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12.5: Circular Waveguides

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  • The details are different, but the modes sustained by a circular wave-guide have much in common with the rectangular wave-guide modes. They may, for example, be classified as transverse electric modes (TE modes) in which there is no component of electric field along the guide axis, or as transverse magnetic modes (TM modes) in which there is no component of the magnetic field along the guide axis.

    12.5.1 TM Modes.

    A vector potential whose transverse components are zero but for which Az is not zero will generate transverse magnetic modes because Hz is necessarily zero since curl(\(\vec A\)) has a zero z-component. Az must satisfy the wave equation (12.2.6) in order that the fields generated by Az satisfy Maxwell’s equations:

    \[\nabla^{2} \mathrm{A}_{\mathrm{z}}+\epsilon_{\mathrm{r}}\left(\frac{\omega}{\mathrm{c}}\right)^{2} \mathrm{A}_{\mathrm{z}}=0. \label{12.56}\]

    It is convenient to use cylindrical polar co-rdinates (r,θ,z) because of the cylindrical symmetry implied by the shape of a cylindrical wave-guide. For a wave travelling along the z-direction, one can write

    \[\mathrm{A}_{\mathrm{z}}=\mathrm{A}(\mathrm{r}, \theta) \exp \left(i\left[\mathrm{k}_{\mathrm{g}} \mathrm{z}-\omega \mathrm{t}\right]\right). \nonumber\]

    From now on the factor \(\exp (i[k_gz − ωt])\) will be understood and not written out explicitly. Using cylindrical polar co-ordinates Equation (\ref{12.56}) becomes

    \[\frac{1}{\mathrm{r}} \frac{\partial}{\partial \mathrm{r}}\left(\mathrm{r} \frac{\partial \mathrm{A}}{\partial \mathrm{r}}\right)+\frac{1}{\mathrm{r}^{2}} \frac{\partial^{2} \mathrm{A}}{\partial \theta^{2}}+\left[\epsilon_{\mathrm{r}}\left(\frac{\omega}{\mathrm{c}}\right)^{2}-\mathrm{k}_{\mathrm{g}}^{2}\right] \mathrm{A}=0. \nonumber\]

    or setting

    \[ \mathrm{k}_{\mathrm{c}}^{2}=\epsilon_{\mathrm{r}}\left(\frac{\omega}{\mathrm{c}}\right)^{2}-\mathrm{k}_{\mathrm{g}}^{2}, \label{12.57}\]

    and multiplying through by r2

    \[\mathrm{r} \frac{\partial}{\partial \mathrm{r}}\left(\mathrm{r} \frac{\partial \mathrm{A}}{\partial \mathrm{r}}\right)+\frac{\partial^{2} \mathrm{A}}{\partial \theta^{2}}+\mathrm{k}_{\mathrm{c}}^{2} \mathrm{r}^{2} \mathrm{A}=0. \label{12.58}\]

    Now let the amplitude \(A(r,θ)\) be written as the product of a function F(r) that depends only on the radius r and the function cos (mθ), where m is an integer. The constant m must be an integer so that \(A(r,θ)\) will be single valued in angle: ie. A(r,0) must be equal to A(r,2\(\pi\)m). The use of the function cos(mθ) is arbitrary. We could just as well use sin (mθ) or a function of the form \(f(θ) = a \cos (mθ) + b \sin (mθ)\), where \(a\) and \(b\) are constants. All of these choices have in common that d2f/dθ2 = −m2f. The various choices of a,b simply amount to a choice of the orientation of the wave-guide mode pattern with respect to the axis θ = 0.

    The equation for the radial function, \(F(r)\), becomes

    \[\mathrm{r} \frac{\mathrm{d}}{\mathrm{dr}}\left(\mathrm{r} \frac{\mathrm{d} \mathrm{F}}{\mathrm{dr}}\right)+\left(\mathrm{k}_{\mathrm{c}}^{2} \mathrm{r}^{2}-\mathrm{m}^{2}\right) \mathrm{F}=0. \label{12.59}\]

    This equation for F(r) can be put in the standard form of Bessel’s equation by the introduction of a change of variable:

    \[\mathrm{x}=\mathrm{k}_{\mathrm{c}} \mathrm{r}, \label{12.60}\]

    then Equation (\ref{12.59}) becomes

    \[\mathrm{x} \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x} \frac{\mathrm{dF}}{\mathrm{dx}}\right)+\left(\mathrm{x}^{2}-\mathrm{m}^{2}\right) \mathrm{F}=0. \nonumber\]

    The solutions of this equation that remain finite at r=0 are

    \[\mathrm{F}(\mathrm{x})=\mathrm{J}_{\mathrm{m}}(\mathrm{x}), \nonumber\]

    where the Jm(x) are Bessel’s functions of integer order because m is an integer. See ”Schaum’s Outline Series, Mathematical Handbook” by Murray R. Spiegel, McGraw-Hill, New York, 1968, Chapter 24. The required form of the vector potential is

    \[\mathrm{A}(\mathrm{r}, \theta)=\mathrm{A}_{0} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \cos (\mathrm{m} \theta), \label{12.61}\]

    where A0 is a constant, and kc is given by Equation (\ref{12.57}).

    The magnetic field components are obtained from \(\vec H\) = curl(\(\vec A\)):

    \mathrm{H}_{\mathrm{r}}&=\frac{1}{\mathrm{r}}\left(\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \theta}\right)=\frac{-\mathrm{m}}{\mathrm{r}} \mathrm{A}_{0} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \sin (\mathrm{m} \theta), \label{12.62} \\
    \mathrm{H}_{\theta}&=-\left(\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{r}}\right)=-\mathrm{k}_{\mathrm{c}} \mathrm{A}_{0} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \cos (\mathrm{m} \theta), \nonumber \\
    \mathrm{H}_{\mathrm{z}}&=0, \nonumber

    (these are all multiplied by the factor exp (i[kgz − ωt]), of course). The notation \(\dot{\mathrm{J}}_{\mathrm{m}}(\mathrm{x})\) means the derivative of the Bessel function with respect to the argument x.

    The electric field components can be calculated from \(\operatorname{curl}(\overrightarrow{\mathrm{H}})=-i \omega \epsilon \overrightarrow{\mathrm{E}}\):

    \mathrm{E}_{\mathrm{r}} &=\frac{-i}{\epsilon \omega}\left(\frac{\partial \mathrm{H}_{\theta}}{\partial \mathrm{z}}\right)=-\frac{\mathrm{k}_{\mathrm{g}}}{\epsilon \omega} \mathrm{k}_{\mathrm{c}} \mathrm{A}_{0} \dot{\mathrm{J}}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \cos (\mathrm{m} \theta), \label{12.63} \\
    \mathrm{E}_{\theta} &=\frac{i}{\epsilon \omega}\left(\frac{\partial \mathrm{H}_{\mathrm{r}}}{\partial \mathrm{z}}\right)=\frac{\mathrm{k}_{\mathrm{g}}}{\epsilon \omega} \frac{\mathrm{m} \mathrm{A}_{0}}{\mathrm{r}} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \sin (\mathrm{m} \theta), \nonumber \\
    \mathrm{E}_{\mathrm{z}} &=\frac{i}{\epsilon \omega} \frac{1}{\mathrm{r}}\left(\frac{\partial}{\partial \mathrm{r}}\left(\mathrm{r} \mathrm{H}_{\theta}\right)-\frac{\partial \mathrm{H}_{\mathrm{r}}}{\partial \theta}\right) \nonumber\\
    &=\frac{-i}{\epsilon \omega} \frac{\mathrm{A}_{0}}{\mathrm{r}^{2}}\left(\mathrm{k}_{\mathrm{c}}^{2} \mathrm{r}^{2} \ddot{\mathrm{J}}_{\mathrm{m}}+\mathrm{k}_{\mathrm{c}} \mathrm{r} \dot{\mathrm{J}}_{\mathrm{m}}-\mathrm{m}^{2} \mathrm{J}_{\mathrm{m}}\right) \cos (\mathrm{m} \theta). \nonumber

    The expression for Ez can be simplified because Jm(kcr) must satisfy the differential equation(\ref{12.59}), therefore

    \[\left(\mathrm{k}_{\mathrm{c}}^{2} \mathrm{r}^{2} \ddot{\mathrm{J}}_{\mathrm{m}}+\mathrm{k}_{\mathrm{c}} \mathrm{r} \dot{\mathrm{J}}_{\mathrm{m}}+\mathrm{k}_{\mathrm{c}}^{2} \mathrm{r}^{2} \mathrm{J}_{\mathrm{m}}-\mathrm{m}^{2} \mathrm{J}_{\mathrm{m}}\right)=0. \nonumber\]

    Using this expression Ez becomes

    \[\mathrm{E}_{\mathrm{z}}=i \frac{\mathrm{k}_{\mathrm{c}}^{2}}{\epsilon \omega} \mathrm{A}_{0} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \cos (\mathrm{m} \theta). \label{12.64}\]

    The fields of Equations (\ref{12.62}) and (\ref{12.63}) satisfy Maxwell’s equations. They must also satisfy the boundary conditions Eθ=0, Ez=0, and Hr=0 at the walls of the wave-guide. Let the inner radius of the wave-guide be R meters. The boundary conditions can be met if Jm(kcR)=0. This condition fixes allowable values for kc and therefore fixes kg through Equation (\ref{12.57})

    \[\mathrm{k}_{\mathrm{g}}^{2}=\epsilon_{\mathrm{r}}\left(\frac{\omega}{\mathrm{c}}\right)^{2}-\mathrm{k}_{\mathrm{c}}^{2}. \label{12.65}\]

    Table(12.5.2) lists the four lowest roots of the equation Jm(x)=0 for Bessel’s functions with m=0,1,2 and 3. These roots determine the wave-vector, kg. In particular, they determine the minimum frequency for which energy can be propagated down the wave-guide. The cut-off frequencies correspond to kg=0, and are given by

    \[\frac{\omega}{\mathrm{c}}=\frac{\mathrm{k}_{\mathrm{c}}}{\sqrt{\epsilon_{\mathrm{r}}}}. \label{12.66}\]

    To take a concrete example, suppose that R=1cm =0.01m. The lowest TM mode corresponds to m=0 and to the first root of the Bessel’s function J0: this is called the TM01 mode. For this case

    \mathrm{E}_{\mathrm{r}} &=-\frac{\mathrm{k}_{\mathrm{g}}}{\epsilon \omega} \mathrm{k}_{\mathrm{c}} \mathrm{A}_{0} \dot{\mathrm{J}}_{0}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right), \label{12.67} \\
    \mathrm{E}_{\theta} &=0 , \nonumber \\
    \mathrm{E}_{\mathrm{z}} &=i \frac{\mathrm{k}_{\mathrm{c}}^{2}}{\epsilon \omega} \mathrm{A}_{0} \mathrm{J}_{0}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) , \nonumber\\
    \mathrm{H}_{\mathrm{r}} &=0 , \nonumber \\
    \mathrm{H}_{\theta} &=-\mathrm{k}_{\mathrm{c}} \mathrm{A}_{0} \dot{\mathrm{J}}_{0}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) , \nonumber\\
    \mathrm{H}_{\mathrm{z}} &=0 , \nonumber

    Here kc = 2.4048/R = 240.48 per meter. This corresponds to a cut-off frequency of 11.48 GHz for \(\epsilon_{r}\) =1. The TM01 mode pattern is shown in Figure (12.5.12(b)).

    Table 12.2.PNG

    Table \(\PageIndex{2}\): The values of x corresponding to the roots of the equations Jm(x)=0 and \(\dot{\mathrm{J}}_{\mathrm{m}}(\mathrm{x})=0\) for the first four Bessel’s functions.

    Figure 12.12.PNG
    Figure \(\PageIndex{12}\): Electric and magnetic field distributions for the TE01 and TM01 modes in a circular wave-guide. The dashed lines represent the magnetic field, the full lines the electric field.

    12.5.2 TE Modes.

    Using the symmetry relations (12.2.3) one can write down the electric field components corresponding to transverse electric modes directly from Equations (\ref{12.62}):

    \mathrm{E}_{\mathrm{r}}&=\frac{\mathrm{mE}_{0}}{\mathrm{r}} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \sin (\mathrm{m} \theta) , \label{12.68}\\
    \mathrm{E}_{\theta}&=\mathrm{k}_{\mathrm{c}} \mathrm{E}_{0} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \cos (\mathrm{m} \theta) , \nonumber\\
    \mathrm{E}_{\mathrm{z}}&=0, \nonumber

    where, as before,

    \[\mathrm{k}_{\mathrm{c}}^{2}=\epsilon_{\mathrm{r}}\left(\frac{\omega}{\mathrm{c}}\right)^{2}-\mathrm{k}_{\mathrm{g}}^{2}. \nonumber\]

    The magnetic field components can be calculated from \(\operatorname{curl}(\overrightarrow{\mathrm{E}})=i \omega \mu_{0} \overrightarrow{\mathrm{H}}\):

    \mathrm{H}_{\mathrm{r}} &=\frac{i}{\omega \mu_{0}} \frac{\partial \mathrm{E}_{\theta}}{\partial \mathrm{z}}=-\frac{\mathrm{k}_{\mathrm{g}} \mathrm{k}_{\mathrm{c}}}{\omega \mu_{0}} \mathrm{E}_{0} \dot{\mathrm{J}}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \cos (\mathrm{m} \theta), \label{12.69} \\
    \mathrm{H}_{\theta} &=\frac{-i}{\omega \mu_{0}} \frac{\partial \mathrm{E}_{\mathrm{r}}}{\partial \mathrm{z}}=\frac{\mathrm{k}_{\mathrm{g}}}{\omega \mu_{0}} \frac{\mathrm{m} \mathrm{E}_{0}}{\mathrm{r}} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \sin (\mathrm{m} \theta), \nonumber \\
    \mathrm{H}_{\mathrm{z}} &=\frac{-i}{\omega \mu_{0} \mathrm{r}}\left[\frac{\partial}{\partial \mathrm{r}}\left(\mathrm{r} \mathrm{E}_{\theta}\right)-\frac{\partial \mathrm{E}_{\mathrm{r}}}{\partial \theta}\right] \nonumber\\
    &=\frac{i \mathrm{k}_{\mathrm{c}}^{2}}{\omega \mu_{0}} \mathrm{E}_{0} \mathrm{J}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right) \cos (\mathrm{m} \theta). \nonumber

    In Equations (\ref{12.69}) the factor exp (i[kgz − ωt]) has been suppressed. The simple form for Hz has been obtained using the fact that Jm(kcr) must satisfy Equation (\ref{12.59}), the differential equation for the radial function F(r). In order to satisfy the boundary conditions Eθ=0 and Hr=0 at the wave-guide walls kcR must be set equal to one of the roots of the equation \(\dot{\mathrm{J}}_{\mathrm{m}}\left(\mathrm{k}_{\mathrm{c}} \mathrm{r}\right)=0\), where R is the inner radius of the wave-guide. The lowest four roots of \(\dot{\mathrm{J}}_{\mathrm{m}}(\mathrm{x})=0\) have been listed in Table(12.5.2) for the first four Bessel’s functions. The lowest cut-off frequency occurs for the first root of \(\dot{J}_{1}(x)\) : this mode is called the TE11 mode. The cut-off frequency for the TE11 mode is 8.79 GHz for \(\epsilon_{r}\)=1 and R= 1cm. Compare this with the cut-off frequency for the TM01 mode, 11.48 GHz. Thus, over the frequency interval 8.79 to 11.48 GHz an air-filled circular pipe having an inner radius of R=1cm can support only a single mode, the TE11 mode. The TE11 mode pattern is shown in Figure (12.5.13).

    Figure 12.13.PNG
    Figure \(\PageIndex{13}\): Electric field lines for the TE11 mode in a cylindrical Waveguide. The electric field lines must be normal to the walls at r=R, where R is the inner radius of the wave-guide. The magnetic field lines are orthogonal to the electric field lines, and Hr=0 at r=R.

    The TE01 mode is of particular interest; the mode pattern is shown in Figure (12.5.12(a)). This mode is very useful for the construction of high-Q cavities of variable frequency. The length of the cavity can be altered by means of a sliding piston. No currents need flow across the gap between the piston and the walls of the cylinder for the TE01 mode: the current lines on the face of the piston are similar to the electric field lines shown in Figure (12.5.12(a)) and are concentric circles. Even if the piston does not make good electrical contact with the cavity walls the field lines in the TE01 mode remain unperturbed by any small gap between the piston and the cylinder walls. This mode is often used to construct microwave frequency meters.

    Wave-guide modes are discussed in detail in the book ”Electron Spin Resonance” by Charles P. Poole, 2cd Edition, John Wiley and Sons, New York, 1983.