1. Introduction

Focusing of surface plasmon polaritons (SPPs) by plasmonic lenses is attracting much interest recently, with applications in microscopy, lithography and sensing to name a few. A variety of focusing schemes were introduced by several authors. It was shown that focusing of SPPs is feasible by the use of a plasmonic lens consisting of a single annular slit [1]. Surrounding the annular slit by a circular periodic grating structure can enhance the intensity at the focal point of the lens [2]. The focus can also be tuned by controlling the incident angle [3]. A plasmonic lens can also generate a narrow beam with reduced diffraction [4]. In addition to circular geometries, plasmonic lenses were also demonstrated as in-plane focusing elements using either metallic [5,6] or dielectric [7] subwavelength structures. A major challenge related to SPP focusing is to achieve tight focusing with high energetic efficiency at the focus. Clearly, the polarization of light plays a major role in mitigating this goal. In this manuscript we propose, optimize, and numerically analyze a plasmonic lens that is illuminated by a radially polarized light. For a plasmonic lens having circular symmetry, an incident radially polarized beam is TM polarized along the entire metal\dielectric interface [8,9]. Therefore, radially polarized light is a natural choice for SPP excitation in such structures, allowing efficient coupling to SPP modes, and preventing radiation into free space of the azimuthal field components which cannot contribute to the focusing process as they are TE with respect to the circularly symmetric structure.

2. Focusing scheme and structure layout

To explain the evolution of focusing in the plasmonic lens we divide the structure geometry into three distinct regions, labeled in Fig. 1(a) as regions A, B and C. We analyze light propagation in each of these regions.

In region A we assume an incident beam that is radially polarized, with a field distribution of a Bessel beam given by: E_R=J₁(αr) where r is the radial coordinate and α is a scaling factor. The Bessel beam is a solution of the wave equation in cylindrical coordinate system, i.e. it is an allowed mode of propagation in free space (in practice, the Bessel beam should be truncated, e.g. by a Gaussian envelope such that it does not carry an infinite amount of energy). The free space wavelength is λ ₀ and light is incident form the left side of the structure. Since focusing is obtained by constructive interference of SPPs at the center of the plasmonic lens, it is desirable to couple as much of the incident light to an SPP mode. By adding a circular grating in region A it is possible to enhance the coupling of incident light to SPPs, as shown and analyzed by several authors (see for example, [10]).

Region B consists of a metallic layer with a thickness of h_B. An annular slit having a width w is carved into the metal. Within the slit, two plasmonic MIM (metal/insulator/metal) modes are supported: an anti-symmetric “TEM-like” mode and a symmetric plasmonic mode (symmetry is defined by observing the symmetry of the field component along the propagation direction, i.e. the z direction for SPPs propagating in the slit) [11]. Generally, the “TEM-like” mode is dominant because it can be excited efficiently by the SPPs propagating along the metal/dielectric interface in region A. Similarly, this mode is coupled efficiently to the SPP modes propagating along the metal\dielectric interface in region C. The transmission of the guided mode to free space is small, because of the abrupt termination of the slit [12].

 

Fig. 1. Structure layout and parameters. (a) structure cross-section. (b) structure top view

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In region C, SPPs that are being coupled from the annular slit can propagate both towards the center of the structure (in-focus) and towards the edges of the structure (counter-focus). The counter-focus propagation can be diminished by adding a circular Bragg reflector surrounding the annular aperture in region C. An animation illustrating the evolution of focusing with and without the Bragg reflector is given in (Media 1) and (Media 2) respectively (single frames are shown in Fig. 2). These animations show that counter focusing occurs only in the case where a Bragg reflector is absent. Therefore, the Bragg reflector significantly enhances the energy density at the focal region.

3. Guidelines for choosing the parameters of the structure

The amount of energy within the focal region is controlled by the structural parameters of the plasmonic lens. Next we discuss these parameters and the considerations in choosing their values. For clarity, we separately discuss each of the three regions:

Region A̱: The grating parameters Λ_A (grating period), f_A (grating duty cycle) and h_A (grating depth) are chosen such that the coupling of the radially polarized incident beam to SPPs is maximized. To match the momentum of the incident wave to that of the SPP propagating along the metal/dielectric interface, a grating period in the order of the SPP wavelength is required. In the following examples we set Λ_A = λ _SPP,A (λ _SPP,A is the SPP wavelength of the metal\dielectric interface in region A). The duty cycle is set to a value of 0.5 such that the first order Fourier coefficient of the dielectric function is maximized, resulting maximal coupling from free space to the SPP mode. The grating depth should be on the order of a few tens of nanometers, and can be optimized specifically for each geometry. The Bessel scaling factor α should be matched to the structure geometry, so that a peak of energy with nearly constant phase will overlap with the slit. In such a case, the illumination profile of the incident beam will match the anti-symmetric TEM mode inside the slit. For other values of α the energy in the focal region will be decreased. The later is no longer true if a grating is applied to region A. In such a case, the incident energy can be collected from outside the slit by coupling to SPPs.

 

Fig. 2. (a) Frame excerpt from (Media 1) showing the energy density, for a structure with a Bragg reflector placed in region C. Counter focus propagation is diminished and energy density at the focal region (around r=0) is high. (b) Frame from (Media 2), for a structure with no Bragg reflector. Counter focus propagation can be observed. Also, the difference in energy density around the focus region is clearly observed. Image produced by an FDTD simulation which will be discussed in section 4. The color scaling is saturated and linear, and identical in both insets.

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Region Ḇ: There are several considerations in choosing the metal thickness h_B. First, the layer thickness should be above the metal skin-depth in order to prevent the incident light from passing directly into region C. In addition, the slit can be considered as a Fabry-Perot resonator, with the mirrors being the interfaces with regions A and C. Therefore, h_B should be chosen so that there is a constructive buildup of the fields inside the slit, satisfying the Fabry-Perot condition (see Ref. [13-15] describing similar mechanisms).

To provide strong coupling of energy from the slit to SPPs in region C, the slit width (w) should be in the subwavelength regiem. For example, it was shown that a slit width of about 0.25λ ₀ will provide optimal coupling to SPPs for the case of an interface between Au and air at the vacuum wavelength of λ ₀=600 nm [16]. In our study, we set the slit width to be w = 180 nm.

In order to optimize the radius of the inner metal disc (i.e. the inner radius of the slit, labeled as d in Fig.1), two competitive processes of SPP excitation and propagation losses need to be considered: As the circumference is increased, the surface area that is generating the SPPs is increased as well. On the other hand, the SPPs need to travel larger distance towards the focal region, thus subjected to higher attenuation. An exact analytic formula describing these mechanisms is given in [1].

Region C̱: A Bragg grating with a period of λ _SPP,C/2 is introduced in order to eliminate the counter-focus SPP propagation. Duty cycle and grating depth of this grating are optimized for maximal reflection and minimal loss.

4. Numerical analysis

4.1 Method and parameters

The analysis of the structure was preformed by numerical finite-difference time-domain (FDTD) simulations. All simulations were preformed with Meep, a free-software FDTD solver [17,18]. A light source with free-space wavelength λ ₀ = 600 nm having a Bessel-like field distribution with a Bessel scaling factor α =1 μm^-1 was assumed. The metal was taken to be Ag, with dielectric constant described by the Drude model:

ε(ω)=ε_∞-(ε₀-ε_∞)×ω_P ²/(ω²+iωγ) with the following parameters: ε_∞=4.017, ε₀=4.896, ω_P=1.419×10¹⁶ rad/sec, γ=1.117×10¹⁴ rad/sec. The dielectric material is SiO₂ (n=1.45), padded infinitely all around the metal (including within the annular slit).

The geometrical parameters are as follows: (following the definitions in Fig.1): In region A, the grating period is Λ_A=λ _SPP,A=381.27 nm (λ _SPP=381.27 nm for an Ag\SiO₂ interface incident with λ ₀ = 600 nm light), the duty cycle is f_A = 0.5 and the grating depth is h_A=30 nm. The parameters for the grating in region C are Λ_C=λ _SPP,C/2= 190.63 nm, f_C=0.2, h_C=20 nm. Duty cycle and grating depth were obtained by numerical optimization, with maximal energy density as a criterion for optimization. The shift of the grating from the aperture edge was chosen to be S_A,C=λ _SPP,C/2. The reasons for this choice will be discusses in section 4.3. Ten grating periods were assumed. The values for the aperture distance from r=0 and the metal thickness (d and h_B) will be given explicitly in the examples illustrated next.

4.2 Polarization of the incident and focused light

We no discuss the role of polarization of the incident field. Specifically, we assume this field to be radially polarized. Radial polarization is attracting growing attention over the last decade [19-26]. For example, it was shown that radially polarized light can be used for achieving tight focusing of light in free space and for enhancing the longitudinal field component in the focal region.

It is well known from transmission line theory that coaxial, perfectly electrical conducting (PEC) cylindrical waveguides supports a TEM mode with no cutoff frequency regardless of the slit dimensions. In our case, the metal is of finite conductivity and the anti symmetric plasmonic mode that exists inside the slit may be coined a “TEM-like mode” (following the terminology used in [11]). This mode is not a pure TEM mode, as it has an electric field component in the propagation direction (i.e., the z direction). Nevertheless, for the typical dimensions discussed above, this mode is guided by the structure. The pure TEM mode (in the PEC limit), has the same field distribution as a radially polarized beam. Therefore, when a radially polarized beam impinges on a metallic (non-PEC) coaxial structure, the “TEM-like” mode is the commonly excited mode. Because both the E_R and H_φ fields of a radially polarized beam and the structure itself have no angular dependence (they do not vary with φ, the azimuthal angle), there is no coupling of the incident field to SPP modes with higher azimuthal order. We are interested in maximizing the electrical energy density at r=0. Unfortunately, the radial electric field component E_R (and also the azimuthal magnetic field component H_φ) vanishes at r=0 due to destructive interference. This can be understood by considering two SPP waves originating from two opposite points along the circumference of the slit and propagating towards r=0. These two SPPs arrive at r=0 with the same amplitude and after accumulating the same phase. However, because of the radial field distribution these two SPPs were originated from the annular slit in anti-phase and therefore interfere destructively at the center of the plasmonic lens, resulting a null at r=0. The E_Z component however, will interfere constructively at r=0 because all SPPs emerging from the slit circumference will end up at r=0 with their z-field components pointing in the same z-direction (see schematic illustration in Figure 3(g)-3(h)). This is exactly opposed to the situation in [1], where the structure was illuminated by a linearly polarized light, and the transverse field component (labeled there as E_X) interfered constructively at r=0, while the E_Z component interfered destructively. The same is true also for circularly polarized illumination. We now keep in mind that the electrical energy density is given by ∣E_R∣²+∣E_Z∣². The question is which field component is dominant: is it the transverse or the z component of the electrical field? By calculating the SPP fields at the metal-dielectric boundary in region C, we find that ∣E_Z∣²/∣E_R∣² = ∣k_R∣²/∣k_Z∣² = ∣ε_M∣/ε_D>1 (where k_i is the SPP k-vector in the i direction and ε_M, ε_D are the permittivities of the metallic and dielectric medias respectively, and ∣ε_M∣/ε_D>1 because -Re(ε_M)>ε_D is a necessary condition for the existence of SPP mode). As a result, the size of the focal spot produced by the electric energy density will be smaller under radial polarization illumination, compared with the case of linear or circular polarization illumination.

The above discussion is summarized in Figure 3(a)-3(f), where the contributions to the energy density of the various field components are plotted. The energy densities are averaged in time over a single period.

 

Fig. 3. (a)-(f) Contributions to the energy density of the various SPP field components. The color scaling of all insets is logarithmic, and non-saturated. For (a)–(c), and for (d)–(f) separately, each color intensity corresponds to the same power density, so it can be seen that the E_R component is weak compared to E_Z and H_φ. We used values of d=1200 nm, and h_B=360 nm. No gratings are applied. (g)–(h) schematic illustration showing the propagation of the E_R and E_Z components from two opposite points on the slit circumference. In (g), because the E_R components are excited in antiphase and accumulate the same phase until they arrive at the center, destructive interference is obtained. In (h), the E_Z components are generated in phase and accumulate the same phase along their propagation towards the center. Therefore, these components interfere constructively. The colors of the arrows in (g)-(h) represent the accumulated phase.

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For most applications, we are interested in the electrical energy density. For such cases, Fig 3(a), 3(b), and 3(d) provide the relevant information. Nevertheless, once could think of applications involving the interaction of light with magnetic materials, with permeability different than 1. For such cases, the magnetic energy density is also of interest. Thus, we also plot the magnetic energy density and the total energy density which is the sum of both electrical and magnetic energy density (Fig. 3(e) and 3(f) respectively). It can be seen that the magnetic energy density has a null at r=0, for reasons that were explained above. Therefore, the spot size of the magnetic field density is larger compared with that of the electric field density.

Another benefit in using radially polarized light is that the incident light has no E_φ component (while linear and circular polarizations does). The E_φ component does not contribute to SPP excitation. Therefore, some of it will be reflected, some will be lost due to absorption in the slit and some will go through the slit and radiate into free space.

The focusing of radially polarized light by the plasmonic lens is somewhat similar to the case of tight focusing of radially polarized light in free space. In both cases, a donut-like distribution is obtained for the transverse field, while the longitudinal field is highly confined within the focal region. Yet, there are two major differences. First, the numerical aperture (NA) of a standard lens is determined by its geometry, refractive index and the refractive index of the surrounding medium. The NA can be higher than one only if the refractive index of the surrounding medium is also higher than one. In contrast, the NA of the plasmonic lens is always higher than one because of the mechanism of surface wave (indicating a 90° focusing angle) and the penetration of the SPP mode into the metal, resulting in a wavelength shorter than the free space wavelength and an effective index higher than that of the surrounding dielectric medium. For example, the numerical aperture of the plasmonic lens discussed above is λ ₀/λ _SPP~1.58 while the highest theoretical possible NA value for a standard lens surrounded by a medium with similar refractive index to SiO₂ is 1.45. Another important difference is the relative contributions of the transverse and the longitudinal components. In the case of free space focusing, the relative contribution of each component is determined by the propagation direction of each plane wave that is refracted from the aperture of the lens. Here, the relative contribution is determined by the dielectric constants of the metal and the dielectric, as discussed above.

4.3 Results

We now estimate the performances of the plasmonic lens. First, we calculated the coupling efficiency of the incident light to SPPs in region C. For now, we ignore the effect of the gratings and we only simulate a structure with an annular slit.

We investigate three figures of merit characterizing the focusing “quality”: 1- “spot size”, defined as the transverse full-width half-maximum (FWHM), 2- “depth of focus”, defined as the FWHM in the longitudinal direction, and 3– “efficiency” (defined later in this section). The spot size is basically the result of interference of counter propagating SPPs. As explained previously, the contributions of the radial and the z component of the field are different both in magnitude and in profile. For the material system considered in this paper (Ag and SiO₂) we found the spot size (SPS) to be ~150 nm, i.e. ~λ _SPP×2.5 close to the FWHM of J₀(k_SPP×r) which can be shown to be the distribution of ∣E_Z∣² similarly to the derivation in [8]. The depth of focus (DOF) is found to be typically ~55 nm, i.e. ~λ _SPP/7.

As mentioned before, when a Bragg reflector is not present, both in-focus and counter-focus SPPs are excited by the slit with similar coupling efficiency. Therefore, we can calculate the coupling efficiency of the incident light to SPPs in region C by the ratio 2×S_SPP,C,out/S_slit where S_slit is the flux of energy incident directly on the slit, and S_SPP,C,out is the SPP flux in region C traveling in the counter-focus direction (the in-focus SPPs result in a standing wave pattern with negligible flux, and therefore we perform the calculation on the counter focus propagating SPPs and make use of the symmetry in coupling to the in-focus and counter-focus SPPs. The latter is true in the limit of large radius, which holds in this case as the radius is order of magnitude larger than the SPP wavelength). With this approach, we found the coupling efficiency to be around ~40% (comparable with the results in [16,27]). Next we considered the effect of the grating in region A. We repeated the calculation with the presence of this grating and found that the coupling efficiency was improved to ~200%. This result indicates strong SPPs excitation on the metal/dielectric interface (the boundary between regions A and B). The effect of this grating on the spot size and the depth of focus was found to be marginal.

We now discuss the effect of the Bragg reflector in region C. The effect is found to be strongly dependant on the shift of the gratings from the slit, SC (see Fig.1). From the classic theory of Fresnel reflection (in the PEC limit), we expect the Bragg grating to reflect the E_R field component in phase, and the E_Z field component in antiphase [28]. Therefore, for a shift of S_C=λ _SPP,C/2, E_R will interfere constructively at the slit edge, while E_Z will interfere destructively. For such a case, where the E_Z field is absent at the edge of the slit, the coupling into SPPs in region C is diminished, and the radiation from the slit into free space is greatly enhanced [29]. Clearly, the enhanced transmission is undesired, because it has a negligible contribution to the energy density at the focus. The situation is opposite when the reflectors are placed with a shift of S_C=λ _SPP,C/4. Now the E_R field will interfere destructively, while the E_Z field will interfere constructively at the slit. This in turn will result an efficient coupling to an SPP mode, while the radiation from the slit is now diminished. The overall enhancement factor when both gratings in regions A and C are applied with a shift S_A,C=λ _SPP,A,C/4 was found to be ~18. A different criterion for estimating the enhancement factor of the gratings is based on measuring the energy inside the focal region (assumed to be a cylinder with a volume of 1/4×π×SPS²×DOF) with and without the gratings. This criterion resulted in an improvement factor of ~16.

 

Fig. 4. Electric power density averaged over a full period of the optical wavelength. Color scaling is logarithmic and non-saturated, and equal in all insets. Calculated for d=1200 nm, h_B=360 nm. (a) no gratings are applied. (b) gratings are shifted from the slit with λ _SPP,C/2. The enhanced radiation surrounding the slit can be clearly observed. In (c) where the shift is λ _SPP,C/4, the radiation is minimal and the energy density at the focus is the strongest.

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4.4 Finite size effects

So far we ignored the effect of the finite size of the device along the transverse direction. However, it turns out that for devices having finite size with respect to the incident beam (or if the device’s size is much smaller than the decay length of the SPP wave), SPPs could be excited also at the edges of the device. In the case with no grating in region C, these SPPs can propagate towards the focal region and interfere with the SPPs emerging from the slit. The interference can be either constructive or destructive, depending on the phase difference between the “slit SPPs” and the “edge SPPs”.

 

Fig. 5. Focusing efficiency as function of the outer disc radius, in units of λ _SPP.. d=1200 nm and h_B=350 nm. All other parameters are as in section 4.1.

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In Fig. 5, the focusing efficiency as a function of the outer metallic disc size in units of λ _SPP is shown. One can see that the focusing efficiency is a periodic function with a period of λ _SPP, corresponding to multiples of 2π phase accumulated by the “edge SPPs” as they travel towards the slit. This effect can be used to maximize the energy density in the focal region.

5. Summary and conclusions

A circularly symmetric plasmonic lens that is illuminated by radially polarized light is proposed and studied in details. Radial polarization is shown to be the natural SPP excitation source for such structures, enhancing the electric energy density in the focus (compared to linearly and circularly polarized light). The focusing efficiency is further improved by the addition of gratings in both layers of the device (top and bottom). These gratings eliminate the counter-focus SPP propagation and enhance the coupling of the incident light to SPPs. The proposed plasmonic lens is attractive for variety of applications, including nanoimaging, plasmonic photolithography, plasmonic “tweezers”, biosensing, and optical nonlinearities, to name a few. For these applications, one needs to consider a modified structure, where the dielectric in region C is either liquid or air. For such asymmetric structures Fresnel reflections will play a role in reducing the focusing efficiency of the device. This drawback can be overcome either by using a lower refractive index substrate (e.g. CYTOP, n~1.31) or a suspended membrane.