1. Introduction

For more than two decades, the extraordinary optical transmission (EOT) of electromagnetic radiation through subwavelength apertures has been an important research topic [1]. Since the initial report [2], the EOT effect has been studied at frequencies from optical [3,4] to microwave [5,6], and has been considered among the hallmark examples of the significant role of surface waves in light-matter interactions. Nearly all previous studies have involved free-space coupling of waves to the subwavelength structures (or coupling from a uniform dielectric medium), such that both the incident wave and the far-field transmitted wave are transverse electromagnetic (TEM) waves. One might expect different behavior if the subwavelength apertures are placed inside a waveguide, since both input and output fields would be restricted to the lone (or few) allowed waveguide mode(s), which need not necessarily be TEM. Some previous works have demonstrated EOT via subwavelength holes inside hollow metal waveguides [7–9], but to our knowledge, the mode selectivity of the EOT effect – that is, the extent to which the input mode affects the transmission spectrum – has not been investigated. A comparison of TEM and non-TEM mode excitation can illuminate the role of surface waves in mediating the transmission through subwavelength hole arrays.

Here, we use broadband terahertz radiation to study the EOT effect for an array of subwavelength holes in a metal screen that has been embedded inside a metal parallel-plate waveguide (PPWG). Unlike previous reports [7–9], our use of a PPWG allows us to excite the subwavelength hole with a TEM mode (which is not possible in hollow rectangular or circular waveguides) [10], as well as non-TEM modes [11,12], permitting a direct comparison between these two distinct situations for the first time. We also highlight the comparison to the normal-incidence transmission through a similar hole array in free-space. We observe that the conventional explanation which relies on the excitation of spoof surface plasmon polaritons (SPPs) [3,13,14] is not descriptive in all cases, particularly where the incident wave has a longitudinal field component (i.e., where it is not TEM). Previous work has demonstrated EOT without the excitation of SPPs in a circular waveguide [8], but our use of the PPWG allows us to directly compare TEM to non-TEM modes experimentally. We utilize another theoretical framework based on electromagnetic mode-matching and circuit theory [15–17] which more accurately predicts the resonant transmission frequencies for both the TEM and non-TEM cases.

2. Design and theory

Our experimental design is depicted in Fig. 1(a). The waveguide is excited using quasi-optic coupling of broadband THz pulses in a conventional THz time-domain spectroscopy system with photoconductive antennas. Each of two PPWG sections is fabricated out of aluminum with length L = 10 mm, and a variable plate separation b. We show results for selected values of b = 0.75, 1.00, and 1.25 mm. The metal screens are made from stainless steel foil of thickness, t = 25 μm (corresponding to roughly λ/40 where λ is the wavelength corresponding to the fundamental transmission resonance), and have a one-dimensional (1D) array of holes with uniform spacing between holes of either a = 0.75 mm or 1.0 mm. We explore two different hole diameters: the larger diameter holes (380 μm) are fabricated by conventional machining, while the smaller diameter holes (300 μm) are fabricated by laser drilling. The width of the waveguide and the metal foil in the lateral (unconfined) direction, as well as the length of the 1D hole array, are all much larger than the excitation beam diameter, thereby ensuring no edge effects. The screen is sandwiched between the two waveguide sections, as illustrated in Fig. 1(b), forming an effective continuous waveguide with the EOT screen located half-way along the length. For comparison, we also fabricate a two-dimensional (2D) array of holes in stainless steel foil of the same thickness and hole diameter of 380 μm, which we characterize by normal-incidence transmission spectroscopy in free-space (i.e., no waveguide).

 

Fig. 1 Panel (a) shows a 3D view of the PPWG (WG1 and WG2) with an array of holes in the middle, where the waveguide plates are pictured translucent to see the screen, b is distance between the two plates, a is the distance between two holes, d is the diameter of the holes, and L is the length of the waveguide section. Panel (b) shows a side view of the experimental configuration where the THz wave is coupled into WG1 from the left and the screen with holes of thickness t is sandwiched in the middle between WG1 and WG2.

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To assess the influence of the incident mode on the EOT spectrum, we compare the transmission through a 2D array in free-space to that of a 1D array inside a PPWG for both TEM and TE₁ input waveguide modes, displayed in Fig. 2. In this comparison, the 2D array has holes in a square lattice with a spacing of 1.0 mm in both directions. The 1D array inside the PPWG also has a 1.0 mm spacing. For the 2D array in free-space, measurements with vertical and horizontal polarizations produce identical results due to the four-fold rotational symmetry of the array as shown in Fig. 2(a). For the 1D array inside the PPWG, shown in Fig. 2(b), we observe that excitation with the TEM waveguide mode (polarization perpendicular to the linear hole array) produces the same resonant frequency as that of the free-space measurements. However, for TE₁ input mode, where the electric field polarization axis is parallel to the 1D hole array, shown in Fig. 2(c), the resonance clearly shifts to a higher frequency. If the EOT physics only depended on the electric field polarization, we would expect the same results for all three of these measurements. Since the amplitude variation of the mode across the hole is small in both cases, we conclude that the presence of a longitudinal magnetic field component (as in the TE₁ waveguide mode) has a significant influence on the EOT resonant frequency.

 

Fig. 2 Power transmission at normal incidence for (top) a 2D array of holes in free-space illuminated by vertical polarization (black) or horizontal polarization (gray), (middle) a 1D array of holes inside a PPWG in TEM mode (blue), (bottom) a 1D array of holes inside a PPWG in TE₁ mode (red). The dashed purple and oranges lines (overlapped) show prediction of RWA minimum and SPP maximum, respectively. This minimum value agrees well with 2D array in free-space and 1D array in the TEM PPWG. Clearly the resonant frequency is shifted for the 1D array in the TE₁ PPWG.

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In the most commonly discussed theoretical descriptions of EOT, the transmission maxima are determined [13] by the excitation of SPPs at the dielectric-metal interface:

We compare these predictions to our situation with the subwavelength hole array situated inside a waveguide. The values of the SPP maxima and RWA minima predicted by Eqs. (1) and (2), shown as dashed lines in Fig. 2, nearly coincide since in the limit that ε_m >> ε_d the dielectric prefactors in Eqs. (1) and (2) are approximately equal. In earlier works at other frequencies, these predictions have been used with some success, although often they exhibit an unexplained offset from the measured transmission features [1,3]. In our case, there is good agreement between the SPP prediction of the transmission minima and the experimental results for both the 2D free-space array and the 1D hole array when excited by the TEM waveguide mode. However, our data demonstrate that the free-space SPP prediction does not apply when the hole array is excited by the TE₁ waveguide mode (see Fig. 2(c)). We emphasize that, in the comparison between the TEM and TE₁ modes, the electric field polarization lies entirely in the plane of the EOT screen’s surface in both cases (i.e., the propagating wave is incident normally, and the electric field vector is transverse, in all of the cases discussed here).

To accurately predict the transmission resonances for both TEM and non-TEM modes, we turn to an alternate description originating from frequency selective surface (FSS) analysis applied to EOT [17]. Instead of invoking SPPs, one can apply a circuit theory perspective, in which transmission is mediated by impedance matching through the subwavelength aperture. As discussed in [17,21], the structure of a 2D array of subwavelength holes in free-space can be divided into unit cells, each containing one aperture, with side lengths corresponding to the periodicity of the array. Then, with the appropriate boundary conditions, each unit cell can be viewed as a fictitious quasi-rectangular waveguide containing a screen with a single aperture. The aperture is a discontinuity in the waveguide that permits excitation of higher order evanescent modes. The transmission process is mediated by matching the impedance of the quasi-rectangular waveguide to that of the aperture. Since the aperture is subwavelength in size and therefore has a large admittance [7], this matching criterion requires that the waveguide mode also have a large admittance. As$f\to f_{c_m}$ (where $f_c$is the cutoff frequency of the corresponding m,n mode), TM waveguide modes have a diverging admittance whereas the admittance of TE waveguide modes vanishes. Thus, the evanescent TM modes facilitate impedance matching and exhibit a transmission maximum as the frequency approaches $f_{c_{mn}}$and a minimum at$f_{c_{mn}}$.

We apply this analysis in a new context to the PPWG system, where we can directly compare TEM and TE₁ waveguide modes to the case of a 2D array in free-space. Each unit cell of our subwavelength hole array is viewed as a fictitious rectangular waveguide with dimensions determined in the x-direction by the hole spacing and in the y-direction by the plate separation. Based on the symmetry of a hole centered in the waveguide, for TEM input excitation, all TM_2p,2q (with m = 2p and n = 2q) modes show this divergent nature of the admittance as cutoff is approached. In contrast, for TE₁ input excitation, it is the TM_{2p + 1,2q} (with m = 2p + 1 and n = 2q) modes [7]. The relevant cutoff frequencies are given by the standard results for a rectangular waveguide, where in our axis labeling, b (the plate separation) corresponds to the mode index m, and a (the hole spacing) corresponds to the mode index n:

The relevant mode facilitating transmission is the lowest order evanescent TM waveguide mode, which for TEM is the TM₂₀ mode, and for TE₁ is the TM₁₂ mode. It is important to note that, for the TEM waveguide mode, this cutoff relationship in Eq. (3) is the same as Eq. (2), which is the transmission minimum predicted by RWA. Thus, for the TEM waveguide mode, this impedance matching description gives the same prediction for the transmission minimum as does the SPP description. However, for the case of excitation by the TE₁ waveguide mode, Eq. (4) does not reduce to the same result as Eq. (2) for the TM₁₂ mode due to the difference of the spatial mode pattern. We emphasize that this analysis only applies to arrays and not to a single isolated hole since the justification for the fictitious rectangular waveguide arises from the periodicity of the array using appropriate boundary conditions [21].

Based on this approach, the transmission can be calculated analytically as a function of frequency using the coupled-mode method (CMM). As described above, we model the system as a rectangular waveguide with a single subwavelength hole. To apply the CMM, we expand the incident field into eigenmodes, then calculate the scattering coefficients from incident to transmitted field by setting appropriate continuity conditions [15,16]. Previously, the CMM in the context of EOT has only been applied to free-space and so would use plane waves as the input and output fields. Here, we adapt the method to use the supported waveguide modes (i.e. TE mode field patterns) for the input (Region 1) and output (Region 3). For the array of holes (Region 2), we employ the discrete waveguide modes of a subwavelength hole, even though they are all evanescent, and match the conditions across these three regions. We use the three least evanescent aperture modes directly excited by the incident mode for good convergence.

In addition to this analytical approach, we also perform numerical computations based on the finite-element method (FEM). We treat the metal waveguide plates as perfect electric conductors (PEC) and the screen as a real (finite conductivity) metal using complex permittivity values from Ref. 20 (although there is little difference if it is treated as a PEC).

3. Results

In Figs. 3 and 4, we compare the experiment, FEM simulation, and CMM theory for a hole diameter of d = 380 μm and various plate separations b for TEM, shown in Fig. 3, and TE₁, shown in Fig. 4, input waveguide modes. Figures 3(a)–(c) and Figs. 4(a)–(c) show the results for a hole spacing of a = 0.75 mm. For comparison, we also display the results for a different hole spacing, a = 1.0 mm, shown in Figs. 3(d) and 4(d). We find good agreement between experiment and prediction for the peak position of the resonant frequencies and minima for both TEM and TE₁ excitation modes. The data showing the larger hole spacing further emphasizes the significance of the impedance matching mediated by TM_m,n modes. In the TEM case, since the relevant mode is TM₂₀, we anticipate no dependence on the hole spacing, as reflected in the lack of a spectral shift between Figs. 3(a) and 3(d). In contrast, for the TE₁ input mode where the relevant mode is TM₁₂, a spectral shift with a change in a is anticipated, and correctly predicted (vertical dotted lines).

 

Fig. 3 Power transmission for a fixed hole separation of (a-c) a = 0.75 mm and (d) a = 1.00 mm for the plate separations of b = 0.75 mm (black, thin line), b = 1.00 mm (red), and b = 1.25 mm (blue, thick line) with TEM mode PPWG excitation. The dashed lines indicate the cutoff of the corresponding TM₂₀ mode.

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Fig. 4 Power transmission for a fixed hole separation of (a-c) a = 0.75 mm and (d) a = 1.00 mm for the plate separations of b = 0.75 mm (black, thin line), b = 1.00 mm (red), and b = 1.25 mm (blue, thick line) with TE₁ mode PPWG excitation. The dashed lines indicate the cutoff of the corresponding TM₁₂ mode. For experiment and theory with a = 0.75 mm and b = 1.25 mm, we see many oscillations which can be attributed to the excitation of higher order propagating waveguide modes (i.e. cutoff of TE₃ is 350GHz).

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This waveguide EOT configuration has potential interest for narrowband tunable filters. In most cases, the geometry of the EOT structure is fixed (i.e., the periodicity of the hole array is not variable), so frequency tuning is challenging. However, in our case the waveguide plate spacing can be easily varied, thus dynamically tuning the transmission resonance. Such tunable filters may find future use in THz communications systems where the PPWG platform has been discussed in the context of various signal processing functionalities [22,23].

4. Conclusion

In conclusion, we describe the first comprehensive study of the role of the incident wave’s spatial mode on the EOT phenomenon. By using a PPWG (in contrast to the rectangular or circular waveguides more typically employed at lower frequencies), we are able to perform a direct comparison between the cases of TEM and non-TEM input modes. This work is reminiscent of recent studies of near-perfect transmission through structures with effective permittivity near zero [24,25]. Here, we focus more specifically on the role of higher order evanescent waveguide modes, which is more closely related to discussions involving surface plasmon-mediated enhanced transmission. We find that the conventional description based on the excitation of surface plasmon modes (or spoof surface plasmons) does not accurately capture the physics of the two distinct mode cases investigated. That is, it does not correctly predict the transmission frequencies in the case when the incident wave is not a TEM wave. Instead, we adapt a formalism based on impedance matching, which more accurately captures the details of the spectra in both cases.