1. Introduction

With the experimental discovery of extraordinary optical transmission (EOT) in the late nineteen nineties [1], it was unveiled that the transmission through small holes could be accomplished by taking advantage of periodicity, largely overcoming the predictions of Bethe’s single aperture theory [2]. It was found that an enhanced transmission peak was always present at a frequency lower than that of the first onset of diffraction, even when the intrinsic resonance of the hole was well above the latter. In the optical regime, this was explained through the coupling of the free space radiation to bound modes commonly known as surface plasmon polaritons (SPPs) [3]. The finding of enhanced transmission at frequencies at which metals do not support SPPs, such as microwaves [4], made it clear that genuine SPPs were not essential to explain EOT. The EOT theory was later unified across a large range of frequencies thanks to the existence of surface modes supported by periodic arrays of holes at microwave frequencies, which were shown in some cases to behave in a similar dispersive manner to the optical SPPs [5, 6]. A large body of work has been undertaken by both the optics and microwave engineering communities on this topic, collected into extensive review articles [7–9] and it still remains important due to the rich physics behind it [10,11].

Although most of the studies of the transmission through small apertures have been based on free-space wave propagation, the EOT phenomenon is also found in the transmission through metallic waveguide diaphragms near the cut-off of higher order waveguide modes, which play the role of the onset of diffracted modes in periodic structures [12, 13]. Moreover, the use of transmission line theory to explain the EOT as an impedance matching problem is based on the transverse electromagnetic (TEM) mode supported by a parallel plate waveguide (PPWG) [14].

Only recently has the EOT effect been studied for transmission through a 1-D periodic array of circular holes embedded in a PPWG by Reichel et al. [15]. In particular, these researchers measured and computed the transmission coefficient when the array was excited both by the TEM mode and by the PPWG TE₁ mode. They found that the EOT peak appears at higher frequencies in the latter case than in the former case.

In this paper we present an approach to understand the origin of the EOT peak frequency shift in PPWG under TE₁ mode excitation. This theory is based on the decomposition of the problem into two problems of EOT through a 2-D periodic array of symmetric holes under oblique plane wave excitation. These EOT problems are in turn accurately analyzed by solving an integral equation involving a periodic Green’s function. The steps followed in the solution of the integral equation provide a way to predict the appearance of Wood’s anomalies and the associated extraordinary transmission through both the 2-D array of holes and the equivalent 1-D array of holes in PPWG under TE₁ mode excitation. Our approach provides new physical insight into the origin of Wood’s anomalies and proposes a highly efficient analysis tool for the numerical study of structures supporting EOT phenomena. It only requires the solution of a small system of equations to obtain accurate results as opposite to other numerical methods such as coupled-mode method (also known as mode-matching). And this is because it adequately accounts for the singular behavior of the field quantities at the edges of the metal screen holes. This singular behavior cannot be handled by the expansion modes used in the coupled-mode method presented in [15].

2. Theoretical model

Let us consider the problem depicted in Fig. 1, where one pure ${\text{TE}}_1^z$ mode propagating in an infinite PPWG impinges on a negligible thickness conducting strip filling the space between the waveguide plates. The strip, located at z = 0, is perforated with a 1-D periodic array of holes. The holes will be assumed to be rectangular slots for the sake of mathematical simplicity, although the physics governing the problem would remain identical if circular holes were to be chosen as in [15]. Both the parallel plates and the strip containing the holes will be assumed to be perfect electric conductors (PECs). In the following, a time-harmonic dependence of the physical quantities of the type e^jωt will be assumed and suppressed throughout. Under these conditions, the complex electric field of the ${\text{TE}}_1^z$ mode incident on the array of slots can be written as [16]

 

Fig. 1 Perspective view of a parallel plate waveguide. The two plates are connected through a negligible thickness PEC strip perforated with an infinite 1-D periodic array of slots. A zoomed view of the hole array with the definition of the geometry parameters is also shown on the bottom-left of the figure.

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Imagine each of the two aforementioned plane waves separately propagates in free space and is obliquely incident on a 2-D periodic array of rectangular slots perforated in a PEC screen at z = 0 shown in Fig. 2, where the unit cell is exactly equal to that of the 1-D periodic structure of Fig. 1. If we superpose these two waves in the way shown in (1), the resulting wave will fulfill the electric wall (EW) boundary conditions imposed by the PPWG of Fig. 1 at y = 0 and y = b. This means the problem of the transmission of the ${\text{TE}}_1^z$ mode through the 1-D array of holes can be viewed as the superposition of two problems of transmission of plane waves propagating in free space which are obliquely incident on a 2-D periodic array of holes in a conducting screen, and the superposition principle can be invoked.

 

Fig. 2 Perspective view of part of the two dimensional periodic array of holes perforated into a negligible thickness PEC screen. The array is illuminated by an obliquely incident plane wave along the direction given by the spherical angular coordinates (ϕ_inc, θ_inc). A zoomed view of the hole array with the definition of the geometry parameters is also shown.

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Now let us concentrate on the problem of a plane wave obliquely incident on the 2-D array of rectangular slots in a PEC screen that is shown in Fig. 2. Let us assume that the propagation direction of the plane wave is given by the angular spherical coordinates (ϕ_inc, θ_inc) shown in Fig. 2. The complex electric field of this wave can be written as

In accordance with the derivations shown in [17] (only valid for the case of normal incidence), in order to determine the scattered tangential electric field, ${\mathbf{E}}_t^{\text{sc}}(x,y,z=0)$, in the rectangular slot δ₀₀ ≡ {(a − w)/2 < x < (a + w)/2; (b − l)/2 < y < (b + l)/2} of the reference unit cell C₀₀ ≡ {0 ≤ x ≤ a; 0 ≤ y ≤ b} of the 2-D periodic array of slots shown in Fig. 2, we need to solve the integral equation

In Equation (5), ${\overline{\mathbf{G}}}_M(x,y)$ stands for the dyadic Green’s function defined by

In order to solve the integral equation of (4), the unknown quantity ${\mathbf{E}}_t^{\text{sc}}(x,y,z=0)$ is expanded in terms of basis functions as shown below

The coefficients of the MoM matrix of (9) can be obtained in the Fourier transform domain as [17]

The constant terms C_i (i = 1, . . . , N_b) of the system of equations (9) can be obtained as

Note that the spectral dyadic Green’s function ${\widetilde{\overline{\mathbf{G}}}}_M(k_x=k_{xm},k_y=k_{yn})$ has poles at frequencies for which

When condition (13) is fulfilled, the elements of ${\widetilde{\overline{\mathbf{G}}}}_M(k_x=k_{xm},k_y=k_{yn})$ tend to infinity, the MoM matrix entries Γ_ij usually tend to infinity and the unknown coefficients f_j tend to zero to keep C_i bounded in (9). As a consequence of this, ${\mathbf{E}}_t^{\mathit{sc}}(x,y,z=0)$ becomes zero in (8), which means the tangential electric field in the holes of the 2-D array vanishes, i.e., the PEC screen with holes behaves as a solid PEC and reflects all the incoming power, which is the origin of a Wood’s anomaly. Therefore, Equation (13) provides the condition for Wood’s anomalies and marks the onset of the m − n-th diffracted mode (also called m − n-th grating lobe). At frequencies just below this divergence, the fields on the slot are largely enhanced, providing a mechanism for the appearance of EOT peaks. This has also been explained in terms of circuit theory reported in [14], showing that an EOT peak always appears at a frequency slightly lower than a Wood’s anomaly owing to impedance matching provided by a resonance created by a frequency dependent capacitance that becomes singular at the Wood’s anomaly. So, a Wood anomaly always presents an associated EOT peak at slightly lower frequencies.

3. Results and discussion

For a PEC screen with a 2-D array of holes under normal incidence conditions (θ_inc = 0), the first Wood’s anomaly is expected to appear at a frequency for which λ₀ = max(a, b). This corresponds to the Wood’s anomaly experimentally detected in [15] for a PPWG under TEM excitation since this problem is equivalent to the EOT problem of a 2-D periodic array of holes under normal incidence [17].

However, for the 2-D array of holes of Fig. 2 under oblique incidence, Wood’s anomalies and the associated EOT peaks in principle tend to appear at frequencies above and below the frequency for which λ₀ = max(a, b). For example, let us consider the transmission curves shown in Fig. 3(a) for a 2-D periodic array of rectangular slots illuminated by obliquely incident plane waves propagating along different directions given by (ϕ_inc = 0, θ_inc) (all the curves in Fig. 3 have been obtained by means of MoM in accordance with Eqns. (4) to (12)). For these particular incidence directions, we would expect Wood’s anomalies for the m = ±1, n = 0 diffracted modes (see (13)) when a/λ₀ = 1/1 ± sin θ_inc. Also, we would expect Wood’s anomalies for the m = 0, n = ±1 modes when b/λ₀ = 1/ cos θ_inc. When a = b, which is the particular case treated in Fig. 3(a), the Wood’s anomalies appear for a/λ₀ < 1 and a/λ₀ > 1, i.e., above and below the frequency of the Wood anomaly for normal incidence (a/λ₀ = 1). Note that total transmission does not occur in the EOT peaks associated with the Wood’s anomalies for which a/λ₀ > 1, which is due to the fact that some diffracted modes have already been launched at those frequencies and are capturing part of the energy of the original incident wave. However, Wood’s anomalies and EOT peaks may not always appear for oblique incidence at frequencies below the frequency for which λ₀ = max(a, b). In particular, Fig. 3(b) shows the transmission curves of a 2-D periodic array of rectangular slots illuminated by obliquely incident plane waves propagating along different directions given by (ϕ_inc = π/2, θ_inc). For these particular incidence directions, we should expect Wood’s anomalies for the m = ±1, n = 0 diffracted modes when a/λ₀ = 1/ cos θ_inc, and for the m = 0, n = ±1 diffracted modes when b/λ₀ = 1/1 ± sin θ_inc.

 

Fig. 3 Transmission spectra of a plane wave impinging on a 2-D array of rectangular slots in a PEC screen. Two polarizations TM^z (a) and TE^z (b) are considered, and different angles of incidence (θ_inc). In both cases, the direction of the incident electric field is contained in the x − z plane so as to excite the fields in the slots. The dimensions of the unit cell were chosen to be l_s/a = 0.4, w_s/a = 0.05 and a = b.

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In the case shown in Fig. 3(b) where a = b, the Wood anomalies and EOT peaks for the m = ±1, n = 0 modes appear when a/λ₀ > 1, but those for the m = 0, n = ±1 modes do not appear when a/λ₀ < 1. The explanation for this latter behavior is hidden in Eqns. (10) and (11). Under the conditions studied in Fig. 3(b), it turns out that k_x0 = 0. This means that when Eqn. (13) is fulfilled for the m = 0, n = ±1 modes, there is a zero-pole cancellation in all the elements of the matrix ${\widetilde{\overline{\mathbf{G}}}}_M(k_x=k_{x0},k_y=k_{y,\pm 1})$ except for the y − y element. Also, under the illumination conditions studied in Fig. 3(b), the functions b_j (x, y) · ŷ are odd functions of x since the planes x = ma + a/2 (m = . . . , −1, 0, 1, . . .) of the 2-D periodic array are all EWs [17]. As a consequence of this, it turns out that b̃_j (k_x0 = 0, k_y,±1) · ŷ = 0. So, when ${\widetilde{\overline{\mathbf{G}}}}_M(k_x=k_{x0},k_y=k_{y,\pm 1})$ and b̃_j (k_x0 = 0, k_y,±1) are both substituted in (10), there is an additional zero-pole cancellation that prevents Γ_ij from being infinite when m = 0, n = ±1 and condition (13) is fulfilled. Note that total transmission does not occur in the EOT peaks associated with the m = ±1, n = 0 modes in Fig. 3(b) when θ_inc ≠ 0, which indicates that the m = 0, n = ±1 diffracted modes have been launched, even though the Wood’s anomaly and EOT peaks for these modes have not appeared.

The MoM formulation of (4) to (12) and the superposition principle have both been used to compute the transmission through the 1-D array of rectangular slots of Fig. 1 under ${\text{TE}}_1^z$ mode excitation. The results are plotted in Fig. 4. For validation purposes, our MoM results have been compared with results obtained by the commercial software HFSS, and excellent agreement has been found. It should be pointed out that our in-house software is about 225 times faster than HFSS for the generation of Fig. 3, which justifies the efforts carried out to develop in-house software in addition to the physical insight provided for the understanding of the Wood’s anomaly phenomena.

 

Fig. 4 Transmission spectra obtained when the TE₁ mode of the PPWG of Fig. 1 impinges on the 1-D array of rectangular slots. Our results (MoM) are compared with HFSS results. The dimensions of the unit cell were chosen to be w_s/a = 0.05 and a = b.

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Taking into account the Wood’s anomalies detected in Fig. 3(b) for the diffracted modes m = ±1, n = 0, we have inserted these two values of m and n in (13) plus the incidence angle values (ϕ_inc = π/2|3π/2, θ_inc = θ₀) (where θ₀ is defined in (2)) for the ${\text{TE}}_1^z$ mode in the PPWG, and we have found that a Wood’s anomaly and the associated EOT peak is to be expected when $a/{\lambda }_0=b/{\lambda }_0=\sqrt{5}/2>1$ (a = b has been assumed in Fig. 4). This exactly coincides with the results obtained in Fig. 4, and it can be verified that this frequency estimation for the Wood’s anomaly roughly matches those experimentally obtained by Reichel et al. [15] under ${\text{TE}}_1^z$ mode excitation.

There is another way to explain the appearance of the Wood’s anomaly for $a/{\lambda }_0=b/{\lambda }_0=\sqrt{5}/2\hspace{0.17em}>\hspace{0.17em}1$, which is based on the fact that the planes x = ma and x = ma + a/2 (m = . . . , −1, 0, 1, . . .) act as EWs and the plane y = b/2 acts as a magnetic wall (MW) in Fig. 1 under ${\text{TE}}_1^z$ mode excitation. This means the periodic unit cell acts as an equivalent rectangular waveguide excited by a TE₀₁ mode. The symmetry conditions imposed by the EWs and the MWs mean that the first higher order TM mode that can be excited in this equivalent waveguide is a TM₂₁ mode. Since TM modes in rectangular waveguides are precisely those which give rise to Wood’s anomalies and EOT peaks [12, 14], it is clear that the first Wood’s anomaly of the PPWG under ${\text{TE}}_1^z$ mode excitation will show up at the cutoff frequency of the TM₂₁ mode, which occurs for $a/{\lambda }_0=b/{\lambda }_0=\sqrt{5}/2\hspace{0.17em}$ when a = b. This is in agreement with previous studies of EOT in hollow metal waveguides [12]. When $a/{\lambda }_0=b/{\lambda }_0=\sqrt{5}/2\hspace{0.17em}$, a combination of TE₁ and TM₁ modes will be excited in the PPWG propagating at oblique angles from the z direction since TM₂₁ modes cannot propagate in a PPWG.

4. Conclusion

In summary, we have presented a theory based on the application of the Method of Moments that explains why the EOT peaks of PPWG with 1-D periodic arrays of symmetric holes under ${\text{TE}}_1^z$ mode excitation appear at frequencies higher than those found for TEM mode excitation. The theory indicates that EOT peaks for ${\text{TE}}_1^z$ excitation should also be expected at frequencies lower than those of TEM mode excitation, but they do not appear because of the symmetry of the holes. The symmetry plane perpendicular to the x axis eliminates the divergent response of the fields on the first onset of diffraction, therefore finding the first peak associated to extraordinary transmission at higher frequencies than at normal incidence.