1. Introduction

Since the extraordinary optical transmission phenomenon was reported by Ebbesen et al. [1] in 1998, the optical properties of metallic films that contain a single subwavelength aperture or two-dimensional aperture arrays have been studied extensively [2–8]. The resonance transmission of the surface electromagnetic modes plays an important role in the emergence of the enhanced transmission [9]. This phenomenon means that more light is transmitted at certain wavelengths than that expected based on either ray optics or on Bethe’s theory, which states that the transmitted light through an aperture is proportional to (r/λ)⁴ when the aperture radius r is much smaller than the incident light wavelength λ [10]. The transmission efficiency and the bandwidth have been the focus of the research into enhanced transmission through subwavelength apertures. Much of the work related to these issues has been carried out in the optical [11–14] and terahertz [15, 16] bands. However, at microwave frequencies, metals are perfect conductors, which means that surface waves are not present on an opaque screen that has been drilled with a single aperture that is not surrounded by periodic corrugations [17, 18]. Thus, a solution to the problem of total broadband microwave transmission through a single metallic aperture is urgently required.

Split-ring resonators (SRRs) are artificially constructed subwavelength structures with scales ranging from microwave to visible wavelengths [19–25]. They are used as components to create metamaterials [26–28], superlenses [29], cloaks [30], and magnetoinductive waves [31, 32]. In fact, most of the unprecedented electromagnetic properties, such as the negative permeability (permittivity) that is a prerequisite for negative-index metamaterials, are derived from the magnetic (electric) resonance properties of SRRs [33], which lead to out-of-phase behavior at frequencies higher than the resonance frequency [34] and does not generally occur in nature materials [35]. Simultaneously, in the resonant state, the large numbers of charges that periodically gather at the gap of the SRR form strongly localized time-varying electric fields. These resonantly localized electric fields have been a recognized phenomenon since SRRs were first proposed [22] and have exhibited promise in applications such as enhanced second-harmonic generation [33, 36].

In this paper, we propose an efficient approach that uses the coupled-mode theory of two resonators to realize total broadband transmission through a subwavelength aperture. The resonators are the strongly localized and periodically varying electric fields that occur at the gaps of the SRRs. By placing two SRRs symmetrically on either side of an aperture and then adjusting the space between the two gaps to a critical size, we obtain broadband (0.4 GHz) and almost total transmission (−0.2 dB) of electromagnetic waves in the C-band (4–8 GHz) through a single metal aperture (r/λ = 0.05 at the center frequency of the band) if the losses related to the materials are neglected.

2. Coupled-mode theory

It is well known that two objects resonating at the same frequency tend to exchange energy efficiently, while simultaneously interacting weakly with other off-resonant environmental objects [37, 38]. In physics, there are many types of resonance-based couplings, such as those based on mechanical, acoustic, electromagnetic, magnetic, and nuclear resonances, which are used to transfer energy. Among these, a typical example is wireless power transfer via strongly coupled magnetic resonances, where the transfer is mediated by the overlap of the near magnetic fields of the two self-resonant coils or dielectric resonators used [37]. In fact, any type of resonance coupling of two objects can be expressed using coupled-mode theory [38], as represented by the following Eq. (1) and (2):

In this work, enhanced transmission of microwaves through a subwavelength aperture is demonstrated in two rectangular waveguides separated by a metal plate with a drilled subwavelength aperture. The electromagnetic energy in the incoming waveguide is localized by the resonant SRR, and is then coupled to another SRR in the outgoing waveguide through an exchange of electric field energy at the gaps of the SRRs, before it is finally restored to the normal TE₁₀ waves that flow from the output of the outgoing waveguide. In Eq. (3), Γ₁ and Γ₂ are caused only by the absorption losses of the SRRs because there is no radiated losses in two closed waveguides; |a₁(t)|² and |a₂(t)|² are the localized electric field energies stored at the gaps of the two SRRs, and Γ_w|a₂(t)|² can be measured from the output waveguide. The entire coupling system can be analyzed using a two-port network. Thus, the coupling efficiency of the two SRRs is denoted by the transmission coefficient [39]. The system meets impedance matching requirements because of its completely symmetrical structure. If the dissipation caused by absorption losses (ohmic losses) is ignored, then the coupling efficiency will be 0 dB at the critical coupling point [39], which is determined by the distance between the two gaps. Therefore, the electromagnetic waves can be totally transmitted, rather than be reflected by the metal plate, through the subwavelength aperture by the two coupled resonant objects located on either side of the aperture.

3. Experiments and simulation

Figure 1 shows a schematic representation of total broadband transmission through the subwavelength aperture by strongly localized electric field coupling of two SRRs. A commercial 1 mm thick FR4 printed circuit board (with dielectric constant of 4.4 and loss tangent of 0.01) with a thin (18 µm) deposited copper plate layer was used to fabricate the SRRs and the metal subwavelength aperture. The simulation model was built and optimized using a finite-element-based frequency-domain electromagnetic solver (HFSS 10). The final scales of the coupled square SRRs are shown in Fig. 1(a), where the geometrical parameters are a = 3.5 mm, w = 0.75 mm, g = 1 mm, and d = 2 mm. Figure 1(b) is a photograph of the coupled SRR array patterns that were fabricated using conventional photolithography techniques. In the experiment, a single SRR unit with dimensions of 6 mm × 6 mm and a coupled SRR pair unit with dimensions of 15 mm × 6 mm were cut down to measure their microwave transmission spectra. Figure 1(c) shows a subwavelength aperture with a radius of r = 3.5 mm that was drilled at the center of the copper plate, which was cut down to a size of 70 mm × 50 mm to enable it to be tightly clamped between the two WR-137 rectangular waveguides, as shown in the schematic in Fig. 1(d). The other ends of the two waveguides are connected to the input and the output of a network analyzer (Agilent PNA-L N5230C).

 

Fig. 1 Schematic of the experimental setup. (a) A unit cell of the designed coupled SRR pair. (b) Photograph of one side of the fabricated coupled SRR array patterns. (c) A coupled SRR pair unit cell inserted in a metal subwavelength aperture. (d) Measurement system using two rectangular waveguides to demonstrate total broadband transmission. The Cartesian frame and the size nomenclature are depicted in (a), (c), and (d).

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4. Results and discussion

Figure 2 shows the transmission spectra of a single SRR unit and a coupled SRR pair unit placed inside the waveguides. In Fig. 2(a), a transmission trough at 6.6 GHz is introduced by the LC resonance of the single SRR. The measured result agrees very well with the simulated results. However, the resonance frequency of the single SRR at 6.6 GHz is split into two resonance frequencies when the two SRRs are placed closely together in a gap-to-gap configuration and coupled through the localized electric fields at the gaps. The lower resonance frequency is 6.05 GHz, and the higher frequency is 7 GHz. The measured and simulated results agree consistently overall, as shown in Fig. 2(b). Our simulations show that the difference between the two resonance modes is the phase difference of the currents in the two SRRs (in phase for the lower frequency mode and out of phase for the higher frequency mode [39], although this is not shown here). In Fig. 2(b), there is a broadband transmission (6.3–6.8 GHz) between the two resonance frequencies of the coupled SRR pair.

 

Fig. 2 Measured (red line) and simulated (blue line) transmission spectra (in dB) of (a) a single SRR unit and (b) a coupled SRR pair unit in the waveguides. The insets show schematics of the experimental setups and the simulation models.

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To visualize the localized electric fields and the coupling efficiency, the electric field intensity distributions at the resonance frequencies are presented in the z = 0 plane, which corresponds to the center of the rectangular waveguide. Figure 3(a) clearly shows that the electric field has been localized around the single SRR at the resonance frequency of 6.6 GHz. On the left of the SRR, the electromagnetic wave in the waveguide is turned into a standing wave because of the strong reflection by the resonance of the single SRR. On the right of the SRR, the electromagnetic wave is very weak. Similarly, in Fig. 3(b) and 3(c), the electric field has also been localized around the coupled SRR pair at their resonance frequencies, and the electromagnetic wave on the left is a standing wave while that on the right is weak. However, in Fig. 3(d), for the coupled SRR pair at the resonance frequency of the single SRR (6.6 GHz), we see that the electromagnetic wave propagates smoothly in the waveguide. We also see that the electric field is strongly localized within the two SRRs. These characteristics of high coupling efficiency and a strongly localized field inspired us to realize total transmission through a subwavelength aperture.

 

Fig. 3 Simulated electric field intensity distributions in the z = 0 plane at (a) 6.6 GHz for the resonance frequency of the single SRR, (b) 6.05 GHz and (c) 7 GHz for the resonance frequencies of the coupled SRR pair, and (d) 6.6 GHz for the center frequency of the passband of the coupled SRR pair. The field intensity values have been normalized.

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Figure 4 shows the measured and simulated results of microwave transmission through a metal subwavelength aperture with and without the localized E-field coupling of the resonant SRR pair. In Fig. 4(a), when only the metal subwavelength aperture is clamped between the two waveguides, both the measured and simulated values of the transmission coefficient are lower than −20 dB, which complies with Bethe’s theory of wave optics. However, when the coupled SRR pair unit is inserted symmetrically into the aperture, the transmission coefficient is as high as −2 dB over the frequency range from 6.1 to 6.5 GHz, as shown in Fig. 4(b). The measured and simulated results are in good overall agreement. If the copper plate and the FR4 are assumed to be lossless, then the simulated result obtained for the transmission coefficient is indicated by the red solid line in Fig. 4(b). We see that the transmission coefficient is almost 0 dB (actually −0.2 dB) at the frequencies of 6.12 and 6.48 GHz in the broadband range, indicating that no reflection is occurring, despite the presence of the metal plate with the subwavelength aperture. The broadband range has been shifted slightly to a lower frequency band (from 6.3 to 6.8 GHz to 6.1–6.5 GHz) when compared with the simulated results shown in Fig. 2(b). This is mainly because of the parasitic capacitance caused by the coupling effect between the metal split ring and the metal plate (or the metal aperture). In summary, by using the coupled localized electric fields at the gaps of the two SRRs placed on either side of the subwavelength aperture, a 0.4 GHz bandwidth total transmission has been achieved.

 

Fig. 4 (a) Measured (black dotted line) and simulated (black solid line) transmission spectra (in dB) of microwaves transmitted through the subwavelength aperture. (b) Measured (blue dashed lines) and simulated (blue solid lines) transmission spectra of microwaves transmitted by E-field coupling of two resonant SRRs located on either side of the subwavelength aperture. The red solid line represents the simulation result obtained when the metal plane and the FR4 are assumed to be lossless. The insets show schematics of the experimental setups and the simulation models.

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To further explore the mechanism of this high efficiency transmission by the electrically coupled SRR pair, we calculated the time-varying electric field intensities at the two gaps in the y = 0 plane at the center frequency (6.3 GHz) of the broadband range. The results showed that (not shown in this paper) one of the electric field intensities decreases until it vanishes altogether, while the other intensity increases until it reaches a maximum. The process will then be reversed until the intensity at the first gap reaches a maximum and the intensity at the second gap reaches zero, and this behavior is repeated cyclically. Figure 5 shows the static field distributions at different phase angles for the incident wave. We clearly see that when the electric field intensity at the gap of one SRR reaches a maximum, the other is at a minimum [at the phase angle of 0° or 90° in Fig. 5(a) and 5(c)], or that the electric field intensities at the two gaps are equal [at the phase angles of 45° or 135° in Fig. 5(b) and 5(d)]. These results indicate that the electric field intensities at the gaps of the two SRRs have a phase difference of 90°, and that the currents in the two metal split rings also have a 90° phase difference. In short, the phase difference between the two resonant SRRs is 90°, which complies completely with the critical coupling state conditions of the two resonators.

 

Fig. 5 Simulated electric field intensity distributions at the two gaps in the y = 0 plane at 6.3 GHz at phase angles of (a) 0°, (b) 45°, (c) 90° and (d) 135°.

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As the last part of this section, coupling state of the two resonators should be discussed comprehensively. In general, a system consisting of two resonators with the same resonance frequency can be divided into overcoupled, critically coupled and undercoupled states [39]. At the overcoupled state, the system has the lower (two resonators are in phase) and higher (two resonators are out of phase) frequency modes, and will result in maximum energy-transfer efficiency at the two resonant frequencies. With the decreasing of the coupling coefficient, the resonant frequency separation also decreases until the two modes converge at the fundamental frequency (the resonant frequency of the single resonator). The phase difference between the two resonators becomes 90°. Thus, the system reaches to the critical coupling state and the maximum transfer efficiency is still achievable at the resonance frequency. If the coupling coefficient decreases continuously, the transfer efficiency begins to fall off precipitously at the fundamental frequency. It means that the system is the undercoupled state. As for the two closely spaced ring or split-ring resonators, the coupling coefficient is generally determined by the space [40] or capacitance [41] between the two resonators. It should be note that the coupled SRR pair presented in this paper is overcoupled in the hollow waveguide [see Fig. 2]; however, it is critically coupled when the metal aperture is put into the waveguide [see Fig. 4]. This is not contradictory because the coupled SRR pair is in two different electromagnetic environments. In Fig. 2, the incident electromagnetic waves act simultaneously on the two resonators. But, in Fig. 4, the exciting waves act only on the first resonator. In addition, the transmission trough at the resonance frequency in Fig. 2(a) turns into the transmission peak in Fig. 4(b). This can be explained using the theories of electric circuits [42].

5. Conclusions

In conclusion, using the critical coupling of the strongly localized electric fields distributed at the gaps of two SRRs placed on either side of a subwavelength aperture, we have realized total broadband transmission of electromagnetic waves operating in the C-band. The proposed coupled resonance transmission has many potential applications in microwave devices, such as couplers and filters. In addition, the central transmission frequency was found to be determined by the scale of the SRRs used, which means that we can easily realize total broadband transmission at a desired electromagnetic region by appropriate design and fabrication of the SRRs.