We report a metallic planar lens based on the coupled nanoslits with variable widths for superfocusing. The influence of the interaction between two adjacent nanoslits on the phase delay is systematically investigated using the finite-difference time-domain (FDTD) method. Based on the geometrical optics and the wavefront reconstruction theory, an array of nanoslits perforated in a gold film is optimally designed to achieve the desired phase modulation for light beaming. The simulation result verifies our design in excellent agreement and the realized metallic lens reveals the superfocusing capability of 0.38λ in resolution, well beyond the diffraction limit.
© 2015 Optical Society of America
The imaging resolution of conventional lenses is restrained by the diffraction limit, which is mainly caused by the absence of the evanescent waves carrying the high-spatial-frequency information in the image . To overcome this limit, many approaches have been suggested, e.g., shifting media based on the theory of transformation optics [2,3 ]. Allowing the enhancement of evanescent components of an illuminated object via the excitation of surface plasmon polaritons (SPPs), plasmonic lenses also have the potential to overcome this resolution limit. A metallic slab superlens can do optical imaging beyond the diffraction limit with a resolution of λ/6 (λ is the working wavelength) , but the object and image are greatly confined to the near-field region, only tens of nanometers away from the superlens. In the last decade, many researchers have employed different metallic nanostructures, such as slits, apertures, holes and surface corrugations, to manipulate the light [5–11 ].
As a typical category, planar lenses based on metallic nanoslits with varying widths have been studied both numerically [12–15 ] and experimentally [16,17 ]. The imaging process is associated with the manipulation of optical phase delay by controlling the widths and positions of nanoslits. Compared with the metallic slab superlens, this type of metallic nanostructured lens can realize imaging at an anticipated distance from the lens via the appropriate design of structural parameters. However, in these previous studies, it was assumed that the metallic wall was thick enough to exclude the coupling of SPPs propagating in any two adjacent nanoslits.
In 2009, taking into account the coupling effect, Verslegers et al designed a kind of metallic nanoslit lens to realize the angle compensation . However, the focal lengths achieved by the numerical simulation differ greatly from the designed values. The main reason for the discrepancy we think is that, due to the aperiodic feature of nanoslits with variable widths constructing the lens, the theory based on the symmetric mode in a periodic metallic waveguide array cannot predict the actual phase delay for all the nanoslits accurately. Therefore, the approach they proposed needs to be optimized. On the other hand, to the best of our knowledge, few studies have reported the realization of superfocusing for the structured planar lenses in the Fresnel region. For these purposes, we propose an optimal design method for the metallic planar lenses composed of coupled nanoslits in a gold film in this research. Using the finite-difference time-domain (FDTD) method, we analyze in detail the influence of the interaction between two adjacent nanoslits on the phase delay of one nanoslit. Based on the analytical results and the phase delay of each individual nanoslit predicted by the symmetric-mode dispersion relation, a metallic lens is purposely designed. The final numerical simulation verifies that the designed lens can achieve superfocusing of light at the anticipated position in the Fresnel region.
2. Phase delay: a numerical & theoretical investigation
We first investigate the influence of the aperiodicity of two coupled nanoslits on the phase delay. Figure 1 is the schematic of a basic structure consisting of two adjacent nanoslits (slit-1 and slit-2) filled with a homogenous dielectric (air) in a gold film with the thickness t of 400 nm. The two nanoslits are spaced apart by a metallic wall with the width of s. w 1 and w 2 represent the width of slit-1 and slit-2, respectively. When a TM-polarized plane wave is normally incident on the structure, the SPPs can be excited and propagate inside the two nanoslits. Moreover, the SPPs fields can penetrate into the gold wall between the two nanoslits over a distance. At the operating wavelength of 650 nm, the penetration depth, δm, also called the skin depth, is about 28 nm, which is given by 
In simulations, the unit length of FDTD cells is set to 2 nm in both x and y directions to ensure the convergence of the computation and model the fine features of the electromagnetic field in the structure. The incident TM-polarized plane wave is defined by setting the electric-field component of Ey. The operating wavelength for all the cases is 650 nm. The permittivity of gold at this wavelength is εm = −12.8915 + 1.2044i . Since the SPPs modes propagating in a nanoslit are transverse magnetic (TM) modes in character and the magnetic field (H) is in the z direction, the phase analysis is performed based on the real part of Hz.
For a specific gold spacing s, the optical transmission in slit-1 is determined by the widths of both nanoslits. Figure 2 plots the FDTD simulated phase delay extracted from slit-1 varying as a function of w 2 when s is kept to 30 nm. The phase delay of slit-1 with variant widths displays different trends. The narrower the slit-1, the more susceptible its phase delay to the slit-2. For example, as the width of slit-2 increases from 10 nm to 100 nm, the phase delay of slit-1 with w 1 = 100 nm changes from 1.27π to 1.35π. And the maximum fluctuation around the phase delay in the case of w 1 = w 2 is only 0.12π. Similarly, for the cases of w 1 = 60, 80 nm, such fluctuations are 0.1π and 0.13π, respectively. For the slit-1 with w 1 = 30, 40 nm, the fluctuations increase to 0.25π and 0.3π, respectively.
The optical propagation in a narrower slit-1, however, changes greatly with the variation of slit-2 width. When w 2 varies from 32 nm to 100 nm, the SPPs in slit-1 with w 1 = 20 nm propagates backwards at the exit, as illustrated in Fig. 3(a) in the case of w 2 = 40 nm, which is denoted by the phase delay of –π; when w 2 is smaller than 30 nm, they pass through slit-1 and radiate into free space as normal [Fig. 3(b)]. For the slit-1 with w 1 = 10 nm, in most cases, the optical transmission is locally off, as shown in Fig. 3(c) in the case of w 2 = 20 nm, which is denoted by the phase delay of π. These two kinds of abnormal transmission badly affect the phase modulation in the lens design. The physical mechanism for them is under investigation. As for the lens design in this research, we try to avoid the appearance of these abnormal phenomena, which can be realized by increasing the spacing between adjacent nanoslits. For instance, as shown in Fig. 4 , for the case of w 2 = 60 nm, when s is larger than 60 nm, slit-1 with w 1 = 10 nm can play a required role in manipulating the phase delay. In general, when w 2 is closer to w 1, the influence of the interaction between the two nanoslits on the phase delay becomes less.
From Fig. 4, we can also observe that the influence of slit-2 on the phase delay of slit-1 has a close relationship with the spacing s. It is because the spacing determines the strength of the SPPs coupling between the adjacent nanoslits. A larger spacing makes the coupling weaker, and so the interaction between nanoslits is smaller. As a consequence, the influence of one nanoslit on the phase delay of an adjacent nanoslit becomes less. Conversely, such influence becomes greater. As illustrated in Fig. 5 , when s = 10 nm, the phase delay of slit-1 with w 1 = 10 nm (w 2 = 60 nm in this case) is 1.62π, which is much smaller than 3.25π for the spacing s = 300 nm where the SPPs coupling doesn’t exist [as shown in Fig. 5(c)]. From Fig. 5(b), we can clearly see that for the case of s = 60 nm, which is slightly larger than 2δm, the SPPs from the two adjacent nanoslits still interact with each other and the coupling is obvious.
On the other hand, the phase delay of each individual nanoslit is predicted by Re(β)t, where t is the thickness of the metal film, and β is the complex propagation constant in the nanoslit which is calculated based on the symmetric-mode dispersion relation in a periodic metallic waveguide array with the following equation Eq. (2) is established based on a periodic structure consisting of identical nanoslits and metallic spacings, in order to achieve the required phase delay for designing a metallic lens whose structure is aperiodic, it is necessary to consider the influence of the aperiodicity of adjacent nanoslits on the phase delay. When such influence is small, the prediction method is valid. Otherwise, it is inaccurate, for example, for the situation where the abnormal optical transmission appears in a nanoslit.
On the basis of Eq. (2), the dependence of phase delay on the nanoslit width at different spacings is shown in Fig. 6 . With the increase of nanoslit width from 10 nm to 100 nm, the phase delay of the nanoslit rapidly decreases, and then gradually flattens out. Moreover, the larger spacing leads to a larger phase delay. When the gold spacing s = 60 nm, the phase delay of a nanoslit approaches its maximum, namely, that of an isolated nanoslit with the same width which is calculated based on the following dispersion relation 
3. Metallic planar lens: an optimal design
For a normally incident light, in order to achieve the focusing of light by using a metallic lens formed by nanoslits, the width of each nanoslit and the metallic spacing must be properly arranged to match a curved wavefront, so that the light coming out from the structure is in phase at the focal point. Theoretically, the required phase delay difference (Δφ(y)-Δφ(0)) as a function of position y (with y = 0 at the center of the designed lens) is calculated by
Based on Eq. (2) and (4) , an original metallic nanoslit lens with uniform 30 nm spacing is built, irrespective of the effect of the interaction between adjacent nanoslits on the phase delay. It consists of a 400 nm thick gold film on a glass substrate and the desired focal length is f = 0.3 µm. As the lens structure is symmetric with respect to y = 0, we just analyze the half part (y≥0) with a total phase delay difference (Δφ(y)-Δφ(0)) of 2π. By using Eq. (2), a series of nanoslits with well-chosen widths are designed to match the required phase delay difference calculated from Eq. (4). Beginning from y = 0, the width sequence of nanoslits is: 20, 20, 22, 24, 28, 36, 64, 10, 10, 10, 10, 10, 10, 10, 12, 12, 14, 16, 18, 24 nm. The FDTD method is utilized to investigate the focusing properties of the originally designed metallic lens. The simulated focal length is 0.235µm, leading to a distinct deviation of 20% from the designed value. This is mainly caused by the poor match of the simulated phase delay differences with the required values at certain positions [as shown in Fig. 7 ]. In particular, the abnormal optical transmission mentioned above appears in the outmost nanoslit and the narrowest nanoslit (10 nm wide) adjacent to the widest one (64 nm wide), which is represented by the phase delay difference of −3π.
Combining the theoretical calculation with the numerical investigation of the influence of the coupling between aperiodic nanoslits on the phase delay, we optimize the original metallic lens by adjusting some nanoslits and gold spacings. Figure 8(a) exhibits the half part of the optimized lens. Numerical results of FDTD demonstrate that the simulated phase delay differences for all the nanoslits match well with the required values [as illustrated in Fig. 7]. The simulated distribution of Re(Hz) is shown in Fig. 8(b), from which we can observe that a curved wavefront is formed when a TM-polarized plane wave propagates through the lens. The focusing behavior is confirmed from the magnetic intensity pattern [Fig. 8(c)]. The realized focal length is 0.298 µm, which perfectly agrees with the designed value. Moreover, the full width at half maximum (FWHM) of the focal spot is 250 nm [Fig. 8(d)], about 0.38λ, well beyond the diffraction limit.
Based on the detailed investigation of the propagating properties of the SPPs mode in coupled nanoslits and geometrical optics, a metallic planar lens composed of elaborately arranged nanoslits and spacings in a gold film is optimally designed. Simulation results of FDTD indicate that the realized focal length agrees excellently with the designed value. And a superfocusing focal point is achieved in the Fresnel region, for which the coupling of SPPs propagating in adjacent nanoslits may be the main reason. The research on the underlying physical mechanism for the realization of the superfocusing by employing the coupled width-variable nanoslits is under way, aiming to achieve a better superfocusing capability. The proposed method in this letter extends the design of plasmonic lenses formed by metallic nanoslits. It is believed that the metallic planar lens would have great potential applications in integrated optics, optical lithography and sensing.
We acknowledge the financial support by the National Natural Science Foundation of China (Grant No. 51375400), the Program for the New Star of Science and Technology of Shaanxi Province (Grant No. 2014KJXX-38), the Aeronautical Science Foundation of China (Grant No. 2013ZC53036), the NPU Foundation for Fundamental Research (Grant No. JCY20130119), the Fundamental Research Funds for the Central Universities (Grant No. 3102014JC02020504), and the Program for the New Century Excellent Talents in University.
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