#### 3.1. Rectangular Waveguide Loaded in the Manner of E-Plane Fully Filled and H-Plane Partially Filled

[11] When rectangular waveguide is loaded in the manner of E-plane fully filled and H-plane partially filled (see Figure 2), the TE mode may exist. According to separating variable method and boundary condition, the eigen function of the rectangular waveguide loaded asymmetrically is given by [*Tang*, 1990]

For the symmetrically filling case, the expression (3) becomes

where *K*_{c1}^{2} = ω^{2}μ_{0}ɛ_{0} − β^{2}, *K*_{c2}^{2} = ω^{2}μɛ − β^{2}, *t* represents loaded slab thickness, and *t* = *a*_{2} − *a*_{1}, *a* stands for waveguide width, *a*_{1} and *a*_{2} the distance from two sides of slab to waveguide wall, respectively, μ_{r} is relative permeability, complex. Equation (4) has many roots corresponding to different modes, and it can be separated into two equations linked to even-symmetrical driven mode and odd-symmetrical driven mode, respectively. They are given by

When the thickness of the loaded slab equals *a* or 0, the propagating constant can be exactly calculated. These two solutions for the full-filled and vacant waveguide modes are given by

where odd *n* corresponds to the even-symmetrical mode, and vise versa.

[12] At first, applying the algorithm depicted in above section computes the propagating constant of rectangular waveguide loaded symmetrically. As an example, standard rectangular waveguide BJ100 (22.86 mm × 10.16 mm) is adopted. The medium thickness *t* is regarded as tracking parameter, and it has a value as

where *NUM *represents the number of total steps of tracking parameter, *NT* is the sequence number and stands for GA encoding parameters of medium thickness as well, and *NT* ranges from 0 to *NUM*. The program begins from *NT* = *NUM *corresponding to waveguide fully filled, where the propagating constant β_{a} could be solved exactly from expression (8). Then β_{a} is regarded as the search center of search area, gradually decrease *NT*, propagating constant β_{NT} is solved favorably in terms of MGA. Since *NT* = *NUM-4*, we can use expression (2) to predict new search center and speed up the computation. When *NT* goes to 0, namely, *t* = 0, all the propagating constants corresponding to different filling thickness are solved conveniently. On the other hand, propagating constant at the point *t* = 0 can be computed analytically according to expression (7), which can be used to testify the algorithm. Tracking step *NUM* can be arbitrary integer, and large *NUM* may avoid the phenomenon of mode skipping, but consume much time.

[13] In the MGA, the fitness function is constructed by

where *val* is the value of the left in equation (5) or (6). If the fitness tends to unity, the *val* will be the ideal value zero, and the equation is solved. In our computation, *val* can be optimized lower than 10^{−6}, almost exact for engineering applications. Figure 3 shows varying rule of propagating constants of TE_{20} mode versus filling thickness for a partial-filled rectangular waveguide. It consists of ten curves, and their EM parameters are shown in Table 1. The material with larger sequent number has higher concentration and larger loss. All the curves begin from the points corresponding to the TE_{20} mode of full-filled rectangular waveguide, and end to same point for TE_{20} mode of vacant waveguide as expected. Obviously, the convergence of the ten curves testifies the validity of the algorithm. Figure 4 shows the curves of the propagating constants of TE_{10}, TE_{30}, TE_{50}, etc. versus the thickness of the magnetic slab, and we notice that the number of the turning points of the curve for higher mode coincides with the first index of the mode, Figure 5 shows magnified part of TE_{B0} mode in Figure 4, in which the turning points is eleven that is accordant with the first mode index. In Figures 4 and 5, *B* in TE_{B0} is hexadecimal number, and stands for eleven. The locus in the figure may vibrate so fierce that the real part of beta become negative somewhere for high-order modes, and this reveals a negative phase velocity occurred. Figure 6 shows the curves of the propagating constants of TE_{20}, TE_{40}, TE_{60}, etc. versus the thickness of the magnetic slab, and Figure 7 presents the local part of Figure 6. Figures 6 and 7 show the same rules as Figures 4 and 5. In Figures 6 and 7, *A* and *C* in TE_{A0} and TE_{C0} are hexadecimal numbers, and stand for ten and twelve, respectively.

Table 1. EM Parameters of Ten Materials | ɛ_{r} | μ_{r} |
---|

1 | 4.780-j0.163 | 1.031-j0.061 |

2 | 5.312-j0.229 | 1.061-j0.125 |

3 | 5.865-j0.299 | 1.092-j0.193 |

4 | 6.442-j0.373 | 1.122-j0.265 |

5 | 7.043-j0.451 | 1.153-j0.342 |

6 | 7.668-j0.533 | 1.183-j0.423 |

7 | 8.320-j0.621 | 1.214-j0.509 |

8 | 8.998-j0.713 | 1.244-j0.600 |

9 | 9.705-j0.811 | 1.275-j0.697 |

10 | 10.442-j0.914 | 1.305-j0.800 |

[14] After phase constant β being solved, the field distribution across the transverse section could be expressed by

where *a*_{1} = (*a* − *t*)/2, *a*_{2} = *a*_{1} + *t*, namely, symmetrically loading*.*Figures 8 and 9 show the field distribution, both coming from a magnetic partial-filled rectangular waveguide. For clarity, the relative amplitude and phase of *E*_{y} and *H*_{x} are regarded as ordinate. We can see that there is a difference between the two circumstances of positive phase velocity and the negative. For the former, the phase monotonously lags from the narrow wall to the center. While for the latter, it continuously precedes from the narrow wall of the waveguide until contact the slab surface, and then, monotonously lags. As for the amplitude, they commonly present like a standing wave, and the *x*-component of the magnetic field are both discontinuous on the slab surface. Obviously, it is reasonable according to the boundary conditions. In Figures 8 and 9 the dash curves represent field *H*_{x}, the solid line field *E*_{y}, and the part between two dot lines the loaded slab. The slab EM parameters are ɛ_{r} = 5.088-j0.103, μ_{r} = 1.209-j0.342, and mode is TE_{90}.

[15] We know that the product of complex μ and ɛ in dissipative magnetic substance is given by

which means the product makes the real part of propagating constant decreased and imaginary part increased for complete-filling status. In the partial-filling status, there is a resonant in transverse section, in other words, presents a standing wave. Along with the decrease of thickness, the slab alternatively losses an area in which the electric field or magnetic field is dominant and results in the changes of propagating constant. That is why the number of turning points always coincides with the mode index.

#### 3.3. Rectangular Waveguide Loaded Asymmetrically in the Manner of E-Plane Fully Filled and H-Plane Partially Filled

[17] The propagating constants of rectangular waveguide loaded asymmetrically with dissipative magnetic slab are more complex, as an example, propagating constant in the manner of E-plane fully filled and H-plane partially filled is calculated. First, compute the propagating constant according to the symmetrical manner to the designated thickness. Second, offset the slab gradually from the center to the location we need and continuously calculate the propagating constant in terms of the algorithm above. For example, when the slab thickness is 5.05 mm, the maximum offset distance should be 8.905 mm, and the tracking computing results are shown in Figure 12. In order to verify the validity of the results, in the maximum offset position, take an image of waveguide structure that forms a new symmetrical-loaded waveguide with width 2*a*, slab thickness 2*t* (see Figure 2b) but the mode index should be double, which means the computing mode after imaging should be TE_{80} if its original mode is TE_{40}. The calculating result of Figure 2b exactly coincides with that of the maximum offset case in Figure 2a, which proves the validity of the algorithm again. In Figure 12, thick curves represent the varying rules of propagation constants for waveguide loaded symmetrically from full-filled mode to vacant mode, and thin curves represent the varying rules of propagating constants when the loaded slab offsets from the center to the narrow wall of waveguide.