1. Introduction

Recent progress in the field of metamaterials and nano-technology ignited interest in the possibility of enhancing absorption by proper structuring of materials. One of the approaches to enhanced absorption is based on the excitation of surface plasmon resonance (SPR) in sub-wavelength metal objects [1]. The SPR can dramatically increase the fields inside the object and in the near zone outside it. The fields can be even stronger for particle ensembles like chains [2] or specially arranged and shaped sets [3]. Similar behavior can be observed for finite length cylinders [4,5] and for 2D cylinder structures and sets [6,7]. Strong resonant fields inside lossy materials mean enhanced power absorption by small objects [8]. Maximizing the absorption inside nanoparticles or in the surrounding medium is an important issue for photovoltaic applications [9], localized heating in biological samples and other nano-technological applications. The investigations presented in most of the cited publications rely on plane wave illumination, which may be unsuitable in applications requiring illumination localization or higher absolute incident power. This is the case, for example, in optical data storage [10], where tightly focused illumination is required.

The field structure of tightly focused beams contains features with spatial dimensions smaller than the wavelength, especially when beams containing singularities are considered [11–13]. Moreover, the focal area of tightly focused beams may also have field components oriented along the beam propagation direction [14]. These features make the tightly focused beams substantially different from plane waves in many respects. Recently, there has been a growing interest in the interaction between nano-particles or nano-cylinders and tightly focused beams [15–18] as well as in trapping of metal particles by focused beams under plasmon resonance conditions [19,20]. The objective of this work is to investigate the enhancement of the absorption of tightly focused structured beams by properly shaped silver nanowires. The ability to control the optical properties of nanoparticles by their shape was investigated under plane wave illumination in [21] while tightly focused illumination was applied to spherical nanoparticles in [16,18]. In the cited literature, the absorption related works considered nanoparticles or nanowires of predefined shapes, some using shape size variation for the sake of optimization. Others, for example [22], were concerned with optimizing grating like nanowire structures.

In this work, we propose a novel physical criterion for shaping the cross-section of nanowires to match the structure of the tightly focused illuminating field. After defining an initial shape for the nanowire we investigate the absorption as a function of the geometrical feature dimensions. We consider the absorption under two types of tightly focused illumination for some corresponding shapes of silver nanowires. The 2D tightly focused field distribution is evaluated using an extension of the Richards-Wolf focusing method [14] for input wavefronts with piecewise quasi constant phase [23,24]. The distribution of the tightly focused field is used for the initial shaping of the nanowire cross sections according to the proposed criterion. The values of the tightly focused incident field on the cross-section boundary are used for the evaluation of the scattered field using the source model technique [25–27].

The next section presents the suggested shaping procedure and its application to the chosen incident beams. Section 3 investigates the influence of various geometric features on the nanowire absorption. The evaluation of the near field and the power flow for a selected case is presented in section 4, which is followed by conclusions.

2. Defining the object shape

“Localized surface plasmons … are non-propagating excitations of the conduction electrons of metallic nanostructures coupled to the electromagnetic field” [1]. Ideally, maximizing the coupling would require matching all the components of the surrounding field to the SPR mode of the given nanostructure. In most situations this is impossible and we restrict the treatment to a given incident field and we attempt to find a nanostructure shape that maximizes the coupling, and consequently, the absorption. Modifications of the nanostructure shape affect its resonant mode and its scattered field which in turn changes the total field, being the sum of the incident and the scattered fields. Thus, for the sake of simplicity, we choose the incident field to determine an initial shape. The other simplification is based on the fact that the concentrations of free charges at portions of the nanostructure surface are related to the locally normal electric field component. This field component is used as a predictor of the degree of coupling between the surrounding field and the SPR mode.

Consider a 2D geometry, which is uniform along the y axis. The nanowire is represented by an infinite cylinder parallel to the y axis. The nanowire cross-section is shaped in a way that maximizes the normal incident electric field on its contour, C. We denote the phasor of the incident field by ${\overline{E}}_{inc}=\widehat{x}{\tilde{E}}_{inc,x}+\widehat{z}{\tilde{E}}_{inc,z}$ and the total contour length by $C_{tot}={\displaystyle {\oint }_{C}dl},$ where $dl$ is a differential element of length along C. Thus we look for shapes maximizing a parameter M, defined by the relation,

Application of the above procedure requires rigorous evaluation of the incident field components, as a function of position, in the vicinity of the geometric focus of a focusing system. The optical system is schematically illustrated in Fig. 1 . An incident plane wave, polarized in the x direction, propagates along the z axis. At the entrance pupil of a focusing optical system it passes a phase mask, which will be discussed later. Restricting the discussion to a 2D geometry we assume that along the y axis the problem is uniform and infinite. The tightly focused field is formed in the vicinity of the geometric focal line O, parallel to the y axis and the infinite dimension of the nanowire is directed along the y axis. The rigorous evaluation of the 2D focused field can be performed using either Kirchhoff or Debye boundary conditions [23,28,29]. These boundary conditions partly depend on the phase mask located at the entrance pupil. When this phase mask is uniform across the pupil, the result corresponds to the tightly focused plane wave (PW). When this phase mask is a π step function, the result corresponds to the tightly focused dark beam (DB) [24]. The evaluation of the tightly focused field was performed using the extension [24] of the Richards-Wolf method [14]. The extension allows focusing of incident wavefronts having piecewise quasi-constant phase. The numerical expressions corresponding to the 2D geometry are given in [23]. For our analysis we assume a numerical aperture of the focusing system, NA = 0.87 and the illuminating wavelength, $\lambda =338nm.$

 

Fig. 1 Optical system schematic. The inset shows the electric field amplitude of the tightly focused plane wave in the xz plane.

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The amplitude value of the electric field of the tightly focused PW is shown in Fig. 2(a) as a normalized color-code. The location of (0, 0) corresponds to the geometrical focal line O as shown in Fig. 1. The focal line is parallel to the y axis, which is normal to the figure plane. As the polarization is not strictly linear through the tightly focused field, visualizing the direction of the electric field is not trivial. Instead, we define here the inclination of the incident field phasor as:

 

Fig. 2 The electric field in the focal area. The location (0, 0) corresponds to the geometric focal line O in Fig. 1. The amplitude of the electric field is color coded and normalized independently for the tightly focused PW (a) and the tightly focused DB (b). The electric field phasor inclination, α, is visualized by the short white lines. The positions and shapes of the contours of the nanowire cross-sections are overlaid in cyan, magenta, green and black curves. The axes x and z correspond to those in Fig. 1, axis y is normal to the figure.

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Fig. 3 The plot of absorption, as a function of nanowire cross-sectional shape parameters for the tightly focused PW (a), and for the tightly focused DB (b). Colors correspond to those of Fig. 2 (for additional detail see the main text).

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The tightly focused field of a DB is shown in Fig. 2(b) with the same nanowire cross-sectional contours placed in the same locations with respect to O, as in Fig. 2(a). The field structure in this case is more complicated due to the presence of an optical singularity. A pair of nanowires with the cross-sections of the form shown by the black contours appears to have highest M value of about 70 a.u.. These nanowires are infinite along the y axis (i.e. along the direction normal to the figure plane). The prolonged sides of the nanowires are oriented along the z axis, which is the beam propagation direction, forming a double-plate. For the field inclination distribution, as in Fig. 2(b), one can suggest another cross-section, shown by the magenta contour. Evaluation of M along the magenta contour gives values between 55 and 40 a.u. depending on the object dimensions as indicated in Fig. 3(b). The magenta cross-sectional contour corresponds to a plate shaped nanowire, with its prolonged side oriented along the lateral x axis. This nanowire, being infinite along the y axis, is further referred to as the lateral plate. Under the DB illumination the M value of the axial plate cross-section, shown with cyan contour in Fig. 2(b), is 30 to 20 a.u., and for the cylinder cross-section, shown with green contour in Fig. 2(b), the value of M is 60 to 45 a.u.. The relatively higher values of M for the case of the cylinder as compared to the axial plate are due to the strong longitudinal field in the vicinity of O (x = 0, z = 0).

3. Matching shapes for absorption enhancement

Having defined the initial shapes for the cross-section we proceed by matching them to the two illumination cases for absorption maximization. The absorption is measured as the integral on the net electro-magnetic power flowing into the volume containing the investigated structure. The volume is defined as a cylinder, centered on the geometrical focal line O having radius equal to the optical system focal distance. The power is calculated for the total field, following the evaluation of the scattered field by the source model technique [25–27]. In this work we are concerned with silver nanowires having dielectric constant ${\epsilon }_{Ag}=-1.07+0.29i$ under the illumination wavelength of $\lambda =338nm.$ It should be noted that the dielectric constant value is close to that of the circular cylinder electrostatic resonance. The effects of heating the material and the resulting change of its optical properties are not accounted for. The dimensions of the investigated silver nanowires are considered large enough to neglect the dependence of the imaginary part of ε_Ag on the cylinder size [30].

Nanowire shape matching is performed by varying some geometrical parameters of its cross-section and calculating power absorption for each parameter value. The varied parameters include cross-sectional shape length, width, distance between the nanowires and cylinder diameter, as described below. For comparison, the absorption of all the shapes is calculated for both illuminations as shown in Fig. 3 as a function of the corresponding geometrical feature size. In case of the axial plate, shown cyan, the only varied parameter is plate length measured along the z axis, while its width, measured along the x axis, is fixed at 0.118λ (40nm). For the double-plate the varied parameters are the distance between the plates (the result marked with a plain black line) and their lengths measured along the z axis (the result marked with black plus marks). Their width, measured along the x axis, is fixed at 0.118λ (40nm); when the distance is varied, the length is constant at 0.7λ (237 nm) and when the length is varied, the distance is kept at 0.7λ (237 nm). The results of the varied distance between the plates exclude a range where there is an overlap between locations of sources used by the source model technique. The length of the lateral plate, being measured along the x axis, is varied and the result is marked with a plain magenta line. The width of the lateral plate, being measured along the z axis, is varied and the result is marked with magenta plus marks. The chosen constant values for the lateral plate were: width = 0.27λ (90nm), length 0.6λ (203nm). The absorption results for the changing diameter of the circular cylinder are shown as a green line. The dimensions of the shapes in Fig. 2 correspond to the chosen constant values as denoted above.

The absorption results under the tightly focused PW illumination are shown in Fig. 3(a). The axial plate shows best absorption topping at about 65%. It is closely followed by the double-plate. The cylinder with the circular cross section shows a relatively high absorption, above 40%, reaching its peak for the diameter of 0.136λ. With increasing size, it scatters more and absorbs less. It is interesting to note that a similar absorption value was reported for metal spheres in tightly focused beams [18]. For the cylinder diameters close to the width of the axial plate, the absorption results are similar to those of very short plates. The laterally oriented plate shows smaller absorption, as it may be expected from the relatively low M values. These results indicate, as expected, that shapes having higher M values tend to have higher absorption and vice versa.

The absorption results for the tightly focused DB illumination are shown in Fig. 3(b). The double-plate shows the best absorption here for a wide range of feature sizes, topping at about 65%. The lateral plate shows absorption above 40%. The shape of the lateral plate, from the view point of matching to the incident field, bears some resemblance to the surface plasmon polariton excitation in a nanowire in [31], where the maximum coupling efficiency was reported to be above 90%. For the cylinder, the highest absorption value of about 35% is reached for cylinder diameter of about λ/2, which is where the cylinder approaches the major streams of the power flow. The axial plate shows relatively low absorption, regardless of its length, as it is situated in the singularity region. In case of the tightly focused DB illumination, the prediction of absorption by M is less accurate although still relevant.

It is important to note that for matched shapes, the share of the absorbed power is above 50% for wide feature size ranges, as shown in Fig. 3(a) and Fig. 3(b). This means that the cross-section area can be significantly reduced without giving up too much absorbed power. Thus the surface temperature of the nanowire can rise much higher, which is an important attribute for some biological applications.

4. Near field results and power flow

It is interesting to assess the field and the power flow in the vicinity of the highly absorbing nanowire. Figure 4(a) shows the total electric field for the case of the axial plate shaped for maximum absorption. The apparent differences with the incident field are field enhancement at the left tip of the plate, and the inclination of the total electric field along the plate sides, which is no longer nearly normal but it makes almost 45° with the surface. The vertical axis on Fig. 4(a) is shifted down to accommodate the inset showing the overall near field distribution. Interference of the reflected wave with the incident wave can be observed in the inset.

 

Fig. 4 The electric field (a) normalized with the same value as Fig. 2(a) and the Poynting vector (b) corresponding to the matched plate in the focused PW. The insets show (a) the zoom out at the whole waist area, (b) the zoom in on the power flow near the tip of the plate.

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The change in the electric field inclination means that the power crosses the object boundary and flows into the plate. The distribution of the power flow in the vicinity of the plate is shown in Fig. 4(b). The plate is positioned as in Fig. 4(a) and includes the inset zooming on the power flow distribution at the plate tip oriented towards the incident wave (the left tip). It can be observed that the power flow distribution pattern is not trivial. The power flow crosses the sides of the plate, flows inside the plate towards the left tip and forms a number of whirlpools and bifurcations in that area. Partly, the power flows into the plate through the leftmost tip. At the zone where the tip interfaces the sides of the plate, the power flows out of the plate. It is comforting to note that the Poynting vector distribution and its topological features have a number of notable similarities to the results presented in Refs [32,33].

The color-scale of Fig. 4(b) suggests that the power flow in the vicinity and inside the plate has a significantly higher magnitude than in free space away from the plate. Thus the overall beam power flow (from left to right) is not apparent from the figure, although it still exists.

5. Outlook and conclusions

The above results show that the suggested physical criterion for nanowire cross-section shaping allows enhancing the absorption up to 65% of the power of an incident tightly focused beam. It is remarkable that properly shaped absorbing nanowires, sized below the classical resolution limit, induce such significant changes in an incident beam. The suggested procedure for the initial shape guess proved itself capable of achieving high absorption results in spite of the simplifications made. An interesting future research direction is synthesis of incident tightly focused fields that maximize the normal electric field to some given nanostructure shape. This can be performed by adjustment of the phase or polarization at the entrance pupil or by changing the NA of the focusing system. Proper adjustment of the illumination and the absorber shape to match the total field may lead to additional enhancement of the absorption.

Straight forward application of our criterion may be useful for devising absorbing bodies intended to heat themselves or their surroundings. The introduced criterion has a potential to be extended to the design of structures for efficient coupling of tightly focused beams into SPPs. Our results can be useful in developing metrological and optical data storage applications that exploit the strong modification of the scattered far field by the enhanced absorption.