II. PROBLEM FORMULATION

The solution domain is split into three distinct regions, as shown in Fig. 1, which are connected by the unknown magnetic currents M. Method-of-moments (MoM) regions R0 and R1 are include a generic SIW circuit with arbitrary number of coupling/radiating slots and posts excited through coaxial/waveguide ports. R2 is the PO region away from the antenna containing a dielectric lens which is stacked on the top wall of the antenna and characterized by the constitutive parameters ε _d and μ _d . Bo2 denotes the boundary surface between R1 and R2, and Bo1 is the boundary between R0 and R1. The coupling slots are located on Bo1, while the radiating slots are placed on Bo2. The normal vector ${\hat n_{i + 1}}$ points into the region Ri. Parallel plates that form the SIW structure are assumed to be a perfect-electric conductor with infinitesimal thickness. Only one excitation is considered here however if several feeding ports are present their effects would be superposed. Excitation port generates the field in R0 and the same time the field in R1 is excited by the coupling slots placed on Bo1.

Fig. 1.

Schematic view of an ILA fed by double-layered SIW slot antenna. Also shown is an entire domain basis function shape used to expand the unknown magnetic current on the slot with corresponding parameters.

In order to model the double-layered SIW-based slot antenna (internal region) MoM along with mode matching technique are necessary. This is done by representing the field in the post-walled structure through the Dyadic Green's function of the parallel plate waveguide [Reference Arnieri and Amendola13, Reference Amendola, Arnieri and Boccia14]. By invoking the field-equivalence principle, the slots are replaced by a metal plate and equivalent magnetic currents (M _inner and M _outer ) are introduced on their respective surfaces. Due to the use of narrow rectangular slots it can be assumed that there are only transverse electric field components across the slots, and the magnetic current is completely directed along the longitudinal direction and its transverse dependence is negligible. Applying the boundary conditions on the aperture of slots requires the tangential electric and magnetic fields to be continuous. The continuity of tangential electric fields leads to the equality M _inner  = −M _outer  = M, as shown in Fig. 1, whereas the continuity of magnetic field across the surface S of the slots leads to the following general integral equation for each slot i [Reference Bayderkhani, Forooraghi, Arnieri and Abbasi-Arand15]:

(1) $${\mathop{n}^{\frown}} \times H_{tot}^{inner} (r) = {\mathop{n}^{\frown}} \times H_{tot}^{outer} (r),$$

(2) $$\eqalign{\hat n \times & \left( {H_{inc}^{inner} (r) + \sum\limits_j^{Ns} {H_{M\_slot\_j}^{inner}} (r)} \right). \cr & = \hat n \times \left( \matrix{H_{inc}^{outer} (r) + \hfill \cr + \sum\limits_j^{Ns} {H_{M\_slot\_j}^{outer}} (r) + \sum\limits_q^{Nq} {\sum\limits_j^{Ns} {H_{jq}^{Po} (r)}} \hfill} \right)r \in {S_i}} $$

In which [Reference Arnieri and Amendola12]

(3) $$\eqalign{& {H_{inc}}(r) = \cr & \quad - j\omega {\varepsilon _0}{\varepsilon _r}\int_{{V_{SRC}}} {{{\bar G}_{PPW}}(r,r^{\prime})} \cdot {M_{SRC}}(r^{\prime})dr^{\prime} + H_S^{{M_{SRC}}} (r),} $$

(4) $$\eqalign{& {H_{M\_slot,j}}(r) = \cr & \quad - j\omega {\varepsilon _0}{\varepsilon _r}\int_{{S_j}} {{{\bar G}_{PPW}}(r,r^{\prime})} \cdot {M_j}(r^{\prime})dr^{\prime} + H_S^{{M_j}} (r),} $$

where r and r′ define the observation and source points, respectively; N _s is the number of slots; Nq is the number of modes used to expand the current on the slot.

M _j is the equivalent magnetic current source defined on surface of slot j (S_j); ${\bar G_{PPW}}(r,r')$ is the Dyadic Green's function of the parallel plate waveguide when all the slots are absent (see Appendix A); $H_S^M $ is the field scattered by the metallic posts due to the current M; M _SRC (r ′) is the magnetic current source distribution defining the port excitation in volume V _SRC located in region R ₀; and H^PO is the magnetic field pre-computed by Ray/PO technique to model the dielectric lens effects, and it has non-zero value only for outer region. To do so, the equivalence principle is applied on the lens surface and the forward as well as backward radiation is calculated. The total field in the external region is then obtained by the summation of incident fields impinging from the antenna and scattered fields from the lens boundary. Surface integration equation was established on the aperture of slots. Finally, the equations with the unknowns are solved to obtain the equivalent surface currents on the aperture of slots. In the application of (2), it should be noted that in each region only corresponding slots should be considered, for example in R0 the radiating slots are vanished.

The equivalent magnetic currents, which are used to model the slots, are expanded in terms of an entire domain sinusoidal basis function. The magnetic currents for a slot j are:

(5) $${M_j}(r^{\prime}) = {\hat v_j}\sum\limits_q {{f_{q,j}}(r^{\prime}) \cdot {V_{q,j}}\quad 1 \le j \le {N_s}}, $$

(6) $${f_{q,j}}(r^{\prime}) = \sin \left( {{k_q}\left( {\displaystyle{{{L_j}} \over 2} + {\nu _j}} \right)} \right),\quad \vert {\nu _j} \vert \le {L_j},$$

(7) $${k_q} = \displaystyle{{q\pi} \over {{L_j}}}\quad {\rm and}\quad r^{\prime} = {r_{jq}} + u{\hat u_j} + v{\hat v_j},$$

where r _jq , V _q,j and L _j are defined as the center of the qth basis function on the slot j, the unknown expansion coefficients and the length of the jth slot, respectively, as shown in Fig. 1, and q spans over the number of modes Nq.

The integral equation (2) is solved using the Galerkin weighting procedure where equations (3)–(7) are substituted into (2). After some mathematical manipulations, the following linear system with N _s  × N _q unknowns is obtained:

(8) $${h_{p,i}} = \sum\limits_{q,j} {Y_{q,j}^{p,i} \cdot {V_{q,j}}\quad \forall (p,i)}, $$

where p and q span over the basis functions, while i and j span, the number of slots, NS.

(9) $${h_{p,i}} = \left\{ {\matrix{ {\int_{{S_i}} {{f_{p,i}}{{\hat v}_i} \cdot {H_{inc}}(r)dr}} & {{\rm for}\;{\rm interior}\;{\rm region,}} \cr 0 & {{\rm other}\;{\rm regions,}} \cr}} \right.$$

(10) $$Y_{q,j}^{p,i} = Y_{q,j}^{p,i} (int1) + Y_{q,j}^{p,i} (int2) + Y_{q,j}^{p,i} (ext),$$

(11) $$\eqalign{& Y_{q,j}^{p,i} (ext) = \cr & \quad \left\{ {\matrix{ j\omega {\varepsilon _0}{\varepsilon _d}\int_{{S_i}} {dr} \int_{{S_j}} dr^{\prime}{f_{p,i}}{{\hat v}_i} \cdot {{\bar G}_{ext}}(r,r^{\prime})\cr \cdot {{\hat v}_j}{\,f_{q,j}} + \int_{{S_i}} {dr} {f_{p,i}}{{\hat v}_i} \cdot {H^{PO}}(r) & {{\rm for}\;{\rm radiating}\;{\rm slots,}} \cr 0 & {{\rm otherwise,}} \cr}} \right.} $$

(12) $$\eqalign{Y_{q,j}^{p,i} (int) = & j\omega {\varepsilon _0}{\varepsilon _r} \cdot \cr & \quad \int_{{S_i}} {dr} \int_{{S_j}} {dr^{\prime}} {f_{p,i}}{{\hat v}_i} \cdot {{\bar {G}}_{PPW}}(r,r^{\prime}) \cdot {{\hat v}_j}{\,f_{q,j}} \cr & \quad + \int_{{S_i}} {dr{f_{p,i}}} {{\hat v}_i} \cdot H_S^{{f_{q,j}}} (r),} $$

where $Y_{q,j}^{p,i} (int{\rm 1})$ and $Y_{q,j}^{p,i} (int{\rm 2})$ are the admittance elements in R0 and R1 regions, respectively, that have the same form of (12). G _ext is the Dyadic Green's function for the external region [Reference Arnieri and Amendola12]. The incident magnetic field radiating into the structure by a port source in the absence of the slots is defined by H _inc (r). In the above formulation, only one port source which is radiated in the interior region is assumed. As mentioned earlier, if several feeding ports are presented, their effects need to be superposed. Consequently, the number of unknowns is given by the number of modes assumed in each slot (Nq) multiplied by the number of slots (N _S ), i.e. Nq × Ns. As a result, the computational efficiency is achieved along with reasonable accuracy using the proposed hybrid method.

III. SIMULATION

In order to validate the proposed technique it was applied to a hemispherical ILA fed by a double-layered SIW-based cavity-backed slot antenna. A comparison between the proposed method and finite-element simulations carried out using Ansoft's HFSS are presented and discussed.

A) Input impedance and radiation pattern

The geometry of the hemispherical ILA fed by double-layered SIW cavity-backed slot antenna is shown in Fig. 2, and the corresponding parameters shown in Fig. 3. The SIW antenna structure consists of two main building blocks consisting of the feeding system and the radiating system, as shown in Fig. 3. The feeding and the radiating systems are implemented on two different substrate layers. The vertical walls of structure are realized using rows of metalized posts embedded in a dielectric substrate (RO4003TM) with a relative permittivity of 3.55, a loss tangent of 0.0027, and a thickness of 10 mil. The radiating system includes a main cavity, where the radiating slots are etched on the top side of the metal, as shown in Fig. 2.

Fig. 2.

Representation of the entire hemispherical lens antenna.

Fig. 3.

Feeding SIW slot antenna including: (a) cavity and radiating slots placed on top of the substrate, and (b) feeding waveguide and coupling aperture placed on the bottom of the substrate.

The cavity is divided into four sub-sections with a radiating slot on the top wall of each sub-section. In order to excite the cavity, a feeding short-circuited SIW waveguide on the rear side of the cavity is used. In order to connect two layers, a coupling longitudinal slot transition located in the common face of two SIW blocks is used. This antenna radiates into a hemispherical dielectric lens with constitutive parameters ε _d  = 2.1 and μ _d  = 1 and radius of R _DL  = 50 mm. The antenna parameters and their corresponding values are listed in Table 1. Figure 4(a) shows the reflection coefficient of the antenna compared with results obtained from HFSS. The phase of the reflection coefficient of the antenna is depicted in Fig. 4(b). The total field pattern of the antenna is shown in Fig. 5. The results indicate that the proposed method agrees very well with the results using HFSS. It should be noted that by tuning the ILA parameters, better matching is achievable. In order to show the dielectric lens effects on the input impedance of the antenna, the impedance with and without dielectric lens is computed and shown in Fig. 6; from this figure, it is found that the input impedance of the antenna is strongly altered due to the presence of lens.

Fig. 4.

Reflection coefficient of the ILA compared with HFSS, (a) magnitude and (b) phase.

Fig. 5.

Total field pattern of the ILA computed using the proposed technique and HFSS in (a) φ = 0 plane and (b) φ = 90 plane.

Fig. 6.

Impact of dielectric lens on the input impedance of the SIW slot antenna obtained by using MoM–PO method (solid lines show the Z11 with lens effects, and dash lines show Z11 without lens effects).

Table 1.

Antenna parameters and corresponding values.

B) CPU time and storage requirements

The hybrid method described in the present paper requires three matrices to be filled and inverted. In the PO region, a matrix needs to be filled in order to compute the PO currents on the lens surface.

The matrix dimension depends on the number of points assumed in order to compute the PO currents. In the MoM region, two matrixes need to be filled, where one of them used to compute the field scattered by the metallic vias and the other one to evaluate the unknown magnetic currents over the slots.

The size of the matrix in the PO region depends on the area that is illuminated by the antenna beam, and the number of mesh cells depends on the computational area. By increasing the number of mesh cells the accuracy of the PO fields and consequently the CPU processing time and memory requirements are increased. The simulated results indicate that a λ _g /10 × λ _g /10 mesh cell size covering the illuminated area is sufficient to ensure the accuracy of the calculated PO fields. In the MoM region, the number of cylinders NC, the number of slots Ns and the number of Z modes Nz define the size of two matrixes.

Table 2 and 3 report the CPU processing time and memory storage used in simulating the hemispherical ILA fed by the double-layered SIW cavity-backed slot antenna, respectively.

Table 2.

CPU time (Intel Core i7 3.55 GHz, 16 GB RAM).

Table 3.

Memory requirement.

Comparison of the CPU time and storage capacity of the proposed technique with HFSS shows substantial reduction is achieved with the proposed technique, i.e. CPU time is reduced by around 95% and memory capacity is reduced by around 96%. Consequently, the proposed technique raises possibility of simulating larger and more complex structures such as multilayered SIW-based antennas with limited computing resources with acceptable numerical accuracy.