## Abstract

In this paper, we propose and numerically demonstrate a new way to realize superfocusing of terahertz waves via the spoof surface plasmons (SSP). With the assist of a modified subwavelength metallic grating, a near-field rapid oscillation can be formed, originating from the Fabry–Perot resonances due to the reflection of SSP waves at terminations. We show that the field pattern of oscillation on textured metallic surface can be engineered by adjusting groove width and grating number. This produces a desired modulation of phase and amplitude for the radiationless electromagnetic interference (REI) focusing. The effective focusing depth through the corrugated metal is evaluated by the full-width-half-maximum (FWHM) beamwidth. At the situation of third-order Fabry–Perot resonance, the FWMH reaches up to 0.069λ at a distance of 0.1λ, improving the beamwidth by more than 540% compared with a single slit. The FWHM is optimized to 0.06λ as the order of Fabry–Perot resonance becomes seven, leading to the superfocusing metric of 1.67. On the basis of this, we further show the focusing ability can be held on the ultra-thin metallic grating. Two-dimensional subwavelength focusing behavior is also numerically verified. Our study may extend the working distance of sensing and super-resolution imaging devices at terahertz frequency.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## Corrections

21 August 2018: A typographical correction was made to the author listing.

## 1. Introduction

Superfocusing of electromagnetic (EM) waves always attracts considerable attention owing to its intriguing potential in developing super-resolution imaging, noncontact sensing, photolithography, information applications, etc. However, the diffraction limit [1], resulting from the propagating loss of evanescent waves, holds back the breakthrough of focusing ability in conventional devices below approximately half of the wavelength. To overcome the diffraction limit and localize EM energy into a deep subwavelength spot, a host of structures have been designed and extensively studied, such as the sharply metallic wire [2], subwavelength size apertures [3], tapered waveguides [4]. Unfortunately, most of these designs tend to have limited working distance, requiring highly precise calibration in the practical usage. By using surface plasmons, the superlens [5–10] at optical wavelength has the ability to magnify evanescent waves at the interface between negative (refers to the values of permittivity) and positive materials. The decay of evanescent waves can be compensated, contributing to a much larger focal length. However, lack of negative materials at other frequency ranges still poses great challenges in extending the applications area of superlens. Inspired by the Fresnel plates, an alternative conception of radiationless electromagnetic interference (REI) was employed to converge the incident EM energy away from the focusing elements [11]. Originating from simultaneously manipulating the amplitude and phase profile of EM waves, this approach excites a rapid transverse oscillation in the near field. This well-deigned oscillation suppresses the far-field radiation, produces destructive interference, and then forces a deep subwavelength scale central lobe with a relatively long distance. Various designs have been proposed to complete the REI focusing process in the microwave and optical frequency range, including plasmonic waveguide arrays [12], interdigitated copper capacitors [13], annular slots [14,15], and dipole slot arrays [16,17]. Recently, the strong Fano resonance and electromagnetically induced transparency [18] that originate from interference between a broad state and narrow state have attracted great attentions due to the unique properties. The Fano resonance has been demonstrated as a novel REI mechanism to focusing EM waves with the enhancement of field intensity [19,20]. However, due to the sophisticated fabrication process and lack of natural plasmonic materials, it is hard to extend most of them into the terahertz (THz) frequency range, where the spectrum resource is very rich.

Based on tailoring the geometrical features of high-conductivity metal, spoof surface plasmons (SSP) [21–24] fills up the blank of plasmonic materials in the THz regime. Such a designed SSP has been widely exploited to manipulate THz waves in subwavelength scale [25–28], including subwavelength waveguiding [29], highly sensitive sensing [30], trapping rainbow [31], and collimating THz laser beams [32]. As for superfocusing of EM energy, many SSP structures have been detailed studied and demonstrated, such as the tapered helical grooves wire [33], tapered corrugated metal wires [25], and tapered domino-plasmon structure [34]. By gradually changing the parameters of SSP structures, the confinement of EM energy can be tailored, leading to the superfocusing of SSP waves. However, these focusing behaviors also face the problem of short focal length, and the focusing depth deteriorates rapidly with the increasing working distance. Note that, the REI focusing based on SSP has not yet been investigated and reported. Recently, SSP-based stripe antennas have been investigated in details, revealing that the highly localized SSP patterns with abrupt phase change can be formed under the condition of Fabry–Perot (FP) resonances [35,36]. It is intuitive to combine the REI principle with SSP, as SSP can produce an essential rapid oscillation within a small fraction of a wavelength in the near field.

In this paper, we propose and demonstrate an effective solution to realize deep subwavelength REI focusing via SSP in the THz range. We reveal that the modified metallic grating allows FP resonances of surface waves, contributing to the near-field rapid oscillation with abrupt change between adjacent lobes. By adjusting the groove width of the metallic grating, the amplitude of each lobe can be modulated. At the situation of third-order FP resonance, a focusing behavior of 0.069λ at a distance of 0.1λ is demonstrated. The focusing performance can be improved by increasing the order of FP resonance and optimizing the groove width. On the basic of this, we further propose the ultra-thin SSP structure to realize two-dimensional focusing of THz waves, which could mitigate the limited working distance of near-field imaging and sensing devices.

## 2. Dispersive relations of SSP waves

The schematic of a subwavelength metallic grating is illustrated in Fig. 1(a), which is the most common SSP structure. Note that, as our principle is based on the resonances of SSP, it can be extended to any proper SSP structures. The periodicity and groove width are described by *p* and *d*. The height of grooves is *h*. To simplify the theoretical derivation, the thickness of the grating in the y direction is assumed infinite and the metal is treated as perfect electric conductor (PEC). The classical theory that can analyze dispersion of SSP is the mode expansion method (MEM) [37,38]. Based on the MEM, a generalized derivation of the dispersive equation is performed. After matching tangential field components at the grating-dielectric interface and enforcing PEC boundary condition on the bottom of grating, we can get an eigenvalue equation describing the relation between transverse momentum *k _{x}* and operating frequency,

*k*,

_{xn}= k_{x}+ 2nπ/p*n*is the diffraction order;

*k*=

_{0}*2πf/c*,

*k*is the wavenumber in air,

_{0}*f*and

*c*are the calculated frequency and velocity of light in vacuum. Based on Eq. (1), we plot dispersive curves of fundamental SSP mode in Fig. 1(b). The parameters of the grating are set as

*p*= 0.06 mm,

*d*= 0.03 mm, respectively, and the groove height

*h*varies from 0.17 mm to 0.23 mm, and finally reaches 0.29 mm. It is observed that the dispersive curves of SSP significantly depart from the light line, and the departure becomes greater as the frequency increases. Note that, above the asymptotic frequency, there is a forbidden band for each dispersive curve, where the fundamental SSP mode cannot exist. The asymptotic frequency decreases with increasing

*h*. To visualize the SSP distribution, the frequency solver of commercial software COMSOL Multiphysics is employed to calculate the evolution of EM waves. In Figs. 1(c) and 1(d), we show the y component of magnetic fields in the case of

*h*= 0.23 mm. The patterns in Figs. 1(c) and 1(d) are corresponding to

*f*= 0.25 THz and

*f*= 0.295 THz, respectively. From these magnetic field distributions, we can see that SSP is a radiationless wave with rapid variation in the transverse direction, and the confinement of energy enhances with increasing frequency.

We further investigate the dispersive relation with the variation of groove width, as illustrated in Fig. 2(a). The height and periodicity of the groove are set as *h* = 0.23mm and *p* = 0.06 mm. As the groove width changes from 0.01 to 0.02, 0.03, 0.04, 0.05 mm, the cutoff frequency slightly decreases. In addition, the dispersion curves with narrower groove width have less departure from light line, and abruptly turn flat with increase of frequency. Therefore, compared with the case of *d* = 0.03mm, the SSP supported by narrower groove has the lower group velocity and less localization near the asymptotic frequency. To give a clear demonstration, the spatial variation of the magnetic field associated with SSP propagating through different gratings is displayed in Fig. 2(b). The magnetic field amplitude at 0.06 mm away from gratings is illustrated Fig. 2(c). The calculated frequency is 0.295 THz and the groove width of the structural changes in the order of 0.03-0.02-0.03 mm. As expected from dispersive relation, the modes have less confinement and higher field intensity in the part of *d* = 0.02 mm. The fact is that the wavelength λ*s* and amplitude of SSP waves can be tuned by engineering the geometry of grating, which is very essential to realize REI focusing.

## 3. Verification of superfocusing effect through SSP

#### 3.1 Combining the FP resonance with REI focusing principle

In order to produce a near-field rapid oscillation, we designed a truncated SSP structure to extract only a few wavelengths from SSP eigenmodes. A schematic of this structure is presented in Fig. 3(a), which is similar with our previous study about SSP probe [39]. The whole structure consists of three parts: two coupling slits, a center grating with uniform groove height, two symmetrical gradient gratings used for cutoff surface waves. The parameters of the structure are described by, *N*, the number of the center grooves, *d _{1}-d_{N}*, the width of each groove in the center grating,

*h*, the height of grooves,

*θ*, the gradient angle,

*d*, the width of gradient grooves and coupling slits,

*p*, the period of all the grooves. The period number of the gradient grating is three, and the gradient angle can be tuned to fulfill the cutoff conditions. The gradient structure can also avoid field deterioration caused by abrupt structure change.

Part of THz waves radiated from coupling slits will launch SSP waves. The excited surface modes travel back and forth along the center gratings, bouncing between two gradient gratings. There will be a well-known interference of SSP waves, when the following phase condition is fulfilled:

in which the*k*

_{ssp}is the traverse wavevector of SSP waves at each groove,

*L*is the total length of the truncated grating,

*Φ*is the phase shift owing to the partial reflection of SSP waves at both terminations,

*m*is a positive integer representing the resonance order. This interference is called the FP resonance, which produces a near-field rapid traverse oscillation with great field enhancement [35].

To realize the REI focusing phenomenon, at least, the standing SSP waves should have three lobes, i.e., two side lobes have destructive interference with the central lobe in the z direction. We first consider this situation by forming a third-order FP resonance. The parameters of the proposed SSP structure are set as: *N* = 3, *d _{1}* =

*d*=

_{2}*d*= 30 μm,

_{3}*h*= 230 μm,

*p*= 60 μm,

*d*= 30 μm,

*θ*= 40°. As the FP resonance occurs whenever a multiple of half the SSP wavelength equals the grating length, the investigated frequency is chosen between 0.3 THz and 0.308 THz, near the asymptotic frequency, where half wavelength of SSP waves is approximately equal to the period of grating. Meanwhile, the SSP waves will be cutoff in the first groove of the gradient gratings. Therefore, three gradient grooves are sufficient to cutoff and reflect SSP waves, and further increasing the gradient grooves has very limited influence on the working performance. The metal is treated as PEC and the surrounding dielectric is air. With polarization of electric field along x-axis, a normal plane wave couples into the truncated SSP structure at the bottom of two slits. The total transmission coefficient of EM waves through SSP structure is presented in Fig. 3(b). It is clear to see an asymmetrical transmission peak related to EM resonance behavior. We illustrate two magnetic field (

*H*) patterns in the insert of Fig. 3(b), respectively corresponding to the peak and dip of the transmission curve. One can easily observe third-order of FP resonance in both cases. The asymmetrical transmission behavior is related to the Fano resonances [19]. The waveguide mode (TEM mode) in coupling slits is regarded as the bright mode required by Fano resonance, as the waveguide mode can always be directly excited and radiate from the structure freely. Compared with the waveguide mode, the SSP-based FP resonance has much higher quality (Q) factor, and only exists when the resonance condition is satisfied (Eq. (2)). In this way, the FP resonance provides the condition for dark mode. The instructive and destructive interference between bright mode and dark mode are respectively corresponding to the transmission peak and dip, leading to enhance or suppress radiation.

_{y}Note that, in the case of suppressed radiation, there is an abrupt π-phase jump at each adjacent groove. This phenomenon produces a rapid traverse oscillation near the center grating, and makes each lobe interference destructively in the z direction. The oscillation is essential for realizing REI focusing of EM waves. To visualize the REI focusing behavior, in Fig. 3(c), the amplitude of magnetic field at 0.3032 THz is illustrated. Meanwhile, we plot the normalized field amplitude at different distances away from grating in Fig. 3(d) (this distance is denoted as dz). As we can see, although the amplitude of the central lobe is larger than two satellite lobes, with the increase of dz, the central lobe is gradually overwhelmed by satellite lobes. When the dz reaches 0.05λ (λ is corresponding to free space wavelength), the central lobe is hardly recognized, which leads to undesired REI focusing performance. This result is attributed to the influence of the radiated waves from coupling silts.

To improve the focusing performance, we optimize the intensity distribution of the central lobe by adjusting *d _{2}*. As illustrated in Fig. 4(a), when

*d*varies from 28 μm to 26 μm, 24 μm, 22 μm, 20 μm, 18 μm, and 16 μm, the value of the other parameters keeps unchanged, the transmission curves have a blue shift. This is because the transverse wavevector of SSP waves slightly reduces with the decrease of groove width, as discussed in Fig. 2. Meanwhile, we find the value of Q factor reaches maximum in the case of

_{2}*d*= 20 μm, indicating the strongest resonance behavior. Further decreasing

_{3}*d*will lead to the cutoff effect of the first and third grooves, resulting in the decrease of FP resonance, as shown in the cases of

_{2}*d*= 18 μm and 16 μm. We calculated the field distribution at 0.3052 THz, corresponding to the transmission dip of the

_{3}*d*= 20 μm curve. The magnetic field distribution near the grating is shown in Fig. 4(b). As we can see, what we realized in Fig. 4(b) is actually REI superfocusing effect. Obviously, the magnetic field changes sign between adjacent peaks indicating that there is an abrupt π-phase shift with respect to the neighbor. The amplitude pattern of magnetic field is shown in Fig. 4(c). It is clear that the central lobe has destructive interference with the two side lobes, resulting in a deep subwavelength focusing spot with a relatively large focal length. The Full width at half maximum (FWHM) of the magnetic field intensity (magnitude squared) remarkably reaches 0.069λ at a distance of 0.1λ, as presented in Fig. 4(d). Compared with the focusing property of a single metallic slit, whose groove width is the same as the width of coupling slit, the FWHM achieved by our REI focusing is reduced 540%. That means the focusing depth is significantly enhanced by 540%. Note that, although the SSP structure looks similar with previous study [40], the function and working mechanisms are totally different. The previous study aimed to provide far-field beaming effect, and there is no superfocusing behavior in the near field. These grooves are employed to diffract beams and generate a leaky mode. The leaky modes from all grooves and the slit are in phase. Therefore, the beaming effect coincides with transmission peaks. The grooves of our structure are used for supporting radiationless SSP waves and forming FP resonances. At the situation of FP resonance, side lobes have destructive interference with central lobe, leading to superfocusing behavior. The superfocusing behavior coincides with the transmission dip.

_{3}#### 3.2 Optimizing the focusing depth

For the purpose of optimization, firstly, we decrease the width of the grating to study the variation of FWHM. When *N* = 3, *d _{1}* =

*d*= 10 μm,

_{3}*d*= 6 μm,

_{2}*h*= 243 μm,

*p*= 20 μm,

*d*= 10 μm,

*θ*= 40°, we perform a simulation at the frequency range of 0.3031 THz to 0.3034 THz. The transmission curve is presented in the Fig. 5(a). It is clear to see the bandwidth of the resonance significantly decreases and Q factor becomes much larger than Fig. 4(a). In the insert of Fig. 5(a), we illustrate the field amplitude pattern at 0.30323 THz. It is clear to observe the REI focusing behavior. The FWHM are 0.039λ and 0.067λ at distances of 0.05λ and 0.1λ, as shown in Fig. 5(b). Compared with the results in Fig. 4, the focusing depth at a distance of 0.1λ is slightly enhanced. Although the grating with smaller grooves can produce more rapid transverse oscillation in the near field, the high order spatial harmonics have a relatively larger decay rate. Therefore, the influence of evanescent waves with very high spatial information is very limited at a distance of 0.1λ. Note that, the structure still has an advantage in the case of short focal length. However, with the decrease of groove width, the corresponding fabrication issues should be taken into consideration.

Then we optimize the REI focusing performance by increasing the order of FP resonance. As the operating frequency near the asymptotic frequency, the number of central grooves *N* is approximately equal to order of FP resonance. Increasing *N* will directly increase the order of FP resonance. To produce symmetric satellite lobes, we only consider the odd order of FP resonance. In the situation of an even order of FP resonance, the SSP waves launched by two coupling slits will have destructive interference. For instance, for the fourth-order FP resonance, the phase distributions of SSP waves from left and right silts are respectively “+ - + -,” and “- + -+”. The superposition of SSP waves results in suppressing the field oscillation. Here, we investigate the case of seventh-order FP resonance, corresponding to *N* = 7. The calculated frequency is adjusted to 0.30323 THz. Other parameters of the proposed structure are set as, *d _{1}* =

*d*=

_{2}*d*=

_{6}*d*= 30 μm,

_{7}*d*=

_{3}*d*= 26 μm,

_{5}*d*= 22 μm,

_{4}*h*= 232 μm,

*p*= 60 μm,

*d*= 30 μm,

*θ*= 40°. The superfocusing of EM energy is clearly observed, as presented in Fig. 5(c). The FWHM (red solid line) in Fig. 5(d) significantly reaches 0.06λ at a distance of 0.1λ. The superfocusing metric [41], which is the ratio of focal length and the focal size, is applied to compare our design with previous investigations. Note that, the poor man’s superlens is 0.5 [41] at optical frequency, the typical REI is 1.34 [11,16,41], the REI focusing integrated with Fano resonance is 1.5 [20], and the proposed structure with seven grooves is 1.67. By increasing the order of FP resonance or optimizing groove width, the superfocusing metric can be further enhanced. Therefore, the focusing mechanism by combing FP-resonance with SSP waves has additional advantage of furtherly enhancing focusing ability. With consideration of the real case, we also replace the PEC with copper. Its permittivity is described by a Drude model

*ε(ω) = 1-ω*, where

_{p}/(ω^{2}-iωγ)*ω*= 1.123 × 10

_{p}^{16}Hz is the plasma frequency,

*ω*is the calculated angular frequency, and

*γ*= 1.379 × 10

^{13}Hz is the collision frequency [35]. Here, we set

*ω*= 2π × 3 × 10

^{11}Hz (corresponding to 0.3 THz). The calculated result is presented in Fig. 5(d), related to blue dashed line. We can notice that the metal loss has negligible influence on the focusing depth. The loss issue only decreases the field intensity. For the lossless case, the intensity of focusing field is 2.5 times larger than incident field. When considering the metal loss, the intensity ratio of the focusing field and the incident field is about 0.196. Therefore, the superfocusing mechanism is also feasible in the real metal.

#### 3.3 Two-dimensional focusing behavior

The simulations discussed above are the two-dimensional cases. The thickness of the truncated SSP structure in the y direction is infinite. With consideration of practical usage, we also analyzed the REI focusing behavior supported by the structure with subwavelength thickness *t*, as shown in Fig. 6(a). A Teflon substrate (its real permittivity being 2.1) is used for supporting the metallic grating. Note that, the ultra-thin grating has been extensively studied for years [42], owing to its great potential in plasmonic devices and circuits. The previous study indicated that the dispersive relation of SSP waves is insensitive to the thickness of metal and two-dimensional highly localized SSP waves can be supported in the limit of zero thickness.

To study the influence of thickness, we vary *t* from 400 μm to 200 μm, 40 μm and 20 μm. Other parameters are chosen as: *N* = 3, *d _{1}* =

*d*= 30 μm,

_{3}*d*= 19 μm,

_{2}*p*= 60 μm,

*d*= 30 μm,

*θ*= 40°. A linearly polarized plane wave with the electric field along x direction is employed to excite SSP waves. The thickness of Teflon substrate is 60 μm. The two-dimensional field patterns on the x-y planes at 0.1λ away from the grating are illustrated Figs. 6(b)-6(e), which are respectively corresponding to

*t*= 400 μm, 200 μm, 40 μm and 20 μm. The calculated frequency is 0.3015 THz. Note that, to compensate the red shift of working frequency caused by decreasing

*t*[42], the heights of central gratings are respectively set as 228 μm, 224 μm, 209 μm and 201.5 μm (corresponding to Figs. 6(b) to 6(e)). It is interesting that the focusing behavior is also held in the y direction, due to localized property of two-dimensional grating. Besides, the focusing ability in the y direction is improved with the decrease of

*t*, while the focusing behavior in the x direction has no big difference. When

*t*reaches ultra-thin scale, for example 40 μm, further decreasing

*t*has very limited influence on the focusing depth. In the insert of Fig. 6(a), we present the magnetic field patterns on x-y plane (

*t*= 40 μm, 0.1λ away from the grating). It is clearly observed that the EM field only keeps the feature of REI focusing in the x direction, i.e., the central lobe has an abrupt π-phase shift at the adjacent with side lobes. Therefore, the FWHMy is always larger than FWHMx. The FWHMx and FWHMy in the case of

*t*= 20 μm are 0.064 λ and 0.12λ, respectively, leading to a spot size of 0.0077λ

^{2}, as plotted in Fig. 6(f). Actually, the ultra-thin metallic grating is quasi-two-dimensional SSP structure. Therefore, REI focusing can only exist in the x direction. To further improve the focusing depth in the y direction, one may replace the metallic gating with some two-dimensional SSP structures [43]. With elaborate adjustment, the two-dimensional REI focusing can be excited to further decrease the size of focusing spot.

## 4. Summary

This paper proposes a new way to realize REI focusing of EM waves in the terahertz regime. Based on the truncated SSP structure, the FP resonance could produce the transverse rapid oscillation, which is critical to the REI focusing. With the interference of SSP waves at a situation of third-order FP resonance, the incident energy can be focused into 0.069λ at a distance of 0.1λ. In addition, we verify that the focusing depth can be efficiently enhanced by increasing the order of FP resonance. As the order of FP resonance becomes seven, the focusing depth is enhanced to 0.06λ, indicating the remarkable superfocusing metric of 1.67. When the thickness of the structure is compressed into ultra-thin scale, the REI focusing property remains feasible, leading to a focusing spot as small as 0.0077λ^{2}. Our work may provide an alternative method for extending the working distance of near-field sensing and imaging applications.

## Funding

This work is sponsored by the National Natural Science Foundation of China under contract 61531002 and 61471007.

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