1. Introduction {#sec1-nanomaterials-10-00003}
===============

The spatial resolution of conventional optical devices is restricted by diffraction to nearly half the operating wavelength, which greatly limits the performance of all the imaging and focusing systems that lie at the heart of modern biology, electronics and optical integrated circuits \[[@B1-nanomaterials-10-00003],[@B2-nanomaterials-10-00003],[@B3-nanomaterials-10-00003]\]. For more than a century, many efforts have been made to overcome the diffraction limit. In recent years, the interaction between matter such as nanostructures and light has been extensively studied by various research groups, offering possibilities for solving the problem. In 2005, taking advantage of the enhancement of evanescent components of an illuminated object via the excitation of surface plasmon polaritons (SPPs) \[[@B4-nanomaterials-10-00003]\], a superlens built with an ultrathin silver film was developed to achieve imaging that breaks the diffraction limit with a resolution of λ/6 (λ is the operation wavelength) \[[@B5-nanomaterials-10-00003]\]. Moreover, super-resolution focusing was also realized based on metal and dielectric nanostructures \[[@B6-nanomaterials-10-00003],[@B7-nanomaterials-10-00003],[@B8-nanomaterials-10-00003],[@B9-nanomaterials-10-00003],[@B10-nanomaterials-10-00003]\]. These imaging and focusing devices with the capability of overcoming the diffraction limit open great opportunities for the development of a new class of nano-optical devices and techniques. However, the working distance of these devices is greatly confined to the near-field region, or the focus locates inside the structure, which limits the feasibility for actual application.

In order to achieve optical focusing at a greater distance, a large number of planar lenses formed by metallic and dielectric nanostructures with nanostructures used to modulate light have been proposed, such as metallic nanoslits \[[@B11-nanomaterials-10-00003],[@B12-nanomaterials-10-00003],[@B13-nanomaterials-10-00003],[@B14-nanomaterials-10-00003],[@B15-nanomaterials-10-00003],[@B16-nanomaterials-10-00003],[@B17-nanomaterials-10-00003]\], metallic cross-shaped apertures \[[@B18-nanomaterials-10-00003]\], metallic antennas \[[@B19-nanomaterials-10-00003]\], dielectric waveguides \[[@B20-nanomaterials-10-00003]\] and dielectric pillars \[[@B21-nanomaterials-10-00003],[@B22-nanomaterials-10-00003],[@B23-nanomaterials-10-00003],[@B24-nanomaterials-10-00003],[@B25-nanomaterials-10-00003],[@B26-nanomaterials-10-00003],[@B27-nanomaterials-10-00003]\]. Few of these planar lenses could overcome the diffraction limit to realize super-resolution focusing. In 2017, by exciting SPPs and enabling them to couple with radiating propagation modes, a conical two-dimensional (2D) plasmonic-zone plate lens was proposed to realize far-field super-resolution focusing \[[@B28-nanomaterials-10-00003]\]. In our latest research, without the excitation of SPPs, we successfully designed a planar lens composed of metallic nanoslits under the illumination of a TE-polarized plane wave (the polarization parallel to the nanoslit) with a capability of super-resolution optical focusing over a larger distance than twice the operation wavelength \[[@B29-nanomaterials-10-00003]\]. In this article, we extensively explore the influence of various factors on the super-resolution focusing performance of the TE-polarized nanoslit-based 2D lens in order to further improve the method and provide a guide for practical applications.

2. Design and Super-Resolution Focusing of a TE-Polarized Nanoslit-Based 2D Lens {#sec2-nanomaterials-10-00003}
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The lens we consider here comprises an array of nanoslits perforated in a gold film on a glass substrate with TE-polarized design \[[@B29-nanomaterials-10-00003]\], which operates in cedar oil under the normal incidence of a plane wave at the wavelength of 405 nm from the glass substrate along the *x* direction, as illustrated in [Figure 1](#nanomaterials-10-00003-f001){ref-type="fig"}. At this wavelength, the relative permittivities of gold and cedar oil are −1.6745 + 5.7286*i* \[[@B30-nanomaterials-10-00003]\] and 2.2952 (*n* = 1.5150), respectively. All the nanoslits have the same length *l*; *t* is the thickness of the gold film, and *f* is the desired focal length.

In our design, the lens structure is symmetric with respect to central nanoslit at *y* = 0. The basic building unit is a nanoslit surrounded by gold walls, as shown in the inset of [Figure 2](#nanomaterials-10-00003-f002){ref-type="fig"}a. Here, *w* is the width of the nanoslit. The thickness of gold film *t* = 200 nm.

When a TE-polarized plane wave is normally incident on a metallic nanoslit, multiple TE modes can be transmitted through the nanoslit. In order to obtain desired focusing performance, the zeroth mode TE~0~ is considered for the lens design. Therefore, the width of all the nanoslits should be above the cutoff width for TE~0~ mode but below that for the TE~1~ mode, which should meet the following condition:$$\frac{2\arctan\sqrt{{- \varepsilon_{m}}/\varepsilon_{d}}}{k\sqrt{\varepsilon_{d}}} < w < \frac{\pi + 2\arctan\sqrt{{- \varepsilon_{m}}/\varepsilon_{d}}}{k\sqrt{\varepsilon_{d}}},$$ where *k* is the free-space wavevector and *ε~m~* and *ε~d~* are the relative permittivities of gold and cedar oil, respectively. The TE~0~ mode of a nanoslit follows the dispersion relationship:$$\frac{w}{2}\sqrt{\varepsilon_{d}k^{2} - \beta_{0}{}^{2}} = \arctan\frac{\sqrt{\beta_{0}{}^{2} - \varepsilon_{m}k^{2}}}{\sqrt{\varepsilon_{d}k^{2} - \beta_{0}{}^{2}}},$$ where *β*~0~ is the propagation constant for the TE~0~ mode. The above equation links *β*~0~ to the free-space propagation constant *k*, the nanoslit width *w*, and the permittivities *ε~m~* and *ε~d~*. Thus, the phase delay introduced by a nanoslit can be predicted by Re(*β*~0~)*t*. [Figure 2](#nanomaterials-10-00003-f002){ref-type="fig"}a illustrates the phase delay of a nanoslit as a function of nanoslit width. It can clearly be seen that the phase delay strongly depends on nanoslit width. Therefore, we can modulate nanoslit width to control phase delay.

On the other hand, according to the principle of optical interference, the focusing of light at the desired focal distance *f* from a lens can be realized, provided that the phase delay of each nanoslit matches the required phase difference (Δ*φ*(*y*) − Δ*φ*(0)) as a function of the position *y*, which is calculated by:$$\Delta\varphi(y) - \Delta\varphi(0) = 2m\pi + \frac{2\pi n_{d}f}{\lambda} - \frac{2\pi n_{d}\sqrt{f^{2} + y^{2}}}{\lambda},$$ where *λ* is the free-space wavelength, *m* is an arbitrary integer and *n~d~* is the refractive index of the working dielectric for the lens.

Based on Equations (2) and (3), the lens immersed in cedar oil with the desired focal length of 1.5 µm was designed, as shown in the inset of [Figure 2](#nanomaterials-10-00003-f002){ref-type="fig"}b. The geometry of the lens is symmetric with respect to the plane *y* = 0. Beginning from *y* = 0 (the lens center), the position sequence of nanoslits is: 0, 0.304, 0.910, 1.376, 1.746, 2.086, 2.408, 2.718, 3.020, 3.316, 3.606, 3.894, 4.180, 4.462, 4.744 and 5.024 µm. The first three nanoslits have widths of 190, 130 and 250 nm, and all the others are 190 nm. The phase delay of each nanoslit is given in [Figure 2](#nanomaterials-10-00003-f002){ref-type="fig"}b. Nanoslit length *l* is assumed to be infinite.

In order to characterize the focusing performance of the designed lens, full electromagnetic field simulations were performed using a commercial finite-difference time-domain (FDTD) solver (Lumerical Solutions, Inc., Vancouver, BC, Canada). In the simulations, the unit length of FDTD cells was set to 2 nm in both *x* and *y* directions to ensure the convergence of the computation and to model the fine features of the electromagnetic field. Perfectly matched layers as the absorbing boundary conditions were employed around the computational domain. The incident TE-polarized plane wave was defined by setting the electric field component of *E~z~* with the amplitude equaling one.

The simulated electric field intensity pattern of the lens is shown in [Figure 2](#nanomaterials-10-00003-f002){ref-type="fig"}c, which obviously verifies optical focusing behavior. The realized focal length is 1.484 µm (3.664*λ*), demonstrating excellent agreement with the design value. The full width at half maximum (FWHM) of the focus is 98 nm, about *λ*/4.13, well below the diffraction limit (*λ*/(2*n~d~*) = *λ*/3.03 = 134 nm). Therefore, the super-resolution focusing of light in the quasi-far field is achieved. Moreover, the length of the line focus is identical with the nanoslit length, on account that it is assumed to be infinite.

3. Effect of Various Factors on the Nanoslit-Based 2D Lens Super-Resolution Focusing {#sec3-nanomaterials-10-00003}
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We firstly explore the effect of nanoslit length *l* on the focusing properties of the nanoslit-based 2D lens, since the actual nanoslit length cannot be infinite. [Figure 3](#nanomaterials-10-00003-f003){ref-type="fig"} gives the focusing properties of the TE-polarized nanoslit-based lens with nanoslit length *l* = 4 µm. It can be clearly seen that the length of line focus is approximately 3 µm, namely 75% of the nanoslit length (as shown in [Figure 3](#nanomaterials-10-00003-f003){ref-type="fig"}a). Meanwhile, the maximum field strength is not at the center of the focal plane but near the sides (as shown in [Figure 3](#nanomaterials-10-00003-f003){ref-type="fig"}b). Moreover, the FWHMs of line focus can be kept nearly invariable at different positions, as illustrated in [Figure 3](#nanomaterials-10-00003-f003){ref-type="fig"}c,d.

Moreover, the length of the line focus approaches the nanoslit length when the nanoslit length becomes larger. [Figure 4](#nanomaterials-10-00003-f004){ref-type="fig"} shows the focusing properties of the TE-polarized nanoslit-based 2D lens with nanoslit length *l* = 15 µm. The length of the line focus is about 14 µm, namely, 93.33% of the nanoslit length. Because the length of line focus is smaller than the nanoslit length, the real value of the focus length of a nanoslit-based 2D lens should be considered in actual application.

From the above analysis, we can see that the nanoslit length has a significant impact on the focus length of a nanoslit-based 2D lens. On the other hand, the focal size (namely, FWHM) that judges super-resolution capability depends on the aperture and focal length of the lens. Utilizing the same design procedure used in [Section 2](#sec2-nanomaterials-10-00003){ref-type="sec"}, a series of nanoslit-based 2D lenses with different apertures was built, which also operate in cedar oil with the designed focal length of 1.5 µm.

[Table 1](#nanomaterials-10-00003-t001){ref-type="table"} illustrates the simulated focusing performances of these lenses. For the lens with a small aperture of 2.942 µm, the real focal length is 1.444 µm. The deviation from the design value is 3.73%, and the FWHM is 160 nm (*λ*/2.53), larger than the diffraction limit of 134 nm (*λ*/3.03). Therefore, for this case, the lens cannot break the diffraction limit. When the lens aperture increases, the super-resolution focusing capability of a lens can be improved and the focal length is also close to the design value, as shown in [Table 1](#nanomaterials-10-00003-t001){ref-type="table"}. From [Figure 5](#nanomaterials-10-00003-f005){ref-type="fig"}a, it can be clearly observed that the FWHM rapidly decreases and gradually flattens out when the lens aperture increases from 2.942 µm to 13.550 µm.

Focal length also has a great influence on the super-resolution capability of the nanoslit-based 2D lens. [Table 2](#nanomaterials-10-00003-t002){ref-type="table"} gives the focusing performance of nanoslit-based lenses with different focal lengths. These lenses were designed with different focal length and nearly identical aperture by using the same design procedure described in [Section 2](#sec2-nanomaterials-10-00003){ref-type="sec"}. From [Table 2](#nanomaterials-10-00003-t002){ref-type="table"}, it can be seen that the FWHM of the lens focus with the focal length of 0.476 µm is 86 nm (*λ*/4.71), much smaller than the diffraction limit of 134 nm (*λ*/3.03), demonstrating an excellent super-resolution focusing ability. However, for the lens with the focal length of 4 µm, the FWHM is 138 nm (*λ*/2.93), larger than the diffraction limit. As a result, the lens cannot break the diffraction limit. The lens with smaller focal length shows better super-resolution focusing ability, as illustrated in [Figure 5](#nanomaterials-10-00003-f005){ref-type="fig"}b.

4. Conclusions {#sec4-nanomaterials-10-00003}
==============

In summary, we explore the effect of various factors on the super-resolution focusing properties of a TE-polarized nanoslit-based 2D lens for guiding practical application. The factor of nanoslit length is firstly considered. The results illustrate that the nanoslit length determines the line focus length of a nanoslit-based 2D lens. The latter is less than the former. Nevertheless, the larger nanoslit length can lead to the smaller deviation of the focus length from it. On the other hand, the focal size of a nanoslit-based 2D lens depends on the two parameters, namely, lens aperture and focal length. When the lens aperture is larger than a certain value, the lens can realize super-resolution focusing. Moreover, the super-resolution capability can be improved as the lens aperture increases. The focal length has a decisive impact on the super-resolution capability of a TE-polarized nanoslit-based 2D lens. A smaller focal length can lead to better super-resolution-focusing performance. When focal length is larger than a certain value, the super-resolution focusing will not be achieved, even though the lens has a large aperture. Our work provides a design guide not only for optimizing a TE-polarized nanoslit-based 2D lens but also for optimizing a TM-polarized nanoslit-based lens \[[@B11-nanomaterials-10-00003],[@B12-nanomaterials-10-00003],[@B13-nanomaterials-10-00003],[@B14-nanomaterials-10-00003],[@B15-nanomaterials-10-00003],[@B16-nanomaterials-10-00003],[@B17-nanomaterials-10-00003]\].

Data curation, S.Z.; Formal analysis, Z.W.; Methodology, Y.Y.; Project administration, W.Y.; Supervision, W.L.; Writing---review & editing, Y.Z. The manuscript was written through contributions of all authors. All authors have given approval to the final version of this manuscript.

This research was funded by the National Natural Science Foundation of China (51622509), National Natural Science Foundation of China (51902240), National Natural Science Foundation of China (51975483), the 111 Project (B13044) and General Armament Department Advanced Manufacture of China (41423020111).

The authors declare no conflict of interest.

![Structure and focusing scheme of a transverse-electric (TE) polarized nanoslit-based two-dimensional (2D) lens. (**a**) The structure of the lens is formed by an array of nanoslits perforated in a gold film located on a glass substrate. (**b**) Schematic focusing of the lens; *t* is the thickness of metal film.](nanomaterials-10-00003-g001){#nanomaterials-10-00003-f001}

![Design and optical focusing of a TE-polarized nanoslit-based 2D lens. (**a**) Phase delay of a 200-nm-deep nanoslit as a function of nanoslit width *w* for the operating wavelength of 405 nm. (**b**) Geometry of the lens (inset) and the phase delay of each nanoslit calculated based on Equation (2). (**c**) Finite-difference time-domain (FDTD) simulated electric field intensity \|*E*\|^2^ of the lens. The plane *x* = 0 is the exit surface of the lens, and nanoslit length *l* is assumed to be infinite.](nanomaterials-10-00003-g002){#nanomaterials-10-00003-f002}

![Focusing properties of the TE-polarized nanoslit-based 2D lens with nanoslit length *l* = 4 µm. Simulated electric field intensity \|*E*\|^2^ distributed (**a**) at the focal plane and (**b**) on the line *y* = 0 in [Figure 3](#nanomaterials-10-00003-f003){ref-type="fig"}a. (**c**) The full width at half maximum (FWHM) of the focus located at the lens center (*z* = 0). (**d**) The FWHM of the focus located at the position of maximum field strength.](nanomaterials-10-00003-g003){#nanomaterials-10-00003-f003}

![Focusing properties of the TE-polarized nanoslit-based 2D lens with nanoslit length *l* = 15 µm. Simulated electric field intensity \|*E*\|^2^ distributed (**a**) at the focal plane and (**b**) on the line *y* = 0 in [Figure 4](#nanomaterials-10-00003-f004){ref-type="fig"}a.](nanomaterials-10-00003-g004){#nanomaterials-10-00003-f004}

![Effect of lens aperture and focal length on the focal size. (**a**) Effect of lens aperture on the lens FWHM. The designed focal length of the lens is 1.5 µm. (**b**) Effect of focal distance on the lens FWHM.](nanomaterials-10-00003-g005){#nanomaterials-10-00003-f005}

nanomaterials-10-00003-t001_Table 1

###### 

The derived focusing performance of nanoslit-based 2D lenses with different lens sizes.

  Lens Aperture (µm)   Designed Focal Length (µm)   Simulated Focal Length (µm)   FWHM (nm)        Max. Intensity (a.u.)
  -------------------- ---------------------------- ----------------------------- ---------------- -----------------------
  2.942                1.5                          1.444                         160 (*λ*/2.53)   0.918
  4.362                1.472                        126 (*λ*/3.21)                1.719            
  5.626                1.480                        114 (*λ*/3.55)                2.264            
  6.822                1.488                        104 (*λ*/3.89)                2.680            
  7.978                1.488                        102 (*λ*/3.97)                3.146            
  9.114                1.484                        100 (*λ*/4.05)                3.677            
  10.238               1.484                        98 (*λ*/4.13)                 4.025            
  11.350               1.484                        96 (*λ*/4.22)                 4.280            
  12.454               1.484                        95 (*λ*/4.26)                 4.607            
  13.550               1.484                        94 (*λ*/4.31)                 4.921            

nanomaterials-10-00003-t002_Table 2

###### 

The derived focusing performance of nanoslit-based lenses with different focal lengths.

  Designed Focal Length (µm)   Lens Aperture (µm)   Simulated Focal Length (µm)   FWHM (nm)
  ---------------------------- -------------------- ----------------------------- ----------------
  0.500                        10.230               0.476                         86 (*λ*/4.71)
  1.000                        10.564               0.984                         96 (*λ*/4.22)
  1.500                        10.238               1.484                         98 (*λ*/4.13)
  2.000                        10.386               1.968                         104 (*λ*/3.89)
  2.500                        10.482               2.472                         118 (*λ*/3.43)
  3.000                        10.474               2.968                         122 (*λ*/3.32)
  3.500                        10.394               3.468                         128 (*λ*/3.12)
  4.000                        10.230               3.968                         138 (*λ*/2.93)
