Fig. 5.1 shows the schematic of the proposed HPCIW with the key design parameters indicated. The top and bottom layers are a pair of parallel metal plates, while the middle layer is an air-core line-defect hexagonal dielectric photonic crystal [146]. The photonic crystal consists of hexagonal arrays of air holes in the dielectric slab. Compared with conventional hollow metallic rectangular waveguide or SIW, the proposed HPCIW removes the metallic parts at the side regions in y direction. It helps to eliminate conduction current flow between the two metallic parallel plates, which is the main source of Ohmic loss for the fundamental HE10 mode, replaced by displacement current flow.
The material for the dielectric slab is desired to be high permittivity and low loss, to create a significant photonic bandgap and to reduce the material
Even (I)
Even (II)
Odd
Fig. 5.2 Band diagram and dispersion curves for the vertically polarized modes in the line-defect photonic crystal waveguide. The even and odd modes here correspond to the symmetric and antisymmetric modes with respect to the 𝑥- 𝑧 plane. The black region below the light line (𝑘 = 𝜔 𝑐⁄ ) shades the slow modes that are not guided by the HPCIW.
absorption loss. Taking into account the available fabrication techniques, Rogers RO3010™ material was chosen for the dielectric slab. Its design Dk (dielectric constant) is 11.2 between 8 and 40 GHz and loss tangent is 0.0022 measured at 10 GHz, according to the datasheet obtained from the manufacturer. Two rows of unit cells are missing in the photonic crystal, so creating an air-core line defect, with width 𝑤 = 3𝑎, where 𝑎 is the lattice constant of the photonic crystal. A line defect with an integer number of missing unit cells helps to maintain the periodicity of the photonic crystal structure if there are waveguide bends in the structure, and thereby reduces other unwanted defects. The radius of the air holes, 𝑟, is chosen to be 0.467𝑎. The reason for choosing this value will be explained later.
The fundamental mode of the proposed HPCIW is a vertically polarized HE10 mode. Like SIW, the height of the HPCIW, ℎ, is adjustable, but it will slightly affect the performance, and will introduce high order modes if the value of ℎ is too large [147]. In this chapter, a HPCIW with a thin dielectric slab, namely, ℎ < 𝑤 2, is considered. Since the electric field of the HE10 mode is evenly distributed along the z-axis, the height of the HPCIW does not contribute to the dispersion relation. Therefore, the problem about dispersion relation is simplified to a two-dimensional (2D) problem in the 𝑥 - 𝑦 plane, and it represents the eigenvalues for the vertically (z-axis) linearly polarized modes confined in the air core of the line-defect photonic crystal waveguide.
The dispersion curves for the guided modes in the 2D line-defect photonic crystal waveguide were numerically studied by using MPB, as shown in Fig. 5.2. The dispersion curves related to different values of 𝑟 are presented, but the band diagram (the gray regions) only corresponds to 𝑟 = 0.467𝑎, which is why the dispersion curves of 𝑟 = 0.467𝑎 are properly aligned with the edge of the band diagram while others are not. For 𝑟 = 4.67𝑎, the mode pattern of the even and odd modes in Fig. 5.2 are shown in Fig. 5.3. As it can be seen, both the odd mode and the even mode (I) can be tightly confined by the line defect, but the even mode (II) is much more lossy with
(a) Odd mode (b) Even mode (I) (c) Even mode (II)
Fig. 5.3 Mode patterns when 𝑟 = 0.467𝑎 . (a) (𝑘-, 𝜔-) = (0.1, 0.4687) ; (b) (𝑘-, 𝜔-) = ³0.4025, 0.4512´; (c) (𝑘-, 𝜔-) = ³0.4925, 0.5159´. Here, 𝑘- = 𝑘𝑎 2𝜋⁄ and 𝜔- = 𝜔𝑎 2𝜋𝑐⁄ . Periodical boundaries are applied.
significant field penetrating into the bulk of the photonic crystal. This is because the dispersion curve of the even mode (II) is close to the edge of the band diagram where the degree of confinement is reduced [130]. The even mode (I) is the desired fundamental mode, and by appropriate coupling in to this field pattern of the HPCIW, avoiding excitation of the odd mode, the waveguide can operate in single mode fashion [17].
Fig. 5.2 shows the dispersion curve of the even mode (I), which is folded at the Brillouin zone edge (𝑘 = 𝜋 𝑎) and turns into the even mode (II) [148]. There is a gap at the Brillouin zone edge between the even mode (I) and the even mode (II) where neither of them can propagate in the HPCIW. The impact of the radius of the air holes, 𝑟, on the dispersion curves is also shown in Fig. 5.2. The radius changes the dispersion curves continuously and predictably. The operating fractional bandwidth of the desired even mode (I) is related to the radius, 𝑟, as shown in Fig. 5.4. Here, the fractional bandwidth is defined as ∆= 𝜔-•− 𝜔
-‘ 𝜔-•+ 𝜔-‘ 2 , where 𝜔-• and 𝜔-‘ are the upper and lower bounds of the dispersion curve for the even mode (I) in the bandgap, respectively. The bandwidth is maximum, ∆= 0.26 , when 𝑟 𝑎 = 0.467. This explains the reason for the choice of 0.467𝑎 for the radius of the air holes. It is noted that a photonic crystal with wider bandgap more strongly confines the field in the line defect, thus reducing the leakage of the wave from the waveguide [130].
The group velocity and GVD of the even mode (I) when 𝑟 𝑎 = 0.467 are plotted vs. normalized frequency in Fig. 5.5. The GVD is defined as 𝜕y𝑘 𝜕𝜔y. Fig. 5.5(a) shows that as the frequency increases, the group velocity first Fig. 5.4 The dependence of the fractional bandwidth on the normalized
increases and then decreases, with the turning point occurring at 0.457𝑎 where the zero GVD occurs, as seen in Fig. 5.5(b).
The design presented here is scalable. One can achieve this by changing the lattice constant of the photonic crystal, and by properly choosing a low- loss and high-permittivity material which suits the target frequency band. In the next section, a design at Ka-band by choosing Rogers RO3010™ as the host material and setting 𝑎 = 4.106 mm will be demonstrated.