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Abstract
A hollow-core circular waveguide with a hyperbolic metamaterial cladding is proposed for single-mode operation in the terahertz spectrum. The strong anisotropy of the cladding is achieved by incorporating an aluminum coating on the inner wall of the cladding air holes. It is evident from the numerical study that the proposed model offers a single-mode window covering from 0.22 THz to 0.34 THz. The radially polarized transverse magnetic TM1 mode offers the lowest loss of 0.23 dB/cm at 0.3 THz within the single-mode window. The proposed fiber also offers around 96.8% of the total power in its wavelength scale fiber core. Low bending loss around 0.28 dB/cm with a sharp bend of 5.0 cm makes this fiber a potential choice for a densely integrated terahertz system.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz spectrum is the slice of the electromagnetic spectrum corresponding to 0.1 THz to 10 THz that links the gap between microwave and optical frequencies [1]. This frontier of electromagnetic waves is also referred to as sub-millimeter waves or far-infrared waves. For the last few decades, the terahertz spectrum has become a intense field of research and innovation because of its promising applications in extremely high resolution imaging [2,3], high speed data transmission [4], label-free detection of DNA molecules [5–7], screening of skin cancer and tooth erosion [8], non-invasive detection [9] and so on. But terahertz appliances are not as trivial in industries as microwave and optical devices due to the lack of efficient terahertz emitter and detector. Compactness and high-density integration of the terahertz components is another key challenge on the way to developing efficient and feasible terahertz systems. Single-mode flexible waveguides with a low loss profile can play an important role in the high-density integration of terahertz systems. The transmission efficiency of the terahertz waveguides is restricted due to the absorption by the background material. Air is the most transparent medium in the terahertz spectrum with minimum absorption. Dielectric fibers and metallic waveguides can satisfy these requirements. Porous core photonic crystal fibers can guide terahertz wave with low effective material loss than solid-core fibers [10–12]. Hollow-core fibers (HCFs) like bandgap fibers [13,14], Kagome fibers [15], anti-resonant fibers (ARFs) [16,17] are some of the best choices for terahertz guidance due to minimum absorption loss in dry air. In bandgap fibers, the photonic bandgap formed due to the periodic structure in the cladding prohibits the mode propagation through it and thus, the wave is bound to propagate through the fiber core filled with air. For non-linear optics and data transmission, a broad bandwidth is a critical requirement. But the low loss transmission window of bandgap fibers is limited to only a few terahertz frequencies as the bandgap formed in the cladding restricts wave propagation for only a certain range of wavenumber. In Kagome and anti-resonant fibers, there is no bandgap formed in the cladding rather the wave is guided through the fiber by inhibited mode coupling. At a certain frequency namely as cut-off frequency, the core mode and cladding modes are resonant and coupled with each other resulting in high transmission loss. Apart from the cut-off frequencies, the core mode is inhibited from coupling with the low-density cladding modes. So, the dominant mode remains propagating in the core [15,18]. The dimensions of this type of dielectric waveguides are larger than the operating wavelength, and consequently, this leads to multimode operation. Waveguides with metal wire embedded in a dielectric host could be another potential medium to guide terahertz waves. In 2013, Anthony et al. experimentally characterizes a Zeonex based fiber with indium rod inclusion. Hybrid electric, HE$_{11}$-like mode shows lowest loss with attenuation co-efficient 0.3 $cm^{-1}$ and 0.5 $cm^{-1}$[19] for two and four-wire inclusion respectively. One of the challenges of this type of fiber is to maintain a specific distance between the metal wires because radiation loss depends significantly on the inter-wire distance. In 2014, another porous dielectric fiber with metal rod inclusion was reported with zero dispersion profile and low attenuation co-efficient of 0.05 $cm^{-1}$ to 0.4 $cm^{-1}$ over a frequency range from 0.2 THz to 1.0 THz [20]. In 2015, Navarro et al. investigated a silver-coated terahertz waveguide. In this proposal, quasi-single mode operation is achieved over 1.0 THz to 1.6 THz for HE$_{11}$ mode with loss profile around 20 dB/m [21] over the mentioned frequency range. But the metallic layer is responsible for the ohmic loss that can be reduced with a dielectric coating in the inner metallic layer of the over-sized waveguide. However, the large dimension of the fiber core facilitates multimode propagation. This leads to a key barrier to the dense integration of the compact terahertz system.
In recent years, metamaterials have become a thriving area of research due to their unusual electromagnetic properties. The unusual properties of metamaterials can be implemented to confine light within sub-wavelength structures. In 2009, Yan and Mortensen theoretically investigated the guiding mechanism of light with hyperbolic metamaterial in the hollow-core waveguide at 10.6 $\mu m$. The study shows that the transverse magnetic (TM) mode is reflected by the metal-wire metamaterial layer while the transverse electric (TE) mode is reflected by the plain metal-dielectric metamaterial layer [22]. The condition of the existence of the guided mode and the dispersion profile of these modes in the hollow-core circular waveguide with anisotropic, uniaxial cladding are extensively studied in the mentioned literature [23,24]. Fiber drawing techniques facilitate the fabrication of wire array [25] and split ring resonator [26] based metamaterial fibers. In 2016, Haisu Li et al. proposed a metal-dielectric hybrid cladding fiber with strong confinement of transverse magnetic (TM) modes. The proposed fiber offers a single polarization window ranging from 0.33 THz to 0.44 THz with a minimum loss of 0.28 dB/cm [27]. The same research group proposed another metal-dielectric hybrid cladding fiber where a few metal rods are intentionally removed to tailor the guided modes [28]. The air holes in the cladding provide linear polarization of transverse magnetic (TM) mode which is desired for many applications and facilitates the coupling with the source. The proposed fiber assures a single-polarization window for x-polarization TM$_2$ mode ranging from 0.36 THz to 0.46 THz. The fiber offers a considerable low bending loss of less than 0.3 dB/m at 0.4 THz with bend radii larger than 5 cm. In the previously mentioned two literature, the indium rod is embedded in the dielectric host to form the hyperbolic metamaterial structure in the cladding [27,28]. In 2020, Shiqi Hu et al. proposed a hyperbolic metamaterial-based plasmonic sensor with alternating bilayers of Ag and TiO$_2$[29]. The proposed sensor offers maximum sensitivity of 9000nm/RIU and figure of merit (FOM) 230.8 RIU$^{-1}$ over the refractive index of the target analytes ranging from 1.33 to 1.4. Jakeya Sultana et al. proposed a terahertz anti-resonant fiber where the concept of inhibited mode coupling and metamaterial cladding is implemented together [30]. The dominant model, LP$_{01}$ shows confinement loss of 0.9 dB/m at 1 THz without metal wire inclusion but the loss drastically reduces to 0.06 dB/m with 140 aluminum wire inclusion in the cladding. In 2021, Yaseer Zaman Chowdhury et al. proposed a hyperbolic metamaterial-based terahertz fiber over operating frequency ranging from 0.24 THz to 1.5 THz and the effect of different metals (silver, gold and indium) on the anisotropic nature of the HMM is investigated [31]. It is evident from the study that gold wire embedded HMM fiber offers comparatively low loss than that of silver and indium embedded HMM fiber while keeping other design parameters fixed for all three metals. But filling the cladding air holes with gold increases the fabrication cost of this fiber significantly.
In this study, we have proposed a hollow-core circular waveguide by exploiting the advantages of hyperbolic metamaterial but in a different manner. Normally, anisotropic hyperbolic metamaterial cladding is formed by incorporating metal wire inclusion in the dielectric host. The conventional metal-dielectric co-drawing process may incorporate irregular deformation of indium rods due to the non-symmetric stress during the drawing process [27]. In addition, using other metals like silver or gold increases the fabrication cost to a significant extent despite these metals offer comparatively low loss than indium rods [31]. In our proposed model, the fiber core is surrounded by several air holes in the cladding and these air holes are coated with thin aluminum films. Aluminum offers better reflectively of 0.995 with comparatively higher resistivity around 2.65 $\mu \Omega$-cm at 513.01 $\mu$m [32] which facilitates better guided mode confinement and reduced surface current on the waveguide wall. The liquid-phase chemistry process can be a potential technique for coating the cladding air holes of our proposed fiber as metal coating deposition of 0.5$\: \mu m$ to 0.7$\: \mu m$ thickness on the dielectric host is reported using this technique [33]. The proposed model shows strong confinement of radially polarized transverse magnetic (TM$_1$) mode with a minimum loss of 0.23 dB/cm at 0.3 THz and a wide single-polarization window of 0.12 THz for high bandwidth telecommunication systems. Another interesting application of this type of waveguide can be in a practical electron accelerator. This type of accelerator needs TM polarized coherent THz pulses to accelerate the relativistic electron bunches [34].
2. Background information
Hyperbolic metamaterials (HMMs) have become a sparking field of research because of their extreme anisotropic material properties with indefinite permittivity tensor and hyperbolic isofrequency surface. By careful designing, the permittivity tensor can be tailored so that the entries of the tensor corresponding to the transverse and longitudinal direction of the wave propagation take the opposite sign. Thus, the material properties exhibited by the hyperbolic metamaterial depends on the orientation of the electric field of the guided mode [22].
Fig. 1. Isofrequency contour of waveguide material: Air core with solid black line and HMM cladding with dotted lines (blue for TE mode and red for TM mode)
Suppose, an anisotropic wire embedded hyperbolic metamaterial exhibits materials properties such that the effective permittivity tensor becomes $\overline {\overline {\epsilon }}=\epsilon _o[\epsilon _t(\boldsymbol {\hat {x}\hat {x}}+\boldsymbol {\hat {y}\hat {y}})+\epsilon _z \boldsymbol {\hat {z}\hat {z}}]$ where $\epsilon _t>0$ and $\epsilon _z<0$. The dispersion relation for TM and TE mode in anisotropic medium yields from Maxwell’s equation as [35]:
where, $k_o$ is the wave number in the free space, $k_t$ is the wave number in the transverse direction of wave propagation and $\beta$ is the component of the wave vector $(k_z)$ in the direction of propagation. From Eqs.(4.1), it’s evident that the isofrequency contour is elliptical $(\epsilon _x\neq \epsilon _y)$ or circular $(\epsilon _x= \epsilon _y)$ for transverse electric (TE) polarization. For the transverse magnetic (TM) polarization, this type of material yields unusual hyperbolic isofrequency contour which facilitates sub-wavelength confinement of light [24,36]. To interpret the guiding mechanism of the terahertz wave in a hyperbolic metamaterial-based waveguide, let us consider that the cladding is formed with non-magnetic uniaxial anisotropic material with transverse and longitudinal permittivity tensor component $\epsilon _t$ and $\epsilon _z$ respectively. The fiber core is filled with air with permittivity $\epsilon _1=\epsilon _o$, permittivity in the free space. The permeability of the core and cladding is considered as the permeability in the free space $(\mu _o)$ due to the non-magnetic approximation. The relation between the wave vectors in the air-core and HMM cladding is depicted in Fig. 1. It’s evident from the Fig. 1 that a TE wave propagating in the air core with a certain wave number is also supported by the HMM cladding as the wave numbers in the transverse direction for both core and cladding are matched for all values of $\beta$ in the air. So, the TE wave propagating in the air core experience the HMM cladding as a transparent medium. But different scenario exists for TM mode. The HMM cladding supports TM mode for wave vector component in the longitudinal direction $(\beta )$ greater than $\sqrt {\epsilon _t} k_o$. Thus, a TM wave propagating in the air core with transverse wave vector $(\beta )$ less than $\sqrt {\epsilon _t} k_o$ will be reflected by the HMM cladding [27].An important point to be noted here is that to restrict the propagation of TM wave only in the fiber core, there is no lower limit of $\beta$ until it approaches zero. Thus, this type of waveguide offers sub-wavelength confinement within a small core.
3. Design of the proposed fiber
The proposed hollow-core HMM cladding fiber is portrayed in the Fig. 2. Zeonex is utilized as the host material whose refractive index is around 1.53 over the terahertz band. Its consistent refractive index profile in the terahertz spectrum with considerably low absorption loss of 0.2 $cm^{-1}$ settles on it a fitting decision for terahertz waveguide [37]. Another purpose for picking Zeonex as host material is its high glass transition temperature $(T_g)$ which encourages the fiber drawing process [38]. In this study, we have designed the proposed fiber at a target frequency of 0.3 THz. Since the HMM cladding can provide confinement in a sub-wavelength scale, we have considered the diameter $(D)$ of the fiber core approximately equal to the operating wavelength. In this study, the diameter of the fiber core $(D)$ is considered 1 $mm$. The cladding comprises 24 air holes around the fiber core. The diameter of the cladding air holes $(d)$ plays a critical role in the optimization of confinement loss and loss ratio of the TE and TM modes. The air holes in the cladding are coated with aluminum. This aluminum layer is responsible for the formation of HMM cladding. The optical properties of the aluminum obtained from Kramers-Kronig analysis are reported in the literature [39,40]. At the operating frequency 0.3 THz, the refractive index of the aluminum is 965.69+ $j$1006.7. The thickness of the dielectric layer between the cladding air holes and the fiber core $(t)$ is considered initially 5 $\mu m$. The effect of the aluminum coating $(t_c)$ and the dielectric layer thickness ($t$) on the loss profile is studied in the succeeding segments. An extra thick cladding is considered deliberately for the investigation of the transparent nature of HMM cladding for TE mode.
4. Synopsis of the transmission properties
To investigate the transmission properties of the proposed waveguide, commercially available full vector finite element method-based software COMSOL Multiphysics v5.0 is used. A cylindrical perfectly matched layer (PML) available in the software is used as the absorbing boundary condition around the cladding. The thickness of the PML is taken as the 10% of the total fiber geometry. The total computational area is divided into 214,128 triangular elements and 13,780 edge elements for better computational accuracy. The Mode analysis study is implemented to investigate the effective mode index as well as the propagating mode shapes for a specific frequency by solving the eigenvalue equation for the electric field mentioned below [41]:
where $k_o$ is the free space wavenumber and $n$ is the effective mode index.4.1 Effect of core diameter variation
In this section, we numerically investigated the effect of core diameter on the normalized propagation constant $(n_{eff}=\beta /k_o)$ for the different transverse magnetic (TM) modes of the proposed waveguide. In our proposed model, the cladding air holes are coated with a nano-scale aluminum layer and this air inside the aluminum layer forms a hyperbolic metamaterial unit. For the investigation of the normalized cutoff frequency, we assume that the diameter of the air holes is 60 $\mu m$ and the radial distance between the center of the two air holes is 140 $\mu m$ in the cladding.
Fig. 3. Effective mode index $(n_{eff})$ as function of normalized core diameter $(D/\lambda )$ for $t=5 \: \mu m$, $T=300 \: \mu m$, $d=60\: \mu m$ and $t_c=500 \: nm$
Analytically the modal cutoff of the different guiding modes is investigated for HMM cladding waveguide in literature [23]. The transverse magnetic modes are denoted according to their modal cutoffs like TM$_1$, TM$_2$, TM$_3$ and TM$_4$. The subscript x and y denotes the orientation of polarization in that specific direction. It’s evident from Fig. 3 that no TM mode exists in the proposed waveguide for normalized core diameter $(D/\lambda )$ less than 0.75. This is the normalized cutoff for the lowest order transverse magnetic mode, TM$_1$. The next modal cutoff for TM$_2$ exists at normalized core diameter, $D/\lambda$=1.15. So, for normalized core diameter$(D/\lambda )$ ranging from 0.75 to 1.15, only TM$_1$ mode is guided in the proposed waveguide resulting in a single-mode window of 0.4. The TM$_3$ and TM$_4$ mode have their cutoffs at $D/\lambda$=1.5 and $D/\lambda$=1.65 respectively. So, it’s evident that our proposed waveguide can support wavelength scale confinement with the single-mode operation. Available terahertz waveguides can not support single-mode operation as their normalized core diameters are greater than 2 $(D/\lambda >2)$[20,21,42,43].
4.2 Effect of cladding air holes
In this section, we investigated the impact of air holes in the cladding on the guided modes. In conventional circular metallic waveguides, the cutoff frequencies of transverse magnetic and the transverse electric modes are dependent on the values of the Bessel function and its derivatives respectively. The cutoff frequencies of TE and TM modes can be expressed as [44]:
where, a is the radius of the circular waveguide and c is the wave velocity in the free space. For TE$_{11}$ mode, $\chi _{1,1} '$ is 1.841. This leads to the modal cutoff for TE$_{11}$ mode at $D/\lambda =0.6$. On the other hand, the modal cutoff of TM$_{01}$ is located at $D/\lambda =0.76$ as $\chi _{0,1}$ turns out to be 2.045 for TM$_{01}$ mode. So, there exists a small single-mode window for TE$_{11}$ mode from values of $D/\lambda$ ranging from 0.6 to 0.76 for the geometry of the waveguide. In the case of TE mode, the wave vector in the air is coupled with the wave vector in HMM; so the TE modes offer comparatively higher loss than the TM modes. For better single-mode operation, it’s always appreciable to maintain a higher loss ratio between TE and TM modes. This requirement can be achieved by tailoring the diameter of the cladding air holes.Fig. 4. Effective mode index $(n_{eff})$ of the TE$_1$ and TM$_1$ mode with different diameter of cladding air hole as a function of frequency.
First, we investigated the guiding properties of the waveguide for cladding air hole diameter$(d)$ 30 $\mu m$, 60 $\mu m$, and $140 \mu m$. The effective mode index $(n_{eff})$ of the TM$_1$ mode depends slightly on the diameter of the cladding air holes. In case of TE$_1$ mode, the effective mode index $(n_{eff})$ varies comparatively below 0.3 THz with cladding air hole diameter. But above 0.3 THz, the effective mode index of TE$_1$ shows slight variation, as can be seen in Fig. 4.
The loss profile of the transverse electric and the transverse magnetic modes has a strong dependence on the size of cladding air holes. The impact of cladding air holes on the loss profile of the proposed fiber is portrayed in Fig. 5. The loss of the transverse electric, TE$_1$ mode decreases with the increasing frequency for a specific cladding air hole diameter. For the small diameter of the cladding air holes, the HMM cladding of the proposed waveguide behaves more like a dielectric cladding and results in week confinement of the TE$_1$ mode in the air core. With the increase of cladding air hole diameter, the HMM cladding behaves more like a metal reflector and the propagation loss reduces to a great extent. For the possible largest diameter of the cladding air holes, the transverse electric mode becomes dominant in the proposed waveguide and it performs like a hollow-core metallic waveguide. The HMM cladding behaves like a metallic reflector for transverse magnetic modes. The anisotropic nature of the HMM cladding increases with the cladding air hole diameter. Hence, the propagation loss of the TM$_1$ mode decreases with the increasing diameter of the cladding air holes. A point to be noted here is that the proposed fiber is dedicated to the single-mode operation and will carry only transverse magnetic mode. The propagation loss for both TE$_1$ and TM$_1$ shows a declining trajectory with the increasing cladding air hole size. So, designing the waveguide is a trade-off between reducing the loss and single-mode guidance. To eliminate the transverse electric modes, the diameter of the cladding air holes is considered such that the ratio of loss between TE$_1$ and TM$_1$ becomes maximum. The proposed waveguide is designed to operate at 0.3 THz. The impact of variation of the cladding air holes on the loss ratio between TE$_1$ and TM$_1$ is investigated at 0.3 THz keeping other design parameters like core diameter, aluminum coating, and dielectric layer thickness fixed.
Fig. 5. Transmission losses of TE$_1$ and TM$_1$ modes as a function of frequency with different diameter of the cladding air holes
For the small cladding air holes, the corresponding losses of both TE$_1$ and TM$_1$ are high because of the week anisotropic properties of the HMM cladding. With the increase of the diameter of cladding air holes, the anisotropic nature of the cladding increases. Thus, the transverse magnetic, TM$_1$ mode experiences the cladding as a reflecting mirror and the propagation loss is reduced with strong confinement. But the loss of the TE$_1$ mode doesn’t change considerably to a certain extent with the increase of cladding air holes size due to the transparent nature of HMM cladding for transverse electric, TE$_1$ mode. At this specific diameter, the ratio of loss between TE$_1$ and TM$_1$ becomes maximum. Further increase of the cladding air holes size facilitates the capacitive coupling between the aluminum films of cladding air holes. Thus, the HMM cladding starts acting like a metallic waveguide, and the corresponding loss of both TE$_1$ decreases rapidly; consequently, the loss ratio also decreases.
Fig. 6. Propagation Losses and the ratio of loss between TE$_1$ and TM$_1$ modes as a function of cladding air hole diameter
Figure 6 depicts the loss profile of the guided modes (TE$_1$ and TM$_1$) with the variation of cladding air hole diameter. The numerical simulation assures that at cladding air hole diameter 60$\mu m$, the loss ratio is maximum. The other design parameters like core diameter$(D)$, aluminum coating thickness$(t_c)$ and thickness of the dielectric layer$(t)$ are kept fixed at 1$\: mm$, 500$\:n m$ and 5$\:\mu m$ respectively throughout this simulation. The maximum loss ratio between TE$_1$ and TM$_1$ mode is around 350 which assures better transverse electric mode elimination than the terahertz waveguide proposed in the literature (loss ratio less than 200) [27].
Fig. 7. Core power fraction $(\eta )$ as function of frequency with different cladding air hole diameter$(d)$
Power fraction is another indicator that assures strong confinement of the transverse magnetic, TM$_1$ mode. Core power fraction implies the fraction of the total power confined in the fiber core. The corresponding core power fraction of the TM$_1$ mode decreases with the increasing frequency as portrayed in Fig. 7. This declining trend of the core power fraction has a dependence on the air openings of the cladding. For comparatively large air holes i.e. at $d=140\:\mu m$, the core power fraction is almost around 97.7% throughout the whole frequency range 0.22 THz to 0.44 THz. For cladding air hole diameter, $d=60\:\mu m$, the declining trend is quite similar and the power fraction is 96.8% at 0.3 THz. But the core power fraction exhibits a comparative rapid declining trend for small cladding air holes. For cladding air hole diameter, $d=30\: \mu m$, the core power fraction is 95.2% at 0.3 THz; this value eventually reaches 94.1% at 0.44 THz.
Fig. 8. Mode field profile at 0.3 THz: a) TM$_1$ b) TE$_1$. The arrows indicate the normalized electric field
On the other hand, it’s evident from Fig. 8 that the mode field pattern of the TE$_1$ mode spreads out in the cladding. Hence, only a very small fraction of the total power will be transmitted by the TE$_1$ mode at the receiving end. From the above discussion, it’s evident that cladding air hole diameter of 60$\: \mu m$ will be a good choice for the proposed fiber from the perspective of the ratio of loss between TE$_1$ and TM$_1$. The transverse magnetic, TM$_1$ mode experiences 0.238 dB/cm transmission loss with power fraction 96.8% at 0.3 THz for this specific air hole diameter.
4.3 Effect of dielectric layer thickness
Fig. 9. Loss (dB/m) and power fraction, $(\eta )$ as a function of dielectric layer thickness $(t)$ at 0.3 THz
In this section, the impact of dielectric layer thickness $(t)$ on the guided mode is investigated. The increase of the dielectric layer between the fiber core and cladding air holes results in a linear increment of the transmission loss of TM$_1$ mode. The core power fraction $(\eta )$ shows a declining trajectory with the increase of dielectric layer thickness. The impact of variation of dielectric layer thickness at 0.3 THz is portrayed in Fig. 9. With the dielectric layer thickness $(t)$ less than 5$\mu m$, the transmission loss is no greater than 0.24 dB/cm and the loss increases monotonically with the increase of dielectric layer thickness. The core power fraction at the target frequency is 96.8%. Further reduction in the dielectric layer influences more power propagation through the core. In the previously proposed indium rod inserted HMM terahertz fiber, the reduction of dielectric layer thickness below 2 $\mu m$ results in an exponential increase of transmission loss due to the Ohmic loss [27]. The monotonic decrease of the transmission loss with the decreasing dielectric layer thickness of our proposed fiber assures the independence of the loss profile from the Ohmic losses. For fabrication convenience, the dielectric layer thickness is considered 5 $\mu m$ for our proposed fiber.
4.4 Effect of aluminum coating thickness
The aluminum coating thickness slightly influences the transmission loss and the core power fraction of TM$_1$ mode. We numerically investigated the loss and the power fraction for various aluminum coating thicknesses. The transmission loss increases with the aluminum coating thickness. The increase of transmission loss due to the variation of aluminum coating thickness from 500 $nm$ to 550 $nm$ is around 0.0014 dB/cm. The transmission loss decreases 0.0092 dB/cm for aluminum coating thickness variation from 500 $nm$ to 450 $nm$.The core power fraction changes in a similar manner to the transmission loss and varies only $\pm$ 0.015% for the variation of aluminum coating thickness $(t_c)$ of 50 $nm$.
Fig. 10. a) Loss (dB/m) b) power fraction $(\eta )$ as a function of aluminum coating thickness $(t)$ at 0.3 THz.
4.5 Ability of single-mode guidance of the proposed fiber
Single-mode guiding ability is one of the most important key requirements of terahertz fibers. The simulation result shows that the TM$_1$ mode has its cutoff frequency at 0.22 THz and the corresponding loss is 0.69 dB/cm at this frequency. The loss of this guided mode decreases with the increase of operating frequency till 0.3 THz. At 0.3 THz, the proposed fiber provides a minimum loss around 0.23 dB/cm with core power fraction $(\eta )$ 96.82%, as can be seen in Fig. 10. Further increase of the frequency exhibits an increase of the propagation loss. The TM$_2$ mode becomes guided at 0.34 THz. So, our proposed fiber can be used for single-mode operation within the frequency range 0.22 THz to 0.34 THz. Within this 0.12 THz of the single-mode window, the loss is minimum at 0.3 THz. At 0.34 THz single-mode window, the TM$_1$ shows transmission loss of 0.242 dB/cm. The transmission loss increases outside the frequency range of a single-mode window. At 0.44 THz, the TM$_1$ mode experience loss of around 0.28 dB/cm, and around 96.4% of the total power is confined within the fiber core. The TM$_2$ mode experiences comparatively more loss than the TM$_1$ mode. At the very end of the single-mode region, the TM$_2$ mode experiences transmission loss around 1.1 dB/cm, and the corresponding loss of this mode decreases with the increase of frequency. At 0.44 THz, the transmission loss of the TM$_2$ mode is 0.44 dB/cm. As can be seen in Fig. 11, it’s evident that at our target frequency of 0.3 THz, the transmission of the proposed fiber is minimum with an appreciable amount of power confinement in the core.
Fig. 11. Effective mode index $n_{eff}$ and loss of the TM$_1$ and TE$_1$ mode as a function of frequency.
4.6 Bending loss
Bending loss of the terahertz fibers plays a significant role in the high-density integration of terahertz systems. Numerically, the bending loss can be evaluated with equivalent refractive index profile defined as $n'(x,y)=n_{material}\: exp\left ( \frac {x}{R_b} \right )$, where $n_{material}$ is the refractive index of the material without bend and $R_b$ is the bend radius [45]. We investigate the effect of bending on TM$_1$ mode for different bend radii at our target frequency of 0.3 THz. Figure 12 shows that the loss of bend fiber increases rapidly for sharp bend radii (less than 5 cm). The loss of the proposed fiber for bend radii 5 cm is 0.28 dB/cm which indicates the increment of 0.04 dB/cm of additional loss for the bend. For more sharp bend, for example, at $R_b=$2 cm, the loss of the bent fiber reaches 0.5 dB/cm and core power fraction reduces to 96.7%. For the bend radii greater than 5 cm, the loss and the power fraction of the proposed fiber are almost constant. A bend of 15 cm introduces only an additional loss of 0.004 dB/cm than that of the straight fiber for our proposed model.
Fig. 12. a) Loss (dB/m) b) power fraction $(\eta )$ as a function of the bend radius $(R_b)$ at 0.3 THz
4.7 Performance comparison
Hyperbolic metamaterials (HMMs) have a diverse range of applications including high-resolution imaging [35], spontaneous emission enhancement [46], waveguide applications [47,48], and so on. To the best of our knowledge, only a few waveguides are reported till now to guide terahertz pulses. The previously proposed HMM-based terahertz fibers show their anisotropic properties due to the presence of sub-wavelength indium rods adjacent to the fiber core (Table 1) [27,28].
Table 1. Performance Comparison of the Proposed Fiber with Other Terahertz Fibers
Our proposed work shows a different way to provide an anisotropic cladding with aluminum-coated air holes for the very first time to the best of our knowledge. Our proposed fiber offers a single-mode window of 0.12 THz for TM$_1$ mode with lowest transmission loss of 0.23 dB/cm at 0.3 THz whereas the previously proposed HMM fibers experience lowest transmission losses around 0.28 dB/cm at 0.44 THz for TM$_1$ mode [27] and 0.25 dB/cm at 0.4 THz for TM$_2$ mode [28]. The bending loss of our proposed fiber is 0.28 dB/cm with a 5 cm bend radius. The previously proposed HMM fibers experience bending loss around 0.3 dB/cm for 5.0 cm bend radius [27,28] and it implies the comparatively better bend-insensitive performance of our proposed fiber than that of the mentioned articles. One of the key features of the HMM-based terahertz fibers is the tight confinement of the transverse magnetic modes and the leaky nature of the transverse electric mode. The loss ratio between the TE$_1$ and TM$_1$ mode is a key index of this feature. Numerical simulation assures that the loss ratio of TE$_1$ and TM$_1$ mode is around 350 for our proposed fiber. But no information regarding this is mentioned in the previously proposed model [28]. Another previously proposed fiber exhibits loss ratio of TE$_1$ and TM$_1$ less than 200 [27]. Hence, our proposed fiber offers better transverse electric mode elimination ability than the previously proposed fibers.
5. Fabrication feasibility
For the practical realization of any terahertz fiber, fabrication feasibility is one of the most significant issues. Previously proposed indium rod-based HMM fibers [27,28] are fabricated using metal-dielectric co-drawing techniques [51]. Our proposed fiber has a metallic coating on the inner wall of the cladding air holes. Conventional fabrication techniques like drilling, stack and draw, 3D printing technology, etc. can be implemented to fabricate this type of dielectric structures [20,52,53]. The challenging part of the fabrication process is to form the aluminum coating. Chemical vapor deposition and atomic layer deposition technique can be implemented to form a uniform metal layer [54,55]. The liquid-phase chemistry method can be a potential technique to fabricate the aluminum layer of our proposed fiber. This method was implemented previously to form a 0.5 $\mu m$ interior copper coating of a hollow-core terahertz waveguide [33]. The very first step of the deposition process is to sensitize the inner surface of the cladding air holes for the strong adhesion of the metal coating on the dielectric background. Normally palladium and stannous-based chemicals like PdCl$_2$ and SnCl$_2$ are used as a sensitizer for the deposition of metals like silver and copper [33,56]. For the aluminum deposition, a solution of Al$_2$(SO$_4$)$_3 \cdot$ 18H$_2$O, NaHCO$_3$ and deionized water with proper pH could be a potential growth solution [57]. One of the most challenging parts of the metal deposition process is the selective filling of the cladding air holes with growth solution. This process leads to the need for techniques to fill the cladding holes with growth solution while blocking the fiber core and that can be achieved by infiltrating UV curable polymer and a multi-step injection-cure-cleave process [58,59]. To maintain the aluminum coating thickness, the flow rate of the growth solution must be maintained with a precise suction system.
6. Conclusion
The proposed fiber offers strong confinement of transverse magnetic modes in wavelength scale hollow-core overcoming the barrier of high material absorption of terahertz waves. The HMM cladding layer acts as a metallic reflector for the transverse mode. The wavelength scale confinement of the terahertz wave assures reduced crosstalk as the evanescent wave is independent of the dimension of the fiber core. The low loss profile and wide single-mode transmission window of our proposed fiber make it a suitable choice for densely integrated terahertz systems. This fiber can be implemented for gas sensing applications. The low refractive index of the gases results in week confinement of light in the fiber core. Thus, the light-gas interaction reduces and the sensitivity becomes low. The proposed fiber offers 96.8% of the total power confined in the core and it’s an indication of the potentiality of this type of fibers as gas sensors.
Acknowledgments
The authors would like to cordially thank Dr. Abdul Khaleque for his valuable suggestions while preparing the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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