Propagating constants of rectangular waveguide loaded with dissipative slab obey complex transcendental equation, of which the property of multiroot and the dependence of modes on those roots often make people puzzled, especially for dissipative magnetic materials in an asymmetrically filling situation. An effective new algorithm combining Genetic Algorithm (GA) with Parameter Tracking Scheme (PTS) and Dynamic Searching Area (DSA) technique is developed. By using the algorithm, the propagating constants of waveguide symmetrically or asymmetrically loaded with dissipative magnetic slab are solved successfully and tightly linked to certain designated modes. The results also reveal some interesting phenomena, such as for some high-order modes and certain thickness of loaded slab, the negative phase velocity may occur.
 Rectangular waveguide loaded with dielectric/magnetic slab is often used in electromagnetic and microwave technique field, for example, phase shifter, circular-polarizer, isolator, short matched terminator, etc. All of above applications need a basic understand to their propagating constants for both transmission modes and evanescent modes so as to do theoretic analysis and structure design. However, the propagating constant meets complex transcendental function, which is difficult to solve, especially when the filling material is dissipative magnetic and asymmetrical. For rectangular waveguide loaded with dielectric slab or post, various methods have been proposed. In references [Auda and Harrington, 1984; Cicconi and Tosatelli, 1977; Ise and Koshiba, 1986; Nielsen, 1969; Sahalos and Vafiadis, 1985], circular posts inside waveguide have been dealt with by using approximate analytical or numerical methods, and rectangular posts have been studied in references [Siakavara and Sahalos, 1991; Yoshikado and Taniguchi, 1989]. As far as we know, the dielectric slab or post is often studied, but only a few papers discuss the dissipative magnetic slab case [Tian et al., 2003]. This paper introduces a new algorithm that combines genetic algorithm (GA) with Parameter Tracking Scheme (PTS) and Dynamic Searching Area (DSA) technique. By using the algorithm, the propagating constants of rectangular waveguide loaded symmetrically and asymmetrically with dissipative magnetic material are calculated successfully. In the computing procedure, some peculiar phenomena are found, including the special regulation of the propagating constant varying with the filling thickness and the occurrence of negative phase velocity. In section 2 the new algorithm, which includes the main ideas of GA and PTS and DSA, is depicted briefly. The propagating constants of rectangular waveguide symmetrically inserted a dissipative magnetic slab, which is in the manner of E-plane fully filled while H-plane partially filled or H-plane fully filled while E-plane partially filled, are computed respectively in terms of the algorithm above in section 3. The propagating constants of rectangular waveguide loaded asymmetrically with dissipative magnetic materials are also computed in this section. Section 4 gives the summary and conclusion.
2. Algorithm Description
 The problem of rooting complex transcendental equation often occurs in electromagnetic theory and engineering. The generally used methods, such as Newton-Raphson iterative method, Muller interpolation method and conjugated gradient method, need both initial values and gradient information, whose selection or determination are not easy and affect the convergence severely. Therefore developing an efficient, stable, initial-value and gradient data free and universal algorithm to this kind of problem is very valuable.
 The main ideas of the new algorithm based on GA, PTS and DSA are as follows: In the process of solving equation F(x1, x2, x3, …, t) = 0, select or append a parameter t, which makes an exact solution present when t is equal to a designated value t0. Then, change t gradually and minutely towards the expected value td, the solution will change gradually and continuously, too. At each step, using GA finds its new solution. When t finally tends to td, the solution is obtained. Because of the continuity of physical parameter, the solution should be changed gradually, this makes us possible to utilize GA in a small area where only a single peek exists, and ensures the algorithm convergent naturally. After the previous solutions are gotten, the subsequent searching center could be predicted according to the difference formula, which speeds up the calculation further.
2.1. Genetic Algorithm
 Genetic algorithm is a fruit of intercross, pervasion and promotion between life science and engineering science. GA simulates the mechanisms of survival of the fittest, natural selection, inheritance and variation in Darwin evolutionism, and essentially is an efficient, parallel, global and random searching arithmetic for optimization. It has two main features: population searching strategy and information exchange between individuals. At present, as a practical and robust optimizing and searching method, GA is utilized generally in electromagnetic field [Johnson and Rahmal-samii, 1997; Weile and Michielssen, 1997], such as the design of absorbing material [Qian et al., 2001], frequency selective surface [Chakravarty et al., 2002], antenna design [Altshuler, 2002], and so on.
 It is well known that for conventional GA, sizing the population is problem-specific and a strong function of the length and cardinality of the chromosome [Goldberg et al., 1992]. For most optimal problems, the length of the chromosomes is a function of the number of parameters to be optimized, the individual parameter range, and the step size to be implemented. Hence, for a multidimensional search space, a large population base and several generations are required to achieve optimal or near optimal results, and this places a considerable burden on the CPU time and computer resource. The algorithm developed in this paper needs to call GA iteratively in every tracking step (see next part), and this will decreases the algorithm efficiency if conventional GA is used. One possible approach to solve this problem is to employ micro genetic algorithm (MGA). Simultaneously, combines MGA with niche and shuffle technique to improve the efficiency. Simply speaking, the MGA starts with a random and small population (generally 5–50), which evolves in a conventional GA fashion and convergences after a few generations. At this point, keeping the best individual from the previously converged generations (elitist strategy), a new random population is chosen and the evolution process restarts. In our case, population convergence occurs when the difference in bits of the chromosome between the best and other individuals is less than 5%. Although the sizing of initial population is small, it can avoid premature convergence [Krishnakumar, 1989]. In the process of MGA, other techniques, such as niche technique [Goldberg and Richardson, 1987], tournament selection [Goldberg and Deb, 1991], shuffle technique, uniform crossover, and elitist strategy are added.
2.2. Parameter Tracking Scheme and Dynamic Searching Area
 The parameter tracking scheme includes three aspects: the determination of the effective tracking parameter, the foundation of the dynamic searching area, and the prediction of the new search area.
 The determination of the correct tracking parameter is very important. According to practical computation and analysis, we confirm the criterions as follows: (1) The purpose of selecting or appending a parameter is to make the equation to be solved having exact solution when the parameter is equal to a certain value. (2) For the case of appending tracking parameter, it should blot out the subordinate part and remain the multiroot part of the equation at the beginning, which may ensure the completeness of solution set. (3) At the beginning, the tracking parameter should make the fast-changing part of the equation remained, which may promote the convergence.
 The foundation of the dynamic searching area includes the flexing of search area and the displacement of search center. The search area may be square or rectangular corresponding to Cartesian coordinate or circular corresponding to polar coordinate, and it depends on the boundary shape. If optimal point approaches the exterior boundary, the point is assumed to be new search center and search area is displaced. If there is enough space between optimal point and exterior boundary of the search area, this point is assumed to be the new search center and search area is displaced, at the same time, the search area should be compressed. The compressing of search area can speed up the computation, and it is reasonable to compress the search area to 4–5 times smaller in practical program. However, the compression must be limited, otherwise GA is disabled because all chromosomes tend to the same when the search area is too small.
 In regard to the prediction of the new search area, the solution of the equation will vary following the change of tracking parameter, and there should be some regularity between them. The linear extrapolation, the two-order difference extrapolation, and the three-order difference extrapolation are the possible options for the prediction of the new searching center. The two-order and three-order difference extrapolations are respectively given by
For clearness, Figure 1 shows the flow chart of the algorithm.
3. Propagating Characteristics of Rectangular Waveguide Loaded With Dissipative Magnetic Material
3.1. Rectangular Waveguide Loaded in the Manner of E-Plane Fully Filled and H-Plane Partially Filled
 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 Kc12 = ω2μ0ɛ0 − β2, Kc22 = ω2μɛ − β2, t represents loaded slab thickness, and t = a2 − a1, a stands for waveguide width, a1 and a2 the distance from two sides of slab to waveguide wall, respectively, μr 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.
 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 βa could be solved exactly from expression (8). Then βa is regarded as the search center of search area, gradually decrease NT, propagating constant βNT 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.
 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 TE20 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 TE20 mode of full-filled rectangular waveguide, and end to same point for TE20 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 TE10, TE30, TE50, 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 TEB0 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 TEB0 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 TE20, TE40, TE60, 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 TEA0 and TEC0 are hexadecimal numbers, and stand for ten and twelve, respectively.
Table 1. EM Parameters of Ten Materials
 After phase constant β being solved, the field distribution across the transverse section could be expressed by
where a1 = (a − t)/2, a2 = a1 + 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 Ey and Hx 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 Hx, the solid line field Ey, and the part between two dot lines the loaded slab. The slab EM parameters are ɛr = 5.088-j0.103, μr = 1.209-j0.342, and mode is TE90.
 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.
3.2. Rectangular Waveguide Loaded Symmetrically in the Manner of E-Plane Partially Filled and H-Plane Fully Filled
 When rectangular waveguide is loaded symmetrically in the manner of E-plane partially filled and H-plane fully filled, the mode TM to Y-direction may exist. Under this situation, characteristic equation can be obtained by the method of alternative mode sets [Harrington, 1961]. Here we omit the similar process, and simply give the results. Figure 10 shows the curves of the propagating constants of TM(1,0)y, TM(1,2)y, TM(1,4)y,etc. versus the thickness of the horizontal magnetic slab, and Figure 11 shows the local part of Figure 10. The locus vibrates so violent that the circles overlap each other. We notice that the number of the turning point also coincides with the first index of the mode, and negative phase velocity is appeared as well for small thickness and higher modes.
3.3. Rectangular Waveguide Loaded Asymmetrically in the Manner of E-Plane Fully Filled and H-Plane Partially Filled
 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 2a, slab thickness 2t (see Figure 2b) but the mode index should be double, which means the computing mode after imaging should be TE80 if its original mode is TE40. 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.
4. Summary and Conclusion
 In order to root complex equations, a new algorithm that combines the powerful function of GA with Parameter Tracking Scheme (PTS) and Dynamic Searching Area (DSA) technique is developed. The problem of propagating constants of rectangular waveguide symmetrically or asymmetrically filled with dissipative magnetic material is studied thoroughly in terms of the algorithm. We get some interesting results as follows:
 1. The varying rules of propagating constants of rectangular waveguide loaded with dissipative magnetic material versus thickness of loaded slab are very complex, especially for rectangular waveguide loaded asymmetrically, horizontally, and in higher modes.
 2. The varying rules of propagating constants versus the concentration of dissipative magnetic material are complex, too. The higher the concentration is, the stronger the vibration takes place.
 3. There is a relationship between propagating constant and the mode, namely, the number of turning points of propagating constant curve is equal to the mode index. For example, there are 12 turning points of propagating constant curve in Figure 7, and it is equal to the first index of mode TEC0. Other curves show similar regularities.
 4. When the rectangular waveguide is loaded asymmetrically, the varying rule of propagating constant versus offset of loaded slab is more intricate. Figure 12 shows the varying locus. For other slab thickness, the curves are similar, too.
 5. For higher modes and thinner slab, the real part of propagating constant may become negative, which means a negative phase velocity occurred. It shows that the amplitude goes down because of attenuation while the phase precedes, which physically like the property of left-handed wave (LHW) [Shelby et al., 2001]. Although it is present in the evanescent mode, it occurs in the natural, homogeneous, isotropic materials, not in artificial substances. Figures 5, 7, and 11 show the phenomena clearly. It is noted that in the case of dielectric filling, no matter how low or high the loss is, no negative phase velocity present, namely, the phenomenon of negative velocity only occurs in magnetic material filled. In many papers, scholars often use split annular resonator to construct the left-handed material (LHM) [Shelby et al., 2001], perhaps it is the magnetic property of annular configuration to form the LHM characteristics.
 6. The algorithm has some advantages of high precision, robustness, efficiency, and universality.