1. Introduction

Wireless electromagnetic (EM) communications have been rapidly developed for their advantages in appreciable convenience, high efficiency, low cost, etc. [1,2]. Nevertheless, current wireless communication systems face a challenge of an unreachable upper data rate limit of $10Gb/s$ due to the narrow bandwidth restriction. Clearly, such a data rate is not adequate for future wireless communications. For example, a faster than $24Gb/s$ data rate is required for simultaneous transmission of the Super-Hi-Vision, a video form with a resolution of $16{\times }1080$p [3]. Therefore, it is essential to explore wider spectrum resources for future communication systems with higher data rates. The terahertz (THz) band thus emerges as a potential candidate due to its wide bandwidth, unallocated application, and other advantages [4,5]. Progresses in THz technologies have been achieved over recent decades [6–10]. Nevertheless, high attenuation of the THz signals in the atmosphere limits the service range and reduces the data capacity by degrading the signal-to-noise ratio of communication systems. [11]. To address the issue, in addition to developing technologies, such as high power sources [9,10], high gain directional antennas [12,13] and/or metasurfaces [14,15], etc., in the THz range, significantly enhancing the weak THz signals at the receiver is also a much feasible, effective, and economical solution.

Meanwhile, applications of surface plasmon polaritons (SPPs) have attracted extensive attentions due to their unique characteristics, particularly local field intensification in the near field regime [16–20]. SPP is a surface wave propagating at the interface of two media with positive and negative permittivities respectively, e.g., air and metal. Such SPPs are intrinsically generated by the resonance between the electric field of the incident EM wave and the collective oscillation of unbound electrons in a plasma frequency of $\omega _{pe}=(4\pi n_ee^2/m_e)^{1/2}$, with the electron density $n_e$, the element charge $e$, and the electron mass $m_e$. The resonance then empowers SPP to effectively enhance the electric field of incident EM waves. This mechanism has been initially widely studied in the visible wavelength regime, since the typical plasma frequencies of metals are in the vacuum ultraviolet range, one or two orders of magnitude higher than the visible bands. Subsequently, the investigation has also been extended to the radio frequency regime by gaseous discharge plasmas with the plasma frequency usually an order of magnitude higher than GHz [21,22]. Yet it is a struggle to find the medium counterparts in the THz band or infrared regime. Fortunately, studies have demonstrated the ability of metal pierced with subwavelength holes to achieve highly localized SPP-like modes, called the spoof surface plasmon polaritons (SSPPs), at frequency bands lower than visible light [23,24,26]. It then provides a possible approach to achieve the enhancement of THz signals at the receiving end.

Dramatic THz electric or magnetic field enhancement based on SSPP has been achieved by squeezing THz waves into extremely narrow, even Angstrom-sized, gaps or nanowires, subsequently providing a suitable platform for cross-area studies such as THz nonlinearity, single-molecular layer detection [27–44], etc. However, these fine structures are neither conducive to the convenient direct detection and transmission of the enhanced signals therein, nor easily manufactured. Thus, they may not be straightforwardly applicable to communication systems. Therefore, it is particularly desirable, both fundamentally and applicably, to find a novel field-intensification approach that facilitates direct detection and further transmission.

In this work then, we propose a split ring resonator (SRR) featuring a subwavelength slit, which is capable to achieve omnidirectional EM signal gains up to tens of dBs in the THz regime. And different from typical LC resonances in previous SRRs where the enhanced electric field is concentrated in extremely narrow gaps [34,36,40–44], the coupling resonant modes excited by the innovatively designed SRR enable the enhanced EM field to exhibit a high-quality waveguide pattern in the circular cavity [48,49]. Besides, such an enhanced EM field distribution matches the spatial resolution of existing THz imaging modalities [7,45–47], allowing direct acquirement for enhanced THz signals without looking for indirect methods. Such major advantages offer remarkable application potential for the proposed SRR as an antenna receiver or sensor to receive or detect weak THz signals.

The rest of this paper is organized as follows. In Sec. 2, the theoretical analysis is presented based on the development of the Bruijn method, applied for obtaining reflection and absorption coefficients in acoustics [50,51] and optics [52,53]. In Sec. 3, the implications of the critical geometric parameters on the EM field enhancement inside the circular cavity are investigated by combining theoretical analysis with numerical simulations. Also, the coupling resonance occurring in the proposed SRR is compared with the typical LC resonance. Furthermore, the incident angle effect is considered to examine the SRR’s practical feasibility. The paper is then concluded in Sec. 4 with a summary and discussions.

2. Framework and analysis

We first introduce and discuss the theoretical model. The metallic SRR consists of a circular cavity with its inner radius $b$, thickness $t$, and a subwavelength slit extending to the external part, as shown by the blue region in Fig. 1, with its width $d$ and depth $h$, as well as metal width $a$ on its both sides. This SRR system is approximately considered as two-dimensional because the structure is uniform and with a size much longer than the incident wavelength in the direction vertical to the figure. The SRR is illuminated above by a TM polarized plane EM wave with an angular frequency $\omega$. The orange arrow implies the incident EM wave with an incident angle $\phi$, and its polarization mode is indicated in green. The white region is for air, which can also be replaced by other dielectrics. We analyze the system in a column coordinate of $r$-$\theta$ with $r=0$ at the center of the circular cavity.

 

Fig. 1. Schematic diagram of the system. The metallic SRR is represented by the blue region. The polarization of the incident EM signal is indicated in light green, and its incident direction is indicated by the orange arrow. The light green wavy line on the top the SRR represents the excited SSPPs. The red arrows inside the slit represent the upward and downward traveling waves in it, with amplitudes $\beta$ and $\alpha$, respectively. The dark green curves in the circular cavity indicate the enhanced waveguide mode, marked for the electric field of the $(1,1)$ mode as an example. The system is analyzed in a cylindrical coordinate system of $r-\theta$.

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In following analysis, the magnetic field of the incident TM polarized EM wave, with only the $z-$ component $H_z^{\xi }$, is applied to represent the field in the $\xi$-th region, with $\xi =c, s, a$ for the circular cavity, the slit, and the region above the slit, respectively. From the Maxwell’s equations, the enhancement factors of electric and magnetic field of an EM wave are the same. For convenience, the temporal oscillation factor $e^{-i\omega t}$ is removed in the linear analysis process, and the amplitude of the incident wave is normalized as unity.

The derivation of the field enhancement factors mainly contains four scenarios. $(i)$ The expressions of $H_z^{c}$, $H_z^{s}$ and $H_z^{a}$ are obtained by analyzing their characteristics. $(ii)$ The relations for amplitudes of the upward and downward propagating waves in the slit as well as various modes in the circular cavity are figured out by applying the field continuity of $H_z^{c}$ and $H_z^{s}$ at the connection of the slit and circular cavity. $(iii)$ The amplitude of the $H_z^{s}$ is derived from the EM continuity satisfied by $H_z^{s}$ and $H_z^{a}$ at the upper surface of the slit. $(v)$ The field enhancement factor in the circular cavity is extracted by substituting the amplitude of the $H_z^{s}$ obtained from $(iii)$ into the relationship obtained in $(ii)$.

For intensifying field, the slit width $d$ should be much shorter than the incident wavelength $\lambda$ to make the effective plasma frequency $\omega _{pe}$ (i.e. the cut-off frequency $\omega _{cut}$) of the slit higher than the incident angular frequency, i.e., $\omega _{pe}=\omega _{cut}=\pi c/d>\omega$. As a consequence, the subwavelength slit can support evanescent modes, a typical feature for SSPP excitation [23,24]. Moreover, such evanescent modes cannot be observed under TE-polarized incidence, again confirming the excitation of SSPPs [25]. This SSPP’s excitation generates high electric fields to drive the incident signal through the subwavelength slit. In comparison with the circular cavity circumference $2\pi b$ of several wavelengths, the effect of such a narrow slit can be reasonably ignored. Considering also that metals can be considered as perfect conductors in THz and/or lower frequency regimes, the field in the circular cavity can thus be well approximated as a superposition of various ideal circular waveguide modes:

At the joint of the slit lower end and the circular cavity $( r=b, \mid \theta \mid \leq d/2b )$, the magnetic field continuity $H_{z}^{s}=H_{z}^{c}$ leads to

Substituting Eqs. (6)–(8) into Eq. (4), one can obtain the relation between $\alpha _0$ and $\beta _0$

And $\alpha _0$ can be solved by applying the EM wave continuity for $H_z^{a}$ and $H_z^{s}$ ($H_{z}^{s}=H_{z}^{a}$ and $\partial H_{z}^{s}/\partial y=\partial H_{z}^{a}/\partial y$) at the upper end of the slit ($r=b+h$, $\mid \theta \mid \leq d/2b$). For the two continuity equations, following the same procedures as for solving Eqs. (4), and (5), one can obtain [39]

Substituting Eq. (11) to Eqs. (6) and (9), we can derive

In order to better reveal the correlation between the geometric parameters and the incident wavelength, dimensionless quantities, $\tilde {b}=b/\lambda$, $\tilde {h}=h/\lambda$, $\tilde {d}=d/\lambda$ and so on, are introduced in Eq. (13) to give

Two approximations are then adopted to simplify the above equation, $\tilde {F}_p\simeq \tilde {I}_p\simeq \tilde {d}/\tilde {b}$ by the ultra-narrow slit width $\tilde {d}$, and $1/N\simeq 0$ due to extremely overdense condition of metal in the THz band and below. Thus, Eq. (14) can be further simplified to

Then, a necessary condition for the high filed enhancement can be apparently observed from Eq. (15) with $J_p^{'}(2\pi \tilde {b})=0$. In other words, the circular cavity circumference $2\pi b$ and the incident wavelength $\lambda$ should satisfy the condition for exciting $p$-modes in a circular waveguide. Unsurprisingly, the field magnification is also influenced by the slit depth and width. And the effects of these critical geometric parameters on the field enhancement are intricate, thus to be further discussed in details.

3. Effects of geometric parameters

We in this section further analyze the relations between the SRR geometric size and field enhancement for normal incidence theoretically, based on the model developed in Sec. 2, and numerically confirmed by the finite element method. In the numerical simulation, the incident EM wave is a TM-polarized plane wave with unit amplitude, and the simulation area is in a range of several wavelengths around the SRR, with a perfect matched layer (PML) boundary condition. The metal permittivity $\varepsilon _m$ in the analysis is approximated by the Drude model

3.1 Dependence on circular cavity inner perimeter

We then investigate the relation between the circular cavity inner perimeter and the EM signal gain, a logarithm form of the field enhancement factor, i.e., $20\lg (|E/E_0|)$ or $20\lg (|H/H_0|)$. Other parameters are the thickness $\tilde {t}=0.1$, the slit width $\tilde {d}=0.01$ and depth $\tilde {h}=0.25$, and the side metal width $\tilde {a}=0.20$ (see also Fig. 1). Theoretical and simulated results for $0.04<2\pi \tilde {b}<6$, where the lower limit of 0.04 is near the minimum circumference with the fixed slit width $\tilde {d}=0.01$, at $f_1=5$ THz, $f_2=0.4$ THz, and $f_3=5$ GHz, are demonstrated in Fig. 2(a) by the red curves and the yellow, black and cyan dots, respectively. Clearly, numerical simulations reproduce resonance characteristics predicted by the theory, though with certain slight deviations caused by the approximations made in the theory and the rounding effect at the entrance of the slit in the simulation. In particular, when $2\pi \tilde {b}<1$, the difference between the theoretical results and the numerical simulations gets more pronounced, mainly due to the assumption made in Eq. (1) of a negligible slit effect on the circular cavity is no longer valid. In addition, the gain of EM signals obtained from the theory and simulations at $5$ GHz and $0.4$ THz are almost comparable, while the gain at $5$ THz is significantly weaker. It should be due to the fact that at $5$ THz the slit width $d=0.01\lambda _3=600$ nm $\sim \lambda _{skin}$ (the skin-depth of the metal), a considerable amount of EM energy penetrates into the metal interior, relative to that left in the slit, and thus weakens the field enhancement. Both theory and simulation indicate that up to tens of dBs signal gain can be achieved when $2\pi \tilde {b}$ is exactly at the $q$-th zero of the $p$-th order Bessel function, consistent with the analysis of Eq. (15) in Sec. 2. The physical insight behind this relation is that such a circular cavity circumference enables the incident wave to excite the circular waveguide eigenmodes of $(p,q)$ , with $p$ and $q$ corresponding to radial and angular mode numbers.

 

Fig. 2. (a) Dependence of signal field gain in the circular cavity on its circumference $2\pi \tilde {b}$. The red curve, as well as the yellow, black and cyan dots, are obtained from Eq. (14), as well as simulations at 5 THz, 0.4 THz, and 5 GHz, respectively. The white/green shaded regions indicate the coupling/decoupling of the Fabry-Perot mode in the slit and the waveguide mode inside the circular cavity. The number of $(p,q)$ at the curve peaks corresponds to the $q$-th zero of the $p$ order Bessel function, i.e., the $(p,q)$ eigenmode in the circular waveguide. Moreover. The inset plot in (a) shows dependence of the received signal gain at $0.4$ THz in the slit-circular connection for scaled down SRR with $\tilde {d}=0.001$, $\tilde {h}=0.025$, $\tilde {t}=0.01$, and $\tilde {a}=0.02$. (b), (c) and (d) are the field distributions for $2\pi \tilde {b}=1.84$, $3.8$ and $0.2$ at $0.4$ THz, respectively. Other dimensionless parameters, except for those of the inset in (a), are the thickness $\tilde {t}=0.1$, the slit width $\tilde {d}=0.01$ and depth $\tilde {h}=0.25$, and the side metal width $\tilde {a}=0.20$. (e) The field distribution corresponding to the LC resonance at $2\pi \tilde {b}=0.11$ shown in the inset in (a).

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The high signal gain at resonance is closely related to the phase difference between the Fabry-Perot mode in the slit and the waveguide mode inside the circular cavity at their interface ($r=b, \mid \theta \mid \leq d/2b$). It is shown that as the circular cavity circumference varies, the two modes shift between in-phase and anti-phase states, i.e., the phase difference is either $0^{\circ }$ or $180^{\circ }$, indicated by the white and green shaded zones in Fig. 2(a), respectively. Such an entanglement between the two modes leads to the gain curves in Fig. 2(a) asymmetric, as the Fano resonance with the Fano parameter $-\infty <F <-1$ [54,55]. Moreover, it can be found that signal gain maxima occur in the in-phase side near the transition from in-phase to anti-phase, while minima are in the anti-phase side near the transition from anti-phase to in-phase. Unambiguously, the field are enhanced by in-phase coupling of the Fabry-Perot mode in the slit and the waveguide mode inside the circular cavity, shown by two typical cases at $0.4$ THz of $2\pi \tilde {b}=1.84$, $3.83$ in Figs. 2(b) and (c), for the $(1,1)$ and $(0,1)$ modes, respectively. It can be seen that for the modes in the circular cavity, the fields are enhanced by factors of $65$ (with a gain $\approx 35$ dB, Fig. 2(b)) and $10$ (with a gain $\approx 20$ dB, Fig. 2(c)), respectively. Moreover, the field distribution in Fig. 2(b) and (c) is almost the same as that of standard circular waveguide modes [48,49], inspiring us that it should be feasible to transmit the enhanced THz signal by connecting the SRR with an ordinary circular waveguide. It is worth pointing out that no resonance occurs at the transition of the in-phase to anti-phase with the circumference of the circular cavity at approximately one-tenth of a wavelength. Nevertheless, the electric field is stronger at this point than that at other non-resonant locations, as shown in Fig. 2(a), although mainly concentrated in the slit with partial leakage into the circular cavity. In the limit with the diameter of the circular cavity shrinking to the same length as the slit width, the SRR structure simply reduces to a slit with a half round end, similar to that in the groove with the field enhancement in the slit reaches the maximum [39]. Clearly, this limit is due to the fact that the other geometric dimensions are fixed at $\tilde {d}=0.01$, $\tilde {h}=0.25$, $\tilde {t}=0.1$ and $\tilde {a}=0.2$. As a result, the depth of the slit $h$ becomes very long as the size of the circular cavity $2\pi \tilde {b}$ shrinks significantly to one-tenth of a wavelength, to cause SRR geometry distortion, as shown in Fig. 2(d). To avoid the effect, we then scale down the SRR size by a factor of $10$ in this regime and indeed observe a typical LC resonance at $2\pi \tilde {b}=0.11$, as shown in the inset in Fig. 2(a). The electric field distribution of the LC resonance is shown in Fig. 2(e). One can see a field enhancement by a factor of $80$ (corresponding to a gain of $\sim 40$ dB), nevertheless in the slit.

Clearly, there are three main differences between the coupling resonance here and the typical LC resonance in SRR [34,36,40–44]. First, the size of the resonator supporting LC resonance is around a tenth of a wavelength, while those supporting coupling resonance are in several wavelengths. Second, the resonant wavelength in LC resonance is independently regulated by the slit and the cavity [43], while coupling resonance is jointly determined by the slit and the cavity. And most critically, the enhanced electric field in LC resonance is mainly concentrated in the slit with only a minor part of the field leaks into the cavity, while the enhanced field in the coupling resonance exhibits a remarkable waveguide pattern in the circular cavity, which facilitates further transmission of the enhanced signal. Furthermore, in terms of field enhancement, the signal gain of the coupling resonance in Fig. 2(b) is on the same order of magnitude as the LC resonance, as shown in Fig. 2(e).

It is the coupling effect that allows the resonance enhancement of the Fabry-Perot mode in the slit to be efficiently transmitted into the circular cavity, and changes the resonance conditions of this mode. We plot in Figs. 3(a) and (b) the magnetic field and phase distributions on the central axis for SRR with parameters are the same as in Fig. 2(b) except the slit depth $\tilde {h}$ changed to $0.5$ to meet the resonance condition for the Fabry-Perot mode in a single slit. We can find that the mode in slit is fully coupled to the circular waveguide mode $(1,1)$ with the phase in the lower half of the slit and upper half of the circular cavity clearly in-phase (both $\sim 120^{\circ }$, as shown in Fig. 3(b)). In this case, the circular cavity acts as an extension of the slit, changing the resonance conditions of the mode in the slit, and resulting in no resonance occurring in the slit (Fig. 3(a)). When the slit depth satisfies the resonance condition at the coupled state, the coupling effect allows the resonant EM modes in the slit to be effectively transmitted into the circular cavity, so that resonance occurs in both the slit and the circular cavity, as shown in Fig. 2(b). To further validate the coupling effect, we change the circular cavity circumference $2\pi \tilde {b}$ in Fig. 3(a) to $2.00$, and the field and phase distributions on the central axis are shown in Fig. 3(c) and (d), respectively. It is obviously an anti-phase case, as shown in Fig. 3(d) where the phase in the lower part of the slit is $160^{\circ }$ while that in the upper half of the circular cavity is $-20^{\circ }$. The two modes are then decoupled with each other, and the EM is thus confined in the slit and hardly transmitted into the circular cavity. As clearly shown in Fig. 3(c), the resonance clearly emerges in the slit, indicating the Fabry-Perot modes in the slit obey their own resonant rules ($\tilde {h}=0.5$) without interfering with the circular cavity. In addition, a comparison between the single mode resonance in Fig. 3(c) and the two modes coupled resonance in Fig. 2(b) demonstrates the coupling effect can enhance the signal field more significantly.

 

Fig. 3. The magnetic field distribution at $0.4$ THz for SRRs with (a) $2\pi \tilde {b}=1.84$, $\tilde {h}=0.50$, and (c) $2\pi \tilde {b}=2.00$, $\tilde {h}=0.50$, respectively; (b) and (d) correspond to the phase distribution on the central axis ($x/\lambda =0$) in Cases (a) and (c), respectively. Other parameters are the thickness $\tilde {t}=0.1$, the slit width $\tilde {d}=0.01$ and the side metal width $\tilde {a}=0.20$, respectively.

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Figure 4(a) shows the numerical $S11$ parameters of the SRR with the same geometric parameters as in Fig. 2(b). As the incident wave frequency increases, i.e., the wavelength decreases, equivalent to enlarge the SRR, $S11$ should increase monotonically. However, a depression in the $S11$ curve is found near $0.4$ THz, which is caused by the resonance enhancement within the SRR (as shown in Fig. 2(b)). This resonance mechanism greatly enhances the electric and magnetic fields in the SRR, with their phases are always $90$ degrees apart, as clearly observed in Fig. 4(b). In the slit (above the green dashed line), the phase of $E_x$ is $-120^{\circ }$, the phase of $E_y$ shifts between $-120^{\circ }$ and $60^{\circ }$, and the phase of $H_z$ is $150^{\circ }$. In the circular cavity (below the green dashed line), the phases of $E_x$ and $E_y$ are $60^{\circ }$ and $-120^{\circ }$, respectively, while the phase of $H_z$ changes from $150^{\circ }$ to $-30^{\circ }$. Such a phase difference results in no significant enhancement of the power density $E\times H$ within the SRR, similar to what has been observed in field enhancement studies in slits [29]. From the energy conservation perspective, the coupling resonance weakens the energy of the reflected wave, causing a depression in $S11$, and then more energy flows into the circular cavity, establishing typical waveguide modes where the electric and magnetic fields are significantly enhanced, but not the energy.

 

Fig. 4. (a) $S11$ parameters of SRR with the same geometric parameters as in Fig. 2(b). (b) The phases of $E_x$, $E_y$ and $H_z$ on the central axis for the case in Fig. 2(b) in black, red, and blue lines, respectively. The green dashed line in (b) marks the junction between the slit and the circular cavity.

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3.2 Dependence on slit depth

Then, we discuss the effect of the slit depth $\tilde {h}$ on the signal gain with the circular cavity circumference $2\pi \tilde {b}$ and thickness $\tilde {t}$, the slit width $\tilde {d}$, and the metal width $\tilde {a}$ fixed at $3.83$, $0.10$, $0.01$ and $0.20$, respectively. In Fig. 5(a), the red curves and the yellow, black and green dots are the theoretical and simulated results at $5$ THz, $0.4$ THz, and $5$ GHz, respectively. Theory is again confirmed by good agreements with the simulation, with slight discrepancies again mainly from the approximations in the theory and the rounding effect in the simulation. And unsurprisingly, the signal gain at $5$ THz is still weak as relatively considerable EM energy penetrates into the metal interior. Both analytic and numerical results in Fig. 5(a) suggest that the signal gain in circular cavities vary periodically vs. the slit depth with a period of $0.5\lambda$, indicating a Fabry-Perot-like behavior. A similar but anti-phase periodic relation is exhibited by the curves of phase difference between Fabry-Perot modes in the slit and waveguide modes in circular cavity vs. the slit depth in Fig. 5(b). And the magnitude of change in phase difference is as small as $0.08^{\circ }$, indicating that the two modes are always coupled in this case, almost independent of the slit depth variation. It is this coupling relation that keeps the field enhancement in the circular cavity varying periodically with the slit depth, closely resembling the familiar Fabry-Perot resonance mode in an isolated slit where the field enhancement vs. the slit depth also has a period of $0.5\lambda$ due to the standing wave interference in the slit caused by the impedance mismatch at both ends of the slit, as reported in near infrared band [20]. Furthermore, our results show that except for the circular perimeter, no other geometric parameters affect the coupling between the two modes.

 

Fig. 5. (a) Dependence of signal field gain in the circular cavity on the slit depth $\tilde {h}$ with the slit width $\tilde {d}=0.01$, the metal width $\tilde {a}=0.20$, the circular cavity circumference $2\pi \tilde {b}=3.83$ and thickness $\tilde {t}=0.10$. The red curve and the yellow, black and cyan dots are obtained from Eq. (14) and simulations at $5$ THz, $0.4$ THz, and $5$ GHz, respectively. (b) The corresponding phase differences between the Fabry-Perot mode in the slit and the waveguide mode inside the circular cavity at their interfaces. (c) The magnetic field distribution at $0.4$ THz for the SRR structure with $\tilde {h}=0.33$ and other parameters the same as those in panel (a).

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Besides, unlike the typical Fabry-Perot resonance condition $\tilde {h}=0.5n$, the maximal signal gain in the circular cavity approximately appears at $\tilde {h}=0.33+0.5n$ with a shift of $1/3$ due to the coupling. And such a coupling is modulated by the circular cavity circumference, so that the signal gain at different resonant positions in Fig. 2(a) is not the same. Then, this modulation mechanism can assist in optimizing or selecting the operating modes within circular cavity by designing the slit depth. For example, if the cavity is expected to operate in the $(0, 1)$ mode, then the slit depth $\tilde {h}$ should preferably be designed to be $0.33+0.5n$ where the signal field is enhanced to $130$ times (with a gain $>40$ $dB$), as shown in Fig. 5(c), rather than $0.25+0.5n$ where the signal field in $(0,1)$ mode is only enhanced by a factor of about $10$, as shown in Fig. 2(c). And for $\tilde {h}=0.25+0.5n$, the optimal operation mode is a $(1,1)$ mode with an enhancement factor of $\simeq 65$, seen in Fig. 2(b).

3.3 Dependence on slit width

We in this subsection explore the dependence of field enhancement in the circular cavity on the slit width, which is closely related to the field enhancement capability of the SRR structure. To search for the maximal signal gain at various slit widths, we show in Fig. 6(a) the curves of signal field gain obtained from the simulation at $0.4$ THz with slit depth $\tilde {h}$ over a period for $\tilde {d}=0.008$, $0.006$, $0.004$, $0.002$ and $0.001$, indicated by blue, orange, green, purple and hollow pink circles, respectively. Other parameters are the circular cavity circumference $2\pi \tilde {b}=1.84$ and thickness $\tilde {t}=0.10$, the width of the metal on sides of the slit $\tilde {a}=0.20$. The corresponding theoretical results are separately shown in Fig. 6(b) with curves in the same colours as in the simulation for better visual clarity. Simulation and theory consistently demonstrate that the resonant points for the slit depth $\tilde {h}$ move from $0.26$ to $0.41$ as the slit width $\tilde {d}$ shrinks from $0.08$ to $0.001$. This is compatible with the metal-dielectric-metal theory [56,57], which suggests that the reduction in slit width increases the SPPs wavelength, subsequently shifting the slit depth resonance position to larger values. However, there is a notable difference between the theory and simulations that the ultimate field enhancement in the simulation occurs at $\tilde {h}=0.28$ for $\tilde {d}=0.004$, but at $\tilde {h}=0.27$ for $\tilde {d}=0.008$ theoretically. This discrepancy is mainly attributed to the assumptions for the slit width in the theory. On the one hand, the assumption in Eq. (1) that $H^c_z$ is a superposition of ideal circular waveguide modes requires the slit to be as narrow as possible. On the other hand, the shrinkage of the slit width to the scale comparable to the metal skin depth $\lambda _{skin}$ deactivates Eq. (2). Thus, the approximations made in theory fail in regions where $d$ is too large or too small. And that is also the reason for the greater difference between the theory and the simulation in the case of $\tilde {d}=0.004$ than of $\tilde {d}=0.006$, indicating $0.004$ may be around the theory lower limit. With the slit width $\tilde {d}$ being further reduced to $0.001$, the difference between theory and simulation is greater and the pink curve in Fig. 6(b) is for reference only. Moreover, the numerical simulations for $\tilde {d}=0.001$, denoted by the hollow pink circles in Fig. 6(a), reveal that except for the resonance point $\tilde {h} = 0.41$, the SRRs with other slit depth are no longer able to enhance the field, i.e., the corresponding gains $<0$. This may be due to the fact that such a narrow-slit causes substantial EM energy not to enter the circular cavity. It should also be noted that the dependence of the field enhancement on the slit depth shown by the hollow pink circles appear to be haphazard because the logarithmic coordinates magnify the difference between the values when the gain is less than $0$. In fact, despite the gain $<0$, the relationship between the field enhancement and the slit depth still exhibits Fabry-Perot-like characteristic, as judged from the contours of the hollow pink circles.

 

Fig. 6. (a) The variation of field enhancement vs. the slit depth $\tilde {h}$ with $\tilde {d}=0.008$, $0.006$, $0.004$, $0.002$, and $0.001$ at $0.4$ THz obtained in simulations, and plotted by blue, orange, green, purple, and hollow pink circles, respectively. Other geometric parameters are the same as those in Fig. 2(b). (b) The corresponding theoretical results, denoted by curves in the same colors in (a). (c) Dependence of signal field gain in the circular cavity on the slit width $\tilde {d}$ with $\tilde {h}=0.28$, $\tilde {a}=0.20$, $2\pi \tilde {b}=1.84$, and $\tilde {t}=0.10$. The red curve and the yellow, black and cyan dots are obtained from Eq. (14) and simulations at $5$ THz, $0.4$ THz, and $5$ GHz, respectively. (d) Magnetic field distribution in the case of ultimate field enhancement at $0.4$ THz in panel (c).

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In Fig. 6(c), the field enhancement in the circular cavity is given directly as a function of the slit width to reveal their relation more explicitly. The geometric parameters are chosen the same as in the ultimate field enhancement case at $0.4$ THz in Fig. 6(a), except for the slit width $\tilde {d}$. The theory and the simulations at $5$ THz, $0.4$ THz, and $5$ GHz, noted by the red curve and yellow, black and green dots, respectively, consistently reveal that the field enhancement first goes up and then goes down as the slit width $\tilde {d}$ decreases. These findings are similar to that in the single slit case [30], as well as in the slit of SRR [36], recently observed in experiment. At the beginning, the cavity effect causes an inverse correlation between field enhancement and slit width. The narrower the slit is, the more profoundly the impedance mismatches between the inside and outside of the SRR, and the better quality of the cavity is. However, overly narrow slits result in less EM wave energy entering the SRR and weakening the field enhancement in the circular cavity. Therefore, there exists an optimal slit width $\tilde {d}_{op}$ that enables the ultimate field enhancement of the SRR. Simulations for $0.4$ THz and $5$ GHz indicate that $\tilde {d}_{op}$ appears near $0.004$, while $\tilde {d}_{op}$ in theory appears at $0.006$. This inevitable difference between simulation and theory arises mainly from the assumptions for the slit width in the theory. It is nevertheless acceptable for not refuting accuracy and advantage of the theory in predicting the effect of structural geometry on the field enhancement. The simulations at $5$ THz are special because at this regime the slit width $d$ is reduced to $<0.01\lambda _3=600 nm\sim \lambda _{skin}$. In the slit with a width on the order of $\lambda _{skin}$, the EM energy penetrating into the metal interior is comparable to that remaining in the slit, which may account for the appearance of the ultimate field enhancement at $5$ THz at a larger $\tilde {d}=0.008$ as well as the slightly different dependence of field enhancement on the slit width. In addition, the search for the maximum signal gain at different slit widths in Fig. 6(a) is exemplified by $0.4$ THz, which may vary slightly at other frequencies hence the strongest ultimate field enhancement at $0.4$ THz in Fig. 6(c).

The ultimate field enhancement at $0.4$ THz available with the proposed SRR is presented in Fig. 6(d), where the $(1,1)$ mode is enhanced by a factor of $150$ (with a gain $>43$ $dB$). And the enhanced field distribution in the range of several wavelength matches subwavelength spatial resolution of existing THz imaging techniques [7,45–47]. It then leads to possible directly detecting the enhanced THz signal without resorting to other complicated methods.

3.4 Strong robustness to incident angle

The practicality-related incidence angle effect on field enhancement is investigated with the geometric parameters fixed at the ultimate field enhancement of the $(1,1)$ mode at $0.4$ THz, with the circular cavity circumference $2\pi \tilde {b}=1.84$ and thickness $\tilde {t}=0.10$, the width of the metal on sides of the slit $\tilde {a}=0.20$, the width of the slit $\tilde {d}=0.004$, and the depth of the slit $\tilde {h}=0.28$. In Fig. 7(a), the red curves and yellow, black, and green dots depict the signal gain variation with the incident angle $\phi$ obtained from the theory and the simulations at $5$ THz, $0.4$ THz, and $5$ GHz, respectively. The simulations at the three frequencies suggest that the signal field enhancement varies slightly in the entire incident angle range, and reaches a maximum at the normal incidence. Since the geometric parameters are fixed at the ultimate field enhancement of $0.4$ THz, the signal gain at this frequency is maximum, between $38-43$ dB. The theoretical signal field gain also exhibits a similar cosine relation, but the variation is so tiny that it shows almost no change with the incident angle, due to no edge effect considered in the theoretical model. Moreover, the theoretical result is lower than the simulation mainly because of the rounding processing introduced in the simulation. In Fig. 7(a), it can be seen that the proposed SRR is capable of achieving signal gains of tens of dBs at arbitrary incident angles in the THz or lower frequency bands. Even with parallel incidence, the magnetic field at $0.4$ THz is still enhanced by a factor of $\simeq 80$ (with a gain of $38$ dB), and the corresponding electric field is enhanced by a factor of about $70$, presented by Figs. 7(b) and (c), respectively. The electric field gain is slightly lower than magnetic field because a portion of electric energy is concentrated in the slit by the SSPPs, resulting highly-localized electric field $>80$, as shown in the inset in Fig. 7(c). Such a strong tolerance to the incident angle gives the proposed SRR great potential to be applied in practice.

 

Fig. 7. (a) Dependence of field enhancement on the incident angle $\theta$. The red curve and the yellow, black and cyan dots are obtained from Eq. (14) and simulations at $5$ THz, $0.4$ THz, and $5$ GHz, respectively. The geometric parameters are the circular cavity circumference $2\pi \tilde {b}=1.84$ and thickness $\tilde {t}=0.10$, the width of the metal on sides of the slit $\tilde {a}=0.20$, the width of the slit $\tilde {d}=0.004$, and the depth of the slit $\tilde {h}=0.28$. (b) The magnetic and (c) electric field distributions for the parallel incidence at $0.4$ THz, The inset in (c) shows the details of the electric field distribution in the slit.

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4. Conclusion

We have achieved remarkable omnidirectional EM signal gain up to $\sim 40$ dB at $0.4$ THz, implemented by the SRR featuring subwavelength slit, anticipated for applications in future communication system to enhance, detect and transmit weak THz signals. These applications of the proposed structure are permitted by the following benefits in technology: (i) equally and significantly enhancing THz electric and magnetic fields simultaneously; (ii) enabling direct detection with current THz imaging technology; and (iii) delivering a viable scheme for enhanced signal transmission. Moreover, a theory model for analyzing the field enhancement properties of the SRRs in THz and lower bands is developed based on the Bruijn method, and potentially contributed to the design of other novel THz plasmonic devices. The effects of SRR geometry on the field enhancement are also explored thoroughly by theory and simulations at $f_1=5$ THz, $f_2=0.4$ THz, and $f_3=5$ GHz, confirming that the proposed SRR is capable of achieving field enhancement at a wide range of desired frequencies by scaling its size accordingly. The theory and simulations together illustrate that the relation between the field enhancement and the circular cavity circumference exhibits a Fano-resonance-like characteristic, due to the entanglement of the Fabry-Perot mode in the slit and the waveguide mode in the circular cavity. As the inner circumference of the circular cavity changes, the two modes switch between in-phase and anti-phase states. And the coupling of the two modes is almost independent of the slit depth $\tilde {h}$ variation. In the in-phase regime, the relation between the signal gain in the circular cavity and the slit depth $\tilde {h}$ behaves Fabry-Perot-like resonance with a period of $0.5$ and a peak of $\simeq 40$ dB occurring near $0.33 + 0.5n$ in the $(0,1)$ mode. In addition, as the slit width $\tilde {d}$ decreases, the field enhancement first increases due to the cavity effect, and then decreases since considerable energy of incident radiation is blocked from the SRR due to the excessive reduction of the slit width. And at $\tilde {d}\simeq 0.004$, there appears an ultimate field enhancement at $0.4$ THz of up to $150$ times (with a gain of $>43$ $dB$) in the $(1,1)$ mode. Moreover, the SRR is highly robust to the incident angle, supporting its practical feasibility. Notably, other geometric parameters not discussed here, such as the circular cavity thickness $t$, the metal width $a$ on the sides of the slit, have little effect on the field enhancement as long as $h\geq t>\lambda _{skin}$ and $\lambda >a\gg d$.

It should also be pointed out that, the LC resonance can also be excited with similar maximal signal gain of tens dBs, if the SRR size shrinks to a level of $\sim 0.1\lambda$, as shown in Fig. 2(a). However, the LC and coupling resonances are in various mechanisms. In the LC resonance, the slit acts as a capacitor and the ring (the circular cavity) acts as an inductor, analogous to a classical LC circuit, and thus the excited electric and magnetic fields are distributed in the slit and the cavity, respectively, similar to that in a circuit. In contrast, in the coupling resonance discussed in this paper, the slit and the ring can either be coupled or decoupled. The resonance, particularly the Fabry–Perot mode, is excited in the slit and coupled to the circular cavity only for the coupled state with appropriate parameters to make the fields in the slit and in the ring in-phase. If the fields in the two areas are anti-phase, they are decoupled with the resonance confined only in the slit. Furthermore, there are several advantages in communication applications for the coupling resonance studied in this work in comparison with the typical LC resonance. $(i)$ Most significantly, the enhanced electric field in LC resonance is majorly confined in the slit with only a minor leakage to the circular cavity, while the coupling resonance shows a remarkable waveguide pattern in the circular cavity, which facilitates further transmission of the coupling resonant signal. $(ii)$ The feature of the coupling resonance can be jointly modulated by the slit and the cavity, to make an easy approach to the desired SRR parameters, while the LC resonance is respectively regulated by the slit and the circular cavity. $(iii)$ The SRR size for the coupling resonance is tens of times bigger than that for the LC resonance, much easily to be manufactured.

In summary, the proposed SRR provides a feasible option for the enhancement of weak THz EM signals with a remarkable advantage of easy detection and transmission. Such advanced features facilitate a comprehensive approach to address the issue of high attenuation in the THz regime, and provide a new approach for next-generation communication systems. Furthermore, our development of the Bruijn method possibly provide a valuable prototype for the design of other subwavelength metallic components suitable for THz EM sensing and EM nonlinearity.