1. Introduction

To date, many significant achievements have been made towards breaking Abbe’s diffraction resolution limit [1], including a number of classic solutions such as near-field scanning optical microscopy by exploiting near-field evanescent waves [2], and stimulated emission depletion microscopy by quenching excited fluorophores so as to sharpening the effective intensity point spread function [3]. The emergent super-oscillatory lens optical microscope was aimed at breaking the diffraction barrier by using the delicate interference of far-field propagating waves, showing resolution better than one sixth of the illumination wavelength [4]. The core component of this technique is a planar multi-annular nanostructured metasurface, which is used as a single ultra-high-numerical-aperture (NA) focusing element, in contrast to traditional lens-based pupil-filtering optical systems [5–7]. This metasurface achieves the required sub-diffraction focusing at a distance of tens of wavelengths away from the mask plane, which essentially outperforms the superlens [8] and the plasmonic lens [9]. With an upgraded metasurface as large as 1 mm in diameter, the working distance for the optimal focal plane is increased to near 200 μm [10], the scenario for a conventional high-NA objective lens.

Currently, design theories for a far-field super-focusing metasurface have been based on the scalar angular spectrum theory [4,11,12], vectorial angular spectrum (VAS) theory [10,13–16], or vectorial Rayleigh-Sommerfeld diffraction integral [17,18]. Meanwhile, the experimental method for detecting the subwavelength focus behind a metasurface uses a high-magnification, high-NA microscopic imaging system [4,12,18]. Nevertheless, fundamental limitations may exist in both theory and experiment [4,10–18]. First, all the above design theories assume a polarization-insensitive scalar function to model the physical process of light transmission through a thin nanostructured metasurface. However, because the minimum feature size is reduced to the subwavelength scale, the polarization-dependent transmission property (a phenomenon known as extraordinary optical transmission) becomes pronounced [19], which has been verified through nanohole array experimental data [20,21]. Moreover, the vectorial nature of the incident light must be considered when the propagation angle of the light beam diffracted from a super-focusing metasurface becomes large [10]. Second, the experimental method has been based on the use of a coherent optical microscope, which indirectly detects the vectorial light field in the front focal plane of the objective lens. Recently, a discrepancy has been observed between the above-mentioned theories and experiments [13,16]. The indirect experimental results do not agree with theoretical results calculated by the vectorial angular spectrum theory and the rigorous electromagnetic (EM) simulation, i.e. the three-dimensional (3D) finite-difference time-domain (FDTD) method; in contrast, they are in better agreement with the predictions of the simplified scalar angular spectrum theory. So, it is necessary to explore the super-focusing and nano-imaging mechanisms of the nanostructured metasurfaces rigorously and quantitatively. Here, we provide an EM exploration to interpret the inherent discrepancy and reveal the fundamental mechanism; our explanation is based on a combination of the 3D FDTD method and an equivalent magnetic-dipole (EMD) vectorial imaging theory.

2. Methods

2.1 Physical model of EM focusing and imaging

To explore the electromagnetic focusing and imaging process, a physical model is established for metasurfaces, as depicted in Fig. 1. The metasurface, a metallic-film-coated nanostructured mask, is composed of many concentric rings, either opaque or transparent. The metasurface was placed in air (refractive index n = 1) or in an oil immersion medium (n = 1.514). The amplitude distribution of the incident vector beam is also assumed to be rotationally symmetric. The EM focusing process of the metasurface can be decomposed into two fundamental steps: light transmission through nanostructured rings or grooves (area I) and light propagation in a homogeneous dielectric medium (area II), as shown in Fig. 1. The vector field of light behind the metasurface mask was rigorously calculated by the 3D FDTD method [16,22], in which the process of light transmission through the subwavelength structure had been physically modeled. The total-field scattered-field (TFSF) boundary and the perfectly matched layer (PML) absorbing boundary condition were applied and the material index parameters for the metallic film were modeled [23]. Particularly, the PML absorbing boundary condition is applied to all FDTD edges (two sides in each direction). If the excitation light source is an x-polarized LPB, the anti-symmetric and symmetric boundary conditions are used in the x and y edges, respectively. This can significantly reduce the computation storage. The TFSF boundary locates inside the 3D FDTD region. This boundary should cover the outermost transparent annulus of a metasurface.

 

Fig. 1 Schematic diagram of an EM focusing and imaging model for metasurfaces. BS and FS are the back surface and front surface of the metal film, respectively. The vector field of light behind the nanostructured metasurface was determined by 3D FDTD simulation and observed by a high-NA coherent optical microscope (infinity correction).

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The vector field of light behind a metasurface illuminated with a specific polarized vector beam was also predicted by the vectorial angular spectrum (VAS) theory. The derived integral formulae are based on the assumption of a scalar transmission function t(r) [4,10,16]. For a linearly polarized beam (LPB, x-polarized) normally illuminating the metasurface mask [13], the y-component of the electric field E is E_y(r, z) = 0. E_x(r, z) is a zeroth-order Hankel transform independent of the observation angle φ, and the longitudinally polarized component E_z(r, φ, z) is a first-order Hankel transform related with φ. The x- and z-polarized electric energy densities are respectively evaluated by |E_x|² and |E_z|², whereas the total electric energy density (or the light intensity) is |E|² = |E_x|² + |E_z|². Along the axial direction (r = 0), the electric field is pure transversely polarized. The integral representation of light field from the scalar angular spectrum theory is the same with the x-polarized component E_x(r, z) from the VAS theory. Thus, the scalar theory merely predicts the transversely polarized electric component [4]. The use of the fast Hankel transform algorithm ensured very high accuracy of the VAS theory and enabled efficient optimization of metasurfaces [10].

The super-focusing light field in the observation plane behind the metasurface mask was mapped out through a high-NA microscopic imaging system [4]. The image detected by the camera was rigorously analyzed according to a vectorial EM imaging theory. When the electric field E in the observation plane behind a metasurface is known, either from the rigorous EM calculation or the scalar/vectorial angular spectrum theory, a high-NA, high-magnification optical microscope was used to map out the intensity distribution |E|², as shown in Fig. 1.

The intensity distribution in a finite volume behind the metasurface is imaged by axially scanning the objective lens using a nano-positioning PZT (piezoelectric ceramic transducer). To rigorously calculate the far-field resulting image on the camera plane, an EMD vectorial imaging theory is used [24]. The essence of this theory is to decompose the EM field in the detection plane into a coherent superposition of EMD waves based on the shift invariance assumption. For an arbitrary off-axis EMD with moment n × p located at a point r_p(x_p, y_p, z_p) in the vicinity of the objective focus, the electric field in the detection plane can be expressed as $E(r_d;r_p)=E_{\text{on}}(n\times p,r_d+\beta q)$, with q = (x_p, y_p, 0). β is the effective magnification of the imaging system and r_d(x_d, y_d, z_d) represents the position vector in the detection plane. E_on denotes the electric field resulting from an on-axis EMD (x_p = y_p = 0), which can be derived from the vectorial Debye-Wolf diffraction integral. The resulting total electric field E(r_d) in the detection plane is the coherent summation of E(r_d; r_p) within a sufficiently large observation plane [24,25]. The light intensity (vectorial imaging result) is calculated by I(r_d) = |E(r_d)|². The practical implementation of calculating the vectorial imaging results was inevitably approximated from a sum over discrete EMDs and intrinsically required very time-consuming computations [25]. A look-up table for integrals was pre-calculated, and then a series of convolutions were performed.

Compared with previous reports [4,13,16], this research has made significant progress in two aspects. First of all, a complete EM model is constructed in contrast to [16], which can be used to thoroughly investigate a variety of micro-/nanostructured super-focusing metasurfaces. As shown in Fig. 1, it is a complete EM model as both the focusing and imaging processes have been rigorously explored. In other words, the electric fields from the incoming excitation source plane to the final detection image plane can be rigorously and quantitatively analyzed using the method proposed in this study. The focusing process of multi-annular metasurfaces was studied by the 3D FDTD method in [16]. The discrepancy found in [16] posed a critical issue on far-field super-focusing metasurfaces. The underlying mechanism is to be explained in this research. Secondly, the detection of the subwavelength focus of a metasurface is first investigated by an EM vectorial imaging theory in contrast to previous publications in this field (e.g [4,12,13,16,18].), in which the microscopic detection process was analyzed using a blurred (low-pass filtered) and magnified light pattern in the detection plane. Current research clearly shows how transversely and longitudinally polarized electric components transmit through a high-NA microscopic imaging system.

2.2 Sample design

The metasurface was designed based on the VAS theory, and optimized with the genetic algorithm and a fast Hankel transform algorithm [10,16]. The initial mask has a predefined total annulus number N and each annulus width is assumed to be equal, denoted by Δr [4]. The optimized metasurfaces have been summarized in Table 1.$\Delta r$ and N represent the fixed ring width (equidistant) and the total ring number of a metasurface, respectively. The metasurface diameter becomes $D=2N\cdot \Delta r$. M₁ and M₂ are selected from [16] and [13], respectively. M₃ is the sample in [4]. To concisely describe the structural parameters of a metasurface, the transmissions t_1~N from the first ring (innermost) to the Nth ring (outermost) are coded by converting every four successive binary digits {0,1} into one hexadecimal digit [10]. For example, the transmissions for the first four rings ‘1100’ have been coded as ‘C’ for M₂ in Table 1. One ‘0’ digit is added to the last three rings ‘011’ for M₁, so it yields ‘6’. In order to clearly show the metasurface structure, the top views of M₁ and M₃ are plotted in Fig. 2. M₁ and M₃ are composed of 9 and 25 transparent annuli, respectively. The black color indicates the opaque region with the metal film and the bright color indicates the transparent region without the metal film.

Table 1. Optimized metasurface samples

View Table

 

Fig. 2 Top view of the metasurface structure. (a) M₁; (b) M₃.

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3. Results and discussions

3.1 Influences of film material and thickness

To clarify the influence of metal-film material and thickness at a visible wavelength, a 14-μm-diameter metasurface was optimized (M₁, Table 1) and investigated by the 3D FDTD method. It is studied in air and has a minimum annulus width of 200 nm. Three typical metal film materials, aluminum (Al), silver (Ag), and gold (Au), are used for comparison. The film thickness for each material is 25 nm, 50 nm, 75 nm, 100 nm, 125 nm, 150 nm, 200 nm. The mesh size for the film region is 15 nm × 15 nm × 5 nm (x, y, z) and 15 nm × 15 nm × 20 nm (x, y, z) for other regions. So, the maximum grid size (20 nm) is 1/32 of the illumination wavelength 640 nm, much less than λ₀/(20n). This grid setting in general meets the requirement of the stability and convergence condition for the 3D FDTD method [22]. To further check the accuracy of the above setting, simulations with a fine grid size of 10 nm (λ₀/64) in all directions have been implemented and the results are found to have no obvious differences for the on-axis intensity distributions, except for the near-field region (half the wavelength). As only the post-evanescent far-field region is investigated in this field [4,10–12], current mesh size is suitable; however, a much finer mesh grid may be required for the study of nanoplasmonic structures [26]. The non-propagating near-field fields are confined along the metal-dielectric interface, so the situation is different from that for far-field super-focusing metasurfaces investigated in this study.

There is one main focus for M₁, located at z_p = 2.65 μm according to the VAS theory. The equivalent-numerical-aperture (NA_eq) is estimated as 0.935. In previous publications [4,12,13,17,18], the position of z = 0 is assumed to locate at the front surface (BS) of the metasurface (the film-medium interface, Fig. 1). However, Fig. 3(a) shows that the focus is axially shifted with the increase of the film thickness. This implies that a systematic error will be induced for the above assumption. In Fig. 3(b), the position z = 0 has been corrected to overcome this drawback and the back surface (BS) of the metasurface (the film-substrate interface, Fig. 1) has been assumed to be the initial zero plane instead. As a result, the EM simulation results are in better agreement with the VAS prediction when the film thickness is from 75 nm to 200 nm. At wavelength of 640 nm, the results for the Al film are best corrected in the entire range from 25 nm to 200 nm. The intensity at the peak position I_p = |E(0, 0, z_p)|² is displayed in Fig. 3(c). For the Al and Ag films, the peak intensities initially increase with the film thickness and then decrease. When the film thickness approaches 100 nm, the peak intensity is maximal. For the Au film, the situation is slightly different and the peak intensity monotonically increases, though not obvious. The spot sizes of the main focus along x, y, and z directions are shown in Fig. 3(d). The full width at half maximum (FWHM) is used to evaluate the spot size in three dimensions. It is evident that FWHM_y and FWHM_z of the focus agree with the VAS predictions (lower and upper data groups). In y and z directions, only |E_x|² contributes to the total electric energy density for a linearly polarized beam (x-polarized). FWHM_y is evaluated to be between 269.4 nm (0.421λ₀) and 298 nm (0.466λ₀), which surpasses Abbe’s diffraction limit of λ₀/(2NA_eq) (342 nm). However, significant differences appear for the spot sizes along x direction (middle data group). The spot sizes evaluated by the rigorous EM simulations (FDTD) are much smaller than the VAS predictions owing to the discrepancy of the longitudinally polarized electric energy density |E_z|². This indicates that imprecise theoretical results are produced using the VAS theory. For the Al film, the VAS-theory-predicted FWHM_x (581 nm) yields a relative error between 22.7% (25 nm film) and 34.1% (125 nm film). Further for the Ag film, the relative error ranges from 14.2% (25 nm film) to 36.5% (150 nm film). The pronounced difference for the total electric energy density (|E|²) between the VAS theory and FDTD simulation demonstrates one limitation of current theories in designing super-focusing metasurfaces.

 

Fig. 3 Influence of the film material and thickness on the main focus of M₁. (a), (b) Axial peak positions before and after correction, respectively. (c) peak intensity. (d) spot sizes evaluated by the FWHM along the x, y, and z directions.

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The skin-depth is the penetration depth in the metal at which incoming light amplitude has been reduced by 1/e. To obtain a large contrast between the aperture brightness and the surrounding metal film, the metal must be optically opaque, which implies that the film thickness should be several times the skin-depth of the metal [27]. Particularly, for the Au, Al, Ag films at the wavelength of 640 nm, the light amplitude, decaying exponentially with respect to the penetration depth z, is shown in Fig. 4. The conductivity σ is elected to be 4.10 × 10⁷ S/m, 3.50 × 10⁷ S/m, 6.30 × 10⁷ S/m for Au, Al, Ag, respectively [28]. It can be known from Fig. 4 that, the skin-depths are less than 5 nm and when the penetration depth z = 25 nm, the light amplitude has been attenuated to be less than 0.002 of the incident light. It is this reason that a minimum film thickness of 25 nm has been selected above and the selected films are therefore believed to be optically opaque. Additionally, the value of 25 nm is found to be valid for the 193 nm deep ultra-violet wavelength [15].

 

Fig. 4 Light amplitude decays exponentially with respect to the penetration depth in the metal of Au, Al, Ag, respectively (the value 1/e corresponding to the skin-depth).

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By further showing the polarization-dependent transmission of the subwavelength metasurface, the electric field structures inside the metal film have been rigorously calculated by the 3D FDTD method. The metasurface M₁ is normally illuminated with an x-polarized uniform LPB. The metal film is 100-nm-thick aluminum. The mesh size in the overall simulation region (FDTD) is 10 nm × 10 nm × 10 nm (x, y, z). The electric field amplitudes are plotted in Fig. 5. The dimensions in Fig. 5 are x, y: −7.275–7.275 μm. The top row corresponds to the transverse x-y plane at a distance of 5 nm away from the back surface of the metal film (z = 5nm). The bottom row corresponds to the front surface of the metal film (z = 100 nm). For an x-polarized uniform LPB, the transmitted electric fields have significant E_y and E_z components, which obviously dissatisfies the assumption of the scalar transmission for a nanostructured metasurface. It can be seen from the top two rows, the E_x component transmits non-uniformly through the metal film, and it is more intense along the x = 0 direction than the y = 0 direction. For the E field and the E_x component, the transmitted amplitudes slightly vibrate along the radial direction within the wide annulus. All these details cannot be investigated in the VAS formula.

 

Fig. 5 Electric field structures in the transverse plane for M₁ illuminated with an x-polarized LPB. Top row: z = 5 nm; middle row: z = 20 nm; bottom row: z = 100 nm (front surface of the film).

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3.2 Experiment and EM explanation

To study the EM mechanism of far-field super-focusing metasurfaces, microscopic imaging experimental data and EM explanation are presented. A single longitudinal-mode laser was used as the light source. The center wavelength of the illumination light was measured to be 532.4 nm. According to the experimental method in [4], the light field behind M₂ was mapped out by a high-NA, high-magnification microscopic imaging system with an apochromatic objective lens of NA 0.95 (infinity correction). The tested sample M₂ was selected from [13] and the experimental method was similar with [13]. However, the effective magnification of the imaging system was recalibrated to be ~188 using a standard grating, and so the resulting measured image was different from the original experiment. The coaxial alignment of M₂ was performed by a 3D step motor stage. The objective was axially scanned with a 20 nm step using a nano-positioning PZT. The distance from the front surface of M₂ was determined using the axial optical sectioning principle of a confocal microscope, in which a CMOS detector was used instead of a physical pinhole. M₂ was optimized with a minimum annulus width of 400 nm and its diameter was increased to 200 μm (Table 1). M₂ was manufactured using electron beam lithography and coated with a 120 nm-thick gold film. The experimental setup in [13] was used and the experimental data was further analyzed using the vectorial EM imaging method.

The theoretical and experimental results are summarized and compared in Fig. 6. The simulation area has been plotted for x: −3–3 μm and z: 25–40 μm. The electric energy densities in the x-z plane calculated by the VAS theory are shown in Figs. 6(a)–6(c). At the transverse plane for z = 33.32 μm, the enhancement of the |E_z|² component breaks the circular symmetry of the |E_x|² component. The |E_x|² and |E_z|² components are plotted in Figs. 6(d) and 6(e), respectively. The total electric energy density |E|² takes a dumbbell shape, as shown in Fig. 6(f). The image in the detection plane calculated by the EM imaging theory is shown in Fig. 6(g) and the experimentally obtained image in Fig. 6(h). The magnification factor of the imaging system was further calibrated in an experiment by using a standard grating, and Fig. 6(h) was recalculated from the experimental data in [13] and different from Fig. 8(b) in [13]. The on-axis intensity distributions are compared in Fig. 6(i) and the transverse intensity distributions along x direction are plotted in Fig. 6(j). Again, the measured data in Fig. 6(j) was recalculated according to the calibrated magnification factor. For the |E_x|² component, the VAS theory calculation obtained FWHM_x = 214 nm (0.402λ₀), which is consistent with the EM imaging theory result (FWHM_x = 216.6 nm, 0.407λ₀). It is evident that the measured spot is broadly consistent with the |E_x|² component but deviates largely from the total electric energy density.

 

Fig. 6 Comparison of the intensity distributions of M₂. (a), (b), (c) VAS theory calculations in the x-z plane. (d), (e), (f) electric energy densities in the x-y plane for z = 33.32 μm. (g), (h) EM imaging result and experimental result, respectively, in the x-y observation plane. (i) on-axis intensity distributions. (j) intensity distributions along the x direction when z = 33.32 μm.

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The above two samples demonstrate that the enhancement of the longitudinally polarized component (|E_z|²) contributes remarkably to the total electric energy density. However, this pronounced energy proportion has decayed to naught in the final image, obtained by a high-NA optical microscope. It should also be noted that, although the results predicted by the VAS theory may deviate from the rigorous FDTD simulations (Fig. 1), the transversely polarized component |E_x|² can generally be predicted by the scalar or vectorial angular spectrum theory.

3.3 Validity exploration of existing theories

To investigate the validity of existing theories, the sample in [4] was examined (M₃, Table 1). It was oil immersed and coated with a 100-nm-thick aluminum film. The illumination light wavelength of the diode laser (single longitudinal mode) was λ₀ = 640 nm. A Nikon oil immersion objective (NA = 1.4 and magnification of 100) was used in the experiment to detect the subwavelength focus behind M₃. The observation distance was 10.3 μm away from the film surface and the initial zero plane was assumed to be located at the front surface (FS, Fig. 1) [4]. The vector field of light behind M₃ was rigorously calculated using the 3D FDTD method with the simulation area defined as x, y: −23.25–23.25 μm and z: −2–17 μm. A uniform linearly polarized beam (x-polarized) was used as the excitation light source. The mesh size was 15 nm × 15 nm × 25 nm (x, y, z). For a more accurate EM simulation of the 40-μm-diameter M₃, it is better to set the maximum mesh size less than λ₀/(20n). The decrease of the mesh size in 3D FDTD simulation would significantly increase the computation storage. Taking into account the practical workstation conditions, the mesh size for M₃ was set to be 15 nm × 15 nm × 25 nm (x, y, z) in this simulation. Alternatively, a small metasurface (14 μm diameter) with similar situations is used to check the validity of the mesh size (15 nm × 15 nm × 25 nm). The metasurface is SOL₂ in [16]. It is also oil-immersed and has the same minimum annulus width of 200 nm. Three FDTD simulations with the mesh size of 15 nm × 15 nm × 25 nm (x, y, z), 15 nm × 15 nm × 20 nm (x, y, z), 10 nm × 10 nm × 10 nm (x, y, z) are conducted, respectively. It is found that the 3D intensity distributions have no obvious changes when reducing the mesh size. The result using the 10 nm mesh size is sufficiently accurate. So, it is believed that the simulation setting for M₃ is suitable according to the numerical theory of the FDTD method and the FDTD test

The electric energy densities calculated by the VAS theory and the 3D FDTD method are shown in Figs. 7(a)–7(c) and Figs. 7(d)–7(f), respectively. The total electric energy density (|E|²) distributions are displayed in Figs. 7(a) and 7(d). According to the scalar angular spectrum theory [4,13,16], the intensity distribution is the same with the transverse component |E_x|² by the VAS theory, as shown in Fig. 7(c). It can be seen that the prediction by the VAS theory broadly coincides with the rigorous EM simulation result; however, significant differences can be observed, e.g. in the axial ranges 5–6 μm (indicated by the dashed square A) and 11–12 μm (dashed square B). Furthermore, the on-axis |E|² distributions (normalized) are compared in Fig. 7(g). The intensity distribution obtained by FDTD is axially shifted from that calculated by the VAS theory. In contrast to [4], the back surface plane is a more suitable position for z = 0 than the front surface (Fig. 1 and Fig. 3). Thus, the plane situated at a distance of 10.3 μm away from the film surface in [4] actually corresponds to the new position z = 10.4 μm in current FDTD simulation (green dashed line). By further considering the axial shifting offset, it is found that this plane is approximately located at z = 10.47 μm, according to the VAS theory (green dashed line). The displacement of 70 nm was determined considering the relative intensity calculated by the VAS theory compared with the FDTD result in the vicinity of the focus. It should be noted that this explanation is consistent with Section 3.1 and Fig. 3. There is not an exact formula or criterion which can be used to accurately determine the specific focal shift and the zero position (z = 0). It can be seen from Fig. 3 that the focal shift still exists after correction. In Fig. 7(g), the axial range for the EM imaging is 9.6–13.35 μm, where the two green rectangles correspond to the dashed green squares in Fig. 7(a). The electric energy densities in the transverse x-y plane obtained by the FDTD simulation are displayed in Figs. 7(h)–7(j) and those calculated by the VAS theory in Figs. 7(k)–7(m). The resulting EM image in the detection plane is shown in Fig. 7(n). It can be seen from Figs. 7(h)–7(n) that the EM imaging result does not coincide with the rigorous FDTD simulation and the VAS prediction; however, it is in better agreement with the |E_x|² component (Fig. 7(h)) and also with the result obtained by the scalar angular spectrum theory (Fig. 7(k)). This quantitative comparison indicates that the experimental method in [4] has the inherent limitation of being a polarization-dependent transmission imaging system. As a result, it cannot map out the total electric energy density (|E|²), but only the transversely polarized electric energy density component (|E_x|²). The scalar angular spectrum theory, however, merely contains this transversely polarized component, which is why the experimental result (Fig. 1(c) in [4]) agrees with the theoretical design (Fig. 1(b) in [4]). The intensity distributions along x and y directions are compared in Figs. 7(p) and 7(q), respectively. According to the scalar angular spectrum theory, the spot size is FWHM_x,y = 209 nm (0.327λ₀). However, in the y direction, the spot sizes are FWHM_y = 183 nm (FDTD, 0.286λ₀) and 185 nm (EM imaging result, 0.289λ₀). In the x direction, the spot sizes are FWHM_x = 370 nm (both FDTD and VAS, 0.579λ₀) and 246 nm (EM imaging result, 0.385λ₀) and for the |E_x|² component, FWHM_x = 246 nm (FDTD, 0.385λ₀). Although the E field calculated by the FDTD method is acquired by the microscopic detection system, the resulting image is simply dominated by the E_x component.

 

Fig. 7 Comparison of the intensity distributions of M₃. (a), (b), (c) VAS theory calculations in the x-z plane. (d), (e), (f) 3D FDTD simulation results in the x-z plane. (g) on-axis intensity distributions. (h), (i), (j) transverse intensity distributions obtained by FDTD simulations when z = 10.40 μm. (k), (l), (m) transverse intensity distributions calculated by the VAS theory when z = 10.47 μm. (n) EM imaging result calculated by the vectorial imaging theory for the observation electric field shown in (j). (p), (q) intensity distributions along the x and y directions, respectively.

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When illuminating M₃ with a circularly or randomly polarized beam, the total electric energy density can be obtained from two orthogonally polarized linearly polarized beam [16]. The intensity distribution behind the metasurface is circularly symmetric and the spot size becomes broader than that illuminated with a linearly polarized beam.

4. Conclusions

In summary, we performed an EM exploration of far-field super-focusing nanostructured metasurfaces. The investigation was based on an EM focusing and imaging model solved by the 3D FDTD method and the EMD vectorial imaging theory. The results predicted by the scalar and vectorial theories were observed to give imprecise predictions compared with the rigorous EM simulation result. The unsuitable zero position was corrected, particularly for the linearly polarized beam. We conclude that the inherent discrepancy between current design theories and the rigorous EM calculation is caused by the assumption of a polarization-independent scalar transmission function for the nanostructured metasurface. The mechanism of the microscopic imaging experiment involved the detection of the transversely polarized electric intensity but the rejection of the longitudinally polarized one. This is the main reason for the inconsistency of the microscopic experimental result with the vectorial theory and rigorous EM calculation predictions, and it clearly explains the phenomenon that the microscopic experimental result is in better agreement with the simplified scalar theory result. This study rigorously reveals the EM mechanism of far-field super-focusing metasurfaces for using in the fields of far-field nanoscopy, nanolithography, and high-density optical storage. The nano-imaging process (point scanning) of a far-field super-focusing metasurface can also be rigorously studied by the proposed method in this research.