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Abstract
We present a conceptual demonstration of a metallic planar lens composed of double-turn waveguides for sub-diffraction-limit focusing. The phase delay of a single double-turn waveguide dependent on its structural parameters is investigated by employing the finite-difference time-domain (FDTD) numerical method. The design utilizes the surface plasmon polaritons (SPPs) that propagate along the metal-insulator-metal (MIM) waveguides to achieve the desired spatial phase modulation in the transmitted field. The simulated focal length achieved is in positive agreement with the design and the full-width at half-maximum (FWHM) is 0.446λ, well beyond the diffraction limit. This superfocusing performance can be maintained very well under the slight change of film thickness and slit width, showing the robustness of the design. The maximum aspect ratio of nanoslits constructing the proposed lens is 3.33, which is far less than the previous reports, alleviating the later fabrication. The metallic planar lens as demonstrated will find its applications in such fields as lithography, integrated optics, and super-resolution imaging.
© 2017 Optical Society of America
1. Introduction
Recently, surface plasmon polaritons (SPPs) have captured much attention for the ability in enhancing the evanescent waves existing in the metal surfaces [1]. Because of its unique optical properties, surface plasmonic devices [2–17] make optical manipulation at the nanometer scale possible in such fields as lithography, integrated optics, and super-resolution imaging. Based on the localized surface plasmon resonances (LSPR) occurring within the nanostructures, various plasmonic lenses composed of nanoapertures patterned in thin metallic films have been put forward and explored extensively, including nanoslits [2–4], nanoholes [5, 6], rings [7, 8], nano-antennas [9], cross-shaped apertures [10], and grooves [11, 12]. Among the suggested methods for realizing these lenses, those formed by nanoslits capture the wide-spread attention, both theoretically [2–4, 13-14] and experimentally [15, 16]. The mechanism of focusing devices is to reconstruct the wavefront of light via the phase modulation in geometrical optics. However, most of plasmonic lenses composed of nanoslits are made up of a single-layer film, which results in the lenses with a large aspect ratio (depth to width, referring to table below), aiming to realize the desired spatial phase modulation more than 2π [2]. In general, increasing the depth [18] or decreasing the width [19] of nanoslits is commonly utilized. In practice, it is exceedingly difficult to make these lenses with the state-of-the-art nanofabrication techniques, e.g. focused ion beam (FIB) milling and electron beam lithography (EBL).
To alleviate the fabrication problem of narrow nanoslits, we propose a new design method for constructing the plasmonic lenses. Transforming the idea of nanoslits composed of a single-layer film, we attempt to adopt multilayer metallic films to construct the desired lenses. In this paper, double-layer films are used to just demonstrate the concept. The desired spatial phase modulation can be achieved by adjusting the distance between the two films and their structural parameters. In this way, we cannot only decrease the aspect ratio of nanoslits but also realize the superfocusing capability.
Based on the finite-difference time-domain (FDTD) numerical method, we undertook a detailed investigation on the propagating properties of SPPs inside subwavelength waveguides. The phase delay of different waveguides with various structural parameters is used in the wavefront shaping for the plasmonic lenses. The focal length and full-width at half-maximum (FWHM) of the focal point are obtained from the electromagnetic simulation results, from which the realized focal length is in positive agreement with the design and the FWHM is 290 nm, 0.446λ, well beyond the diffraction limit of 457 nm (~0.703λ). Furthermore, the influences of film thickness and slit width on the focusing performance are surveyed to verify the excellent superfocusing robustness of the designed lens.
2. Phase delay of a single double-turn waveguide
We first investigate the properties of light propagation in the double-turn waveguide structure. The lens composed of gold films is immersed in a homogenous dielectric, air. To weaken the coupling effect [2], the thickness of the two metal films is set at least twice the skin depth (about 56 nm according to the Eq. (1)). Here, we chose metal films with the thickness of 100 nm. A two-dimensional (2D) model of the double-turn waveguide structure is illustrated schematically in Fig. 1(a), assuming the slit length to be infinite in the z direction. As shown in Fig. 1(a), the waveguide consists of an entrance with the width of w1, an exit with the width of w2, and a connection segment with the depth of t in between (for simplicity, assuming t = w1 = w2 = w). The two films are named as the film-1 and film-2, respectively. In order to optimize the light propagation in the main waveguide, it is indispensable to place two reflecting planes in the turning corners, and their inclined angles are both 45° as assumed. Moreover, d is the distance between the central lines of slit-1 and slit-2. When a transverse-magnetic (TM) polarized plane wave is normally incident from the film-1 side, the SPPs can be excited and propagate inside the nanoslits in the specific waveguide modes until reaching the exits. Because the waveguide is composed of three pieces of nanoslits, the propagation constant β of the TM mode can be acquired by [20],
with k1 = (β2-εdk2)1/2 and k2 = (β2-εmk2)1/2, where k is the wavevector of light in dielectric space, and w is the width of nanoslits. εm and εd are the permittivity of metal and dielectric, respectively. The real and imaginary parts of the propagation constant represent the phase velocity and propagation loss of SPPs inside the metallic nanoslits, respectively. The operating wavelength for all the cases is 650 nm. The permittivity of gold at this wavelength is εm = −12.8915 + 1.2044i [2]. The multilayer-film system is coated on the glass substrate, with its refractive index of 1.46. Figure 1(b) shows the schematic illustration of a plasmonic lens based on the geometrical optics and the wavefront reconstruction theory.
Fig. 1 Schematic illustration of structural parameters. (a) A double-turn waveguide. (b) A designed plasmonic lens based on the geometrical optics and the wavefront reconstruction theory.
By employing the FDTD numerical method, we investigated the propagation properties of TM-polarized plane wave in the waveguide. Since the SPPs modes in double-turn waveguides are TM modes and the magnetic field is in the z direction, the phase analysis was performed based on the real part of Hz. In simulations, the unit cells were set to 2 nm in both x and y directions to guarantee the accuracy and convergence of the computation and model the fine features of the electromagnetic field in the structure. Figure 2(a) shows the Poynting vector in the waveguide when the value of d and w is 150 nm and 30 nm, respectively. From Fig. 2(a), we can observe that the light propagates gently inside the waveguide even at the turning corners. Figure 2(b) shows the FDTD simulation results of light transmission in free space passing through the proposed waveguide structure. Since only the real part of Hz contributes to the phase retardation, the imaginary part was ignored in the design process. The real part of βD, expressed as Re(βD), plays a dominant role in achieving the desired phase modulation [2], where D is related to the structural parameters including d, t, t1 and t2. From Fig. 2(b), the output port of the waveguide can be modeled as a point source.
Fig. 2 (a) The Poynting vector in the waveguide with d = 150 nm, w = 30 nm. (b) The simulated magnetic field distribution of Re(Hz) out of a waveguide structure.
For a practical plasmonic lens based on a metallic nanoslits array, the total phase difference more than 2π is advantageous [2]. We undertook a detailed investigation on the propagating properties of SPPs inside subwavelength waveguides. Based on the FDTD numerical method, Fig. 3 plots the extracted phase delay varying as a function of d when w increases from 10 nm to 50 nm. The phase delay of waveguides displays an increasing trend as d increases and w decreases. Once w keeps constant, the required phase delay can be attained by only adjusting d. For example, as d increases from 30 nm to 370 nm, the phase delay of the main waveguide with w = 30 nm changes from 1.19π to 3.21π.
Fig. 3 Phase delay caused by the parameters d and w, assuming t = w1 = w2 = w. The phase delay of double-turn waveguides displays an increasing trend as d increases and w decreases. The narrower the nanoslits, the greater influence of d on its phase delay.
Figure 4 shows that the increase of phase delay in the double-turn waveguides is obvious compared to the nanoslits with the same width and depth. From Fig. 4, it is clear that the double-turn waveguides tremendously increase the efficiency of spatial phase modulation under a small aspect ratio. For instance, in the case of w = 30 nm and d = 400 nm, the phase delay of double-turn waveguide is about three times as large as that of nanoslit with 30 nm width on a 230nm-thick single-layer gold film. Therefore, the double-turn waveguides can modulate the phase delay more efficiently.
Fig. 4 The increase of phase delay in the double-turn waveguides is obvious compared to the nanoslits with the same width and depth, as the parameter d increases from 30 nm to 450 nm. The structural parameter w ranges from 10 nm to 50 nm. In the case of w = 30 nm and d = 400 nm, the phase delay of double-turn waveguide is about three times as large as that of nanoslit with 30 nm width on a 230nm-thick single-layer gold film.
On the basis of the analysis above, the phase delay of double-turn waveguides has been investigated in detail under the specific condition of t = w1 = w2 = w. To illustrate the effect of the parameter t on the phase delay, the waveguides’ phase delay with different t ranging from 20 nm to 40 nm is also simulated, as illustrated in Fig. 5(a), from which the phase delay of double-turn waveguides displays an increasing and similar trend as d increases. The smaller t, the more obvious of the phase delay to d, which agrees well with the Eq. (1).
Fig. 5 Phase delay caused by the parameters d, t, w1 and w2. (a) The t ranges from 20 nm to 40 nm, assuming w1 = w2 = 30 nm. (b) The w2 ranges from 20 nm to 40 nm, assuming t = w1 = w = 30 nm. (c) The w1 ranges from 20 nm to 40 nm, assuming t = w2 = w = 30 nm. (d) The θ ranges from 0° to 60°, assuming t = w1 = w2 = w = 30 nm.
Figure 5(b) presents the FDTD simulation results of the phase delay for varying w2 under the specific condition of t = w1 = w = 30 nm. It is obvious that the phase delay displays an increasing and similar trend as d increases and w2 decreases. With the increase of d from 30 nm to 450 nm, the phase delay of double-turn waveguide approximately increases from 1.5π to 4π when w2 = 20 nm. What surprises us is that three lines is approximately in superposition, especially w2 = 30 nm and w2 = 40 nm when t = w1 = w = 30 nm, as illustrated in Fig. 5(b). Figure 5(c) presents the similar regularity. So we can come to the conclusion that the variation of w1 and w2 has little influence on the spatial phase modulation. This is significant for the practical nanofabrication processing.
Eventually, we undertook a detailed investigation on the structural parameter θ, as shown in Fig. 5(d). To simplify the research process, the specific condition of t = w1 = w2 = w = 30 nm is taken into consideration. From Fig. 5(d), we can observe that all lines are approximately in superposition. In other words, the parameter θ has little influence on the spatial phase modulation. The lines increase gradually with the increasing of d, corresponding to θ = 0°, 15°, 30°, 45°, and 60°, respectively.
It cannot be negligible that the structural parameter θ does impact on the optical transmission of the double-turn waveguide. We undertook a detailed investment on the effect of the variation in the inclined angle θ based on the FDTD numerical method. From the Fig. 6, the larger the structural parameter θ is, the more the transmitted loss gets. The reason for this positive relationship is likely to the size of clear aperture at the turning corner. Because a larger inclined angle θ leads to a smaller clear aperture, the interference happens violently between two beams of SPPs along with both sides of the nanoslit at the turning corner. As a result, the transmitted loss increases with the increasing inclined angle θ. What’s more, there are two reasons why we choose the inclined angle θ = 45°. One is to avoid the incident light directly illuminating the middle channel, the inclined angle θ should be set at least 45°. The other is to decrease the SPPs’ transmitted loss, the clear aperture at the turning corner should be chosen as large as possible.
Fig. 6 The magnetic field intensity in double-turn waveguide with different structural parameter θ, (a) θ = 0°, (b) θ = 15°, (c) θ = 30°, (d) θ = 45°, (e) θ = 60°.The structural parameters t = w1 = w2 = w = 30 nm and d = 400 nm remain constant for all the cases.
From the above simulation results of the phase delay, the efficient spatial phase modulation can be realized by double-turn waveguides. The structural parameters t and d play a dominant role in the process of spatial phase modulation. The lines of phase delay remain smooth and steady very well under the slight change of w1, w2, and θ, showing the robustness of the design. The double-turn waveguide can be utilized to construct a conceptual planar lens.
3. Lens design
As we know, the SPPs fields can penetrate into the gold wall between the adjacent nanoslits over a distance, δm, which can be calculated by [21],
where k0 is the wavevector in free space, defined as k0 = 2π/λ. The ε’m and εd are the real part of the permittivity of metal and dielectric, respectively. At the operating wavelength of 650 nm, the skin depth δm is about 28 nm.From the above analysis, we know that every waveguide’s exit can be modeled as a point source. According to the geometrical optics, in order to achieve the focusing at a particular location, the position and size of the apertures constructing plasmonic lenses have to be set reasonably, so that we can receive a desired wavefront curvature generated by the double-turn waveguides with different sizes and positions (as shown in Fig. 1(b)). The parameter α is the angle between the optical axis and the outmost converging ray.
Because the lens is symmetric by the x axis, we just take the half part (y ≥ 0) of the plasmonic lens into consideration. For a plasmonic lens with the focal length f, the phase delay related to the distance y between the centerline of emergent slit and the optical axis can be calculated as,
where λ is the operating wavelength, n is an arbitrary integer.According to the Fig. 3 and Eq. (3), we have the ability to construct a desired plasmonic lens. Under the assumption of t = w1 = w2 = w = 30 nm, a series of metallic planar lenses are designed. Table 1 shows the structural parameters of a designed plasmonic lens based on the geometrical optics and the wavefront reconstruction theory. The aimed focal length is 1 μm.where y1, y2 is the centerline of slit-1, slit-2, respectively.
Table 1. Structural parameters for the designed plasmonic lens with a focal length f = 1 μm
The structural parameters of the designed plasmonic lens are kept the same as those shown in Table 1. Figure 7 presents the FDTD simulation results of the magnetic field pattern and intensity profile for the designed metallic planar lens. In simulations, the plasmonic lens is coated on the glass substrate. The sample is then illuminated from the substrate side with the TM-polarized plane wave. Figure 7(a) shows the half geometry of the designed lens. As shown in Fig. 7(b), we can observe obviously the focusing behavior in the Fresnel region at x = 994 nm, which perfectly agrees with the design. From Fig. 7(c), the FWHM of the focal spot is 290 nm, about 0.446λ, beyond the calculated Rayleigh diffraction limit (rR) as [22],
where λ is the operating wavelength, NA is the numerical aperture of the optical system, which can be further calculated by,where n is the refractive index of the immersion medium, α is the angle between the optical axis and the outmost converging ray. In our research, the lens is immersed in a homogenous dielectric, air, so n = 1. Based on Eqs. (4) and (5), the Rayleigh diffraction limit of the desired plamonic lens can be calculated to 457 nm, about 0.703λ. Moreover, the maximum aspect ratio (depth to width) of nanoslits constructing the proposed lens is only 3.33, much less than those reported in previous literatures (see Table 2).Fig. 7 The FDTD simulation results of the metallic planar lens for the focal length f = 1 µm under the assumption of t = w1 = w2 = w = 30 nm. (a) The half geometry of the lens composed of double-turn waveguides (yellow). (b) The FDTD simulation result of the magnetic field intensity. (c) The derived |Hz|2 at the focal plane.
Table 2. The aspect ratio and deviation of the focal length between the theoretical design, numerical simulation, and experimental measurement (unit: μm)
In practice, due to the fabrication errors, the practical thickness of the metallic film and the width of nanoslits are not exactly the same as the design. Therefore, some important factors to evaluate the focusing robustness of the designed lens have to be considered. From the Figs. 8 and 9, the focusing performance is maintained very well when the thickness of film-1 and film-2 decreases solely from 100 nm to 50 nm, respectively.
Fig. 8 The focusing performance of the plasmonic lenses with different film thicknesses. The simulated magnetic field intensity for (a) the original lens with t1 = 100 nm, (b) the adjusted lens with t1 = 75 nm, (c) the adjusted lens with t1 = 50 nm. In all cases, t2 keeps a constant value of 100 nm. The insets show the corresponding field intensity distribution at the focal plane. The white dashed lines express the exit surfaces of the lenses.
Fig. 9 The focusing performance of the plasmonic lenses with different film thicknesses. The simulated magnetic field intensity for (a) the original lens with t2 = 100 nm, (b) the adjusted lens with t2 = 75 nm, (c) the adjusted lens with t2 = 50 nm. In all cases, t1 keeps a constant value of 100 nm. The insets show the corresponding field intensity distribution at the focal plane. The white dashed lines express the exit surfaces of the lenses.
Moreover, we explored the influence of the deviation in nanoslit width on focusing performance by manipulating merely w1 and w2, respectively. Figures 10 and 11 give the simulated magnetic field intensity, from which we can observe that the focusing performance is maintained very well when w1 and w2 change solely from 20 nm to 40 nm. The wider w1 and w2 are, the stronger magnetic field intensity in the focusing points will be.
Fig. 10 The focusing performance of the plasmonic lenses with different slit widths. The simulated magnetic field intensity for (a) the adjusted lens with w1 = 20 nm, (b) the original lens with w1 = 30 nm, (c) the adjusted lens with w1 = 40 nm. In all cases, w2 keeps a constant value of 30 nm. The insets show the corresponding field intensity distribution at the focal plane. The white dashed lines express the exit surfaces of the lenses.
Fig. 11 The focusing performance of the plasmonic lenses with different slit width The simulated magnetic field intensity for (a) the adjusted lens with w2 = 20 nm, (b) the original lens with w2 = 30 nm, (c) the adjusted lens with w2 = 40 nm. In all cases, w1 keeps a constant value of 30 nm. The insets show the corresponding field intensity distribution at the focal plane. The white dashed lines express the exit surfaces of the lenses.
These simulated results verify the excellent superfocusing robustness of the designed lens. The deviations introduced in the film thickness and nanoslit width have little influence on the ultimate focusing capability of plasmonic lenses, which is significant for the practical nanofabrication.
4. Conclusions
According to the detailed investigation of the double-turn waveguides, a metallic planar lens based on the waveguide array in gold film can be reasonably designed. The phase delay of SPPs in the waveguide is used to shape the required wavefront for a specific focal length. Based on the FDTD numerical method, the superfocusing properties of the designed lens in the Fresnel region are demonstrated. From the simulation results, the focal length achieved is in positive agreement with the design and the FWHM is 0.446λ, well beyond the calculated diffraction limit, about 0.703λ. The aspect ratio of the waveguide is quite smaller than the previous reports. In addition, the thickness reduction and slit width variation of the designed plasmonic lens have negligible influence on the superfocusing performance. Both features alleviate the future fabrication of metallic planar lens in the nanometer lengthscale.
Funding
National Natural Science Foundation of China (Grant No. 51375400, 51622509); the Specific Project for the National Excellent Doctorial Dissertations (Grant No. 201430); the Fundamental Research Funds for the Central Universities (Grant No. 3102017jg02007); the 111 Project (Grant No. B13044); Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (CX201606); the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (Z2017013).
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