**3. Waveguide port approach for composite geometry**

#### **3.1 Equivalent problems for composite geometry**

**Figure 4** shows the geometry of the problem, consisting of a composite structure composed of *k-1* homogeneous regions *Di*, *i =* 1,2, … , *k*�1, located in the free space region *D*<sup>0</sup> and exposed to waveguide excitation from the waveguide region *B*, which will be considered as *k-*th region of the problem. The region *Dk* is a finite section of the waveguide, confined by the PEC walls, the port surface *Sa*, and the dielectric surface *Sd <sup>k</sup>*, through which the structure is fed. The port surface *Sa* separates the region *Dk* (*B*) from the semi-infinite waveguide region *A* with incident waveguide excitation. In addition, each region *Di* is excited, in general, by the incident field E*inc <sup>i</sup>* , H*inc i* .

To formulate the waveguide port excitation problem through the port surface *S<sup>a</sup>*, we first consider the aperture coupling problem between the waveguide regions *A* and *B* [9]. Thus, we cover the port surface *S<sup>a</sup>* with a PEC sheet and introduce equivalent magnetic currents �**<sup>M</sup>** and **<sup>M</sup>** on both sides of *<sup>S</sup><sup>a</sup>* to divide the excitation problem into two different equivalence problems: the internal problem for region *A*, and the external problem for region *B* (*Dk*), as done in Section 2.1. Then, the internal equivalent problem is identical to that formulated in Section 2.2 and solved in Section 2.3. The external equivalent problem requires consideration of equivalent problems for each boundary surface in regions *Di*, *i =* 1,2, … , *k*, including the port surface *Sa*.

#### **3.2 Formulation of the external equivalent problem for composite geometry**

An external equivalent problem for composite geometry is reduced to a set of equivalent problems for each conducting and dielectric boundary *S<sup>c</sup> <sup>i</sup>* and *S<sup>d</sup> <sup>i</sup>* of free space region *D*<sup>0</sup> (*i* = 0), composite structure regions *Di* (*i* = 1, … ,*k-1*), and finite

**Figure 4.** *Waveguide port problem for composite geometry with waveguide excitation.*

waveguide region *Dk* (*i* = *k*). In turn, each surface *Sd <sup>i</sup>* comprises a set of boundary surfaces *sij* ¼ *Di* ∩ *Dj* (*i* 6¼ *j*), being the interfaces between the regions *Di* and *D <sup>j</sup>*.

Per the equivalence principle [24], the total EM field inside the *i*-th region *Di* can be expressed as the sum of the incident field E*inc <sup>i</sup>* , H*inc <sup>i</sup>* and that induced by the total surface currents distributed over its boundary surface *Si* and radiating into a homogeneous medium with constitutive parameters *ε<sup>i</sup>* and *μ<sup>i</sup>* of the region *Di*. The total electric currents J*<sup>i</sup>* on the boundary surface *Si* consist of conducting currents **J** *c i* , flowing on the inner sides of conducting boundaries *S<sup>c</sup> <sup>i</sup>* , and equivalent electric currents **J** *d <sup>i</sup>* , flowing on the inner sides of dielectric boundaries *Sd <sup>i</sup>* . Magnetic currents in the region *Di* are equivalent currents **M***<sup>d</sup> <sup>i</sup>* , flowing on dielectric boundaries *S<sup>d</sup> <sup>i</sup>* . In addition, in the waveguide region *Dk* there are equivalent magnetic currents **M** on the port surface *S<sup>a</sup>* .

Unknown electric and magnetic currents can be found using the boundary conditions at the conducting boundaries of the composite structure:

$$\left[\mathbf{E}\_i^{inc} + \mathbf{E}\_i^{sc} \left(\mathbf{J}\_i^{\epsilon}, \mathbf{J}\_i^d, \mathbf{M}\_i^d\right)\right] \Big|\_{\text{un}}^{\text{S}\_i^{\epsilon}} = \mathbf{0}, \quad i = \mathbf{0}, \mathbf{1}, \dots, k - \mathbf{1} \tag{21}$$

dielectric boundaries of regions *Di* (*i*, *j* ¼ 0, 1, … , *k*,*i* 6¼ *j*):

$$\left[\mathbf{E}\_i^{inc} + \mathbf{E}\_i^c \left(\mathbf{J}\_i^c, \mathbf{J}\_i^d, \mathbf{M}\_i^d, \mathbf{M}\delta\_{ik}\right)\right]\_{\tan}^{|\epsilon\_{\vec{\eta}}} = \left[\mathbf{E}\_j^{inc} + \mathbf{E}\_j^c \left(\mathbf{J}\_j^c, \mathbf{J}\_j^d, \mathbf{M}\_j^d, \mathbf{M}\delta\_{jk}\right)\right]\_{\tan}^{|\epsilon\_{\vec{\eta}}},\tag{22}$$

$$\left[\mathbf{H}\_{i}^{inc} + \mathbf{H}\_{i}^{c} \left(\mathbf{J}\_{i}^{\varepsilon}, \mathbf{J}\_{i}^{d}, \mathbf{M}\_{i}^{d}, \mathbf{M}\delta\_{ik}\right)\right]\_{\tan}^{|\epsilon\_{\overline{\eta}}|} = \left[\mathbf{H}\_{j}^{inc} + \mathbf{H}\_{j}^{c} \left(\mathbf{J}\_{j}^{\varepsilon}, \mathbf{J}\_{j}^{d}, \mathbf{M}\_{j}^{d}, \mathbf{M}\delta\_{jk}\right)\right]\_{\tan}^{|\epsilon\_{\overline{\eta}}|},\tag{23}$$

and on the port surface *Sa* and the conducting boundary *Sc <sup>k</sup>* of the *k*-th region:

$$\left[\mathbf{E}\_k^{inc} + \mathbf{E}\_k^{sc} \left(\mathbf{J}\_k^c, \mathbf{J}\_k^d, \mathbf{M}\_k^d, \mathbf{M}\right)\right] \Big|\_{\tan}^{S^t + S\_k^c} = \mathbf{0} \qquad \text{just outside } S^t \tag{24}$$

$$\left[\mathbf{H}\_k^{inc} + \mathbf{H}\_k^{sc} \left(\mathbf{J}\_k^c, \mathbf{J}\_k^d, \mathbf{M}\_k^d, \mathbf{M}\right)\right] \Big|\_{\tan}^{\mathbb{S}^u} = \mathbf{H}\_A^{\mathbb{S}^u} \big|\_{\tan} \quad \text{just inside } \mathbb{S}^d,\tag{25}$$

where *δik* is the Kronecker delta, which shows that magnetic currents **M** radiate only in a waveguide region *Dk*. The magnetic field on the right-hand side of (25) is expressed by Eq. (3). The scattered EM fields in (21)–(25) can be expressed in terms of electric and magnetic currents **J***<sup>i</sup>* and **M***<sup>i</sup>* in the dielectric region *Di* as

$$\mathbf{E}\_i^{\rm sc}(\mathbf{J}\_i, \mathbf{M}\_i) = -L\_i^{\rm Ef}(\mathbf{J}\_i) - L\_i^{\rm EM}(\mathbf{M}\_i) \tag{26}$$

$$\mathbf{H}\_i^{\rm c}(\mathbf{J}\_i, \mathbf{M}\_i) = -L\_i^{H\rm f}(\mathbf{J}\_i) - L\_i^{HM}(\mathbf{M}\_i) \tag{27}$$

where *LEJ <sup>i</sup>* , *LEM <sup>i</sup>* , *LHJ <sup>i</sup>* and *LHM <sup>i</sup>* are linear integro-differential operators of EM fields applied to currents radiated in the *i*-th region. It can also be shown [19–22] that the equivalent currents on opposite sides of the dielectric boundaries are related as:

$$\mathbf{J}\_i^d = -\mathbf{J}\_j^d, \quad \mathbf{M}\_i^d = -\mathbf{M}\_j^d \quad \text{on} \ s\_\# \tag{28}$$

Equation (21)–(25) together with relations (26)–(28) and expansions (3) represent the general (EFIE-PMCHWT) form of integral equations for a composite structure with an arbitrary excitation, including the waveguide port.

#### **3.3 MoM solution of the external equivalent problem for composite geometry**

To solve the coupled system of integral Eqs. (21)–(28), we use the MoM to discretize the geometry of all boundary surfaces of the regions *Di* (*i*=1, … ,*k*) into the planar patches and to consider the following expansions for the unknown currents:

$$\mathbf{f}\_{k}^{\epsilon} = \sum\_{n=1}^{N^{\epsilon} + N\_{k}^{\epsilon}} I\_{n}^{\epsilon} \mathbf{f}\_{n}, \quad \mathbf{M} = \sum\_{n=1}^{N\_{a}} M\_{n}, \quad \left[ \mathbf{f}\_{i}^{\epsilon} \right]\_{i=0}^{k-1} = \sum\_{n=1}^{N^{\epsilon}} I\_{n}^{\epsilon} \mathbf{f}\_{n}, \tag{29}$$

$$\left[\mathbf{J}\_{i}^{d}\right]\_{i=0}^{k} = \sum\_{n=1}^{N^{d}} I\_{n}^{d} \mathbf{f}\_{n}, \qquad \left[\mathbf{M}\_{i}^{d}\right]\_{i=0}^{k} = \sum\_{n=1}^{N^{d}} M\_{n}^{d} \mathbf{f}\_{n}, \tag{30}$$

where **f***<sup>n</sup>* are the suitable BFs, *I c <sup>n</sup>*, *Mn*, *I C <sup>n</sup>* , *I d <sup>n</sup>* and *M<sup>d</sup> <sup>n</sup>* are the unknown expansion current coefficients, and *N<sup>a</sup>*, *N<sup>c</sup> <sup>k</sup>*, *N<sup>C</sup>* and *N<sup>d</sup>* are the numbers of BFs on the surfaces *Sa* , *S<sup>c</sup> <sup>k</sup>*, *S<sup>c</sup> i* � �*<sup>k</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> , if any, and *Sd i* � �*<sup>k</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> , respectively. Expansions (29) and (30) take into account relations (28) for unknown equivalent currents on opposite sides of the dielectric boundaries. They also consider the ratios for adjacent currents at material junctions, which are the boundaries between several media [22].

Substituting (29) and (30) into (21)–(25) taking into account (3), (5), (26)–(28) and testing the resulting equations with weighting functions **w**1ð Þ**r** , **w**2ð Þ**r** , … , **w***m*ð Þ**r** , defined in the range of the respective boundary operators, we obtain the following MoM system of linear algebraic equations:

*ZJ c J c mn* � � *<sup>Z</sup><sup>J</sup> c M mn* � � <sup>0</sup> *<sup>Z</sup><sup>J</sup> c J d mn* h i *<sup>Z</sup><sup>J</sup> c Md mn* h i *ZMJ<sup>c</sup> mn* � � *<sup>Z</sup>MM mn* <sup>þ</sup> *<sup>Q</sup><sup>W</sup> mn* � � <sup>0</sup> *<sup>Z</sup>MJ<sup>d</sup> mn* h i *<sup>Z</sup>MMd mn* h i 0 0 *Z<sup>J</sup> CJ C mn* h i *<sup>Z</sup><sup>J</sup> CJ d mn* h i *<sup>Z</sup><sup>J</sup> CMd mn* h i *ZJ dJ c mn* h i *<sup>Z</sup><sup>J</sup> dM mn* h i *<sup>Z</sup><sup>J</sup> dJ C mn* h i *<sup>Z</sup><sup>J</sup> dJ d mn* h i *<sup>Z</sup><sup>J</sup> dMd mn* h i *ZMdJ c mn* h i *<sup>Z</sup>MdM mn* h i *<sup>Z</sup>MdJ C mn* h i *<sup>Z</sup>MdJ d mn* h i *<sup>Z</sup>MdMd mn* h i 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 *I c n* � � ½ � *Mn I C n* � � *I d n* � � *M<sup>d</sup> n* � � 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 ¼ *Vc m* � � *V<sup>M</sup> <sup>m</sup>* <sup>þ</sup> *<sup>V</sup><sup>W</sup> m* � � *V<sup>C</sup> m* � � *Vd m* � � *VHd m* � � 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 (31)

where the matrix elements are defined as *Zαβ mn* ¼ � **<sup>w</sup>***m*, *<sup>L</sup>*^*αβ***f***<sup>n</sup>* D E, *<sup>L</sup>*^*αβ* is the respective boundary integral operator, superscripts *α*, *β* ¼ *J c* , *M*, *J <sup>C</sup>*, *J <sup>d</sup>*, *M<sup>d</sup>* � �; *V<sup>c</sup> <sup>m</sup>* <sup>¼</sup> **<sup>w</sup>***m*, **<sup>E</sup>***inc k* � �, *V<sup>M</sup> <sup>m</sup>* <sup>¼</sup> **<sup>w</sup>***m*, **<sup>H</sup>***inc k* � �, *V<sup>C</sup> m* � � *<sup>i</sup>* <sup>¼</sup> **<sup>w</sup>***m*, **<sup>E</sup>***inc i* � �, *V<sup>d</sup> m* � � *ij* <sup>¼</sup> **<sup>w</sup>***m*, **<sup>E</sup>***inc <sup>i</sup>* � **<sup>E</sup>***inc j* D E, *<sup>V</sup>Hd m* � � *ij* ¼ **w***m*, **H***inc <sup>i</sup>* � **<sup>H</sup>***inc j* D E are the voltage elements due to the incident wave in *<sup>i</sup>*-th and *<sup>j</sup>*-th

media, and the elements *Q<sup>W</sup> mn* and *V<sup>W</sup> <sup>m</sup>* are the same as those expressed by (18) and (19) and determine the additional inclusions in the matrix and voltage elements due to the waveguide ports.

The MoM system (31) generalizes the solution (15) of the canonical waveguide port problem to the case of composite geometry. In the structure of the MoM matrix of this solution, blocks of waveguide excitation, complex structure, and couplings between these objects through dielectric interfaces are clearly seen.
