## Abstract

An tapered hyperbolic metal waveguide is suggested for the nanofocusing of terahertz waves. We numerically show that, at the frequency of 1 THz, the focal spot can be as small as only 5 nm, which is smaller than that of a plate waveguide by 2 orders of magnitude. Correspondingly, the longitudinal component of the energy flow density is stronger than that of a plate waveguide by 3 orders of magnitude for the same input. It is shown that these significant improvements come from the small imaginary part of the effective index of the hyperbolic metal waveguide.

© 2014 Optical Society of America

## 1. Introduction

Terahertz (THz) radiation, normally refer to electromagnetic waves at frequencies ranging from 0.1 to 10 THz (or in wavelengths from 30 *μ*m to 3 mm), is one of the hottest and most compelling research topics. Due to its high spectral sensitivity and non-destruction to the samples (e.g. amino acids and DNA base molecules), THz technology has huge application potential in biological spectroscopy, sensing, and imaging [1]. However, traditional diffraction-limited systems can only achieve a focal spot of about *λ*/2. To take advantage of THz radiation in nanoscale applications, there are two main ways to circumvent the diffraction limit. One way is to directly impinge light on the metallic nano holes or nanogaps [2–6]. Recently, the sub-skin-depth THz field enhancement by a metallic nano slit (*λ*/30,000) have been achieved owing to the effect of the nanogap-capacitor charged by the light-induced currents [3]. However, the light intensity (i.e. the longitudinal component *S _{z}* of the energy flow density) enhancement is smaller than the electric field enhancement, for there is a nearly

*π*/2 phase difference between the enhanced electric field and the slightly changed magnetic field [3]. Another way is to use tapered surface plasmon polariton (SPP) waveguides [7]. Surface plasmon polaritons are electromagnetic waves confined to and guided along a metal-dielectric interface, and have been known for decades [8,9]. The discovery of strong localization of guided SPP mode far smaller than the diffraction-limited size has leaded to a rapid development of nano-photonics [7,10–16]. Actually, various tapered SPP waveguides in the optical region have been reported [17–20], among them a nanoscale spot with an intensity increase of 3 orders of magnitude has been achieved at the tip of a conical metal wire waveguide [17]. Recently, tapered SPP waveguides have been used to achieve sub-wavelength concentration and giant field enhancement for the THz and mid-infrared waves [21–26].

A bare metal wire [27,28] can be used to guide the THz waves, a tapered metal wire waveguide can be used to successfully confine THz waves to a spot of about *λ*/100 [21]. But, radial polarization character of this mode field makes it difficult to be coupled from the commonly used linearly polarized THz source [29]. In addition, owing to the high loss and strong dispersion in nanoscale structure, the focusing ability of the tapered THz metal wire waveguide is limited [30]. Coaxial waveguide (CWG), parallel plate waveguide (PPWG), and slot waveguide (SWG) have been used for the nanofocusing of THz waves and mid-infrared waves [22–25]. However, they all have limitations either in the intensity enhancement or in the size of focal spot.

In this letter, we propose an adiabatically tapered hyperbolic metal waveguide (ATHMWG). The waveguide has the advantage of low loss, hence we are able to achieve both giant intensity enhancement and extremely small focal spot for THz waves at the same time. In particular, we numerically show that, at the frequency of 1 THz, the focal spot can be as small as only 5 nm (~*λ*/60,000), while maintaining an extremely strong intensity enhancement at the tip.

## 2. Waveguide geometry

The waveguide geometry consists of a pair of silver structures in air (Fig. 1(a)). The cross-section of the waveguide is defined by the two branches of a hyperbola. The hyperbola can be determined by the semi-transverse axis *a* and the angle *θ* that is between the asymptote and the x axis. The parameter *a* gradually varies along the optical axis z from *a*_{1} to *a*_{2}, as qualitatively shown in Fig. 1(b), and *θ* = 45° remains constant. Figures 1(c) and 1(d) are the input and the output planes, respectively. At the input plane, *a*_{1} = 1 *μ*m, and at the output plane, *a*_{2} = 1 nm.

## 3. Eigen-mode analysis

For simplicity, we consider the case of eigen-mode incidence at the input plane. We assume that, as the parameter *a* changes from 1 *μ*m to 1 nm along the propagation direction, the THz eigen-mode energy in the input plane can be adiabatically focused to the output plane without mode scattering and reflection. Hence the field in each cross-section is virtually eigen-mode, the distribution of which can be calculated by finite element method (FEM) with COMSOL Multiphysics. Its transverse electric field distribution at the input plane and the output plane are shown in Fig. 2. The frequency is chosen to be 1 THz, and the permittivity of silver *ε _{m}* = −2.39 × 10

^{5}+ j1.04 × 10

^{6}is derived from the fitted Drude model [31,32]. In the far-infrared and THz regions, the fitted Drude model [31,32] is widely and commonly used to obtain the permittivities of metals [3,28,30,33–41]. The electric field distribution at the input plane distribution (Fig. 2(a)) resembles that at the output plane (Fig. 2(b)), and most part of the electric field appears in the gap area. It arises from the continuity of the dominant normal displacement field at the material interface and the high permittivity of the silver (~10

^{6}).

Usually, there are two kinds of polarizations for the electric field of a surface plasmon, linear polarization and radial polarization. The linearly polarized surface plasmon usually exists along the smooth flat metal surface and can be excited by periodic structures such as metal gratings [42–44]. The radially polarized surface plasmon can exist along a bare metal wire [28] and be excited by the end-fire technique in combination with an axicon [45]. But now, the polarization of the electric field within the gap is some kind of “elliptic polarization”, as shown in Fig. 2. This feature comes from the orthogonality of the ellipse-hyperbola coordinate system, where the coordinate lines of the confocal ellipses are orthogonal to that of the hyperbolae.

In micro scale, the method of conformal mapping [46] can be used to analytically calculate the mode field of the THz waveguide [47], same way as one solves the electrostatic problem in two dimensions [48]. With this method, one can derive the electric field distribution in the gap region. The gap region is first mapped to a fan-shaped area by inverse Joukowski mapping [49] and the potential function can be calculated there. Then, by remapping the fan-shaped area back to the original gap region, one can get the target potential distribution. Finally, as the gradient of the potential, the electric field can be obtained:

*c*=

*a*/cos

*θ*is the distance from the center to either of the foci.

Out of curiosity, we compared the analytical expression of Eq. (1) with the simulation result when the gap draws close. We surprisingly found that they are almost the same, even at nano scale. This result shows that the electrostatic approximation of the transverse electric field remains valid at the nano scale. In our *θ* = 45° case, at the output plane *a* = 1 nm, the full width at half maximum (FWHM) on *y* axis can be calculated as $2\sqrt{6}a\approx 5\text{nm}$ and the simulation result also gives a ~5 nm FWHM focal spot in Fig. 2(b).

Under the same adiabatic assumption, the effective index (*n _{eff}*) of the tapered hyperbolic metal waveguide varies with the optical axis [17]. The effective index

*n*at each cross-section of the ATHMWG can be regarded as the same as that for the eigen-mode of the corresponding hyperbolic metal waveguide (HMWG), and can also be calculated by the finite element method (FEM). Figure 3 shows the

_{eff}*n*(real part 3(a), imaginary part 3(b)) of the HMWG and the PPWG with

_{eff}*a*varying from 1 nm to 1

*μ*m. One can see that when HMWG and PPWG have the same value of Im(

*n*), the semi-transverse axis

_{eff}*a*of the HMWG is much smaller than that of the PPWG. For example, when Im(

*n*) = 0.0682, for the HMWG the semi-transverse axis

_{eff}*a*is 1 nm, and for the PPWG that is 192 nm. It has been studied that the tapered parallel plate waveguide [22] can concentrate the THz radiation to 2

*a*≈300 nm, the spot has a practical limited size as about the skin depth. But the ATHMWG has the potential to reach a much smaller focal spot, about 1/10 of the skin depth (~60 nm) of silver at 1THz radiation.

## 4. Sub-skin-depth nanofocusing

In order to adiabatically focus the THz waves in the ATHMWG, one may design a tapered curve by employing the eikonal approximation [50] or WKB approximation [51], which is valid under the condition

*k*is the complex wave vector component in the propagation direction,

_{z}*k*

_{0}is the wave vector in the free space,

*δ*

_{1}is the gradient of the waveguide and

*δ*

_{2}is the adiabatic parameter describing how slowly the wavelength changes along the propagating direction. To satisfy the adiabatic assumption, we set the upper limit of

*δ*

_{1}and

*δ*

_{2}to be 0.05, which can be expressed aswithin the range from

*a*= 1

*μ*m to 1 nm. Previous works [22,33] show that the WKB approximation works well in THz region as long as

*δ*

_{1}and

*δ*

_{2}are both smaller than 0.05.

By using the data of *n _{eff}* with different

*a*in Fig. 3 and Eq. (3), one can obtain an adiabatically tapered curve of the waveguide (relation between

*a*and

*z*), which makes the waveguide satisfy the WKB approximation. The curve is shown in Fig. 4(a) quantitatively and in Fig. 1(b) qualitatively. There is a critical point on the curve, where

*a*= 0.4677

*μ*m. The curve linearly decreases with

*z*at first, and then it starts to be funnel shaped after passing the critical point. Figure 4(b) shows the corresponding relation of

*δ*

_{1}and

*δ*

_{2}with

*a*, respectively. Before the critical point

*a*= 0.4677

*μ*m,

*δ*

_{1}

*> δ*

_{2}, and

*δ*

_{1}is fixed to 0.05; after the critical point,

*δ*

_{1}

*< δ*

_{2}, and

*δ*

_{2}is fixed to 0.05. Through this design, we are able to ignore both the reflection and the mode scattering approximately.

Under the adiabatic condition, the total mode energy flux *P*(*z*) in each cross-section of the tapered structure can be expressed as:

*P*(

*z*) is defined as$P\left(z\right)={\displaystyle \iint {S}_{z}(x,y,z)}dxdy$, which is convergent in each cross-section, ${S}_{z}(x,y,z)$ is the eigen-mode mean (time average) energy flux distribution in the cross-section of the ATHMWG, and

*n*(

_{eff}*z*) is indicated in Fig. 3. The modal area

*A*(z) in the cross-section is defined as the ratio of the total mode energy flux and the center energy flux density, which is

*A*(

*z*) can be obtained by the FEM method, the mean electromagnetic energy flux density (light intensity) ${S}_{z}(0,0,z)$along

*z*direction can be given as the following equation

In order to show the advantage of the ATHMWG in THz nanofocusing, a comparison with an adiabatically tapered parallel plate waveguide (ATPPWG) of the light intensity enhancement along the *z* direction, namely ratio of center intensity *S _{z}* and entrance center intensity

*S*(0,0,0), has been made. Likewise, the gradient

_{z}*δ*

_{1}and the adiabatic parameter

*δ*

_{2}of ATPPWG are set to satisfy Eq. (3). As we can see in Fig. 5, for the ATHMWG the intensity enhancement continuously increases and reaches about 10

^{4}at the output plane, while for the ATPPWG the intensity enhancement gradually decreases after reaching its peak at

*a≈*75 nm. The enhancement factor at the output plane of the ATHMWG is ~10

^{3}times larger than the peak enhancement of ATPPWG, which is a big promotion for THz nanofocusing.

It is worth mentioning that the sub-skin-depth is the practical limit for some geometries [22], for example, at terahertz region the tapered coaxial waveguide and the tapered plate waveguide both have relatively great propagation loss (the imaginary part of the effective index is comparable to its real part) when the gap approaches the sub-skin-depth. However, the focal spot at the output plane (5 nm) of the ATHMWG is much smaller than the sub-skin-depth (60 nm), since its propagation loss is much smaller than those two geometries. Our result has the similar electric field enhancement and focal spot scale comparing to the recent relevant work by Seo et. al. [3], but with a 3 orders of magnitude larger longitudinal Poynting vector *S _{z}* of the energy flow density.

## 5. Discussions

In order to show the validity of the WKB approximation in our ATHMWG, we performed a full field 3D simulation by COMSOL. Owing to the accuracy of the software in 3D simulation, we only simulate the forepart of the ATHMWG (*a* ranges from 1*μ*m to 10 nm), and the back end (*a* ranges from 10 nm to 1 nm, where the gradient is smaller than 2.5 *×* 10^{−4}) is totally within the application scope of WKB approximation. Comparison of the center intensity enhancement of the ATHMWG from *a* = 1 *μ*m to 10 nm between the WKB approximation and the 3D simulation is shown in Fig. 6. The enhancement profile of the 3D simulation result agrees well with that from the WKB approximation but has a slight deviation, which is of about 15% at *a* = 10 nm. The drop of the enhancement comes from the reflection and mode scattering of the tapered structure. The WKB approximation is still valid since the center intensity enhancement is of a same magnitude. In order to reduce the reflection and mode scattering, further performance improvement could be made by adopting a better tapered curve and, meanwhile, varying the parameters of the hyperbola along the propagation direction.

The coupling efficiency of the ATHMWG is relatively high compare to that of other sub-wavelength waveguides. Based on a 3D simulation, in which we directly couple a Gaussian beam from free space into the HMWG (the input plane of which is the same as that of the ATHMWG), the eigen-mode coupling efficiency turns out to be about 20%, where the waist of the incident Gaussian beam is one *λ* (*λ* = 300 *μ*m) and the input gap width 2*a*_{1} = 2 *μ*m. Besides, if we extend the waveguide backwards making the pre-input gap width increase to 2*a*_{0} ~2*λ*/3, by simply applying a linear gradients between *a*_{0} and *a*_{1} (gradient *δ*_{1} = 0.05 to assure that WKB approximation is valid), we can improve the input plane coupling efficiency to about 33%.

In order to show the low dispersion of the ATHMWG, we get the real parts of the effective mode indices of the HMWG with respect to wavelengths. Here we define *n _{p}* as the phase effective index and

*n*as the group effective index of the HMWG.

_{g}*n*=

_{p}*c/v*= Re(

_{p}*n*),

_{eff}*n*=

_{g}*c/v*=

_{g}*n*-

_{p}*λ*(d

*n*/d

_{p}*λ*), where

*v*and

_{p}*v*represent the phase velocity and the group velocity, respectively. As is shown in Fig. 7, when the HMWG has a minimum gap (

_{g}*a*= 1 nm, same as the output of the ATHMWG), the changes of

*n*-1 (upper solid line) and

_{g}*n*-1 (upper dashed line) with respect to wavelengths are negligible. Similarly, when the HMWG has a maximum gap (

_{p}*a*= 1

*μ*m, same as the input of the ATHMWG), the changes of

*n*-1 (lower solid line) and

_{g}*n*-1 (lower dashed line) with respect to wavelengths are also negligible. We simply name the value of

_{p}*n*-1 and

_{g}*n*-1 of the minimum gap the “upper one”, and the value of

_{p}*n*-1 and

_{g}*n*-1 of the maximum gap the “lower one”. As we can tell in Fig. 7, the variation of the “lower one” are 2 orders of magnitude smaller than that of the “upper one”. The value of

_{p}*n*-1 and

_{g}*n*-1 of the ATHMWG happen to be between such two extreme conditions, and should be smaller than the “upper one”, which is already small enough to avoid dispersion. This advantage of low dispersion implies that the sub-skin-depth focusing ability of the ATHMWG could be easily extended to THz pulse focusing.

_{p}The ATHMWG might be realized either by high-resolution electron-beam lithography (EBL) [25,52] or atomic layer lithography [40]. The EBL can achieve a resolution of 12 nm [52], while atomic layer lithography can achieve a resolution of one nanometer according to a recent publication [40]. Since nanofabrication technologies have been rapidly developed, there would be growing excitement in using these techniques to manufacture more complicated structures with nanometer or sub-nanometer precision, thus we believe that our structure is achievable in the near future.

If one uses a simple linear function *a*(*z*) with *δ*_{1} *=* 0.05 between *a*_{1} = 1 *μ*m and *a*_{2} = 1 nm, instead of the curve shown in Fig. 4, there will definitely be an impact on its performance. Simulation results show that the intensity enhancement at the output plane in the linear case is ~1.0 × 10^{4}, almost the same as that of the adiabatically tapered case. However, the eigen-mode transmittance drops from 54% (adiabatically tapered case) to 48% (linear tapered case). The drop of the eigen-mode transmittance is due to the relatively larger reflection and mode scattering of the linearly tapered structure. It is remarkable that the final field distribution changes insignificantly at the focal region of the output plane. Thus, for linearly tapered condition, conformal mapping still can be used to derive the electric field distribution within the gap region at the output plane, and a 5 nm focal spot remains achievable.

The classical electrodynamics holds well down to nanometer length scales, and quantum mechanics is required only when smaller gaps of a few angstroms are considered [5]. It is suitable to use only classical electrodynamics in our structure where the smallest gap size (2*a* = 2 nm) is much larger than the critical size. The 2D mode analysis module of COMSOL Multiphysics had been used to calculate the effective indices of the HMWG with different *a*. For the rapidly damping field inside the silver, a 10 nm grid size was used along the normal direction near the silver-air interfaces, and the grid size became larger as the process shifts away from the interfaces to both the silver and air region. It was small enough to resolve the skin depth (~60 nm). The grids around the nanogap were refined in order to resolve the field. The radii of the calculation region were large enough to mimic the open boundary condition. The extremities of the calculation region were assigned a perfect electric conductor (PEC). A convergence analysis was implemented to ensure both the real and imaginary parts of the complex effective indices and *A*(*z*) varied by less than 1%. The silver-air interface was assign to an impedance boundary condition in the 3D simulation for the eigen-mode coupling efficiency, which is accurate enough at the input plane with a 2 *μ*m gap. The two PML-backed interior ports (one of the new features brought by COMSOL Multiphysics 4.4 version, and the result has been verified to be consistent with that done by the scattering field method) at both input and output plane are used for the simulations of ATHMWG scattering, the coupling efficiency, and the linearly tapered HMWG transmittance correspondingly.

## 6. Conclusions

We have proposed a new type of metal waveguide called the adiabatically tapered hyperbolic metal waveguide (ATHMWG) with low loss in nanoscale for THz nanofocusing. It can achieve sub-skin-depth focal spot with great intensity enhancement. The simulation results show that when the gap is 2*a* = 2 nm at the output plane, a 5 nm focal spot could be achieved, and the intensity enhancement reaches ~10^{4}. The ATHMWG may lead to many potential applications in super-resolution THz imaging, high S/N spectroscopy, nanolasers, and even in non-destruction cell imaging [53,54]. It is worth mentioning that, we found this element could also be used for nanofocusing of light at visible, near-infrared, and mid-infrared frequencies, since this element essentially has a relatively small imaginary part of the effective index.

## Acknowledgment

This work was financially supported by the National Natural Science Foundation of China with Grant Number 61275103.

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