A spiral plasmonic lens with directional excitation of surface plasmons

Conventional plasmonic lenses are composed of curved slits carved through metallic films. Here, we propose a new plasmonic lens based on a metallic slit with an auxiliary groove. When the lens is illumined normally, only inward surface plasmon polaritons (SPPs) can be generated and then focused into a hot spot at the center of the lens. The focusing effect is theoretically investigated by varying the groove parameters and incident polarizations. It is found that this phenomenon exists for both the circular and linear polarizations of incidence. Under optimal groove parameters, the intensity of the focal spot in our lens can be 2.5 times of that in one without grooves for both linearly and circularly polarized illuminations.

. Directional excitation of SPPs by an asymmetric plasmonic nanoslit. (a) Side view of a plat Au film with a slit that has a width of w 1 = 150 nm and is along the y direction. The thickness of the film is h 1 = 250 nm. (b) The same as (a) but for a structure with an auxiliary groove with a width of w 2 − w 1 and depth of h 2 . When the slits are illuminated from the lower side by a 2D Gaussian light beam with x-polarization, wavelength λ, width of w g = 7 μm, and |E| = p in center, leftward and rightward SPPs can be excited in the upper side with the same [different] intensities in (a) [(b)]. (c-e) ln(E R ), I right , and excitation factor F E = I right ln(E R ) (shown in color) as a function of w 2 and h 2 , respectively. Here, λ = 700 nm and extinction ratio E R = I right /I left , where I right and I left are the rightward and leftward SPP intensities in (b), respectively. The excitation factor F E reaches a maximum at optimal parameters of w 2 = 288 nm and h 2 = 120 nm. (f) Logarithmic simulated profile of |E z | 2 /p 2 for the optimal structure obtained in (e). Although our structure is similar to the one in ref. 22, its underlying physical mechanism is different from that in ref. 22 [see ref . 24]. We apply a finite-element method to simulate the structure, where the dielectric constant of gold ε Au is from experimental data 25 . For incident wavelength λ = 700 nm, ε Au = − 15.2 + 1.24i, resulting in wavelength λ SPP = λ(ε Au + 1)/ε Au ] 1/2 = 677 nm for SPPs at the air-Au interface. slit that has the same cross-sectional configuration as that in Fig. 1(b), but is curved into a right-handed Archimedes' spiral shape in the x-y plane. The slit is illuminated from the back side by a Gaussian light beam with wavelength λ = 700 nm, width of w g = 7 μm, and |E| = p in center. (b) 3D plot of the area that is outlined by the dashed box in (a). (c) Simulated profile of |E z | 2 /p 2 at a plane 50 nm above the Au film in (a). Here, a 7 μm × 7 μm area is shown and the incident light has y-polarization. , respectively. Here, S x is the time-average power flow along the x direction, h = 0.3 μm, z = 0 is at the upper surface of the Au film, g = 2 μm, and x = x 1 + g and x = x 2 − g are at the left and right boundaries of the slit, respectively. An extinction ratio E R and excitation factor F E are also defined as: We note that a large excitation factor F E relates to both a large I right and small I left .
In Fig. 1(c-e), ln(E R ), I right , and F E are plotted as a function of w 2 and h 2 , respectively. We can see that excitation factor F E reaches a maximum when w 2 = 288 nm and h 2 = 120 nm. For such optimal parameters, extinction ratio E R also reaches a maximum (2144), and intensities of rightward SPPs I right is close to its maximum(I right /I right,max = 0.8), simultaneously. We note that I right reaches a maximum at w 2 = 310 nm, h 2 = 100 nm, while E R is not high enough (E R = 13). Hence, in the followings, we focus on structures with w 2 = 288 nm and h 2 = 120 nm.
Figure 1(f) shows the simulated profile of |E z | 2 /p 2 for the above optimal structure. SPPs propagating to the right can be clearly observed, while leftward SPPs can hardly be seen. Such unidirectional launching of SPPs occurs in a broad frequency range. The extinction ratio E R > 10 for wavelengths from 670 nm to 759 nm, and E R reaches a maximum (2144) at λ = 700 nm [ Fig. 2]. It is interesting to note that, if the slit is replaced by a groove (so that a groove-groove structure is formed) and illuminated from the top 22,23 , both rightward and leftward unidirectional SPPs can occur at different wavelengths. But in our slit-groove structure, only rightward unidirectional SPPs can occur 24 .
A plasmonic lens with directional excitation of SPPs and its performance at different incident polarizations. The above asymmetric Au slit is along the y direction, which can be used to generate SPPs propagate along + x direction. If the slit is curled into a right-handed Archimedes' spiral shape in the x-y plane and the auxiliary groove is located at the inside of the spiral [ Fig. 3(a,b)], SPPs can be further focused into the central area of the spiral. The inner boundary of the slit is given by: where r 0 = 2 μm, and r and θ are cylindrical coordinates. When Eq. (3) is satisfied, the antipodal points (i.e. points with θ = θ 1 and θ = θ 1 + π) at the spiral slit can generate inphase SPPs at the center of the spiral (r = 0) when the incident light has a linear polarization. We note that, when the slit is curled into a circle, this constructive interference cannot be achieved unless radially (rather than linear) polarized light is impinged on the structure 23,26,27 . Since such a radially polarized light beam should be centered at the center of the lens 26,27 , it cannot work in the lens array studied below. We do simulations for the above plasmonic lens, which is illuminated normally from the back side by a Gaussian light beam. The incident light beam has a wavelength λ = 700 nm, width of w g = 7 μm, |E| = p, and center at the origin (r = 0). Figure 3(c) shows the simulated profile of |E z | 2 /p 2 at a plane 50 nm above the Au film for the incidence with y-polarization. We can see that SPPs are excited at certain parts of the lens (π/4 < θ < 3π/4 or 5π/4 < θ < 7π/4, approximately). For the other parts, the angle is small between the slit and the E-field of the incident light, so that SPPs can hardly be launched. Besides, due to the introduction of auxiliary grooves, only inward SPPs are excited, which propagate to the center of the lens and form a hot spot. Similar phenomena can also be observed for incidence with x-polarization [ Fig. 3(d)].
We have also studied the incidence with circular polarizations [ Fig. 3(e,f)]. Light with left-handed polarization can be decomposed into x-and y-polarized light (E left = E(e x + ie y )), which can generate SPPs with inphase E z at the center of the lens. Such constructive interference forms a hot spot at the center of the lens [ Fig. 3(f)]. Similar analyses have also be done for the right-handed incident light with E right = E(e x − ie y ). A dark spot appears at the center of the lens [ Fig. 3(f)] due to the destructive interference. But a bright ring occurs around the dark spot. The spiral plasmonic lens exhibits different characteristics for left-and right-handed polarized incidence, and thus can serve as a device for discerning the handedness of circular polarization 4,5,28 . Comparison with imperfect plasmonic lenses. As comparison, we have also studied lenses with other structures as shown in Fig. 4. We can see that if the auxiliary groove is at the outside of the spiral [ Fig. 4(a)], SPPs can propagate only outward while inward SPPs can hardly been observed. As a result, no hot spot is observed at the center of the lens [ Fig. 4(c)]. When the auxiliary groove is not introduced [ Fig. 4(b)], inward and outward SPPs can be excited simultaneously, forming a hot spot at the center of the lens [ Fig. 4(d)]. However, due to the existence of outward SPPs, the intensity of the hot spot is only 1/2.5 times of that in Fig. 3(c). The same difference (1/2.5 times) also occur for circularly polarized illuminations. We note that for the inner side of the spiral, the thickness of the Au film decreases from h 1 to h 1 − h 2 when comparing the structure in Fig. 3(b) to that in Fig. 4(b). This can also influence the intensity of the hot spot in Fig. 3(c), so that the observed enhancement (2.5 times) deviates from 2 times.

Plasmonic lens array.
A plasmonic lens array may find applications in maskless lithography and multiple pixel 9 . Figure 5 demonstrates the performance for arrays of our and conventional plasmonic lenses. Here, an 8 μm × 8 μm area is simulated with using periodic boundary conditions and plane-wave incidence. For the conventional lens array, the interference of SPPs is obvious in the area outside lenses [ Fig. 5(b)]. Such interference, which may be disadvantageous to applications, are not found in our plasmonic lens array [ Fig. 5(a)]. Because of the elimination of field outside lenses, the field intensity at the center of our lens is much stronger than that of conventional lens. Thus, our lens array could have improved performance in the above-mentioned applications.

Discussion
We have presented a spiral plasmonic lens based on a metallic slit with an auxiliary groove. When the lens is illumined normally from the back side, no outward SPPs can be excited, while only inward SPPs can be launched and then focused into the center of the lens. This phenomenon occurs for all the circular and linear polarizations of incidence. When optimal groove parameters are applied, the intensity at the focal spot of our lens can be 2.5 times of that without an auxiliary groove for both linearly and circularly polarized illuminations.