Radio Science

Analysis of finite-length gyrotropic hard-surface waveguide

Authors

  • Tero Uusitupa,

    1. Electromagnetics Laboratory, Department of Electrical and Communications Engineering, Helsinki University of Technology, Espoo, Finland
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  • Ari Viitanen

    1. Electromagnetics Laboratory, Department of Electrical and Communications Engineering, Helsinki University of Technology, Espoo, Finland
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Abstract

[1] 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.

1. Introduction

[2] 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 equation image) are considered and the propagating fields depend on z as equation image, where the propagation factor β is a real number. Longitudinally slotted waveguide has been analyzed by Aly and Mahmoud [1985] and Mahmoud [1991, pp. 98–100]. Further, mode transformation inside a HS waveguide has earlier been studied by Viitanen [2000], 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 [2001], 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.

Figure 1.

Hard-surface waveguide filled with gyrotropic material.

[3] 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.

2. Theory

[4] The constitutive relations and material parameters for ferrite are

equation image
equation image

where equation image is the Larmor precession frequency (equation image is the strength of the static magnetic flux density in z-direction), equation image, γ is the gyromagnetic ratio and equation image is the saturation magnetization [Collin, 1966, pp. 286–299]. Here, boldface notation is used for vectors. equation image stands for unit dyad (matrix) and equation image, 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.

[5] First step of the analysis is to find the propagating eigenfields depending on z as equation image and the corresponding propagation factors. The electric and magnetic fields are written in terms of transverse and axial parts as equation image and equation image and are inserted into the Maxwell equations equation image and equation image. Assuming that wave propagates in equation image direction, equation image. Thus, Maxwell equations for ferrite are

equation image
equation image

That is

equation image
equation image
equation image
equation image

From (8) one gets

equation image

Inserting e into the equation (6) and solving h one obtains

equation image

Inserting h into the equation (7) leads to the scalar equation

equation image

Similarly, using equations (5) and (8) and the expression for h, the following equation is obtained

equation image

Defining

equation image

equations (11) and (12) can be written as

equation image
equation image

In the special case equation image these two equations reduce to

equation image
equation image

It is seen that if equation image, the above equations become decoupled. In this nongyrotropic case there is a separate Helmholtz equation for equation image and equation image, i.e., pure TM and TE waves can propagate. If equation image, equations are coupled. Thus, a propagating wave solution is a hybrid mode, i.e., equation image and equation image are not independent of each other. In general case, the equations (14) and (15) can be written as

equation image

where

equation image
equation image
equation image

3. Propagating Wave Solutions

[6] Assuming a propagating wave (equation image), Maxwell equations lead to Helmholtz equation (18). The solution of Helmholtz equation in a circular waveguide geometry can be written in a form equation image, where equation image is the Bessel function with mode index n. Instead of equation image, also cosequation image or sinequation image are valid solutions for the azimuthal dependency. Because equation image and equation image are both zero at the HS boundary, these longitudinal fields can be written as

equation image
equation image

Note that this kind of trial solution is not sensible in a PEC (Perfect Electric Conductor) waveguide, because boundary conditions are different for equation image and equation image. Using (22) and (23) with (18), one gets

equation image

Thus, it must hold

equation image

where equation image is the eigenvalue of the matrix. For λ one gets

equation image

The solutions equation image and equation image cannot be independent if equation image, i.e., equation image. Require that

equation image

Using this with (25), one finds out that for a propagating wave solution it must hold

equation image

From the HS boundary condition, equation image = equation image = 0 at equation image, it follows that

equation image

where equation image are the zeros of the Bessel function.

[7] 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

equation image

From (28) it follows that

equation image

In practice equation image is very close to one, if ω is high enough. If equation image and equation image, equation image. This special case of small gyrotropy has been studied by Uusitupa and Viitanen [2001]. Also one can show that if equation image,

equation image

Hence, as equation image, at the same time equation image and equation image. The sign of equation image:s denominator depends only on equation image, because equation image is assumed positive equation image. For waves propagating in equation image-direction equation image. Thus, in this case, denominator is positive and the sign of equation image is determined only by its numerator. Again assuming equation image, the numerator of equation image can be written as

equation image

Thus, equation image has positive and negative value only if

equation image

i.e, both + and − waves can exist. On the contrary, if equation image, equation image is always positive (+-wave cannot exist). Also inspecting the formula for equation image reveals that if equation image, +-wave cannot propagate (but − -wave still can if equation image).

4. Transverse Fields and Power Densities

[8] 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.

[9] Figure 2 shows some notations related to isotropic and gyrotropic HS waveguides. Let the incident TE and TM fields travel in equation image direction and the reflected travel in equation image direction. Thus, for an isotropic HS waveguide, incident TM fields are

equation image
equation image
equation image

and the reflected TM fields are

equation image
equation image
equation image

Incident TE:

equation image
equation image
equation image

Reflected TE:

equation image
equation image
equation image

From (9) and (10) it follows that in a gyrotropic HS waveguide eigenfields propagating in equation image direction can be written as

equation image
equation image
equation image

where asterisk stands for conjugation and

equation image

and

equation image
equation image

For power density one gets

equation image

If the wave is propagating in equation image direction, the above expressions (47)(53) can be used by replacing equation image by equation image and equation image by equation image (changing the sign of equation image changes the sign of equation image, as seen in formula (31)). How do the field expressions change by this replacement? It is seen that coefficients equation image and equation image are invariant to propagation direction (equation image). On the other hand, it is seen that the sign of equation image is changed, but the sign of equation image remains unchanged. Hence, the sign of equation image is changed, as it should. Because only the sign of equation image is changed, the propagation direction does not affect the rotation direction of the field vector equation image. That is how it should be, because the gyrotropic medium is nonreciprocal.

Figure 2.

Isotropic and gyrotropic waveguides.

5. Reflection and Transmission of Fields at Different Interfaces

[10] 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)

[11] First, a junction of an isotropic and gyrotropic waveguide section is studied. The interface is located at equation image. Reflected and transmitted field amplitudes are solved assuming a TM or TE incident wave from the isotropic section equation image. Along the isotropic section there is incident TM (equation image) and TE (equation image) fields, and also reflected TM (equation image) and TE (equation image) fields (Figure 3a). The transmitted fields in the gyrotropic section are + and − waves having amplitudes equation image and equation image, respectively.

Figure 3a.

Isotropic-gyrotropic interface.

[12] Assume first a TM incident field. Requiring the total transverse fields continuous at equation image, i.e.,

equation image
equation image

leads to an array of equations. In matrix form equations can be presented as

equation image

The coefficients are

equation image
equation image
equation image
equation image

From the matrix equation (56) unknowns equation image, equation image, equation image and equation image can be solved. The solution is

equation image
equation image
equation image
equation image

where

equation image

[13] With TE incident field, a similar array of equations is obtained. Only the right-hand side is changed, when comparing to (56):

equation image

In TE case the solution for amplitudes is

equation image
equation image
equation image
equation image

Collecting the results from (61)(64) and (67)(70) together, one can write

equation image
equation image

where equation image and equation image 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)

[14] Now a waveguide junction gyrotropic-isotropic is studied. Again, the interface is located at equation image. 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 equation image direction (Figure 3b). The total transverse fields are required continuous at equation image, i.e.,

equation image
equation image

Now there is ±-fields propagating in both directions. Earlier it was pointed out that the direction change can be taken into accout by replacing equation image by equation image and equation image by equation image. For example, using the field formulas of the previous section, the continuity condition of magnetic field, equation (74), splits into two orthogonal parts. At equation image, the uz X equation image -part becomes

equation image

and the equation image -part becomes

equation image

Using the earlier definitions for equation image and equation image these can be written as

equation image
equation image

In a similar way, the continuity condition for electric field equation image leads to

equation image
equation image

Finally, collecting the continuity equations (77)(80) together, in matrix form they are

equation image

The solution for the amplitudes is

equation image
equation image
equation image
equation image

Using the above definitions for reflection factors equation image, equation image, equation image and equation image, the reflected amplitudes can be written as

equation image

and the transmitted amplitudes as

equation image

The matrices equation image and equation image give the reflected and transmitted amplitudes in case where the incident ±-field travels in equation image direction. Assume now that the incident and transmitted fields travel in equation image direction, instead of equation image (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 equation image is simply replaced by equation image. Hence,

equation image

which leads to

equation image

The minus sign in equation image refers to the assumption that the incident and transmitted fields travel in equation image direction. equation image is not affected by the change in propagation direction of the incident field.

Figure 3b.

Gyrotropic-isotropic interface.

6. Analysis of a Gyrotropic HS Waveguide Section

[15] 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 (equation image), transmitted field may be purely TE (equation image). Unlike in Uusitupa and Viitanen [2001], here the effects of gyrotropic medium are taken into account exactly.

Figure 4.

Gyrotropic section between isotropic sections.

[16] The goal is to find out the transmitted amplitudes equation image and equation image and the reflected amplitudes equation image and equation image. Multiple reflections inside the gyrotropic section are taken into account. Let us define a phase shift matrix for the gyrotropic section as

equation image

Hence, one can write for the transmitted wave amplitudes

equation image

where equation image stands for equation image 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

equation image

The reflected amplitudes are

equation image

The background of different terms is illustrated in Figure 5. Again, adopting the sum formula, one gets

equation image
Figure 5.

Reflection terms (left) and transmission terms (right).

7. Example Mode Transformers

[17] 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.

[18] In this section examples of different mode transformers are shown. All the structures are like in schematic Figure 4. The transmitted and reflected amplitudes, equation image, equation image, equation image, equation image, 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

[19] In the first example a well-matched case is studied, i.e., the permittivities are same in the isotropic and the gyrotropic waveguides, equation image. The incident field is purely TM. Assume a ferrite material with equation image, equation image T equation image GHz ) and relatively low saturation magnetization equation image T, i.e., equation image GHz. Assume a mode with equation image and core radius equation image mm. The length of the gyrotropic section, to change from TM to TE, is now equation image 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 (equation image 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.

Figure 6.

Transmission power of TE and TM mode (normalized powers shown). Within the range equation image mm, the longer the gyrotropic section, the more is the incident TM power transformed to the TE power. With equation image19 mm, incident TM is entirely transformed to TE. Rev1: In this example, parameters are equation image, core radius equation image1.7 mm, equation image GHz, equation image GHz and equation image.

7.2. Example 2

[20] In this case, the gyrotropic section is between two air-filled waveguides (equation image). The parameters of the ferrite are equation image and equation image GHz, which should be very typical values [Collin, 1966, p. 299]. Assume a mode with equation image and core radius equation image mm. The length of the gyrotropic section, to change from TM to TE, is now equation image 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.

Figure 7.

Transmission power of TE and TM mode and the total reflected power (all powers normalized). In this example, gyrotropic section is between air-filled waveguides. Parameters are equation image, equation image, core radius equation image mm, equation image GHz, equation image GHz, equation image.

[21] In Figure 8 it is shown how changing the bias magnetic flux density equation image 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 equation image and equation image are relatively close to zero at the quasi-HS boundary.

Figure 8.

Changing equation image (i.e. equation image) shifts the transmission spectrum. Other parameters are equation image, equation image, core radius equation image mm, equation image GHz, equation image.

[22] Assume a dense corrugation with thin corrugation grooves. The HS condition is

equation image

where equation image is the wavenumber in depth-direction inside the corrugation groove, h is the groove depth and equation image is the wavenumber in the material used in the corrugation groove. In principle, this condition cannot hold exactly for all the waves simultaneously, because equation image. 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].

[23] 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 equation image (equation image) and equation image close to equation image.

8. Conclusion

[24] 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 equation image 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.

Ancillary