Electromagnetic fields and mode transformation effect are theoretically investigated inside a hard-surface (HS) circular waveguide. The waveguide section causing a mode transformation is filled with gyrotropic material, e.g., magnetoplasma or ferrite. In this paper, the gyrotropic material is ferrite. The effect of gyrotropy is taken into account exactly, i.e., no approximation of small gyrotropy or small anisotropy is made. There are different ways to implement HS boundary. One can use longitudinal strips at the surface of a dielectric coating or longitudinal corrugation. In principle, if the corrugation is in axial direction and the depth of the corrugation is effectively a quarter wavelength, a boundary condition equal to hard surface is obtained. Inside the gyrotropic waveguide section the eigenwaves are elliptically polarized and are propagating with different propagation factors. The difference in propagation factors causes a mode transformation. Reflection and transmission formulas are derived for a structure of type isotropic-gyrotropic-isotropic in a general case. Transformer examples are given.
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 The tuned corrugated wave-guiding structures are often used in microwave applications where special kind of properties for field propagation are needed, for example, in antenna horn feed for parabolic reflector antennas [Clarricoats and Olver, 1984, pp. 5–10]. In corrugated waveguide there can propagate TE, TM and circularly polarized fields and in hard-surface waveguide also TEM fields [Kildal, 1990]. In this study the corrugation is in axial direction, thus, forming the boundary condition for hard-surface waveguide within certain frequency range. Time harmonic fields (depending on t as ) are considered and the propagating fields depend on z as , where the propagation factor β is a real number. Longitudinally slotted waveguide has been analyzed by Aly and Mahmoud  and Mahmoud [1991, pp. 98–100]. Further, mode transformation inside a HS waveguide has earlier been studied by Viitanen , where the HS waveguide was filled with chiral material. In this paper the waveguide is filled with ferrite material biased by static magnetic field which is oriented along the waveguide axis. Unlike earlier in Uusitupa and Viitanen , in this paper the effect of the gyrotropic medium is taken into account exactly. No approximation of small gyrotropy or small anisotropy is made. Naturally, the cost is more complicated analysis. However, the benefit is less restricted modelling formulas. One benefit of using ferrite instead of chiral material is that the material parameters of ferrite can be controlled electrically by DC current. Schematic Figure 1 shows a HS waveguide with some notations used.
 The contents of the paper is roughly as follows. In Sections 2 and 3 exact formulas for eigenwaves propagating in ferrite filled hard-surface waveguide are introduced. In Section 4 transverse fields are given. These fields are used in Section 5, where different waveguide interfaces are analyzed. Knowing model formulas for the interfaces, a finite-length gyrotropic HS waveguide section is studied in Section 6. Finally, in Section 7 mode transformer examples are given.
 The constitutive relations and material parameters for ferrite are
where is the Larmor precession frequency ( is the strength of the static magnetic flux density in z-direction), , γ is the gyromagnetic ratio and is the saturation magnetization [Collin, 1966, pp. 286–299]. Here, boldface notation is used for vectors. stands for unit dyad (matrix) and , where footnote 't' refers to 'transverse part'. For magnetoplasma constitutive relations are of similar form i.e. one can use dyadic permittivity model. In terms of theoretical analysis, these two media, ferrite and magnetoplasma, are quite similar and can be obtained by duality from each other. In this study the focus is on ferrite material.
 First step of the analysis is to find the propagating eigenfields depending on z as and the corresponding propagation factors. The electric and magnetic fields are written in terms of transverse and axial parts as and and are inserted into the Maxwell equations and . Assuming that wave propagates in direction, . Thus, Maxwell equations for ferrite are
It is seen that if , the above equations become decoupled. In this nongyrotropic case there is a separate Helmholtz equation for and , i.e., pure TM and TE waves can propagate. If , equations are coupled. Thus, a propagating wave solution is a hybrid mode, i.e., and are not independent of each other. In general case, the equations (14) and (15) can be written as
3. Propagating Wave Solutions
 Assuming a propagating wave (), Maxwell equations lead to Helmholtz equation (18). The solution of Helmholtz equation in a circular waveguide geometry can be written in a form , where is the Bessel function with mode index n. Instead of , also cos or sin are valid solutions for the azimuthal dependency. Because and are both zero at the HS boundary, these longitudinal fields can be written as
Note that this kind of trial solution is not sensible in a PEC (Perfect Electric Conductor) waveguide, because boundary conditions are different for and . Using (22) and (23) with (18), one gets
Thus, it must hold
where is the eigenvalue of the matrix. For λ one gets
The solutions and cannot be independent if , i.e., . Require that
Using this with (25), one finds out that for a propagating wave solution it must hold
From the HS boundary condition, = = 0 at , it follows that
where are the zeros of the Bessel function.
 In a gyrotropic HS waveguide the propagating waves are right-hand and left-hand elliptically polarized waves (+ and − waves). From (26) one finds out the dispersion relation for + and − waves
In practice is very close to one, if ω is high enough. If and , . This special case of small gyrotropy has been studied by Uusitupa and Viitanen . Also one can show that if ,
Hence, as , at the same time and . The sign of :s denominator depends only on , because is assumed positive . For waves propagating in -direction . Thus, in this case, denominator is positive and the sign of is determined only by its numerator. Again assuming , the numerator of can be written as
Thus, has positive and negative value only if
i.e, both + and − waves can exist. On the contrary, if , is always positive (+-wave cannot exist). Also inspecting the formula for reveals that if , +-wave cannot propagate (but − -wave still can if ).
4. Transverse Fields and Power Densities
 Transverse fields and power densities are given for isotropic and gyrotropic HS waveguides. These fields are needed in the next section, where junctions of different waveguides are analysed.
Figure 2 shows some notations related to isotropic and gyrotropic HS waveguides. Let the incident TE and TM fields travel in direction and the reflected travel in direction. Thus, for an isotropic HS waveguide, incident TM fields are
and the reflected TM fields are
From (9) and (10) it follows that in a gyrotropic HS waveguide eigenfields propagating in direction can be written as
where asterisk stands for conjugation and
For power density one gets
If the wave is propagating in direction, the above expressions (47)…(53) can be used by replacing by and by (changing the sign of changes the sign of , as seen in formula (31)). How do the field expressions change by this replacement? It is seen that coefficients and are invariant to propagation direction (). On the other hand, it is seen that the sign of is changed, but the sign of remains unchanged. Hence, the sign of is changed, as it should. Because only the sign of is changed, the propagation direction does not affect the rotation direction of the field vector . That is how it should be, because the gyrotropic medium is nonreciprocal.
5. Reflection and Transmission of Fields at Different Interfaces
 In this section different waveguide interfaces are studied. Reflection and transmission matrices for isotropic-gyrotropic (IG) and for gyrotropic-isotropic (GI) interface is obtained. These matrices are used in the next section when analysing a gyrotropic waveguide section of finite length.
5.1. Isotropic-Gyrotropic (IG)
 First, a junction of an isotropic and gyrotropic waveguide section is studied. The interface is located at . Reflected and transmitted field amplitudes are solved assuming a TM or TE incident wave from the isotropic section . Along the isotropic section there is incident TM () and TE () fields, and also reflected TM () and TE () fields (Figure 3a). The transmitted fields in the gyrotropic section are + and − waves having amplitudes and , respectively.
 Assume first a TM incident field. Requiring the total transverse fields continuous at , i.e.,
leads to an array of equations. In matrix form equations can be presented as
The coefficients are
From the matrix equation (56) unknowns , , and can be solved. The solution is
 With TE incident field, a similar array of equations is obtained. Only the right-hand side is changed, when comparing to (56):
where and are the reflection and transmission matrices at the IG-interface. Using the power density formulas (37), (40), (43), (46) and (53), one can investigate power transmission and reflection. For example, one can check whether the incident power is equal to the total scattered power of reflected and transmitted fields.
5.2. Gyrotropic-Isotropic (GI)
 Now a waveguide junction gyrotropic-isotropic is studied. Again, the interface is located at . Reflected and transmitted fields are solved assuming a combination of 'plus' and 'minus' incident waves from the gyrotropic section. Assume first that the incident field is travelling in direction (Figure 3b). The total transverse fields are required continuous at , i.e.,
Now there is ±-fields propagating in both directions. Earlier it was pointed out that the direction change can be taken into accout by replacing by and by . For example, using the field formulas of the previous section, the continuity condition of magnetic field, equation (74), splits into two orthogonal parts. At , the uz X -part becomes
and the -part becomes
Using the earlier definitions for and these can be written as
In a similar way, the continuity condition for electric field leads to
Finally, collecting the continuity equations (77)…(80) together, in matrix form they are
The solution for the amplitudes is
Using the above definitions for reflection factors , , and , the reflected amplitudes can be written as
and the transmitted amplitudes as
The matrices and give the reflected and transmitted amplitudes in case where the incident ±-field travels in direction. Assume now that the incident and transmitted fields travel in direction, instead of (Figure 3b again,but orientation of z-axis reversed). It turns out that the new equation array is exactly same as before, if the “old” unknown is simply replaced by . Hence,
which leads to
The minus sign in refers to the assumption that the incident and transmitted fields travel in direction. is not affected by the change in propagation direction of the incident field.
6. Analysis of a Gyrotropic HS Waveguide Section
 In this section a structure shown in Figure 4 is studied. In general, incident field is a combination of TM and TE fields. Along the gyrotropic section this combination is changed. For example, if incident field is purely TM (), transmitted field may be purely TE (). Unlike in Uusitupa and Viitanen , here the effects of gyrotropic medium are taken into account exactly.
 The goal is to find out the transmitted amplitudes and and the reflected amplitudes and . Multiple reflections inside the gyrotropic section are taken into account. Let us define a phase shift matrix for the gyrotropic section as
Hence, one can write for the transmitted wave amplitudes
where stands for unit matrix. The first term corresponds to the direct transmission, the second term transmission after double-reflection etc., see Figure 5. Using the sum formula of infinite geometric progression, one gets
The reflected amplitudes are
The background of different terms is illustrated in Figure 5. Again, adopting the sum formula, one gets
7. Example Mode Transformers
 Along the gyrotropic section the total field is made up of + and − waves, which travel with different propagation factors. As a result, TM field is gradually changed to TE, for example. In general, an incident combination of TM and TE fields can be changed to another combination.
 In this section examples of different mode transformers are shown. All the structures are like in schematic Figure 4. The transmitted and reflected amplitudes, , , , , are computed using formulas (92) and (94). Knowing the amplitudes, the power densities are computed using formulas (37), (40), (43) and (46).
7.1. Example 1
 In the first example a well-matched case is studied, i.e., the permittivities are same in the isotropic and the gyrotropic waveguides, . The incident field is purely TM. Assume a ferrite material with , T GHz ) and relatively low saturation magnetization T, i.e., GHz. Assume a mode with and core radius mm. The length of the gyrotropic section, to change from TM to TE, is now mm. Figure 6 shows transmitted TM and TE power normalized by incident TM power. The curves are shown as a function of frequency f, with different lengths d ( means there is no gyrotropic section at all, i.e., there is no mode transformation). The normalized reflected power in this case is close to zero (one can see that the transmitted powers approximately add up to 1 which is the normalized incident power). In this case the gyrotropy and the reflections are so small that the structure could be well analysed in an approximative way, i.e., assuming small gyrotropy and anisotropy, and neglecting the reflections.
7.2. Example 2
 In this case, the gyrotropic section is between two air-filled waveguides (). The parameters of the ferrite are and GHz, which should be very typical values [Collin, 1966, p. 299]. Assume a mode with and core radius mm. The length of the gyrotropic section, to change from TM to TE, is now mm. Figure 7 shows transmission power of TM and TE mode as a function of frequency, normalized by incident TM power. The normalized reflected power in this case is very high in average, but at some frequency ranges tolerable. One can see two transmission peak frequencies, where incident TM power has changed to transmitted TE power.
 In Figure 8 it is shown how changing the bias magnetic flux density shifts the transmission spectrum. Hence, one can tune the transmission peak to the operation frequency range of the HS boundary. Naturally, a real-life HS boundary is more like a quasi-HS boundary, with a certain operation bandwidth. Within this bandwidth, longitudinal fields and are relatively close to zero at the quasi-HS boundary.
 Assume a dense corrugation with thin corrugation grooves. The HS condition is
where is the wavenumber in depth-direction inside the corrugation groove, h is the groove depth and is the wavenumber in the material used in the corrugation groove. In principle, this condition cannot hold exactly for all the waves simultaneously, because . To make the HS condition more independent on the β, high permittivity material should be used inside the corrugation. Also by shaping the corrugation groove profile properly, one can increase the range of use of the corrugation. For example, one can use dual-depth slots [Clarricoats and Olver, 1984, p. 154; Sridhar and Srivastava, 1982].
 In summary, the bandwidth of the mode transformation is limited by the medium contrast between the isotropic and gyrotropic sections, and by the limited range of use of the HS boundary. Keeping this in mind, an optimum ferrite for this application has relatively small () and close to .
 Gyrotropic HS waveguide section, situated between isotropic sections, can be used as a mode transformer. For this kind of structure, transmission and reflection formulas were derived in a very general case, i.e., the effect of the gyrotropic medium was taken into account exactly. If the gyrotropic medium is ferrite, one can control the gyrotropic parameter by static bias magnetic field, i.e., by DC current. Thus, one can tune the transmission peak frequency to the operation frequency range of the HS boundary. The bandwidth of the mode transformation is mostly limited by the medium contrast between the isotropic and gyrotropic sections, and by the limited range of use of the HS boundary. Example mode transformers were given.