## Abstract

We report the generation of a subwavelength focal spot for surface plasmon polaritons (SPPs) by increasing the proportion of high-spatial-frequency components in the plasmonic focusing field. We have derived an analytical expression for the angular-dependent contribution of an arbitrary-shaped SPP line source to the focal field and have found that the proportion for high-spatial-frequency components can be significantly increased by launching SPPs from a horizontal line source. Accordingly, we propose a rectangular-groove plasmonic lens (PL) consisting of horizontally-arrayed central grooves and slantingly-arrayed flanking grooves on gold film. We demonstrate both numerically and experimentally that, under linearly polarized illumination, such a PL generates a focal spot of full width half maximum 274 nm at an operating wavelength of 830 nm. The method we describe provides guidance to the further structure design and optimization for plasmonic focusing devices.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The diffraction limit has been thought to be the main obstacle in modern optics because it does not allow localization of electromagnetic field in regions noticeable smaller than half the wavelength. This imposes a fundamental limit on the resolution of imaging as well as the level of miniaturization of optical devices. Near-field strategies, such as scanning near-field optical microscopy [1,2], field-concentrators [3,4], and superlens [5–7] were explored to recover quickly fading evanescent waves to reduce the size of a focal spot below the diffraction limit. In the far-field regime, approaches including angular spectrum synthesis [8] and optical eigenmode technique [9] have been proposed to generate the subdiffractive spot. The concept of super-oscillations was also exploited to yield a focal spot of arbitrarily small size at the expense of the spot intensity and the emergence of sidebands [10–12]. On the other hand, new opportunities for confining light to rather small dimensions are provided by utilizing surface plasmon polaritons (SPPs), which are collective electromagnetic excitations propagating at a metal/dielectric interface [13,14]. The ability to focus SPPs with subwavelength resolution is essential to a wide range of applications including nano-optics [15,16], super-resolution imaging [17] and nanolithography [18], etc. Various plasmonic lenses (PLs), most comprising a single or a series of concentric rings on a metal film, have been proposed to couple free space light into SPPs and redirect them towards the center of the circle. The focal spot size by such circular-shaped PLs is limited to λ_{0}/2, where λ_{0} refers to the incident wavelength [19,20]. When a radially polarized beam is utilized to match the rotational symmetry of the lens, the plasmon spot size can be reduced to about λ_{0}/2.78 [21,22]. Recently, sub-100 nm focusing is achieved exploiting short-wavelength SPPs, which provides the possibilities for super-resolution in vivo bio imaging [23]. Despite such great promise, an analytical means to optimize the design of PLs for a smaller spot is still lacking.

In this paper, we propose to reduce the size of a plasmon focal spot by increasing the proportion of high-spatial-frequency components in the plasmonic focusing field. Based on such concept, we demonstrate a rectangular-groove PL that achieves sub-one-third wavelength focusing of SPPs excited by linearly polarized illumination. The proposed lens consists of horizontally-arrayed central grooves sandwiched between flanking slantingly-arrayed grooves. Key parameters such as the focal length and the subtended angle at the focus by the central grooves are optimized by SPP point source model. We demonstrate both numerically and experimentally that such a PL generates a focal spot of full width half maximum (FWHM) 274 nm at an operating wavelength of 830 nm. Moreover, rotationally symmetric focusing of SPPs using two proposed PLs without radially polarized illumination is also demonstrated.

## 2. Design principle

#### 2.1 Theory

In Fourier optics, an object can be mathematically taken as a weighted superposition of many spatial frequencies. Larger the proportion of high-spatial-frequency components contained in a focusing field, smaller the size of a focal spot will yield. For monochromatic waves, high spatial frequencies are related to plane waves traveling at large off-axis angles. One important example is the use of annulus pupils to give a narrow central lobe in the point-spread function [25]. Nevertheless, components with intermediate spatial frequencies are also required to form a localized focus. As a limiting case, counter-propagating SPP waves will create a standing-wave like pattern that produces the finest but extended interference fringes [18]; this is because only a single spatial frequency component is involved. For plasmonic focusing, high-spatial-frequency information is carried by SPPs propagating with large convergent angles. In this section, we concentrate on how to raise the proportion of these high-angle SPPs in the focusing field to achieve a subwavelength focal spot.

First, we derive an analytical expression to account for the angular-dependent contribution to the focus formed by SPPs converging from different directions. When an SPP coupler (such as a groove or a ridge) are normally illuminated with a plane wave, every point on the structure functions as an SPP point source that radiates SPPs with the same initial phase. This SPP point source model has been served as an effective tool for calculating resulting SPP field by adding all of the fields of the SPP point sources and taking into account their phases and amplitudes [26–29]. Consider these point sources are continuously distributed along the curve of an arbitrary even function *f*(x) on the *x*-*y* plane [Fig. 1(a)]. For a *y*-polarized incidence, we can evaluate the electric field at focus (0, *F*) by elementary SPP point sources as $dE=A\frac{\text{cos}(\alpha )}{\sqrt{r}}\text{exp}(ik\text{r})\text{exp}$ ($-\frac{r}{2L}$)*d*s (time dependent factor exp(−i*ωt*) is omitted), here *E* refers to the z-component of the field which dominates the total intensity profile for an SPP mode; *A* the source strength per unit length; *k* the wavenumber of SPPs; *L* the decay length of SPPs; radius *r* and azimuthal angle *α* specify the source position with respect to the focus and –*y*-direction, respectively; *d*s is an element of length along the curve; the cosine term accounts for the directivity of the source emission. The $1/\sqrt{\text{r}}$ term is from the approximation of the 1st order Hankel function of the first kind [30]. Our calculation verifies that this expression holds for *r* > 1*µ*m in our study. Since the physical process of coupling free-space light to SPPs is not included in the model, this expression is inapplicable to deal with any problems related to the coupling efficiency. Noting that $d\text{s}=\sqrt{{\left(dr/d\right)}^{2}+{r}^{2}}d$ with d*α* being the angle subtended by *d*s to the focus, we have:

*α*, therefore |

*dE*/

*dα*| dependence on

*α*quantifies the relative contribution of the sources locating at a particular azimuthal angle. Since only the SPPs that arrive at the focal region could contribute significantly to the subwavelength focus. The spatial-frequency information carried by these SPP waves mainly depends on the azimuthal angles at which SPPs travels. The larger the azimuthal angle

*α*, the higher the spatial frequency. Therefore |

*dE*/

*dα*| quantifies the relative weight of a particular spatial-frequency component in the focusing field. Throughout this paper, we will be dealing with the SPP mode confined on the interface of air and 50-nm-thick gold film. The plasmon wavelength and decay length of the mode are λ = 814 nm (λ

_{0}= 830 nm) and

*L*= 30

*µ*m, respectively.

Next, we seek a proper geometrical arrangement of point sources (seek a function *f*(x)), whereby sources situating at large azimuthal angles significantly contribute to the focal field. To this end, let us consider a familiar case where the distribution of SPP point sources is a 5-*µ*m radius semicircle centered at (0, *F*) (i.e., $f\left(x\right)=-\sqrt{{5}^{2}-{x}^{2}}+5$, in unit of micron). Due to the cosine term in Eq. (1), the corresponding |d*E*/d*α*| decreases dramatically with increasing *α* [Fig. 1(b)]. This result reflects the TM nature of SPP modes and explains why the focal spot size (along the direction orthogonal to the incident polarization) of a circular-shaped PL limited to one-half of the plasmon wavelength under linearly polarized illumination [19,20,23]. To circumvent this problem, the curve *f*(x) has to fulfill the following geometry requirements: for large azimuthal angle *α*, more point sources should be included within d*α* to compensate for the vanishing cosine term. Intuitively, the simplest and possible solution is a linear function *f*(x) = tan*β*•*x*, here angle *β* formed by the line and the + *x*-axis. Figure 1(b) depicts calculated |*dE*/*dα*| as a function of *α* for different values of *β*. In the calculation, focal length *F* is fixed at 5 *µ*m. For a given *α* exceeding 20°, |*dE*/*dα*| increases with decreasing *β*. In particular, the curve of *β* = 0 is highlighted for a pronounced peak at *α* = 80°, meaning that the contribution to the focal field is dominated by high-spatial-frequency components when point sources distributed along a line perpendicular to the incident linear polarization. As a result, function *f*(x) = 0 is an applicable solution. Besides the geometrical arrangement of point sources, the focal length *F* also plays a role in shaping the angular-depended contribution. Figure 1(c) depicts |*dE*/*dα*| as a function of *α* for different *F* when *f*(x) = 0. As can be seen, the peak is more pronounced for curves with small values of *F*. This is because longer focal length design leads to a higher propagation loss, especially for SPPs that come to the focus along large azimuthal angles.

#### 2.2 Structure and parameter description

Based on the theoretic investigation above, we propose a PL to achieve super-focusing of SPPs with linearly polarized light. The lens consists of three layers of rectangular nano-grooves, which are carved into a 50 nm-thick gold film with a glass substrate [Fig. 2(a)]. Grooves in each layer lie on (also aligned along) a polyline, which consists of a horizontal line segment and two flanking slanted line segments ( ± 45-degree angle to the *x*-axis). Mathematically, these polyline are given by a piecewise function [Fig. 2(b)]:

*i*= 1,2,3 is the layer index;

*F*the focal length;

*d*= 2λ/3 and $d\text{'}=\sqrt{2}\text{\lambda}/3$ are the layer spacings;

*α*the half-angle subtended by horizontal line segments. All grooves have the same width of λ/2 to the advantage of a high coupling efficiency [31,32]. The groove lengths (for both central and flanking grooves) are determined in the way that the distance between the focus and inner and outer edges of the

*k*th groove are

*F*+ (

*i*–2)

*d*+ (

*k*–1)λ/3 and

*F*+ (

*i*–2)

*d*+

*k*λ/3, respectively. Here

*k*= 3,6,9,12,… for

*i*= 1;

*k*= 1,4,7,10… for

*i*= 2;

*k*= 2,5,8,11… for

*i*= 3. When a

*y*-polarized plane wave illuminates the lens, each constituent groove serves as a local SPP coupler and such a configuration ensures that all groove-excited SPPs have a mutual phase-lag ≤ 2π/3 at (0,

*F*), thus SPPs will interfere constructively at the focal point. According to the discussion above, if an infinite number of central grooves are positioned on infinitely long horizontal lines (

*α*= 90°), a subdiffractive spot will be formed because the proportion of high-spatial-frequency components in the focusing field is significantly enhanced. Calculations using SPP point source model reveals that the transverse FWHM of the focus decreases with the design focal length

*F*[Fig. 2(c)]. For

*F*≤ 2

*µ*m, the FWHM saturates at 272 nm, which is almost λ

_{0}/3.05. One advantage of such lens design is that: By removal of existing grooves or adding additional grooves to the structure, we can modulate its angular-dependent contribution to the focus, thus the focal spot size can be engineered. In principle, transverse FWHM of the focus can be reduced if low order grooves are removed and can be increased if additional low order grooves are added. Here, the 1st to 4th grooves are selectively removed from the full structure to reduce the transverse FWHM of the focus. After exploring all possible combinations and taking into account the reduction in focal intensity (due to less coupling from the grooves) and the sidelobe level, we find that removal of the 2nd to 4th grooves is an applicable way to significantly reduce the spot size [Fig. 2(c)]. For PLs (

*F*= 5

*µ*m,

*α*= 90°) with and without 2nd to 4th grooves, the FWHMs of their focal spots are 278 and 262 nm, which are λ

_{0}/2.99 and λ

_{0}/3.17, respectively. Meanwhile, the intensity at the focal spot is reduced by ~25%. In fact, removal of the 1st groove could further reduce the spot size at the price of significant sidelobes and lower focal intensity. For instance, a PL (

*F*= 2

*µ*m,

*α*= 90°) without the 1st groove generates a focus with a FWHM = 240nm (λ

_{0}/3.5), but the focal intensity is reduced by up to 40% and the sidelobe level is ~0.68 relative to the central lobe because of the standing-wave like pattern, similar to that reported in [23,24]. Therefore there is a trade-off between the spot size and the efficiency of focusing.

For practical concerns, infinitely arrayed central grooves should be truncated by setting *α* to be a value smaller than 90°, and the missing SPP waves that travelling along azimuthal angles beyond *α* are compensated by utilizing additional flanking grooves [Fig. 2(b)]. Figure 2(d) shows the transverse FWHM of the focus for a proposed PL (*F* = 5*µ*m) with or without flanking grooves under different truncation parameter *α*. Indeed, comparing with the PL without flanking grooves, the presence of flanking grooves reduces the size of focal spot, especially for small *α* designs. As mentioned before, the focus size can be further reduced by ~20 nm if 2nd, 3rd, and 4th grooves are removed from the structure for a given parameter *α* [Fig. 2(d)]. It is worth noting that, for PL with flanking grooves, structures with larger *α* designs leads to a smaller focal spot, which verifies that the horizontally arrayed central grooves are superior to slantingly-arrayed flanking grooves in providing high-spatial-frequency components, as predicted in Fig. 1(b). We also note that the different of using slanted straight lines (*β* = 45°) and circular curves in the flanking grooves is little within the framework of SPP point source model [Fig. 1(b)]. In fact, for PLs with larger *α* designs which is essential to achieve subdiffraction focusing, a circular curve approximates a line segment since its arc central angle (i.e.,π/2–*α*) is small.

As a limiting case, the smallest spot size by focusing SPPs of high-spatial-frequency is ~λ/4, where counter-propagating SPP waves produce a standing-wave like pattern (sidelobe level is 1 relative to the central lobe). To suppress the strong sidelobe intensity, intermediate spatial frequencies must be involved in the focusing field. Therefore, the focusing mechanism here is to focus SPPs of high-spatial-frequency with large amplitude as well as intermediate spatial frequencies in moderate amplitude. It is well known that by combining different spatial frequency components, sub-diffraction focusing can be achieved. A famous example is the super-oscillation, which always oscillates with very small amplitude since it is one kind of destructive interference of light with different spatial frequencies at some points by carefully choosing the amplitude of every frequency. A higher weight of the high-spatial frequencies for super-oscillation is unnecessary. Therefore, the working principle of our proposed PLs is different from super-oscillation. For a proposed PL, the ratio of power in the focal spot to total power in the focal plane is estimated to be ~20%, which is significant higher than that of a typical superoscillatory lens.

#### 2.3 Results and discussion

In this section, the performance of a proposed PL (*F* = 5µm, *α* = 80°) without 2nd, 3rd, and 4th grooves is investigated [Fig. 3(a)]. The 3D finite-difference time-domain (FDTD) method is utilized to calculate the intensity of SPPs at the plane of z = 0. In the simulation, rectangular grooves (depth = 40 nm) are etched on a 50 nm-thick gold film which is sandwiched between air and a glass substrate. Mesh sizes of 10, 10 and 5 nm are employed for the *x*-,*y*- and *z*-axes, respectively. A *y*-polarized plane wave is used as the excitation source, which illuminates from the air side and covers the entire PL structure. The dielectric constants of gold and glass are set to *ε*_{gold} = 26.6 + 1.67*i* (interpreted from [33]) and *ε*_{d} = 2.25, respectively. Perfectly matched layer absorbing boundary conditions are used at all boundaries. In Fig. 3(b), FDTD simulated SPP intensity (|*E*|^{2} = |*E*_{x}|^{2} + |*E*_{y}|^{2} + |*E*_{z}|^{2}) distribution is shown, which exhibits a sharp focal spot due to the optimized lens design. The transverse and longitudinal line cuts through the focus are depicted in Fig. 3(b), revealing a focal spot FWHM of 274 nm, which is λ_{0}/3.03 and coincides with the calculated value of 274 nm by SPP point source model [Fig. 2(d)].

Figure 3(d) shows the scanning electron microscopy (SEM) image of the device, which was fabricated using a focused ion beam milling system on a 50 nm-thick gold film sputtered onto a microscope slide glass substrate. A *y*-polarized beam from a CW Ti-Sapphire laser (operating wavelength *=* 830 nm) was normally incident upon the sample from the air side. Through a cylindrical lens L_{1}, the incident field was focused into a line (width ~15*µ*m, length ~1cm) to cover the entire structure. The SPP intensity distribution was detected using a leakage radiation microscope (LRM) equipped with an oil-immersion objective O_{1} (100 × , NA = 1.4), three auxiliary lenses L_{2}~L_{4} (focal length = 120 mm), and a charge coupled device [Fig. 3(g)]. A spatial filter was introduced to filter out the directly transmitted light through the gold film [34]. Because only the in-plane components (|*E*_{r}|^{2} = |*E*_{x}|^{2} + |*E*_{y}|^{2}) of the SPP field participate to the LRM image [35,36], we do not expect direct observation of the subdiffractive confinement for SPPs in the absence of the dominant *E*_{z} component (|*E*_{z}/*E*_{r}| = |*ε*_{gold}/*ε*_{d}|~12), which could be detected using near-field technique, such as apertureless SNOM, or using scattering technique. However, the existence of such a small focus can be well confirmed by comparing the measured image [Fig. 3(e)] and simulated image of SPP in-plane components [Fig. 3(f)]. Instead of a sharp focus, a two-lobe pattern along the *x*-direction is formed due to the destructive interference of the *E*_{x} component of SPP waves propagating along large azimuthal angles. Note that all fine details in Fig. 3(e) are precisely reproduced in Fig. 3(f). A comparison plot of transverse line cuts through the centers of the images is given in Fig. 3(h), where these two intensity profiles are in good agreement. The slight asymmetry of the measured profile is due to the misalignments of the laser source and the sample. All these results validate that a focal spot with a FWHM of λ_{0}/3 is achieved. Since the out-of-plane component *E*_{z} dominates in the SPP mode, therefore, the created plasmonic focal field is an evanescent optical “needle” with the electrical field substantial normal to the metal/air interface, which may find applications in near-field imaging, sensing, lithography and nanoparticle manipulations.

It should be mentioned that both the SPP point source model we use and Eq. (1) are only valid for *y*-polarized illumination; the designed PLs are unable to focus SPPs under *x*-polarized illumination. For circularly-polarized illumination, the resultant SPP field is equal to the sum of two SPP fields: the field excited by *y*-polarized illumination and the field excited by *x*-polarized illumination (with a relative phase difference = ± π/2). The undesired SPP field excited by the latter may jeopardize our goal of subdiffraction focusing. Therefore the proposed PLs are unsuitable for circularly-polarized illumination.

## 3. Rotationally symmetric focusing of SPPs

As well known, a rotationally symmetric focus can be created through using a combination of radially polarized illumination and circular-shaped PL due to the perfect match between polarization and structure symmetry [21,22]. Nevertheless, such symmetry-matched scheme requires a perfectly alignment of the beam center and the center of the lens; utilizing radially polarized source also makes the scheme less practical. Here we demonstrate that rotationally symmetric focusing of SPPs can be achieved in an optimized way from two proposed PLs facing each other [Fig. 4(a)]. The focal lengths of the two PLs are purposefully designed with mismatch of λ/2 to compensate for the phase-lag of SPPs originating from different PLs [24]. For each PL, an additional 1st groove is placed besides the original one with spacing of λ to slightly squeeze the resultant focal spot in the *y*-direction. For rotationally symmetric focusing, the in-plane components of the SPP field is a doughnut-shaped pattern (not shown) due to the destructive interference of the *E*_{x} component and *E*_{y} component at the center. Therefore, the LRM image is unable to show the rotationally symmetric focusing of SPPs. However, FDTD simulated SPP intensity verifies that a clear and distinctive rotationally symmetric focus is generated by these PLs [Fig. 4(b)]. Line cuts through the focus are depicted in Fig. 4(c), revealing transverse and longitudinal FWHMs of 303 and 308 nm, respectively, almost as small as that of a circular-shaped PL under radially polarized illumination.

## 4. Summary

To summarize, we have reported the controlled generation of a subwavelength focal spot for SPPs by increasing the proportion of high-spatial-frequency components in the plasmonic focusing field. Based on SPP point source model, we find that when SPPs are launched from a horizontal line source, SPPs traveling along large convergent angles can be enhanced. Subsequently, we propose a PL comprising horizontally-arrayed grooves sandwiched between flanking slantingly-arrayed grooves on a gold film. Employing such a PL with linearly polarized illumination, we achieve a spot size of one-third of incident wavelength. We also successfully realize rotationally symmetric focusing of SPPs using two proposed PLs without radially polarized illumination. Finally, we emphasize that our methodology purely relates to the wave nature of SPPs, therefore the presented design method should be compatible for shorter wavelength plasmons to yield an ultra-small spot, which is crucial for nanofocusing applications involving SPPs.

## Funding

National Natural Science Foundation of China (NSFC) (61205051, 11504307, 11574011); Foundation of Fujian Educational Committee (JAT170002); Fundamental Research Funds for the Central Universities (20720180010, 20720160042).

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