large incident angle. The simulation results are consistent with the theoretical results. This method provides a new access to sub-diffraction focusing with a large incident angle, which may find practical applications in super-resolution microscopy and optical precise fabrication.