1. Introduction

Enhanced optical transmission (EOT) of light through perforated metal films with periodic subwavelength hole arrays (HAs) has been extensively studied in the past decade [1–3]. The associated transmission spectrum consists of a number of relatively sharp Fano-type resonances that are closely related with the Bravais lattice reciprocal vectors in the HA structure factor. Recently there has been growing interest in studying EOT through aperiodic [4] and fractal [5] HAs fabricated on metallic films. It has been shown that prominent reciprocal vectors in k-space of such HA structures still give rise to sharp resonances in the transmission spectrum, even when the hole geometry lacks translational symmetry. The leading model for explaining the EOT phenomenon is based on interference of surface plasmon-polariton (SPP) excitations that are launched from the apertures on the film surface. Therefore it may be possible to study SPP transport on perforated metallic film by conducting a thorough investigation of the EOT spectrum. In particular, the quality factor and strength of the resonances in the transmission spectrum may be related with the SPP correlation length on the metallic films that are affected by the various HA structures.

Quasicrystals (QCs) constitute one member of a group of aperiodic structures with intermediate order geometries between fully periodic Bravais lattice and fully disordered structures. QCs do not have a unit cell, nor do they exhibit translation symmetry. Nevertheless, they possess local rotational symmetries that are not constrained to those of regular crystals having Bravais lattice, as well as long-range order that leads to X-ray diffraction patterns having unusual point group symmetries. This QC property led to the 2011 Nobel Prize in Chemistry being awarded to Dr. Dan Shechtman from the Technion. In addition to X-ray diffraction, special interest has been also paid to the electronic and optical excitations in QCs. It has been realized that the excitation dispersion relations in these media possess multifractal pseudogaps that impede transport, and may cause localization. This localization is not as strong as ‘Anderson localization’ that leads to a complete halt of the transport process, since QCs may still show relatively high conductivity based on imperfections [6]. However it has been known for some time that transport in QCs is unusual; for example the conductivity in atomic QCs increases with disorder, and has an ‘inverse Matthiessen rule’, where it increases with the temperature [7].

Recently it has been recognized that reduced transport could occur not only in QCs or disordered systems, but also in deterministic aperiodic systems (DAS). Unlike random media or QCs, DAS are described by simple mathematical prescriptions such as ‘inflation rules’, which encode a fascinating complexity. One such structure, for example is based on the Fibonacci series. DAS belong to a special class of aperiodic structures that have physical properties that are distinct from periodic, disordered, random, or QC structures. For example, they show highly localized excitation states characterized by high field enhancement and low energy transport. In fact, experimental evidence for light localization in DAS formed from dielectric multilayers that follow one-dimensional Fibonacci series has been achieved by several groups [8–10]. Dallapiccola et al. recently reported near-field optical microscopy measurements on two-dimensional (2D) Fibonacci nano-particle structures [11, 12]. They found that a larger near-field intensity enhancement could be obtained in Fibonacci structures as compared to periodic square arrays of metal nano-particles. We note, however, that evidence for weak localization of SPP excitations in DAS structures is still missing. It is important to study SPP localization in DAS structures in order to show that the suppression of transport in such structures is a universal phenomenon.

In this work we used terahertz (THz) time-domain spectroscopy to investigate the optical transmission through HA structures in the form of 2D Fibonacci perforated metal films, and compared the obtained EOT spectra to the spectra of corresponding square lattice periodic HAs. We found that the Fibonacci HA structures also show Fano-type resonances that correspond to prominent reciprocal vectors in the structure factor in k-space, similar to HA structures based on QCs [4]. However the Fibonacci-related resonances are much weaker than the resonances associated with the underlying hole grid arrays in these structures, indicating a smaller SPP correlation length. In addition we also systematically studied the resonance strength of both Fibonacci and periodic HAs as a function of the number of holes, and found that the Fibonacci related resonances show a much weaker intensity, consistent with the lower correlation length of the SPP in these structures.

2. Experimental details

A one-dimensional Fibonacci series is usually constructed by stacking together two different optical materials, A and B, which are designed using the following deterministic generation scheme: S_{j + 1} = {S_j-1, S_j} for j ≥1, where S₀ = {B}, S₁ = {A} and S_j is a structure obtained after j iterations of the generation rule; here A and B are seed letters. 2D Fibonacci structures have been developed previously by several groups [13–15] using an algorithm that generates Fibonacci series in two orthogonal directions (x,y). In this work we generated 2D Fibonacci structures using the method proposed by Dal Negro et al. [15], in which we applied two complementary one-dimensional Fibonacci recursive generators: g_x: A→AB, B→A and g_y: A→B, B→BA in two orthogonal directions (x,y). We designed and fabricated HA structures with circular apertures on 5x5 cm² area of 75 µm thick free-standing stainless steel foils using this generation method, where A is an aperture, and B is a space lacking an aperture (i.e. a missing hole). The details of the HA generation method can be found elsewhere [11]. An important property of such a structure is that is based on an underlying grid HA that also contributes to the structure factor. We fabricated HA patterns of periodic, 2D Fibonacci, and disordered structures with various numbers of holes in arrays of up to 90x90, with varying hole diameters using a computerized generation scheme.

In Fig. 1(a) , we show an example of a 2D Fibonacci HA with a total of 173 holes having 0.44 mm diameter, where the nearest neighbor distance, a = 0.8 mm. A four-fold rotational symmetry is apparent; however, the structure is still incommensurate. The 2D fast Fourier transform (FFT) spectrum of the fabricated structure is shown in Fig. 1(b), where several reciprocal vectors are assigned. Unlike periodic and random HA structures, the Fourier transform spectrum of the fabricated 2D Fibonacci structure contains two types of prominent reciprocal vectors: (i) two sets of four vectors, G₁ and G₂, respectively, where G₁ = 2π/a( ± 1:0) and 2π/a(0: ± 1), and G₂ = 2π/a( ± 1: ± 1), which correspond with the underlying grid holes on which the Fibonacci structure is based, and (ii) three additional sets of eight vectors, F₁ to F₃, which belong to the Fibonacci structure factor. The vectors G₁ and G₂ originate from the nearest neighbor and next nearest neighbor holes of the form ( ± a:0), (0: ± a), and diagonal ( ± a: ± a), respectively. On the other hand, the vectors F₁ to F₃ mostly originate from the Fibonacci sequences in the two perpendicular directions (x,y). It is clear that the four-fold rotational symmetry seen in Fig. 1(a) is maintained in the reciprocal space for the G vectors; however the eight-fold rotational symmetry of the F-vectors seen in Fig. 1(b) is not trivial.

 

Fig. 1 Hole array geometries and corresponding FFT spectra. (a) Fibonacci hole array and (c) random hole array on a square grid array; aperture diameter d = 0.44 mm, nearest neighbor spacing a = 0.8 mm with 173 holes. The structure factor of the Fibonacci (b) and random-hole-on-grid array (d) calculated using a 2D FFT. The Fibonacci-related reciprocal vectors, (F)⁽ⁱ⁾ and grid-related reciprocal vectors, (G)⁽ⁱ⁾ that correspond to Miller indices (1,0) and (1,1), respectively are denoted.

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We used THz time-domain spectroscopy (THz-TDS) to measure the optical transmission spectra, t(ω) of the perforated metal films, where the THz frequency ν = ω/2π. Photoconductive devices were utilized for both emission and coherent detection of the THz electric field. Two off-axis paraboloidal mirrors were used to collect and collimate the THz radiation beam from the emitter and focus the beam to the detector. The samples were attached to a solid metal plate with a 5 cm x 5 cm opening that is significantly larger than the THz beam size and placed in the path of the collimated THz beam. The detected transient photocurrent, PC(τ) was recorded as a function of the translation stage path that determined the time delay, τ between the ‘pump’ beam that hits the emitter and the ‘probe’ beam that arrives at the detector. PC(τ) was subsequently Fourier transformed and normalized to a reference transmission, yielding both the electric field transmission magnitude and phase, t(ω) in the range ~0.1 THz to 0.5 THz. The resulting Fourier transformed data may be described by the relation:

In this expression, E_incident and E_transmitted are the incident and transmitted THz electric fields, respectively, and |t(ω)| and φ(ω) are the magnitude and phase of the amplitude transmission coefficient, respectively. The THz-TDS technique is unique in that it allows for a direct measurement of the transient THz electric field transmitted through the structures, yielding both amplitude and phase information. From these spectra both real and imaginary components of the dielectric response, ε(ω) can be directly obtained without the need for Kramers-Kronig transformations, where somewhat arbitrary assumptions about asymptotic behavior are typically made.

3. Experimental results and discussion

It has been shown that the optical transmission from HA structures is composed of a continuum spectrum related to the transmission through the individual holes, superimposed by Fano type resonances of constructive (C) and destructive (D) interferences [16, 17] that gives transmission maxima and minima, respectively, which correspond to the prominent reciprocal vectors in the HA structure factor in k-space [4, 18]. Therefore we expect five resonance features to dominate the optical transmission spectrum, t(ω) of the 2D Fibonacci structure shown in Fig. 1(a); namely two resonances due to the two G-vector sets, and three resonances that belong to the three F-vector sets (Fig. 1(b)). The dispersion relation for the G-vectors should be close to ω = ck/n, where ω is the light angular frequency, k is the wave-vector amplitude, and c is the speed of light; in this relation we assume that the refraction index n~1 for the SPP excitations. According to this dispersion relation and the G-vectors in reciprocal space (Fig. 1(b)) we expect in the t(ω) spectrum two G-related grid-resonances at frequencies f(G₁) = 0.375 THz and f(G₂) = 0.530 THz.

Figure 2(a) shows the transmission spectrum from the Fibonacci HA structure presented in Fig. 1(a), but from an array that contains 300 apertures. Several Fano-type resonances are apparent in t(ω). We can clearly identify a dominant transmission minimum, D at f(G₁); whereas the transmission minimum that corresponds to G₂ (namely f(G₂)) is outside the experimental spectral interval of our measurements. It is interesting that the D(G₁) follows the simple dispersion relation with k = G₁, whereas the transmission maximum, C(G₁) does not follow it. Using the same type of dispersion relation we expect Fibonacci-related transmission minima D’s to appear at frequencies f(F₁)≈0.21 THz, f(F₂)≈0.28 THz, and f(F₃)≈0.34 THz. As is seen in Fig. 2(a) transmission minima do appear in the transmission spectrum at those frequencies (denoted by arrows), but they are much weaker than the G₁-related minima.

 

Fig. 2 THz time-domain spectroscopy studies of HA structures described in Fig. 1. (a) THz transmission spectrum, t_F(ω) of the 2D Fibonacci HA (300 holes) and its corresponding t_RG(ω) spectrum of random-on-grid HA (RG) array. The transmission minima F₁-F₃ associated with the Fibonacci structure, and those associated with G₁ related to the underlying grid are assigned. (b) The spectrum of the transmission ratio t_F(ω)/t_RG(ω). (c) THz transmission spectrum t_F(ω) of Fibonacci HA structures fabricated with the same nearest neighbor distance a = 0.8 mm and number of holes (800 holes), but different hole diameters d as denoted. The Fibonacci-related transmission minimum frequencies are indicated by dashed vertical lines. (d) Real and imaginary parts of the effective dielectric constant of the Fibonacci HA structure with d = 0.68 mm obtained from the transmission amplitude and phase spectra. The transmission minima are denoted.

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For our studies we also fabricated corresponding HA structures in which the holes are randomly generated on a square grid having the same number of holes and nearest neighbor distance, a as the 2D Fibonacci HA structures; these structures are denoted ‘random on grid’, or RG. Figure 1(c) shows an example of such an RG aperture structure in which 173 holes are randomly placed on a square grid with 0.8 mm spacing. The 2D FFT spectrum of this RG aperture structure contains only two sets of the G reciprocal vectors (G₁ and G₂, respectively) that arise from the grid structure, with no other prominent reciprocal vector visible; this confirms that the aperture distribution in this sample is indeed random.

To better reveal the F-resonances in the transmission spectrum we divided the transmission spectrum, t_F(ω) of the Fibonacci structure by t(ω) of the RG structure, t_RG(ω), to obtain the ratio spectrum t_F/t_RG, as shown in Fig. 2(b). The transmission minima related with the G₁ and F₂ reciprocal vectors are still seen in the t_F/t_RG spectrum. However two additional F-resonances, at D-frequencies f(F₁)≈0.21 THz and f(F₃)≈0.34 THz, which could not be discerned in t_F(ω) (Fig. 2(a)) are also clearly identified here. Also an additional D-feature at ~0.15 THz that also originates from a specific F-vector in the Fibonacci structure factor (Fig. 1(b)), which could not be identified in Fig. 2(a), can be clearly identified in Fig. 2(b). We also note that the ratio t_F/t_RG fluctuates around the value, t_F/t_RG = 1 (solid line in Fig. 2(b)). Agrawal et al. [5] postulated that there exists a sum rule in the transmission spectra of HA structures related with the number of holes, where the integrated transmission strength remains constant regardless of the aperture arrangement on the metal film surface. This sum rules can be vividly seen in the fluctuation of the t_F/t_RG spectrum around the value t_F/t_RG = 1, thus confirming the HA transmission sum rule.

Another method of enhancing the F-resonances in the EOT spectrum is to change the hole diameter, d [4]. Figure 2(c) shows t(ω) of Fibonacci HA structures having the same nearest neighbor distance, a but different d, ranging from 0.3 mm to 0.68 mm. These diameters correspond to the aperture waveguide cut-off frequency, f_c ranging from 0.58 THz to 0.26 THz, respectively [18]. The variation of d strengthens the F-resonances at frequencies close to f_c, since the Fano resonance coupling between a discrete EOT feature and the transmission continuum due to the individual holes becomes stronger when the discrete frequency is close to f_c [16, 17]. In Fig. 2(d) we calculated the real and imaginary components of the dielectric constant, ε(ω) of the Fibonacci HA from the transmission amplitude and phase (not shown) given in Fig. 2(a) (d = 0.68 mm). We have previously shown that ε(ω) can be well described by a superposition of a non-resonant (continuous) and resonant (discrete) components given by the formula proposed by Agrawal et al. [18]. In comparison to ε(ω) of periodic HAs [18], ε(ω) of Fibonacci HAs is much more complicated because of the multitude of resonances associated with the G-vectors as well as the F-vectors. We note the enhanced intensity of the F₂ resonance close to f_c of the individual holes. We also note that the F-resonance at 0.15 THz more clearly appears in this HA structure in agreement with our finding in the t_F/t_RG spectrum of Fig. 2(b).

The localization properties of the SPP excitations in the Fibonacci HA structure can explain the weak F-resonances in the EOT spectrum. To show this, we compare the F-resonance with the G-resonance intensities in Fibonacci HA structures for different numbers of holes, N, while keeping constant the lattice constant a = 0.8 mm and diameter d = 0.44 mm; and periodic HA having the same number of holes, d, and a. Figure 3(a) shows t(ω) of a Fibonacci HA structure with 800 holes; the characteristic G- and F-resonances are clearly seen. The spectrum t(ω) of the corresponding random HA having the same N and d is also shown in Fig. 3(a). The F₂-resonance is chosen for studying the resonance intensity, I, defined as the area under the transmission of the F₂ peak with respect to the transmission of the corresponding random HA structure. The F₂ resonance intensities defined in this way are plotted vs. N in Fig. 3(b). We also calculated the resonance peak intensity from the FT intensity associated with the corresponding reciprocal vectors in the structure factor; Fig. 3(b) inset demonstrates this calculation for the FT F₂ intensity. We found that the intensity of both resonances increases nonlinearly with N. In addition it is clear that the experimental intensities are smaller than the calculated intensities from the FT. For the G₁-resonance we need a multiplication factor of ~4 to match the experimental to the calculated intensities. A factor of 2 is justified because the polarized THz beam that subtends only two G-vectors out of possible four; another factor of two shows that in reality there are some SPP losses in the HA structure. However for the F₂ resonance we need a factor of 24(!) to match the theoretical calculation. We therefore conclude that the F-resonances are much weaker than the G-resonances in the Fibonacci HA structure, and do not fit the FT calculation even relative to the G₁ resonance. This shows that the F-resonances are inherently weak, and it therefore requires a more sophisticated explanation.

 

Fig. 3 The dependence of THz transmission resonance strength on the number of holes for the Fibonacci and periodic HA structures. (a) THz transmission spectrum of a typical Fibonacci HA structure, and (c) periodic HA lattice, compared to the spectrum of the corresponding random HAs (all structures have 800 holes). The Fibonacci resonance F₂ and periodic resonance G₁ strengths are shaded. The integration of (b) F₂ and (d) G₂ resonance strengths is plotted versus the number of holes in the structure. The corresponding strengths in the structure factor in k-space are calculated (blue symbols). (b) The inset shows an intensity profile which is formed by cutting the reciprocal lattice shown in Fig. 1(b) along the line that connects the origin with the F₂ reciprocal vector.

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For this we fabricated a number of periodic HAs with different numbers of holes, but the same periodicity a = 0.8 mm and hole diameter d = 0.44 mm. As in the case of Fibonacci HAs, we extracted the intensity of transmission resonance by integrating the area limited by the G₁ peak envelope and the corresponding random HA, as shown in Fig. 3(c). The periodic G₁ transmission intensity dependence on the number of holes is shown in Fig. 3(d). We also calculated the transmission strength of the peak G₁ follow the same method as described above for the Fibonacci HA. We notice that the FFT spectrum was calculated from arrays of points located in the centers of holes. Therefore the structure factor in Figs. 1(b) and 1(d) does not contain the contribution of the continuum spectrum, which is due to the transmission through individual holes in the experiment. This is the reason that we compare the theory with the experimental transmission strength extracted from the area difference between the transmission spectra of the Fibonacci (or periodic) HAs and their corresponding random HAs. Interestingly, the experimental data shown in Figs. 3(c) and 3(d) follow the calculation trends very well. We note that the periodic resonance strength substantially increases with the number of holes, N, for relatively small values of N; but start to saturate for values of N that are smaller than the value of N for which saturation take place with the Fibonacci F₂ transmission strength.

It is clear that the experimental periodic G₁ transmission strength is nearly two orders of magnitude stronger than the Fibonacci F₂ transmission strength, and the Fibonacci experimental strength is an order of magnitude smaller than its calculation value. A possible reason that might lead to this large difference between the G- and F-resonant strength is the proximity of the different resonances to the cut-off frequency, f_c. While the G₁ resonance frequency at 0.375 THz is close to f_c ~0.4 THz, the F₂ resonance frequency at 0.27 THz is further away (see Fig. 2(c)). The strongest transmission strength for the F₂ resonance occurs for HA structures having d = 0.68 mm, which is ~5 times stronger than it is for structures with d = 0.44 mm. Nevertheless, the transmission strength F₂ at d = 0.68 mm is still an order of magnitude weaker than the transmission strength of the G₁ resonance at d = 0.44 mm. We therefore conclude that the G-resonances are much stronger than the F-resonances even if the proximity to the cut-off frequency f_c is taken into account. This is in contrast with the behavior of hot electromagnetic spot intensity in Fibonacci structures using nano-particle arrays [11].

We can understand the weak transmission of F-resonances in the Fibonacci HA by introducing a correlation length, R to the SPP waves. The transmission resonances are formed by SPP waves that are launched from the individual apertures and subsequently interfere. With no attenuation, the interference would be as strong as the intensity of the reciprocal vectors in the Fourier space. However attenuation may reduce the interference strength because the electric field amplitude decreases with the distance. In solid state physics this effect has been studied by introducing a ‘correlation length’, R to the Bloch wave function, Ψ^B_k(r)~exp(-ik•r); so that the ‘attenuated’ Bloch function in real materials is Ψ^at_k(r)~exp(-ik•r)exp(-r/R), where the second exponential decay term describes the amplitude attenuation with the distance, r. To see the effect of the ‘attenuation term’ on the strength of the reciprocal vectors in k-space due to smaller interference, we calculated the structure factor of a periodic square HA using Ψ^at_k(r) rather than Ψ^B_k(r), as depicted in Fig. 4 . It is clearly seen that the intensity of the FT peak above the background diminishes substantially at small R (Fig. 4(a)). To further study this effect we calculated the integrated intensity of the G₁ transmission resonance in a periodic HA as a function of R (Fig. 4(b)). It is seen that the integrated intensity of the transmission G₁-resonance increases quadratic with R.

 

Fig. 4 Calculation of the correlation length, R. (a) DFT spectra calculated at two different correlation lengths around (1,0) peak of a periodic structure with a period of 1 mm. (b) The integration of (1,0) peaks as a function of R in a log-log scale. The fit line shows that the DFT intensity increases quadratic with R. (c) Transmission spectra of a Fibonacci HA with d = 0.68 mm hole diameter (dash blue line) and a periodic HA with d = 0.4 mm (red line). The blue solid line shows a Fibonacci spectrum normalized to the same hole area with the periodic HA. The dash dark line is a cut-off line above which we calculate the transmission strength of the Fibonacci F₂ and periodic G₁ peaks. (d) Hole-to-hole correlation function calculation, g of periodic, RG and Fibonacci HAs with 800 holes and the nearest neighbor distance of 0.8 mm.

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We therefore propose that the weak F-resonances in the 2D Fibonacci structure are related to smaller SPP correlation length in this structure. In fact we can extract the correlation length ratio, R_G/R_F of the SPP waves in the periodic and Fibonacci HA structures, from the obtained G- and F-resonances in the transmission spectrum of the respective structure. Figure 4(c) shows the F- and G- resonances in the two respective HA structures at optimal conditions (namely when the resonance frequency is closest to the cut-off frequency, f_c). It is clear that the G-resonance intensity is larger than that of F-resonance by a factor of ~18. As a consequence of the quadratic dependence of the resonance intensity with R (Fig. 4(b)) we conclude that the correlation length ratio, R_G/R_F ≈4.2. The reason for the smaller SPP correlation length in the Fibonacci structure may be related to the underlying grid on which the Fibonacci HA is fabricated, where the interference between the SPP waves launched from the Fibonacci-related aperture is disturbed by the waves launched from the grid-related apertures.

For completeness, we also analyzed our data using the concept of hole-to-hole correlation functions, as was recently suggested by Przybilla et al. [19]. Figure 4(d) shows the hole-to-hole correlation function calculation of three different HAs with 800 holes. The spectra are discretely distributed at certain distances. We found that these structures give the same pair distances, which are defined by the lattice vectors of the grid. We therefore conclude that the concept of ‘local’ and ‘global’ disorder, as introduced in ref [19]. cannot be applied to aperiodic structures.

4. Conclusion

We studied anomalous transmission in the range of 0.1 THz to 0.5 THz through two-dimensional deterministic Fibonacci HA structures. We found that multiple Fano-type transmission resonances are also formed in this complex aperiodic structure at frequencies that closely match prominent reciprocal vectors in their structure factor in k space. However the Fibonacci transmission resonances are very weak compared to the resonances from periodic HA structures. We systematically studied the transmission strength of the clearest Fibonacci-related resonance F₂ and compare it with the G₁ resonance of the corresponding periodic HA structure, as function of the number of holes in the HA structures. We found that the transmission strength of the Fibonacci-related resonance is more than an order of magnitude weaker than that in the periodic structure. We explained the weak resonance in the Fibonacci HA structures by attributing it to a smaller SPP correlation length.