# 3D Fields Inside a Cylindrical Dielectric Resonator

## Introduction

The mode nomenclature of cylindrical dielectric resonators has received a wide interest, since it finds great application in several microwave devices including cavity filters, resonators and other structures. Nevertheless, between various published investigations, mode indexing for this geometry is not always based on common guidelines. This fact is especially true for microwave filter analysis, where publications are using a diverse mode nomenclature to analyse the modes excited within the resonators. For example, mode EH11δ is the same as mode HEM12δ, although they use different indexing.

To solve this problem, one approach would be to identify the internal electric (E) and magnetic (H) fields of the cylindrical resonator, since their distribution is always unique at each resonance. But, then how can we all agree on mode indexing?

Mode identification is a vital task that microwave engineers need to consider. Nowadays a vast amount of literature sources, describe in many different ways how to classify modes, based on their field pattern. To avoid this confusion, one source which, I believe, deserves particular attention and is quite popular among many articles, is that of Kajfez [H1]. In relation to DRAs and in relation to the topic of this website, a similar mode nomenclature is also discussed in the publication by Mongia [K1].

An introduction to mode nomenclature for CDRs has already been given in this website. I will attempt to extend this knowledge by relating the distribution of the axial components of the E and H-fields within a resonator, i.e. Ez and Hz, with the basic field equations within DR antennas, as presented in Table I in [K1] (quoted below). This discussion will help to attain a greater understanding of the fields at resonance, which leads to successful mode assignment.

(table below taken by [p.234, K1] - Fields Inside an Isolated Cylindrical Dielectric Resonator)

The analysis will be mainly assisted by viewing internal 3-D plots of computer simulated isolated cylindrical resonators for various modes and relate them with simple mathematical expressions.

## Analysis of Internal Field Distributions

In my opinion, the Equations Table represents the core concept in terms of mode description and the DRA's radiation properties, expressed in mathematical equations.

### TE01 mode:

#### Equatorial plane (z=0)

Starting from the first mode, namely TE01δ, the Hz is expressed as Hz = J0(hr)cos(βz), where ‘J0’ the Bessel function of first kind and of order zero, ‘h’ is the radial wavenumber, ‘r’ is radial distance from the centre of the circle, ‘β’ is the axial (z-directed) wavenumber and ‘z’ is the distance from the resonator’s mid-plane. TE01δ mode possesses azimuth (φ) symmetry. As it has been mentioned in another article, the E-field lines in the equatorial plane of this mode are forming a ‘ring’ pattern.

Plots of the Hz component in equatorial and meridian plane are shown on the tabbed panels below.

In the equatorial plane, z=0, hence Hz=J0(hr) at half way plane through the resonator (please note that the x=y=z=0 point in the 3D plots is located at the bottom face of the resonator – ignore this! The field equations on the equations Table assume a xyz coordinate system that is centred at the point of resonator’s symmetry point, i.e. half way through resonator’s thickness and in the centre of the circle). So, the magnitude of the Hz component varies similarly with that of Bessel function of the first kind of order zero.

Plots of Bessel functions of the first kind are shown below:

Indeed, by observing the J0(x) plot above and the 3D plot in the tabbed panel, it can be seen that, for example, when r=0 (centre of circle) Hz is maximum (red colour in the 3D plot, indicates maximum field strength), which agrees with the case when x=0 in J0(x), i.e. J0 gets maximum.

As you travel away from the centre of the circle while staying at a constant z-plane, r gets bigger, so the argument ‘hr’ in J0(hr) gets bigger too. According to the J0(x) plot, as ‘x’ departures away from x=0, the value of J0(x) reduces. Indeed, checking the 3D plot, we see that the magnitude of the Hz component reduces, which again agrees with the J0(x) plot shown above.

By considering all directions away from the centre towards the periphery of the circle, the magnitude of the Hz component in TE01 mode within the resonator appears as a ‘bell’ like shape, as seen in the 3D plot.

#### Meridian plane (φ=0)

In meridian plane we need to consider a plane that splits through the resonator. As an example, a 3D plot for the plane at φ=0 is shown, again for the Hz component of TE01 mode.

The expression Hz=J0(hr)cos(βz) needs to be analysed by considering simultaneous observation of the arguments ‘hr’ and ‘βz’. At locations close to the circle’s centre (r->0) J0(hr) attains its maximum value (see Bessel plots above for reference). This fact, combined with an observation point close to resonators mid-plane, i.e. z=0 (cos(βz)=1), results a maximum overall value for the Hz component. Indeed, by observing the 3D plot of TE01 mode at this point we can see that Hz achieves a maximum.

Away from r=0 and z=0, the value of Hz attenuates: As you move along z-axis towards resonators faces (constant ‘r’) from z=0, Hz varies with cos(βz). Similarly, away from circle’s centre (increasing r and constant z) it varies with J0(hr). Simultaneous variations of ‘r’ and ‘z’ result in the 3D plot seen.

### Higher Order Modes:

Similar analysis can be applied for other modes for the remaining of the equations on the Table. Note that, for higher order modes, the Bessel functions within the field equations express of a higher order too, such as J1(x), J2(x) and so on. Hence, expect that the 3D plots of the Hz or Ez components will be presenting a similar change in shape too.

For example, by observing Hz (=J1(hr)cos(βz) ) of mode HEM12 in the equatorial plane (z=0), we can notice the presence of two ‘bells’. These two ‘bells’ are related with the plots of Bessel function of order one, J1(x). However, notice the dependence of Hz-field due to cos(φ) (or sin(φ) ) terms, too.

## General Observations for Field Equations

It’s quite interesting to notice:

- The value of the first index in the mode nomenclature (see other article for definition, mn(p+δ) ) indicates the order of the Bessel function of the first kind. For example, TM01 mode is dependent on J0(hr), HEM12 mode is dependent on J1(hr) and so on.
- For TE modes, Hz always dominates Ez. The opposite applies for TM modes. This is logical, since the internal field distribution is connected with the operation of a virtual vertical magnetic or electric dipole respectively for these two modes (TE0n mode radiates like a vertical magnetic dipole, hence the dominant z-directed component is Hz. On the other hand, TM0n mode radiates like a vertical electric dipole and the dominant z directed component is Ez. See Table for further description).
- HEM modes have azimuth variation and this is indicated by the presence of cos(nφ) factor (see Table). It can also be seen, that this factor is interchangeable with sin(nφ), which implies the same mode but rotated by 45 degrees when viewing a constant z-plane.
- HEM modes are degenerate. This means that you can excite (using an appropriate feed scheme) two such modes in the same frequency, but with orthogonal polarisations.
- The strength of the Ez or Hz fields within the resonator on the equatorial plane is varying in a similar manner as the plot of the corresponding Bessel function for a given mode. (????????????? ->>At the circumference of the resonator, the field strength is approximately close to zero, agreeing with the zero crossing point of the Bessel plot,see Bessel plots above ???????????, - <<--this statement is under revision)

## 3D Plots of TE01 and HEM12 modes of Cylindrical Resonator

- TE01 - Hz equatorial ( Hz=J0(hr)cos(βz) )
- TE01 - Hz meridian
- HEM12 - Hz equatorial ( Hz=J1(hr)cos(βz)cos(φ) )
- HEM12 - Hz meridian

Above: magnitude of Hz component within a cylindrical DR for TE01 mode in the equatorial plane (z=0).

Above: Hz component within a cylindrical DR for TE01 mode in the meridian plane (φ=90 deg).

Above: magnitude of Hz component within a cylindrical DR for HEM12 mode in the equatorial plane. (amplitudes are shown, otherwise if the phase was taken into consideration the left 'peak' would be 180 degrees reversed, i.e. it would be pointed downwards, to match shape of J1(x) plot)

Above: Hz component within a cylindrical DR for HEM12 mode in the meridian plane (φ=0). (Plot is shown with phase variation for the Hz to match the shape of J1(x) plot)