1. Introduction

The phenomenon of resonant transmission of electromagnetic radiation through flat or corrugated thin metal films has been a topic of research interest for many decades [1, 2]. Resonant light transmission through these classically opaque metal films has been attributed to excitation of coupled surface plasmon modes on both sides of the thin metal films. However recently the phenomenon of extraordinary optical transmission (EOT) in metallic films perforated with arrays of subwavelength apertures and their associated dielectric response have attracted significant interest [3, 4]. Much of this interest arises from the possibility of subwavelength field localization, creating opportunities in a variety of nanophotonics applications [5, 6]. It has been demonstrated that the EOT phenomenon is inherently related to the propagation and interference of surface plasmon-polaritons (SPPs) along metal-dielectric interfaces [7–9]. The optical properties of these plasmonic media, including frequency response, resonance linewidth, and enhanced transmission efficiency are dependent on the specific geometrical parameters of the structure, which may lead to the existence of well-defined reciprocal vectors (RVs) in k-space (structure factor). Thus, it is important to fully understand the EOT properties and dielectric function response of aperture arrays having differing types of orientation order, and, correspondingly diverse structure factors.

It was recently demonstrated that the sharp EOT resonances associated with the array structure factor is in fact analogous to the traditional x-ray diffraction from regular patterns such as crystals. Crystalline structures possess both long range order (LRO) and short range order (SRO), forming a series of well-defined RVs in the reciprocal space in the form of a Bravais lattice. The RVs result in sharp Bragg peaks in the x-ray diffraction pattern, just as they correspond to sharp EOT resonances from the associated aperture arrays. In contrast to crystals, quasicrystals are made of building block (BB) units that are arranged in a nonperiodic but highly ordered pattern, and exhibit LRO in the presence of some SRO [10]. Nevertheless their structure factor also contains well-defined RVs, which consequently lead to sharp EOT resonances [11, 12]. However, when the BB units form structures that lack LRO, the amplitude of the Bragg peaks in x-ray diffraction diminish, and, correspondingly, we expect EOT resonances of lesser strength. However, some order on local length scales may still exist in such structures, and this results in “diffuse” scattering due to the diffuse RVs (RV_D) in the structure factor. Such “diffuse” scattering is common in diffraction studies of liquid and amorphous materials that lack LRO [13, 14].

In this work, we study the terahertz (THz) transmission properties and dielectric response of metallic films perforated with subwavelength aperture arrays that possess SRO but lack LRO (hereafter referred to as ‘SRO structures’) and compare their optical properties with those of random aperture arrays. Surprisingly, we find that EOT resonances are still evident from these SRO structures. Their THz dielectric response is characterized by the superposition of a broad principal resonance due to randomization in the orientation of the BB units, which can be easily controlled by limiting the degree of orientation randomization, and discrete resonances that are due to the presence of a virtual lattice based on the BB units.

2. Experimental details

A schematic of the design procedure for the SRO structures studied here is shown in Fig. 1(a) [15]. The BB unit is a square of side length a, which is placed at the vertices of a virtual square lattice of side length b (here b=2a). This procedure is then repeated across the entire film surface. A subset of the designed aperture array is shown in Fig. 1(b). The individual BB units in the SRO structure are rotated randomly by angle θ that can have any value between 0 to π/2. The SRO structure representation in k-space, shown in Fig. 1(c), exhibits two significant features. The first feature is well-defined RVs arranged in a square Bravais lattice, similar to those in regular plasmonic lattices. However, the RVs here are associated with a virtual lattice in real space having lattice constant b. The second feature is the concentric diffused band at RV_D that is related to the random rotations of the square BB units. We fabricated the SRO structures on 75 µm thick free-standing stainless steel metallic films, with varying values of a, b, orientation angle, and aperture diameter, D. The random arrays in our studies were designed to have the same average spacing, a, and D (hence, the same fractional aperture area) as the corresponding SRO structures.

 

Fig. 1. (a). Schematic of the design procedure for fabricating a SRO structure. Randomly rotated squares of side length a are placed at the vertices of a virtual lattice having lattice constant b. (b) A 5×5 unit cell subset of an actual SRO structure, where the black circles correspond to the apertures. (c) The reciprocal space representation of the SRO aperture in (b), where the individual reciprocal vectors, RVs, are assigned.

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The optical properties of the various aperture arrays were studied using a conventional THz time-domain spectroscopy (THz-TDS) setup [16], where photoconductive devices were used for both THz generation and coherent detection. The films were placed at normal incidence at the center of two off-axis parabolic mirrors that were used to collect, collimate, and refocus the THz beam from the emitter to the detector. Reference transmission spectra were measured without the sample for normalization purposes. The detected transient photocurrent was Fourier transformed and normalized to the reference transmission spectra, yielding the amplitude transmittance spectra in the range ~0.05 to 0.5 THz. The THz-TDS technique is unique in that it allows for a direct measurement of the transient THz electric field transmitted through the structures. Thus, we are able to determine independently both the magnitude and phase of the amplitude transmission coefficient, t(ν), using the relation

In this expression, E_incident and E_transmitted are the incident and transmitted THz fields, respectively; |t(ν)| and φ(ν) are the magnitude and phase of the amplitude transmission coefficient, respectively; and ν is the THz frequency. |t(ν)| (labeled simply, t(ν)) and φ(ν) spectra are used in our studies to obtain both the real and imaginary components of the effective dielectric function spectrum, ε(ω), of the array medium, without resorting to traditional Kramers-Kronig transformations.

3. Experimental results and discussion

In Fig. 2(a), we show the THz transmission spectra, t(ν) for the SRO structure shown in Fig. 1(b), with a=1 mm, b=2 mm, and two different aperture diameters D, along with the spectra for the corresponding random aperture arrays. t(ν) through the random structures is characterized by a broad continuum resulting from the individual, uncorrelated apertures. However, the transmission through the SRO structures is characterized by a principal peak (labeled RV_D) centered at a frequency ν_D~0.3 THz. This frequency corresponds to the circular “diffused” band observed in the reciprocal space representation at RV_D (Fig. 1(c)) corresponding to k=k_D, where ν_D=ck_D/2π (c is the speed of light in vacuum). Superimposed on this principal resonance are discrete resonances (labeled RV_i) that occur at frequencies, ω_i (=2πν_i) directly corresponding to the lattice RVs in the reciprocal space that occur at k_i (ν_i=ck_i/2π). The presence of these sharper resonances in t(ν) is surprising because there are no real apertures in the SRO structure that correspond to a square Bravais lattice with lattice constant b. It is important to note that, in contrast to periodic plasmonic lattices where only three discrete bands have been observed, we are able to simultaneously observe here five discrete resonances. We attribute this to the presence of the ‘diffuse’ band, and more specifically to the resonant interaction between this band and the discrete resonances in the SRO structures.

From the spectra in Fig. 2(a), we directly obtained ε(ω) for the SRO structure and its corresponding random structure, as shown in Fig. 2(b). ε(ω) for the random structure exhibits the dispersion properties of a plasma, similar to that of a metallic medium, but with an effective plasma frequency approximately equal to the cutoff frequency of a cylindrical aperture. However, ε(ω) of the SRO structure is modulated in the vicinity of both the principal and discrete resonant frequencies. It is easy to verify that, in fact, this modulation is superposed on an envelope having a ‘plasma-like’ response. In fact, it is very similar to that of the corresponding random array medium.

We approximated the dielectric properties of perforated metallic films to exhibit a complex ‘effective’ dielectric response that can be described using an ‘effective plasma frequency’ model. The dielectric constant, ε_m(ω) of the SRO structures described here is expressed as:

Here, ε_∞ is the high-frequency dielectric constant; $\tilde{\omega }$_p is the ‘effective’ plasma frequency related with the individual uncorrelated apertures; ω_Tj are the resonant frequencies that correspond to the discrete peaks observed in t(ν); ω_Lj is an effective ‘longitudinal frequency’ for the resonant modes; and γ_j are the respective damping constants. ω_l and ω_h in the last term of Eq. (1) are the lower and upper frequencies, respectively, of the broad principal peak in the EOT spectrum. Thus Eq. (2) incorporates the properties of an effective plasma (first term), discrete resonant modes (second term) and a principal broad resonance (third term) due to RV_D.

 

Fig. 2. (a). The THz transmission spectra through SRO structures with two different aperture diameters D=500 µm (red curve) and D=600 µm (blue curve), along with spectra for the corresponding random aperture structures (dotted lines). Resonances are assigned corresponding to RVs in reciprocal space shown in Fig. 1(c). (b) Spectra of the real and imaginary components of the effective dielectric constant obtained from (a) for SRO structure with D=600 µm (red curves). The modeled ε(ω) of the effective medium (blue line) is based on a fit using Eq. (2) with parameters given in the text.

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The effective plasma term arises from the transmission through the uncorrelated apertures. Using values of ε_∞=80, $\tilde{\omega }$_p/(2π)=0.29 THz (equal to the cutoff frequency for an aperture diameter of D=600 µm), and γ=0.76 rad/psec, this term fits well the broad band t(ν) spectrum of the random array. In contrast, the ‘resonant modes’ term in Eq. (2) is related to the structure factor of the underlying Bravais lattice with lattice constant b, where ω_Tj are the resonant frequencies. Therefore ω_Tj in Eq. (2) corresponds to the Wood’s anomalies, which are given in the standard SPP model [3] by

where m and n are integers.

 

Fig. 3. (a). The reciprocal space representation of a SRO aperture with orientation angles limited by θ_max=π/6; the RVs are as in Fig. 1(b). The localization of the diffuse band at RV_D is clearly evident. (b) THz transmission spectra through such SRO structures with a=0.75 mm and D=400 µm (red curve) and a=1 mm and D=500 µm (blue curve).

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These frequencies, ν_Tj=ω_Tj/2π, occur at ~0.15 THz, ~0.21 THz, ~0.30 THz, ~0.33 THz, and ~0.42 THz that correspond to indices (m, n)=(±1,0), (±1,±1), (±2,0), (±1,±2)=(±2,±1), and (±2,±2), respectively. The last term in Eq. (1) that describes the principal broad resonance is a continuous integral of quasi-resonant modes between the lower and upper frequencies, ω_l (=2π∙0.15 THz) and ω_h (=2π∙0.45 THz), respectively. This corresponds to the broad diffuse band at RV_D about k_D, with beginning and end k-vectors at k_l and k_h, where ω_i=ck_i. The oscillator strength, which is proportional to ω_L ($\tilde{\omega }$_T) in the last term, has a Gaussian profile distribution with center frequency at ν_D=0.30 THz, and a linewidth of 0.09 THz. This is determined by analyzing the cross-sectional profile of the broad RV_D in the SRO structure factor (Fig. 1(c)). The fit obtained for the two experimental ε(ω) components using Eq. (2) with the parameters given above is shown in Figs. 2(b) (blue lines). The good agreement between theory and experiment for the SRO arrays validates the approach used here for ε(ω) and emphasizes the relation between the structure factor and the optical response of the perforated films.

We also found that the transmittance amplitude and linewidth of the principal resonance in t(ν) of the SRO structures can be modified by changing the degree of randomization in the orientation of the BB units. When the maximum random rotation angle, θ, of the BB units in such structures is limited to values smaller than π/2, then the diffused bands in reciprocal space become localized, which manifests itself as symmetrical arcs along the circumference at k=k_D. This is evident in Fig. 3(a), where we show the reciprocal space representation of a SRO structure in which θ was limited to values between 0 to π/6. The localized structure around k_D now exhibits six-fold symmetry, corresponding to the symmetry of the allowed angular distribution in θ. In Fig. 3(b) we show t(ν) spectra of two SRO structures having θ_max=π/6 with two different aperture diameters D and lattice constants a. It is seen that the principal resonance due to the diffused streaks in the reciprocal space is narrower here compared to the corresponding resonances in Fig. 2(a). This is in agreement with the localization of the diffuse bands in k-space.

4. Conclusion

In conclusion, we measured the dielectric response and transmission through metallic films perforated with subwavelength aperture arrays that possess SRO, but lack LRO. We demonstrate that extraordinary transmission is also possible in such structures, despite the absence of LRO. We show that light transmission through aperture arrays can be treated in a manner similar to x-ray diffraction, which was extensively studied in crystalline and amorphous materials. When the array pattern exhibits LRO, sharp spots in the diffraction spectra result in sharp transmission resonances. On the other hand, when the array exhibits only SRO, the diffraction spectra have associated diffused streaks that result in broader transmission resonances. Further we demonstrate that the optical response of the SRO structures can be described as a superposition of a broad continuum associated with the individual aperture response, a principal resonance associated with the diffuse bands in k-space, and discrete resonances associated with the well-defined RVs in the structure factor.