1. Introduction

In the past decade, Bethe’s theory for the diffraction of light through a small hole in a metal film [1] has been challenged by the discovery of extraordinary optical transmission (EOT) through hole-arrays [2, 3]. Recently, Bethe’s theory was extended to the array configuration and it showed 100% transmission at resonant wavelengths [4, 5]; therefore, total transmission for an array of holes is not a contradiction to Bethe’s original work. It should be noted, however, that the original motivation of Bethe’s work was not for arrays, but for “the effect of a small hole in a cavity” and “the effect of a small gap in a wave guide” [1].

Here we revisit the waveguide system and show that total electromagnetic transmission is possible from only a single aperture in a waveguide wall; this is very surprising considering the the usual application of Bethe’s theory waveguides [6]. For the same configuration as considered here, the usual finding is that the transmission can be made arbitrarily small as the aperture size is reduced [6], with the limitation on effective hole-size coming from electromagnetic penetration into real materials [7, 8]. We demonstrate, both by theory and by comprehensive numerical calculations, that total transmission remains when the aperture size is reduced. In the past, we have suggested that the resonance can occur in the waveguide system (Sec. IIID of [5]), and here we extend the array theory of that work to the waveguide system and present supporting numerical calculations. A recent study of resonances in the array and waveguide systems provided an analysis of array and single aperture systems using numerical mode-matching calculations and a parametric circuit model [9]. The circuit description provides a nice interpretation of these results and can be traced back to early works of aperture arrays in screens [10]. Here we discuss the circuit interpretation of our model as well. The comprehensive numerical simulations are also extended to include the material response (including loss) at microwave and the visible-infrared frequencies.

In addition to its surprising relation to Bethe’s original work, this phenomenon is interesting for applications across several disciplines. First, at the resonance frequency, all of the electromagnetic energy is squeezed into the aperture, allowing for enhanced interaction with matter at an extreme subwavelength scale. This has possible applications in many areas of physics, ranging from the generation of plasmas to cavity quantum electrodynamics. Second, the transmission resonance wavelength is sensitive to the polarizability of the aperture, so that small perturbations to the aperture will lead to large resonant frequency shifts that may be used for sensor applications. Third, the transmission resonance may be used as a compact waveguide filter. The results presented in this paper can be readily tested with existing methods, at least for the microwave regime where standard waveguide components exist.

2. Analysis

2.1. Analytic theory

Figure 1(a) shows the geometry of the structure under consideration: a waveguide with a transverse metal screen containing a hole at the center. For simplicity, the waveguide and the hole are considered to be square; however, this may be easily generalized to other configurations, for example, a rectangular waveguide with a circular aperture. The waveguide has side-length a and the aperture has side-length a_h. The lower-left corner of the screen is chosen to be the origin of the x-y-z coordinate system, with the z-axis pointing along the waveguide. A TE₁₀ mode is incident from the negative z-direction.

The y-component of the electric field at the iris can be written as a Fourier decomposition of the waveguide modes:

on the incident side and

on the transmitting side. The time-harmonic convention we use here and throughout the paper is e^iωt where i is the imaginary unit. Due to the boundary conditions, the fields on each side should be equal. For the excitation of modes with small m and n, the TM₁₂ mode for example, the field of the mode have a slow variation in space. Therefore when considering coupling to these modes in the small aperture limit, the field can be considered constant over the aperture region [5]. It is also noted that the y-polarized excitation will excite only the TE_x waveguide modes, they do not have electric components in the x-direction and magnetic components in the y-direction. Moreover, the field should be zero at the surface of the PEC. By applying a Fourier decomposition to this field profile, the coefficients in Eq. (2) can be found:

for m,n ≠ 0 and

for m ≠ 0.

The magnetic field is calculated at the aperture to apply Bethe’s theory. Of critical importance to the proposed resonance phenomenon, the x-component of the magnetic field of the TM₁₂ mode diverges as the frequency approaches the cutoff, for any finite electric field. The magnetic field is calculated from the electric field by using Faraday’s law and evaluated at the center of the waveguide at the position of the aperture $(\frac{a}{2},\frac{a}{2},0)$. This is done for frequency f → f _c12 ^-, where the cutoff frequency is $f_{c12}=\frac{\sqrt{5}c}{2a},$ and c is the speed of light. Since $H_x(\frac{a}{2},\frac{a}{2},0)\to \infty$ as f → f _c12 ^-, we may approximate the modal expansion solely by the excitation TE₁₀ mode and the divergent TM₁₂ mode. This approximation requires that the infinite series contribution from remaining modes has negligible contribution. We have not found a supporting proof to that approximation; however, the dominant role of the TM₁₂ mode is supported well by our numerical calculations, as will be described below. The x-component of the magnetic field is:

 

Fig. 1. (a) Schematic of square waveguide with a metal screen and a square aperture at the center of the screen. (b) Theoretical transmission spectrum of the TE₁₀ mode through a perfect-electric conductor (PEC) screen with a square aperture at the center in a 10 cm wide square waveguide. Three results are shown in this graph with aperture width 1 cm, 1.5 cm and 2 cm, in blue, red and black. (c) Comprehensive numerical simulation showing transmission of the TE₁₀ mode through equivalent structures as in (b), except for 1 mm metal screen-width. Simulations were done using a PEC metal (lines) and lossy aluminum (circles) as the material of the waveguide and the screen.

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where i is the imaginary unit and t ₁₂ = 2t ₁₀, or Eqs. 3 and 4, has been used and Z₀ is the impedance of free-space. A similar expression is found for the side of reflection.

Bethe’s aperture theory can now be applied directly using the magnetic fields on the aperture [11]. This requires using self-consistency so that the transmitted power is equal to the power that the magnetic dipole emits [5]. The transmittance of the of TE₁₀ mode is:

where the magnetic polarizability is given by ${\alpha }_m=\frac{\pi a_h^3}{16}$ for a square aperture with sides of a_h. There is a resonant peak with 100% transmission at the frequency where the term in brackets of Eq. (6) goes to zero – all of the energy from the incident waveguide mode is transmitted through the hole. This total transmission occurs close to the cutoff frequency. Clearly, at the cutoff wavelength of the TM₁₂ mode, that is f = f _c12, the transmittance T is zero. Equation 6 is the main analytic result of this work for applying Bethe’s theory to the waveguide structure, allowing for a complete-transmission resonance. Figure 1(b) shows the theoretical transmission through a hole in an infinitesimally thin metal in a square waveguide with solid lines. The following dimensions were used: waveguide side a = 10 cm and aperture side a_h of 1 cm, 1.5 cm and 2 cm. This result can be readily tested experimentally with well-established microwave techniques [12].

2.2. Comprehensive numerical simulations

To test the theoretical result predicting 100% transmission with comprehensive numerical solutions to Maxwell’s equations, we used both the finite-integration method and the finite-difference time-domain method (both within CST Microwave Studio). Figure 1(c) shows the result of finite-integration (FI), which is more efficient than finite-difference time-domain simulation (FDTD) due to the high-quality of the resonance. We simulated a structure with the same dimensions as in Fig. 1(b), with a finite metal screen thickness d = 1 mm. A TE₁₀ mode was generated at the input, 35 cm from the screen, towards the screen and the transmitted wave was measured at the output, 35 cm after the screen. Table 1 shows that the transmission peaks obtained from the FI simulations are very close to the prediction by the theory. To show that the result can be extended to a waveguide with the same dimensions, but using a real metal including losses, we repeated the simulation by changing the material of the waveguide and the screen to aluminum (Al). This result is also shown in Fig. 1(c), and as expected, the nearly total transmission peak remains since aluminum is a good conductor for microwaves. The resonance frequency of the PEC and the real-metal simulations also agree well.

Table 1. Transmission peak frequencies (GHz)

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We repeated the simulations using a FDTD method. For this method, the transmission was less than unity due to the truncated integration time in the transient. However, the accuracy of the relative peak frequency increased (Table 1). Nevertheless, as it is most concerned in this paper, the strong resonance near the cutoff frequency of the TM12 mode remains. Aside from computational error, some additional discrepancy is expected from the finite screen thickness and the fact that the theory considered only 2 modes, which requires further investigation. Nevertheless, the main result is retained: a resonance with total transmission is found near the cutoff frequency of the TM₁₂ mode.

To investigate the limitations to directly extend these EOT results to visible and near-IR frequencies, additional simulations were performed using the usual Drude model for silver and an appropriately scaled-down structures (results not shown). In the shortest wavelength example attempted, considering practical fabrication limits, we chose the dimensions of the waveguide as follows: waveguide width a = 550 nm, hole-size a_h = 125 nm, metal screen thickness d = 40 nm, and length of 1500 nm. As expected, there was still a transmission peak at 513 nm with > 65% transmission and a quality of approximately 30. There is also the expected minimum transmission feature exactly at the cutoff wavelength (508.5 nm) of the TM₁₂ mode for that structure. We have seen similar resonances for 1550 nm resonant structure. These are promising early results and further investigation of the visible-IR regime is required to find the optimal configurations for demonstrating this effect.

3. Discussion

3.1. Origin of resonance phenomenon and comparison with other effects

This phenomenon is distinct from past works on EOT. Since the theory is formulated for a single hole in an infinitely thin PEC screen, it does not rely on surface plasmon polaritons (SPPs) [2, 13], spoof-SPPs of thick films [14], array effects [4], or Fabry-Perot resonances [15, 16], which can play a role in the EOT of arrays and single holes in a screen. The hole considered in this work is well below cutoff, and therefore this phenomenon is also distinct from resonant transmissions happening near the cutoff frequency for the hole [17]. Resonant total transmission of millimeter-wave through a slit in a waveguide is previously reported and employed [18], but the physical explanation is not given. The physics of total transmission in the waveguide screen aperture can be explained by considering the role of the TM mode below the cutoff frequency. Since this mode is bound to the screen, it stores the photons scattered by the aperture like a cavity, only to re-scatter them into the lowest-order propagating mode on the other side of the screen.

There is a simple impedance matching description of this process. The evanescent TM mode is required because its magnetic-to-electric field ratio diverges when approaching cutoff – it has a large admittance. This means that, for small apertures, the TM mode is coupled most-efficiently to the large admittance of the aperture. EOT phenomena originated from impedance matching are discussed previously by considering a perfect metal [9] and plasmonic metama-terials [19]. We now show that this impedance matching technique is equivalent to Bethe’s theory in the small hole limit. Mathematically, the root of the bracketed term in Eq. (6) is a formulation of impedance matching. For the perfect metal case, the size of the hole can be in-finitesimally small when compared to the wavelength, and there will still be 100% transmission at the resonant frequency.

As described in this theory, the TM₁₂ mode dominates the transmission process for small apertures. To validate that the enhanced transmission is accompanied by the strong excitation of the TM₁₂ mode, the simulated electric field distribution is analyzed away from the aperture. Figure 2 shows the z-component of the electric field spatial distribution at z = 30 cm at the transmission side for the resonant frequency, which matches the distribution of a TM₁₂ mode. This numerical result confirms the role played by the TM₁₂ mode as an energy reservoir in the resonant transmission process. There is no propagating power associated with this evanescent mode. The TM₁₂ mode is not the only one that has a divergent magnetic-to-electric field ratio near cutoff. In fact, for the hole being at the center of the waveguide, all TM_2m+1,2n bounded mode (where m,n = 0,1,2…) have this property near their cutoff frequencies; however, for higher-order modes there will be spurious coupling to the lower order propagating modes by the aperture. Therefore, we consider only the TM₁₂ mode.

Movies of the numerical calculations are provided for dynamic visualization of the transmission resonance phenomenon. (Media 1) and (Media 2) show two-dimensional slices of the steady-state electric field (y-component) using on-resonance and off-resonance frequency values, as calculated using CST Microwave Studio. (Media 1) shows that total transmission is obtained on-resonance, which is the result of the excitation of the TM₁₂ at the aperture. (Media 2) shows that off-resonance, there is negligible transmission. (Media 3) shows the same results, but for a time-domain calculation using the FDTD method. (Note that the vertical z-direction is compressed by a scaling factor of 9). This shows long-lived ringing of the electric field from the resonance.

3.2. Field enhancement

Figure 2 also shows the y-component of the electric field spatial distribution at the aperture (z = 0 cm) at the resonant transmission frequency. The maximum field strength in this contour shows a 16-fold enhancement over the maximum of the incident field. By energy conservation, only a 5-fold average enhancement is expected. The additional field enhancement is the result of the near-field distribution in the aperture differing from the incident field distribution, both in shape and scale. A sharply-peaked near-field distribution was predicted by early aperture theory [20], and recently confirmed with near-field measurements [21]. This additional field enhancement will benefit the many applications involving electromagnetic-matter interactions. Most importantly, these numerical simulations confirm that the waveguide structure with an aperture has the ability to concentrate the electromagnetic energy effectively.

3.3. Potential applications

Aside from the interesting physics of EOT in a waveguide, there are many potential applications such as filtering, enhanced electromagnetic-matter interactions, sensing and extremely localized heating and plasma generation. In addition, the structure may be used as a compact high-quality waveguide filter. Our device can be used as a resonant filter in the terahertz regime, which have shown near-unity transmission in arrays [22]; however, only a single aperture is needed here. Moreover, it is possible to tune the pass-band frequency and the bandwidth of this filter by changing the shape of the aperture, as described in past works [22, 23]. The dependence of the transmission peak on hole shape is expected to be more sensitive in our structure due to the very narrow bandwidth. The sensitive dependence of the transmission peak on the aperture polarization naturally lends this transmission mechanism to sensing applications. For example, a spectrum shift away from the transmission peak can be easily detected when there is a slight change in the polarizability at the aperture, due to the attachment of certain molecules or the introduction of a gas. A useful feature of the proposed structure is that it uses a guided TE10 mode as input, which can be excited and manipulated by standard methods. Overall, sensing technology is a rapidly developing area of research and much work has been done on apertures in metals: for example single apertures [24] and apertures arrays [25] have been investigated for sensing very small quantities of fluorescent molecules.

 

Fig. 2. Comprehensive numerical simulation showing the z-component of the electric field at z = 30 cm for the resonant frequency and the y-component of the electric field in the hole. The side of the aperture is 2 cm. The profile at 30 cm matches the TM₁₂ mode. The color-scale of this field has red as its the maximum (blue as the equal magnitude negative minimum) and the absolute maximum field strength is 6.2 times the incident field maximum. The y-component of the electric field at the screen has a color-scale where red is the maximum field strength with a 16-fold enhancement compared with the incident field maximum. We provide additional animations showing the y-component of the electric field in the yz-plane for steady-state on-resonance (Media 1), steady-state off-resonance (Media 2) and time-domain (Media 3).

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Due to the fact that a total transmission is achieved at the resonant frequency for this structure, there can be a huge field concentration in the hole. Previous works have proposed ways to squeeze electromagnetic waves well-below the optical wavelength. Some of those works exploited interesting modes for different material properties [26, 27]. Here, a perfect electric conductor is assumed, so the mechanism that squeezes the electromagnetic waves is geometric in nature.

It should be noted that there is a practical consideration when using the proposed configuration for extremely small apertures. The total transmission happens at a frequency very close to the cutoff frequency of the TM₁₂ mode – the smaller the aperture is, the closer the resonant frequency to cutoff. Near cutoff, if the waveguide is too short, the bounded TM₁₂ mode will be disrupted and the total transmission will be hindered. Therefore, for high-quality resonances, a longer waveguide is required.

4. Conclusion

A theory of total optical transmission through a small hole in a metal screen in a waveguide was presented, which was the physical result of resonant energy storage in the evanescent TM₁₂ mode near cut-off. Theoretically, this 100% transmission remains for a PEC metal, no matter how small the hole. The theoretical result was confirmed by comprehensive electromagnetic simulations, and the near-total transmission was retained when including loss at microwave frequencies and strong transmission was observed using a Drude model at optical frequencies. The direct consequence of this total transmission was the extreme squeezing of the resonant field into the hole. Based on this effect, a number of potential device applications were possible, including sensors, filters, field concentrators for electromagnetic interaction with matter (including trapping, plasma generation, linear and nonlinear spectroscopy) and local heating. This high-quality resonance with strong local field could have profound impact on cavity QED as well [28] and the strong field concentration may be useful for near-field optical scanning.