1. Introduction

Surface Plasmon Polaritons (SPPs) [1] are electromagnetic waves propagating at the boundaries between metals and dielectrics that offer the potential for developing compact photonic devices for light manipulation at the nanoscale. Potential applications lie, not only in their ability to control light at surfaces, but also to shape lightfields away from the surface and into the far-field. For example, a far-field beaming effect [2–4] that relies on SPPs has been demonstrated. A range of alternative mechanisms have also been investigated for beam shaping applications. Nanoslits on a curved surface [5], as well as beam manipulation using nano-metallic films consisting of arrays of apertures with varied widths, including nanoholes [6], nanoslits [7, 8] and square apertures [9] have been proposed as an alternative to focusing or collimating by means of refractive lensing. Localized resonances in slits and apertures can also be used for beam shaping applications. Such resonances depend strongly on the geometry of the slit or aperture and the optical properties of the surrounding media [10]. These resonances are accompanied by wavelength-dependent phase shifts in the transmitted fields, enabling manipulation of wavefields which could, potentially, be employed for applications such as beam focusing. Although the use of square apertures [9] has been proposed as a means of fabricating planar lenses using this mechanism, the only experimental research undertaken to date has involved one-dimensional structures. There is clearly an interest in extending this research to two-dimensional structures and the research presented here is aimed at providing background research into the extension.

Here, in parallel with designing two-dimensional structures, we demonstrate that a one-dimensional focusing element can be obtained by tailoring the phase across a device consisting of an array of slits in a metallic film with slit widths varying around the resonant dimensions for a particular design wavelength. This work which builds on our preliminary studies [11] differs from that conducted previously by other groups in that we investigate TE-polarized plane-wave illumination where the incident electric field is parallel to the slits. Slits illuminated with the TE orthogonal polarization exhibit a cut-off wavelength λ_c for transmission which is absent in the case of the orthogonal TM-polarization and thus, provide a useful analogy for investigating two-dimensional structures consisting of apertures that also have a characteristic cut-off wavelength accompanied by a resonance. The resonance mechanism producing maxima in the transmission spectrum is fundamentally different to that seen in the orthogonal TM-polarization that has been investigated by other authors [7,8], where surface plasmon modes are excited on the inner surfaces of the slits. In the latter case, the structure supports a TEM mode. In two-dimensional hole arrays, however, simple apertures such as squares and circles, do not support a TEM mode and, hence, the polarization investigated here provides a closer analog to 2D devices. This is particularly significant when considering the computational intensity of simulating 2D rather than 3D structures. The research reported here thus forms the basis for investigating 2D devices consisting of nanometric apertures in metal films.

Here we show that a tailored spatial phase modulation can be introduced into an incident optical wavefield by varying aperture sizes across the structure around resonant dimensions for a particular design wavelength for TE-polarized illumination. This enables control of wavefields that can be employed for beam deflection, beam focusing or for producing a wavelength-specific spatial field change [10]. Here, we present results of Finite Element Method (FEM) simulations and experimental results, demonstrating a one-dimensional focusing element.

2. Computational background

One of the key considerations central to the design of these devices is the optimization of the phase shift through the device whilst maintaining a threshold transmission. Device design was performed by simulating the passage of light through single slits in a silver (Ag) screen with the basic geometry as defined in Fig. 1 . The slit widths C and wavelengths λ were varied to study the transmission and phase through the single slit structures. These studies give an insight into understanding the impact of modulating structural geometries such as C, as well as changing the incident wavelength on the resultant phase shift through a device comprising spatially varying slits.

 

Fig. 1 (a) Schematic illustration of a single slit aperture of slit width size C formed on a 400 nm thick Ag film below a glass substrate. A TE-polarized (electric field in the z-direction) plane wave is incident from above the slit array and glass substrate in the y-direction as indicated. Calculated (b) normalized transmission (total transmitted power normalized to incident power) and (c) phase shift for a range of single slit apertures each formed on a 400 nm thick Ag film over a range of slit width sizes and wavelengths. The cut-off wavelength for a perfectly conducting TE₁ parallel-plate waveguide mode is shown.

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Two-dimensional electromagnetic field simulations using the finite element method (FEM) (COMSOL Multiphysics [12] with the RF Module) were performed. We modeled a 400 nm thick Ag film on a glass substrate (n = 1.52), perforated with a single slit with the medium inside and above the film having a relative permittivity of 1. (Fig. 1(a)) Optical constants of Ag were obtained from [13]. The simulation region is surrounded by Perfectly Matched Layers (PML) to prevent any unwanted back reflections. A plane wave with transverse electric (TE) polarization illuminates the sample. Here, we define the TE-polarization as depicted in Fig. 1. Numerical simulations were performed for TE-polarized light normally incident on the structure with the plane wave propagating in the y-direction, and the electric field polarized in the z-direction (i.e. parallel to the grooves).

The FEM simulation in Fig. 1 depicts the (b) calculated total transmitted power normalized to incident power and the (c) phase shift (3 μm below the centre of each aperture) relative to that in the absence of the screen for a range of single slit apertures where the slit width and wavelength were varied. The cut-off wavelength (at approximately 2C) for a perfectly conducting TE₁ parallel-plate waveguide mode is also shown. The transmission of light through these devices can be seen to decrease rapidly at longer wavelengths for a fixed slit width. It can be inferred that, for a particular slit width, such a structure possesses localized resonances that depend strongly on the incident wavelength. Similarly for a chosen design wavelength, these resonances are sensitive to the slit width. Such resonances are also accompanied by phase shifts in the transmitted fields that vary rapidly with incident wavelength as observed in Fig. 1(c). The choice of slit widths, as well as the applied incident wavelength, strongly affects the device performance and prudent selection of such parameters is clearly imperative for optimized performance. In particular, slit widths for the design wavelengths must be above that of the cut-off slit width size where C_cut-off ≈λ/2, below which poor transmission results.

The normalized transmission (Fig. 2(a) ) and the phase shift of the transmitted field (3 μm below the centre of each aperture) relative to that in the absence of the screen (Fig. 2(b)) for a range of single slit apertures in 400 nm thick Ag film at λ = 800 nm are plotted in Fig. 2. This clearly shows the variation in transmission and phase at a fixed wavelength over a range of slit widths. By varying the slit width across the device around resonant dimensions for this particular wavelength, therefore, a tailored spatial phase modulation can be introduced into the incident optical field.

 

Fig. 2 Calculated (a) normalized transmission (total transmitted power normalized to incident power) and (b) phase shift for a range of single slit apertures each formed on a 400 nm thick Ag film (ε_silver = −29.98-i0.19) where λ = 800 nm and 300 nm ≤ C ≤ 700 nm.

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3. Wavefield control using slit arrays

As discussed in the previous section, for a plane wave incident on a structure containing slits that have a width that varies across the device, the phase shift through each individual slit is a sensitive function of the slit size. A tailored spatial phase modulation can be introduced into the optical field by varying the widths of the slits across the device around the resonant dimensions for a particular wavelength. For example, a curved phase front can be introduced into the transmitted field to produce a cylindrical lens. According to the equal optical path principal [14], the required phase distribution as a function of the distance x from the centre of the structure to produce a focal length f is given by:

To illustrate this approach to modulating the phase, several planar focusing elements were modeled and fabricated. These structures were designed to behave as lenses according to Eq. (1) and assuming the results of the single slit aperture simulations obtained in Section 2. The typical geometry of a fabricated device consists of slits separated by 750 nm (center-to-center) in a 400 nm thick Ag film on a BK7 glass substrate (n = 1.52) as shown in Fig. 3(a) . Two-dimensional FEM simulations were performed for a TE-polarized plane wave normally incident on the structure with the plane wave propagating in the y-direction, and the electric field polarized in the z-direction (i.e. parallel to the grooves). A maximum mesh element size of 40 nm was chosen, to enable the modeling of fine features of the field at the metal-air interfaces inside the slits, while maintaining a reasonable simulation domain size.

 

Fig. 3 (a) Schematic illustration of a typical focusing element comprising a 400 nm thick Ag film, perforated with spatially varying slits C with equal slit interspacing d formed on a glass substrate. A TE-polarized plane wave at λ = 800 nm is incident from the left of the slit array in the y-direction. (b) The phase shift (green dotted line) and transmission (solid black line) across a focusing element with 7 slits, (with slit sizes 412 nm ≤ C ≤ 616 nm indicated by the asterisks) as a function of the distance x from the center of the structure. The position of each slit is indicated by the crosses. (c) x cross-section through the focal spot and the 2D FEM simulation corresponding to the (d) Poynting vector. Modeled device dimensions are 4.6 μm × 29.0 μm.

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As an example, the phase shift and transmission across a device containing 7 slits with widths 412 nm ≤ C ≤ 616 nm is shown in Fig. 3. The device, schematic shown in Fig. 3(a), is illuminated with a normally incident plane wave with λ = 800 nm. The results of the single slit simulations suggest a phase variation as a function of the distance x from the center of the structure as shown in Fig. 3(b), should produce a focal length of 15 μm. The 2D FEM simulations in Fig. 3 depicting (c) the x-axis cross-section through the focal spot of the Poynting vector and the (d) magnitude of the Poynting vector for a lens clearly demonstrates the cylindrical focusing of a wave. The modeled device gives a focal length of ~7.3 μm with a Full Width at Half-Maximum (FWHM) of ~1.3 μm at λ = 800 nm. There is a significant discrepancy between this value and that predicted using the results of single aperture simulations and Eq. (1). The finite size of the overall device and the resulting diffraction [15] which clearly play a significant role have not been included in the original design using Eq. (1).

To fabricate a focusing element, electron beam deposition was used to deposit thin Ag films onto a glass substrate. The desired array patterns were subsequently milled through the Ag films by scanning a Focused Ion Beam (FIB) (30 keV Ga+ ions, 30 pA current) across the Ag surfaces. In our design, the choice of slit width ranges from 360 nm to 616 nm on Ag films of h = 400 nm. The fabricated lens dimensions are 6.6 × 19.9 μm (Fig. 4(a) ). An in-house confocal microscope set-up with a Nikon 100× 0.95 NA microscope objective was employed to experimentally determine the focal length of the fabricated lens. A tungsten-halogen white light source was used to illuminate the sample through the glass substrate side. As this particular focusing device was designed for λ = 800 nm, light was passed through a 800 ± 40 nm bandpass filter and polarized parallel and perpendicular to the long-axis of the slits to obtain TE and TM-polarization measurements. Note that as the lens is designed for performance under normal incident illumination, an off-normal incidence would alter the phase profile across the slits, according to the angle of incidence, and in turn, shift the focal length and reduce the transmission of light through the structure. Further investigation into the effect of off-normal incident illumination on these devices is of on-going interest.

 

Fig. 4 (a) SEM image of the fabricated device. 2D FEM simulations for a focusing element for (b) TE and (c) TM-polarization. Modeled device x and y dimensions are 6.4 μm × 30.0 μm. Experimental results corresponding to a lens fabricated with the same design parameters as (b) and (c), showing transmission images taken in the xz-plane, obtained at different y-positions along the focal plane of the sample for TE (d)-(i) and TM-polarization (j)–(o). These images correspond to the transmission across the xz-plane along the y-axis in simulations. The fabricated lens x and z dimensions are 6.6 × 19.9 μm.

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Figure 4(d)-(i) shows the transmission images obtained at different y-positions along the propagation direction of the output beam of the structure, where y refers to the distance from the surface of the sample. Two dimensional FEM simulations for this focusing element are shown in Fig. 4(b) and (c), depicting sample illumination under TE and TM-polarization respectively. For TE-polarization, the simulation predicts a focal length of ~11.25 μm. The measured intensity maximum occurs at y = 12.5 μm. The focusing effect of the fabricated lens can hence be observed in Fig. 4(g) with an experimentally determined focal length of ~12.5 μm. For TM-polarized illumination, the experimental transmission images at varying y are in fair agreement with the simulations shown in Fig. 4(j)-(o). It is important to note that the focal spot observed in Fig. 4(g) clearly differs from the transmission image of the sample under TM-polarized illumination in Fig. 4(m), where no confinement of the beam in the x- (vertical) direction can be seen. Experimental results shown are consistent with 2D FEM simulations.

While diffraction interferes with the resultant focusing as observed, the agreement between simulations and experimental results nevertheless indicates the robustness in the design of devices using this approach. Research into resolving the impact of diffraction on the lens performance is ongoing.

4. Summary

We have shown that the phase of transmitted light through devices consisting of arrays of spatially varying near-resonant slits can be controlled by varying the structural geometry and incident wavelength. Results of FEM simulations shown in the previous section indicate that prudent consideration of the incident wavelength and device geometries such as the slit width is imperative in the optimization of the device performance, namely, the beam deflection and phase excursion across the structure for a given transmission profile. We have shown that by tailoring the phase shift across the device, compact focusing elements, which may find application in next-generation imaging and data storage systems can be produced. Current research is aimed at extending these concepts to two-dimensions.