The effect of dielectric cladding on the waveguiding characteristics of an array of metallic pillars on a metal plane in the sub-terahertz band is explored. Firstly, a 2D structure made up of a metallic grating of infinite lateral width with various dielectric overlays is analytically studied to get more insight into the problem. Then the ideas inferred from the 2D structure are applied to the realistic 3D structure that has a finite lateral width. It is shown that by proper design of the dielectric medium surrounding the metallic structure the modal field confinement can be enhanced in a broad frequency band resulting in a low bending loss. Especially, by integrating the pillars into a silicon channel of finite size and evacuating the spaces between them a highly confined spoof surface plasmon is supported and a considerable reduction in the bending loss over a broad bandwidth is observed. Due to small cross-sectional size, low bending loss and ease of fabrication, the proposed waveguide is a promising choice for millimeter-wave and terahertz integrated circuits; particularly those based on the silicon technology.
© 2017 Optical Society of America
Waveguides play an important role in the development of terahertz (THz) technology . Besides being used for the guided transport of THz signals , they are building blocks of various devices such as sources , detectors , and passive components  and are also key components in some THz sensing and spectroscopy systems . With the advent of commercial THz systems, the need for integrated THz circuits is felt necessitating the development of waveguides having high field confinement, low bending loss, and proper geometry for circuit realization. Periodic metallic structures capable of propagating confined surface waves, also referred to as spoof surface plasmons (SSP), are among the most promising waveguides for this purpose [5–10]. These waveguides support highly confined surface waves or SSP’s and their loss in the sub-THz regime is around 0.01~0.02 mm−1 [1,5].
Based on the idea of periodically altering the metal surface, various spoof plasmonic waveguides suitable for millimeter wave (mm-wave) and THz planar circuit realization have been proposed. A periodic array of metal tapes between two parallel metal plates, called parallel-plate ladder waveguide [11,12], an array of metallic pillars protruding out of a metallic surface, called Domino waveguide [13,14], a double-periodic metallic grating of domino-like elements for dual band operation , grooved metal strip , and a metallic surface with L-shaped grooves  are some examples.
By the use of low-loss dielectrics, several hybrid metallic-dielectric structures for THz waveguiding are also proposed. In  a hybrid plasmonic waveguide composed of a plastic ribbon waveguide integrated with a metal grating is demonstrated for subwavelength confinement in the THz band. A periodic dielectric structure integrated with a graphene layer is also proposed as a tunable THz surface waveguide . In order to localize certain frequencies at different locations on a domino waveguide, the space between metallic teeth is filled gradually with a dielectric . Due to low-loss and low-dispersion characteristics of high-resistivity (HR) silicon in mm-wave and THz bands  and its available fabrication processes, planar silicon waveguides are another promising candidate for THz integrated circuits [22,23]. Highly-doped Si has also been used as a conductor in THz waveguides. As an example, a 1D array of V-shaped grooves etched on a highly-doped silicon substrate is proposed for THz surface plasmon propagation .
Spoof plasmonic waveguides can provide subwavelength confinement in a limited bandwidth where a low bending (radiation) loss is achievable. But, obtaining a low bending loss in a wide bandwidth remains a challenge. As presented in , bend loss is 10% for the bending radius of one wavelength, but it increases drastically when the bending radius becomes smaller compared to a wavelength. In this paper, we show that by using a hybrid plasmonic structure, i.e., a hybrid metallic-dielectric periodic structure, low bending loss is achievable in a broader bandwidth as compared to conventional all-metallic structures. Benefitting from the mentioned features of both HR silicon and the periodic metallic structures, we propose a periodic array of metallic pillars embedded in a silicon ridge for highly-confined transfer of mm-wave and THz waves with low bending losses over a bandwidth of nearly one octave.
In order to enhance the field confinement of the mentioned periodic metallic waveguide, we study two modifications: burying the metallic pillars in a silicon layer of finite size and filling the spaces between the pillars, i.e. the grooves, with air. It should be stressed that these two modifications can readily be implemented by the available fabrication processes; such as planar dielectric waveguide technology, deep reactive ion etching (DRIE), electroplating, and micromachining [22–26]. Hence, an important advantage of the proposed structure from a practical point of view is its compatibility with common planar fabrication processes, particularly those related to the silicon technology. The metallic pillars can be realized by highly-doped silicon and the surrounding dielectric structure can be micro-machined or etched HR silicon. Therefore, the proposed waveguide is a potential candidate for mm-wave and terahertz integrated devices and circuits.
The outline of this paper is as follows. First, we analyze a 2D hybrid metallic-dielectric grating having an infinite lateral width to investigate the effect of different configurations of the dielectric coating on its guiding characteristics. Main ideas for the design of the final structure are inferred from this analysis. Then, applying the results of the 2D analysis to the design of a finite lateral width structure, we investigate its waveguiding performance and assess its confinement and bending loss at different frequencies. Finally, we study the performance of sharp waveguide bends based on the proposed structure using a full-wave simulation.
2. Analysis of the 2D structure
The proposed waveguide is composed of a periodic set of metallic box-shaped pillars, also called dominos, on a metal plate as illustrated in Fig. 1. The outermost region covering the whole structure is assumed to be filled with a relative dielectric constant of ε1 while the intermediate region overlaying the pillars and the grooves are filled with dielectrics of relative permittivity ε2 and ε3, respectively. The dielectric structure is assumed to be composed of only air and HR silicon. By using a customized dielectric cladding made up of HR silicon and air, we achieve more degrees of freedom in the design of the waveguide without increasing its attenuation constant considerably.
The waveguiding properties of the mentioned metallic structure are dependent on the geometrical parameters, such as height, period, lateral width, and the spacing of metal pillars that are defined by h, d, l and a, respectively. Parameters h2 and l2 denote the height and width of the intermediate dielectric layer covering the metallic pillars as depicted in Fig. 1.
The proposed structure has a finite lateral width. However, to shed some light on the problem and to facilitate the design procedure we start from the analysis of a 2D structure having an infinite lateral width, i.e., the structure with no variation along the y-axis and l = l2 →∞. In the different regions of the 2D structure, the electromagnetic fields of the spoof surface plasmon propagating in the x-direction can be expanded into transverse-magnetic (TM) Bloch waves. In the outermost semi-infinite region, the transverse component of the magnetic field can be written as:
The grooves between the metallic teeth are like a short circuited parallel-plate waveguide whose plate spacing is a. Since a is assumed to be very smaller than a wavelength, we consider only the TEM mode in this region. Considering the ground plane at z = h, for the region inside the grooves (0<z<h) we have:
Applying the continuity of tangential field components at z = 0 and z = -h1, utilizing the orthogonality of Bloch waves over a period, and doing some algebraic manipulations, we obtain the following expression for the dispersion relation of the SSP supported by the 2D structure under investigation:Eq. (4) reduces to the dispersion relation of all-metallic gratings given in .
The field confinement of the grating is related to the exponential decaying rate of the zero-order Bloch wave in the overlaying dielectric layer, i.e., α2,0. Hence, it is dependent on how much the phase constant of the spoof surface plasmon (kx) is higher than the wavenumber in the surrounding dielectric layer. To get a high confinement, the grating should be designed for a large α2,0. As a measure of the confinement, we can define the decaying length as the distance above the grating at which the normalized amplitude of the magnetic field reduces to 1/e. Considering only the zero-order Bloch wave, we can approximate the decaying length by 1/α2,0.
3. Filling the grooves with a different dielectric
If the region above the metallic grating is homogeneously filled with a dielectric while the grooves are possibly filled with a different one, i.e. ε1 = ε2 ≠ε3, Pn becomes zero and Eq. (4) reduces to
Now, there are three different choices for the dielectric constant ratio; ε1/ε3 = 1, ε1/ε3<1, and ε1/ε3>1. By assuming the period and the spacing of the grating as d = 50μm and a = 40μm, respectively, the dispersion diagrams for different dielectric constant ratios are plotted in Fig. 2. In this figure, the light line in the overlaying dielectric region is plotted by a dashed line. At each dielectric constant ratio, the grating height, h, is chosen so as to have the resonant frequency at about 300GHz. In this way, the frequency band of operation as well as the grating period is kept fixed for all structures. As can be seen in this figure, for the dielectric constant ratios less than unity (ε1/ε3<1), the dispersion curves are closer to the light line (kx ≈k0) and consequently, the guided wave will be less confined to the grating surface. In this case, only over a narrow frequency band is a high confinement achievable.
On the other hand, when the dielectric constant ratio becomes higher than unity, that is, when the grooves are filled with a lower dielectric constant, it is expected that the dispersion curves separate from the light line at lower frequencies. Hence, for a fixed resonant frequency, a high confinement can be obtained over a wider bandwidth. This is one of our proposed modifications for increasing the confinement of the hybrid metallic-dielectric grating in a wide bandwidth. But if the outer space is air (ε1 = 1), it is not realizable.
Therefore, a grating of period d = 50μm and a = 40μm submerged in silicon is considered. Now ε1/ε3>1 can be realized by filling the grooves with air. The dispersion diagram of this structure for two cases of filling the grooves with air and silicon is calculated and illustrated in Fig. 3. In each case, the grating height is designed so as to set the resonant frequency at 300GHz. It is evident that the hybrid grating with air-filled grooves possesses a higher phase constant and consequently a better field confinement all over the operational bandwidth. This fact is also evident from Fig. 4 in which the normalized decaying length of both structures is illustrated. The grating with air-filled grooves has a shorter decaying length, equivalent to a higher field confinement, especially at lower frequencies.
4. Dielectric overlay of finite thickness
In this section, we assume that the metallic grating is coated by a finite-thickness layer of silicon (ε2 = ε3 = 11.9) and the remaining space is air (ε1 = 1). The period is d = 50μm and the grating height is h = 65μm. The spacing of grooves is assumed to be a = 40μm. We study the dispersion diagram of this structure for different thicknesses of the silicon overlay (h1).
In Fig. 5, the dispersion curve of the structure having a finite overlay of thicknesses h1 = 10, 50, 250μm along with that of infinite thickness (h1→∞) are illustrated. In this figure, the light line in air and silicon is also shown. At higher frequencies where the SSP is highly confined, the thickness of silicon overlay has a negligible impact on the dispersion diagram. However, at lower frequencies its effect is more notable. By increasing the overlay thickness, the phase constant also increases and more confinement is anticipated. When the layer is so thin that the wave even at high frequencies is not wholly confined in the dielectric layer, it also changes the resonant frequency as can be seen in the dispersion diagram for h1 = 10μm. Figure 6 shows the normalized decaying length for various overlay thicknesses. It is evident that a silicon overlay of thickness h1 = 250μm results in a better confinement over 50-300GHz compared to the structure with infinite silicon layer. Hence, the thickness of the silicon overlay has a considerable impact on the confinement of the hybrid grating and can be optimized based on the frequency band of operation.
5. Finite dielectric overlay and air-filled grooves
Up to this point, we have investigated two approaches for the design of the dielectric coating of the 2D grating under investigation. Firstly, it is shown that the bandwidth and confinement improve by filling the grooves with a lower dielectric constant, e.g. air. Secondly, a better confinement is achieved by using a finite-thickness dielectric overlay. In this section a combination of the mentioned approaches is studied.
For the sake of comparison, dispersion curves of a metallic grating with the period of d = 50μm and the resonant frequency of 300GHz embedded in various dielectric structures are illustrated in Fig. 7. These dielectric structures include a homogeneous silicon space (ε1 = ε2 = ε3 = 11.9, h1 = ∞ and h = 65μm), a finite-thickness silicon overlay (ε1 = 1, ε2 = ε3 = 11.9, h1 = 50μm, and h = 65μm), an infinitely-thick silicon overlay with air-filled grooves (ε1 = ε2 = 11.9, ε3 = 1, h1 = ∞ and h = 165μm), and a finite-thickness silicon overlay with air-filled grooves (ε2 = 11.9, ε1 = ε3 = 1, h1 = 50μm, and h = 165μm). According to Fig. 7, filling the grooves by air enhances the confinement regardless of the thickness of the silicon overlay. Another noticeable point is that the overlay thickness becomes less important at higher frequencies.
6. 3D waveguide of finite lateral width
In this section, the ideas inferred from the analysis of 2D structure are extended to the 3D structure having metallic pillars of finite width l and height h shown in Fig. 1. This domino-like metallic structure, called domino waveguide (DW) hereafter, is embedded in a silicon ridge of width l2 and height h2 = h + h1. Here we choose l = 50μm, l2 = 200μm, and d = 50μm. The width of grooves is also assumed a = 40μm and the height of dominos is chosen in order to have a resonant frequency of 300GHz.
For the mentioned 3D structure, four different configurations can be considered: conventional DW in a homogeneous silicon space (C) (ε1 = ε2 = ε3 = 11.9 and h = 57μm), conventional DW in a homogeneous silicon space with air-filled grooves (C-AF) (ε1 = ε2 = 11.9, ε3 = 1 and h = 82μm), DW integrated into a dielectric (silicon) ridge (DR) (ε1 = 1, ε2 = ε3 = 11.9, h1 = 100μm, l2 = 200μm, and h = 57μm), and DW in a dielectric (silicon) ridge with air-filled grooves (DR-AF) (ε2 = 11.9, ε1 = ε3 = 1, h1 = 100μm, l2 = 200μm, and h = 82μm). In all of these structures, the height of the metallic pillars, h, is designed to set the resonant frequency at 300GHz. For the sake of brevity, we use the abbreviated names of these structures, i.e. C, C-AF, DR, and DR-AF, henceforth.
All the mentioned waveguide configurations are analyzed using a full-wave simulator and their dispersion characteristics are extracted. It is done by performing an Eigen-mode analysis on one unit cell of the waveguide including one metallic pillar in the surrounding dielectric structure. The input and exit faces of the unit cell which are normal to the direction of propagation are assigned as periodic boundaries with a certain phase shift. The lateral faces of the unit cell are closed by PEC boundaries far enough from the metallic pillar. The dominant mode of this periodic structure is the hybrid SSP mode of the waveguide which is negligibly affected by the lateral PEC boundaries. The dispersion relation of this guided mode is extracted by sweeping the phase difference of periodic boundaries from 0 to π and solving for the resultant Eigen-frequencies. By including the material losses in this simulation, the attenuation constant of the waveguide due to metallic and dielectric losses can also be calculated.
The results of this full-wave analysis are presented in Fig. 8. It is evident from this figure that filling the grooves with air results in a higher phase constant with respect to the light line, equivalent to a higher field confinement, in a wider bandwidth both for C-AF and DR-AF waveguide configurations. Moreover, dielectric ridge (DR) embedded structures outperform the conventional domino waveguide (C) structures in terms of broadband field confinement. These claims are also proved by assessing the mode field distribution in the cross-section of the aforementioned waveguides. The normalized electric field intensity at the cross section of these waveguides at 200 GHz is illustrated in Fig. 9. Effectiveness of the two proposed modifications, i.e., embedding the structure in a dielectric ridge and filling the grooves with air, in improving the modal field confinement is evident in these figures, too. The former can be examined by comparing Fig. 9(a) and Fig. 9(c) and the latter by comparing Fig. 9(a) and Fig. 9(b). The effect of both modifications on improving the field confinement is clearly observed by comparing Fig. 9(a) and Fig. 9(d).
While the intuitive idea behind the use of a dielectric ridge for improving the confinement is clear, that of the second modification, i.e. filling the grooves with a lower dielectric constant, needs more clarification. In fact, by filling the grooves with a lower permittivity the amplitude of the electric field inside the grooves is enhanced due to the discontinuity of the normal electric field at the interfaces of the groove. This component of electric field is responsible for the transfer of power along the periodic structure in the exterior region of the grooves. Therefore, the second modification increases the interaction of the SSP mode with the periodic structure by enhancing the penetration of electric field inside the grooves. As a result, the modal field confinement is improved over a wider range of frequencies. This explanation is visualized in Fig. 10 where a 1D plot of electric field along the z-axis in the vicinity of a groove for C and C-AF structures is presented. In this figure, the amplitude of the electric field is normalized to its maximum outside the groove and the z-axis is normalized to the depth of the groove (h) for the ease of comparison. In comparison with C structure, an enhanced penetration of electric field inside the groove and a higher confinement of field to the surface of the groove in C-AF structure is observed in this figure.
7. Performance of waveguide bends
Due to enhanced filed confinement, a lower bending loss in a broad frequency band is also anticipated for the domino waveguide with dielectric coating and air-filled grooves, i.e., the DR-AF structure. To examine this hypothesis, we have simulated a waveguide bend of radius Rbend = 255μm (nearly λ/4 at 300GHz) based on the structures investigated in Fig. 8. Top view of the simulated bend is shown in Fig. 11(a). The metallic pillars, the grooves and the finite-size silicon coating surrounding them are apparent in this figure. The waveguide is excited in its SSP mode efficiently by a customized waveguide adapter proposed in . This adapter converts the TE10 mode of a rectangular waveguide to the guided SSP mode of the waveguides under investigation. By using this method, the scattering parameters of the bend can be calculated and its radiation loss can be extracted. It should be noted that in this analysis, dielectrics and conductors are assumed lossless and radiation is the only source of loss. The bending loss is extracted by comparing the insertion loss of the waveguide bend with that of a straight waveguide of the same length. In other words, the difference in insertion loss (S21 parameter) of the bend and the straight waveguide is attributed to the radiation loss of the bend.
The normalized electric field intensity in the plane of the bend at 200GHz is presented in other parts of Fig. 11 for C, C-AF, DR, and DR-AF structures. Again, improvement of the confinement and noticeable reduction of bending loss in the waveguides with air-filled grooves is apparent by comparing Fig. 11(b) and Fig. 11(c). Comparing the field distributions of Fig. 11(d) and Fig. 11(e) with their counterparts in Fig. 11(b) and Fig. 11(c), we can deduce that by limiting the size of silicon coating a lower bending loss is attainable. Therefore, the bending loss can be reduced both by covering the metallic grating by a finite layer of silicon and filling the spaces in between pillars with air.
For each of the mentioned structures the bending loss is calculated and plotted versus frequency in Fig. 12. The bend radius, Rbend = 255μm, is nearly one quarter of free-space wavelength at 300GHz. For C structure, the bending loss is more than 10dB at 180GHz reducing below 1dB at 275-300GHz. Hence, for this structure the fractional bandwidth over which the bending loss is less than 1dB is less than 9%. This bandwidth is increased to nearly 16% by filling the grooves with air in C-AF structure.
The DR structure has less than 1dB bending loss at frequencies above 200GHz equivalent to the fractional bandwidth of about 41%. In this structure, the bandwidth can be increased even more by filling the space between the metal teeth with air. Hence, the DR-AF has 1dB bending loss at 175-300GHz which is equivalent to a bandwidth of 52%. As seen in Fig. 12, the bending loss of DR-AF waveguide is less than 0.4dB at frequencies higher than 180GHz. It should be emphasized that this level of bending loss has been obtained by a hybrid plasmonic waveguide with a cross-sectional dimension of less than λ/5 and bending radius of nearly λ/4, where λ is the free-space wavelength at 300GHz. These dimensions are much lower than those of silicon channel waveguides which are around λ/2 as reported also in .
By proposed waveguide structures C-AF, DR and DR-AF, the confinement is increased with respect to the conventional domino waveguide. The enhancement of confinement is certainly achieved at the cost of higher conductor losses. To study this issue, the normalized propagation length, i.e. the distance along which the transmitted power reduces to 1/10 of its initial value, for each of the waveguide structures under investigation is calculated using the Eigen-mode analysis introduced in Section 6 and depicted in Fig. 13. In this calculation, the conductors are assumed to be silver with the conductivity of 6.3 × 107 S/m and for HR silicon the conductivity of 2.5 × 10−4 S/m and relative permittivity of 11.9 are considered. The propagation length is normalized to free-space wavelength (λ).
At lower frequencies, due to poor field confinement and spread of electromagnetic energy in the dielectric medium surrounding the metallic structure, the propagation loss is dominantly due to dielectric losses. Hence, the structure consisting of dominos integrated into a dielectric ridge (DR) has the longest propagation length at lower frequencies since some part of power is transferred via free-space surrounding the dielectric ridge in this structure. This fact can also be seen in Fig. 13. At higher frequencies where the mode field is highly confined to the region near the metallic pillars, the dominant loss mechanism is the conductor losses. In this regime, the structure with lowest confinement will have the longest propagation length as a result of less conductor losses and vice versa. As can be seen in Fig. 13, the conventional structure (C) consisting of metallic dominos in a medium homogeneously filled with silicon possesses the longest propagation length at higher frequencies in agreement with our expectations. In both C and DR structures, by filling the spaces in between the pillars by air, the confinement is enhanced and a higher field confinement is achieved resulting in a higher conductor loss. Therefore, structures with air-filled grooves (AF) will have a shorter propagation length compared to their counterparts as can also be seen in Fig. 13. However, at sharp bends the radiation loss dominates the material losses. As a result, the overall loss of bends made from AF structures is expected to be much lower than the others.
So far, we have shown that enhancement of field confinement in C-AF, DR, and DR-AF structures leads to increased propagation (conductor) losses and reduced bending (radiation) losses. Since in sharp bends the radiation loss is the dominant loss mechanism, the overall transmission through the bends is expected to improve by enhancing the confinement. This hypothesis is verified by simulating the wave transmission along a 90 degree bend of radius . The bend connects two straight waveguide sections of length 1mm and is excited by waveguide adapters. Rectangular waveguide ports of the adapters are used to extract the S-parameters of the bend. The overall power transmission coefficient along the bends considering dielectric, conductor and radiation losses is calculated and illustrated in Fig. 14. A schematic of the bend and the adapters is also shown in this figure. It is evident that low-loss sharp bends can be realized using the proposed structures. Especially, the bend based on the DR-AF structure has an overall transmission loss of less than 2.5 dB in a bandwidth of nearly one octave. Therefore, the proposed modified domino waveguides can be a promising candidate for the realization of millimeter-wave and terahertz integrated devices.
In this paper, a hybrid metallic-dielectric waveguide composed of an array of metallic pillars of sub-wavelength size over a ground plane coated by a customized dielectric structure is examined for the sub-THz band. First, the 2D structure having infinite lateral width is studied analytically and the effect of various dielectric overlays on the dispersion diagram and the confinement is investigated. The results of 2D analysis are utilized to derive basic ideas for enhancing the confinement in the realistic 3D structures.
It is shown than covering the metallic grating by a finite-thickness layer of a dielectric while keeping the grooves empty (air-filled) results in a high confinement over a wide bandwidth. It is also stated that by using high-resistivity silicon as the dielectric layer, very low losses will be added to the waveguide in the THz band. Moreover, the structure is realizable using the well-known planar dielectric fabrication methods. The idea of filling the grooves with air while covering the metallic grating by a finite-size silicon layer proves effective in both 2D and 3D structures.
Finally, it is shown that the bending loss of the metallic pillar array with air-filled grooves and embedded in a silicon ridge with the cross-sectional dimension of about 0.2λ × 0.2λ is less than 1 dB in fractional bandwidth of 52% for bend radius of Rbend = 0.25λ where λ is the free-space wavelength at the high end of the frequency band. Therefore, by the proposed hybrid plasmonic waveguide, low bending loss and high confinement are achievable over a wide frequency band with comparatively small waveguide dimensions. Thus, the proposed structure is promising for mm-wave and THz guided-wave applications, especially the integrated circuits.
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