1. Introduction

In 1999, Pendry and associates proposed two fundamental constituents for the construction of a left- handed metamaterial (LHM) – an artificial medium exhibiting a negative refractive index, wherein the electric field E, magnetic field H, and wave vector k follow the left-hand rule. One of the constituents consists of arrays of thin metal wires, which gives negative permittivity (ε) [1]. The other constituent comprises metallic structures called split-ring resonator (SRR) that provide negative permeability (μ) [2]. By combining these two, a metamaterial can be obtained that shows a negative index of refraction within a certain wavelength range. This metamaterial appears to the incident electromagnetic radiation as a homogeneous medium with an effective ε and μ, instead of a collection of components, i.e., wires and SRRs, provided the wavelength is much larger than the size and spacing of the individual components. Recently, a magnetic response from periodic arrays of U-shaped single-slit split-ring resonators (SSRRs) was demonstrated at optical and telecommunication wavelengths [3–5]. The SSRR, which forms the 'magnetic atom' of the LHM, can be viewed as an LC circuit with the metal represented by the inductance L of a single winding coil and the gap by a capacitance C between the ends of the coil [3]. An oscillating current can flow through the LC circuit due to time-varying E and H fields. At resonance, the magnitude of the oscillating current in the LC loop is the strongest and it gives rise to a large magnetic moment in the gap, which is crucial to obtain negative permeability [3–6]. This LC resonance frequency (or magnetic resonance frequency ω_r) can be tuned since it scales inversely with the lateral size of the SSRRs until the limits of size scaling are reached [5].

It has been observed that when a linearly polarized beam of light is normally (or obliquely) incident onto a periodic array of gold (Au) or silver (Ag) SSRRs, effectively two kinds of structure-dependent resonances arise in the spectrum of the transmitted light [3–9]. One of the resonances, occurring at ω_r, is related to the current flowing through the SSRR when the light is horizontally polarized, i.e., the electric field is along the horizontal bottom part of the SSRR. The current can be excited by the time-varying H- or E-field coupling depending on the incident polarization and the orientation of the sample [6]. Other electrical resonances, addressed as 'Mie resonances' in [3,4] including the previous LC resonance, have been explained by plasmonic eigenmodes of increasing order at the front air-metal interface along the entire U-shaped structure by Rockstuhl et al [10]. The spectral positions and the magnitudes of the different resonances are known to depend on the polarization, ratio of the lateral arm width to thickness, and the intrinsic properties of the metal [10,11]. However, there still remains a lack of simple predictive models that would give a clear understanding of the origin of these resonances and how the optical properties of nanostructures in general change with regard to geometrical shape variations, especially in the visible and NIR regime.

The electrical LC analogy can also be invoked for the more general crescent-shaped nanostructures, the nano-crescents (NCs), which are nano-particles that have been extensively studied over the last few years owing to their field enhancement properties [12–16]. Although the NCs have not been used for the construction of magnetic atoms in LHM, it is very likely that they could be used since both, the NCs and the SSRRs, have a common U-shaped geometry. In this paper, we study the optical transmission properties of NCs with rectangular inner boundaries ('NCR' geometry) at normal incidence, in the visible and near-IR wavelengths, using numerical simulations. In particular, we analyze the principal resonant modes arising in the transmission spectrum of periodic arrays of such NCR structures and compare them to those in the conventional, rectangular-edged SSRR transmission spectra. One can observe an extra tunable resonance peak in the transmission spectrum of the NCRs that does not exist for the SSRRs with similar dimensions and periodicity. All the resonance conditions in the visible and near-IR region are characterized with the help of the associated near-field distributions of the electric and magnetic fields, so as to analyze their origin and understand their physical properties. Our results suggest that sometimes it becomes necessary to also consider the plasmonic mode pattern at the metal-glass interface, along with the air-metal interface, to explain certain far-field transmission resonances occurring with a U-shaped metal nanostructure. Thus, for a general three dimensional U-shaped metallic structure, the transmission resonances can be explained as a result of coupling between plasmonic modes excited at all the metal-(air/dielectric) interfaces, namely front, back, top, bottom, inner surfaces as well as lateral sides.

2. Modeling

The propagation of light (plane waves) through periodic arrays of gold (Au) SSRRs and NCs on dielectric substrates is examined by the full, three-dimensional (3D) vector finite-element method (FEM) using COMSOL Multiphysics. The advantage of using FEM is that it can model and mesh irregular, curved geometries like NCs with great accuracy. The dielectric substrate chosen, in our case, is glass (n_sub = 1.5 unless specified otherwise). The refractive index of Au in the visible and near-IR range is obtained from literature [17]. Figure 1 shows the unit cells of the two types of U-shaped structures considered, namely the SSRR and NCR. Both structures are first made in 2D and then extruded a distance t in the z direction. The NCR has an elliptical outer boundary and a rectangular inner boundary. It can be constructed in 2D by immersing a rectangle of width w_x into the ellipse from the top until the desired value of depth d_r is reached. Then, the intersection of the two 2D geometries is taken and the resultant is extruded to give a 3D NCR. A similar approach is followed to construct a 3D SSRR, but here the outer boundary is a rectangle of dimensions l_x x l_y.

 

Fig. 1 Unit cells of two different types of periodic U-shaped nanostructures: (a) single-slit split-ring resonator (SSRR), (b) nano-crescent with rectangular inner boundaries (NCR). Incident light is linearly polarized along the x axis and propagates in the + z direction.

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The parameters of the SSRR array are kept the same as those in the published experimental work of Klein et al [5]: l_x = 110 nm, l_y = 95 nm, w_x = 34 nm, d_r = 45 nm, t = 45 nm, and a square lattice with lattice constant a = 240 nm (Fig. 1). This is done so as to confirm the results of our computations and to facilitate the comparisons between resonances. The whole NCR geometry is completely inscribed into the SSRR's outer dimensions and the gap width w_x is kept the same as in the SSRR. Plane waves are incident onto the periodic nanostructures from air (n = 1) and it propagates in the + z direction. The discussion in this paper is limited to horizontally (x) polarized light. Periodic boundary conditions are applied at the sides of the unit cell, i.e., yz and xz planes, to simulate a square periodic array of Au SSRRs/NCRs. Please note that the above dimensions of l_x, l_y, w_x, and t are kept the same throughout the manuscript for, both, SSRRs and NCRs.

3. Analysis of the transmission spectrum

3.1. Dependence of transmission resonances on lateral depth

The lateral depth parameter (d_r) is varied and the transmission and reflection spectra are calculated as a function of wavelength for horizontally polarized incident light. Figure 2 shows the comparison of the transmittance and reflectance spectra for periodic NCR structures of three different d_r values and an SSRR array (d_r = 45 nm) in the visible-NIR range.

 

Fig. 2 (a) Comparison between transmission spectra of periodic NCR (crescent shaped structures with a rectangular inner void) and conventional SSRR structures of same lateral depth (d_r = 45 nm). Figures (b) and (c) show the transmission and reflection spectra for NCRs with different values of d_r. The numbers 1, 2 and 3 are labels for the different resonances arising in the transmission/reflection spectra.

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One can observe three distinct and significant modes (or peaks) at the same wavelengths in the transmittance and reflectance spectrum, labeled as 1, 2, 3, of NCRs (Fig. 2). But for the SSRR, only two modes are present, namely 1 and 3 [Fig. 2(a)]. Mode 1 is the resonance that occurs approximately at frequency ω_r predicted by the electrical LC theory. Since the light is normally incident onto the glass substrate with the SSRRs/NCRs, an oscillatory current is induced by E-field coupling [6,7] and flows throughout the 'U' in a closed loop. At mode 1 condition, the magnitude of the current is maximal and the net magnetic moment in the gap is large. This explains the low transmission at mode 1. In the case of the SSRR, the wavelength of this mode can be tuned further by varying d_r while keeping the overall size fixed. Enkrich et al [7] experimentally demonstrate a gradual blue shift in mode 3 by decreasing d_r. When d_r = 0, i.e., the gap vanishes, mode 1 coincides with mode 3 giving rise to a single broad resonance peak (not shown). The blue shift can be explained in simple terms by following the LC analogy, in which the inductance and the capacitance of the SSRR change as the lateral depth is varied. Since both the NCs and the SSRRs are U-shaped, we expect to observe similar effects for mode 1 in the NCR structures. Hence, as d_r is decreased from 60 nm to 30 nm for the NCR, resonance 3 moves from 1010 nm to 770 nm. Its magnitude also remains effectively constant [Fig. 2(a)], consistent with the results in [7].

Resonance 3 can be explained by the formation of a higher-order mode at that wavelength and it will be considered in detail later (in section 3.3). There is virtually no change in the wavelength of mode 3 (580 nm) as d_r is altered from 30 nm to 60 nm. The most interesting aspect about the NCR geometry is the presence of a new peak, i.e., mode 2, that is otherwise absent in case of the SSRR. It can be observed that as d_r ranges from 60 nm to 30 nm, the magnitude of mode 2 gets stronger and its peak becomes narrower. The spectral tuning of mode 2 also closely resembles that of mode 1 outlined above, as it experiences a gradual blue shift with decreasing d_r.

3.2. Dependence of transmission resonances on the refractive index of substrate

The refractive index of the substrate is varied from 1.5 to 1 and the NCR depth d_r is fixed at 45 nm. Figure 3 shows the changes that occur in the different transmission modes when substrate index is brought closer to that of air.

 

Fig. 3 Behavior of transmission resonances(denoted as 1, 2 and 3) of NCR (t = d_r = 45 nm) with a variation in the refractive index of substrate (n_sub).

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It can be observed that there is a gradual blue shift in all modes accompanied by a weakening of mode 2 and 3, as the substrate index approaches 1. Also, mode 2 gradually tends to merge with mode 3. These are interesting observations as it can be agreed that the wavelengths at which resonance conditions (i.e. mode 1, 2, and 3) exist depend not only on the geometrical dimensions of the NCR (or SSRR) but also on the optical properties of the substrate. Also, studying plasmonic modes only at the front metal-air interface is insufficient to explain this effect. A similar blue shift in the resonances is observed for the case of SSRR (i.e. mode 1 and 3) but it is not shown in this paper as it does not convey any new information. In the next section, we try to put forth a valid physical explanation behind the existence of mode 2 in NCR structures considering observations in sections 3.1 and 3.2.

3.3. Discussion

In order to understand the origin of the new resonance (i.e., mode 2) arising in the NCR structures, the magnetic and normal electrical near-field distributions are analyzed at the three mode conditions described above, for the NCR with d_r = 45 nm. The vector-valued H field is projected onto fictional planes 1' and 2' cutting through the cross-section of the vertical arms and the horizontal part of the NCR respectively, as shown in Fig. 4(a) . The H-field projections provide valuable information about current flow direction in the respective parts of the U shaped structure. The E_z plot, on the other hand, is basically a snapshot of the E_z distribution (total E field in the z direction). The character of the various plasmon modes at the front air-metal interface (xy plane) can be visualized by the E_z distribution as it is the component perpendicular to the metal surface [10]. The E_z plot is calculated on a xy plane, 20 nm before the surface of the NCR/SSRRs in air. The color scale that depicts the amplitude of E_z – positive (red) and negative (blue) – provides useful phase information of the near field.

 

Fig. 4 (H)-field (array of red cones on left-side pictures) and E_z-field (colored map in right-side patterns) distribution of the NCR with thickness t = d_r = 45 nm at different resonance conditions. (a) Two slices 1' (xz plane) and 2' (yz plane) onto which the vector magnetic field is projected. Points A and B denote the sharp edges of the NCR structure, whereas C is the cross-section of the bottom part. (H)-field projections on planes 1' and 2', and E_z distribution (from left to right) are shown for (b) mode 1, (c) mode 2, and (d) mode 3, respectively. The sizes of the red cones are proportional to the magnitude of projected (H) field. The white arrows merely represent the current flow direction in the respective arms of the NCR as deduced by the (H)-field projections for modes 1 and 2. The E_z component is color-coded, with red corresponding to positive and blue to negative values, respectively.

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At mode 1, circulatory magnetic fields are observed on planes 1' and 2', which is a signature of an oscillating current flowing in the respective arms of the NCR [Fig. 4(b)]. By the right-hand thumb rule, the direction of the current can be deduced and it is shown by white arrows in the E_z plots. The E_z pattern shows a single maximum that oscillates from one vertical arm (field is concentrated at the tip) to the other and resembles a dipole. However, the E_z plot for mode 3 [Fig. 4(d)] reveals the presence of a higher-order quadrupole mode with two diagonally opposite maxima and minima. The results for both modes 3 and 1 are consistent with the work in recent publications in which the plasmonic character of the resonance is discussed [8,10,11]. Mode 1, due to its dipole resemblance, is called the lower-order plasmonic mode and mode 3 is a higher-order plasmonic mode. At mode 2 [Fig. 4(c)], however, circulatory magnetic fields with a closed loop around the vertical arms (1') can be seen, but not around the horizontal bottom arm (2'). Thus, there is no evidence of a unidirectional current in the bottom arm of the NCR. As stated before, this resonance (mode 2) is absent in conventional SSRRs with rectangular edges. Also, the E_z plot at mode 2 looks nearly identical to that at mode 1. This suggests that the formation of plasmonic eigenmodes at the air-metal interface alone, especially just the front interface of the metal nanostructure as in the work of Rockstuhl et al [10,11], cannot explain all the resonances arising in a U-shaped nanostructure.

The H-field projection on plane 1' at mode 2 condition is compared for the NCR structures with d_r = 30 nm, 45 nm, and 60 nm (Fig. 5 ).

 

Fig. 5 (H)-field distribution on plane 1' (as shown in Fig. 3(a)) for NCR structures with different lateral depths (d_r): (a) 30 nm, (b) 45 nm, and (c) 60 nm at mode 2 condition.

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Mode 2 occurs at 670 nm (d_r = 30 nm), 710 nm (d_r = 45 nm), and 730 nm (d_r = 60 nm) for the cases analyzed (as seen in Fig. 2). It is observed that the strength of this resonance becomes stronger as d_r decreases. It is found that the H-field strength in the gap is largest for d_r = 30 nm and it decreases as the lateral depth increases. This effect is consistent with the change in transmission magnitude of the NCR structures with greater lateral depths. But still, the reason behind the reversal of currents, as we move from mode 1 to mode 2, despite identical E_z distributions at the front surface of the U-shaped nanostructure is unanswered at this point.

As the incident light is x-polarized, E_x is the superposition of the incident and the scattered electrical field, while E_y and E_z are generated purely by the scattering of the incident light. The component E_z is responsible for plasmon generation for the front (air-metal interface) and back (metal-glass) surfaces, E_y for the top-bottom (air-metal) surfaces, and E_x for the lateral surfaces (air-metal). Before discussing the NCR geometry, it will be interesting to first consider the case of a conventional SSRR. Figure 6 contains the E_z plots at points z₁ (in air) and z₂ (in glass) along the z axis for a SSRR with d_r = 45 nm. These points are situated at 20 nm before and after the metal nanostructure. We clarify that the E_z plots in Fig. 4 are indeed calculated at z = z₁.

 

Fig. 6 Plasmonic mode distribution (total E_z field) at the front (metal-air) and back (metal-glass) interface at different mode conditions of SSRR (d_r = t = 45 nm). Labels z₁ and z₂ denote xy planes situated at a distance of 20 nm from the SSRR, in air and glass, respectively.

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It is observed that for SSRR, the plasmonic distribution maintains the same character (i.e., dipolar for mode 1 and quadrupolar for mode 2) at the front and back interfaces for a given incident wavelength, but they are phase-shifted by π radians with respect to each other. Also, the E_y (E_x) distribution at the top-bottom (lateral) surfaces of the SSRR is interesting at the two mode conditions (Fig. 7 ). The distribution is plotted over xz planes (for E_y) and yz planes (for E_x) set 20 nm away from the air-metal interfaces. Since the lateral sides are symmetric, the E_x plots are the same for both lateral surfaces at a given wavelength (Fig. 7(b)). Looking at E_y and E_x plots, it can be clearly observed that the plasmonic modes occurring at the front air-metal interface and the back metal-glass interface couple via the top surface for mode 1 (910 nm). However, the coupling is via the bottom surface for the higher order mode 3 (610 nm). This suggests that even though the plasmon distribution at the front air-metal interface (E_z plot) is enough for us to make a distinction for different resonance conditions of SSRRs, it does not fully explain the physics behind the occurrence of peaks in the far-field transmission spectrum. The resonances are, in fact, a result of an intricate coupling process between plasmonic modes excited at all the surfaces of the 3D-metal nanostructure. And, it is the manner of coupling that is unique to every transmission resonance.

 

Fig. 7 Plasmonic mode distribution at the top-bottom and lateral surfaces of the SSRR: (a) E_y plots for the top (y = y₁) and bottom (y = y₂) surface of the SSRR at different far-field resonance conditions. Planes y₁ and y₂ are 20 nm away from the respective metal surfaces. (b) Respective E_x plots for the lateral surfaces of the SSRR at the same resonance conditions.

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The above results are consistent with the E_z plots for the NCR at different mode conditions (Fig. 8 below).

 

Fig. 8 Plasmonic mode distribution (total E_z field) at (a) the metal-air (z₁) and metal-glass (z₂) interfaces at different mode conditions of the NCR structure (t = d_r = 45 nm). Kindly note the mismatch in the mode pattern at z₁ (dipolar) and z₂ (quadrupolar) for mode 2 condition.

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The only peculiarity, however, is the plasmonic mode distribution at the mode 2 condition (at 710 nm). It can be seen that the distribution is dipolar for the front air-metal interface but quadrupolar for the metal-glass interface. This mismatch, which arises from the NCR geometry, to some extent explains why the current directions are different in the vertical arms of the NCR structure despite the E_z plots being identical, at mode 1 and mode 2, at the front air-metal interface [Fig. 4(b) and 4(c)]. The current flowing through the different arms of the NCR (and SSRR) actually depends on the net E-field distribution – vector sum of E_x, E_y and E_z – near all its surfaces and not just on the E_z distribution at the front surface. It is in turn a function of the plasmonic modes occurring at all the surfaces and their associated coupling which is evidently different for mode 1 and mode 2. Even the observed blue shift in all the resonances accompanied by a change in the magnitudes of mode 2 and mode 3 when the substrate index is varied (Fig. 3) from 1.5 to 1, is in line with this reasoning since the properties of different plasmonic modes excited at the back metal-substrate interface changes.

As it was shown in case of SSRRs (Figs. 6 and 7), the manner in which coupling takes place between the dominant plasmonic modes at the front (air-metal interface) and back (metal-glass interface) surfaces, is what completely characterizes any transmission resonance. If we compare the geometries of SSRR and NCR, the top two flat surfaces (SSRR) are reduced to sharp edges (NCR), and the lateral and bottom surfaces (SSRR) merge to form one single elliptical surface (NCR). This changes the plasmonic eigen mode conditions of the lateral elliptical surface and deeply influences the manner in which the coupling takes place between those at the front and the back surfaces of the NCR. The change in the manner of coupling manifests itself as mode 2 for the NCR geometry. As the rectangular inner surfaces of the NCR is modified into a circular or elliptical surface to form a natural crescent structure, we end up destroying the hitherto mode 2 condition and the transmission spectrum from such a structure is completely different to that of a NCR (not presented here). This observation backs our aforementioned qualitative explanation regarding the effect of geometrical shape of the 'U' on the coupling dynamics. However, more intense investigation is required to precisely clarify what all parameters of the geometrical shape can be modified to freely tune the positions of different transmission resonances.

4. Conclusions

We have discussed transmission resonances arising in periodic nano-crescent structures with rectangular inner boundaries (NCR) at normal incidence and have compared them with conventional SSRRs with all rectangular edges for horizontal polarization. In the former, a new resonance condition is found that can be spectrally tuned by varying the lateral depth parameter of the NCR. The magnitude of the peak increases as the lateral depth decreases. All the resonances have been characterized using the near-field distributions, i.e., the H-field and E_z (component of the E field in the propagation direction) plots. We also revisited the interpretation of transmission resonances arising from periodic subwavelength metal SSRRs on glass at normal incidence. It is found that in order to completely explain the existence of far-field transmission resonances, it is essential to consider the plasmonic distribution at all the surfaces of the U-shaped geometry and how they are coupled to each other. The manner in which coupling takes place between plasmonic modes at all the surfaces, ideally including those at the inner walls, is what completely characterizes any transmission resonance and it is unique for a given transmission resonance. Our computational results suggest that increasing order of plasmonic modes at the front air-metal interface alone cannot explain all the resonances that arise in the transmission by periodic arrays of arbitrary U-shaped metal structures. In some cases, e.g., for the NCRs, it is essential to also consider the E_z distribution at the metal-glass interface since the E_z plots at the front air-metal interface are identical at different resonance conditions. The modeling results specifically reveal the effect of the lateral surface shape of the U-shaped NCR in the manipulation and coupling of the plasmonic modes at different interfaces giving rise to a new resonance in the transmission spectrum. It is also a significant step towards building a comprehensive theory explaining how and why the resonances evolve and shift, when the different parameters related to the geometry of the 'U' vary.