DISPERSION EQUATIONS FOR CURVED IRIS-LOADED WAVEGUIDES OF RECTANGULAR CROSS-SECTION

B. N. Morozov, E. V. Padusova

(Presented by the scientific seminar of the Radio Engineering Faculty)

Introduction

Recently, at the Tomsk Polytechnic Institute, on the initiative of Professor Dr. A. A. Vorob’ev, experimental and theoretical studies have been carried out on the properties of curved and closed waveguides, which may find application in cyclic electron accelerators with a controlling magnetic field [1]. In the chamber of such an accelerator, an electromagnetic wave having a longitudinal component of the electric field must propagate in the azimuthal direction with a phase velocity equal to the speed of light.

One of the methods of slowing waves is the loading of a waveguide with a periodic iris structure. Below, two variants of iris loading of a curved waveguide of rectangular cross-section are considered. The choice of variant may be dictated by the structural features of the waveguide cyclic accelerator, as well as by the magnitude of attenuation and coupling impedance.

Derivation of the Dispersion Equation for a Waveguide Iris-Loaded Along the Curved Walls

Figure 1 shows the waveguide system that is the subject of the investigation, and indicates the adopted notation. The waveguide is assumed to be perfectly conducting, and the interaction space (of size \(2a\) in the radial direction) and the resonator space (the region between the metallic ribs) are filled with an ideal dielectric.

In the cylindrical coordinate system \(r, \varphi, z\), it is expedient to classify electromagnetic waves along the rectilinear coordinate \(z\) [2], i.e., to consider “electric” waves \((H_z = 0)\), or “magnetic” waves \((E_z = 0)\), or, finally, hybrid waves—a combination of the two. In the present section, waves of the “magnetic” type are considered, for which the components of the electromagnetic field are expressed through the Hertz magnetic vector by the relations:

\[ E_r=-j\frac{\kappa}{r}\frac{\partial \Pi_H}{\partial \varphi},\qquad E_\varphi=j\kappa\frac{\partial \Pi_H}{\partial r},\qquad E_z=0, \]

\[ H_r=\frac{\partial^2\Pi_H}{\partial r\partial z},\qquad H_\varphi=\frac{1}{r}\frac{\partial^2\Pi_H}{\partial \varphi\partial z},\qquad H_z=\frac{\partial^2\Pi_H}{\partial z^2}+\kappa^2\Pi_H, \tag{1} \]

where the function \(\Pi_H\) is a particular solution of the scalar wave equation

\[ \Delta\Pi_H+\kappa^2\Pi_H=0. \tag{2} \]

Under the boundary conditions

\[ \Pi_H=0 \quad \text{for } z=0 \text{ and } z=b, \tag{3} \]

where \(b\) is the height of the waveguide, \(\kappa=\dfrac{2\pi}{\lambda}\).

Fig. 1. Curved waveguide iris-loaded along the side walls.

In the interaction space we shall consider waves propagating in the positive azimuthal direction. In view of the periodicity of the structure with respect to \(\varphi\), the wave character of the process in the azimuthal direction is represented by the factor \(e^{-jn_s\varphi}\), where \(n_s=n_0+\dfrac{2\pi s}{\Psi}\) and \(s=0,\pm1,\pm2\) is the number characterizing the forward and backward spatial harmonics. We shall consider only the case in which one half-wave fits in the \(z\) direction, which does not affect the generality of the final result. Then the expressions for the field components are written in the form:

\[ E_\varphi=j\kappa\sin k_z z \sum_{s=-\infty}^{\infty} \chi\left[A_s I'_{n_s}(\chi r)+B_s N'_{n_s}(\chi r)\right]e^{-jn_s\varphi}, \]

\[ H_\varphi=-j\cos k_z z \sum_{s=-\infty}^{\infty} \frac{n_s}{r}k_z\left[A_s I_{n_s}(\chi r)+B_s N_{n_s}(\chi r)\right]e^{-jn_s\varphi}, \tag{4} \]

\[ H_z=\kappa^2\sin k_z z \sum_{s=-\infty}^{\infty} \left[A_s I_{n_s}(\chi r)+B_s N_{n_s}(\chi r)\right]e^{-jn_s\varphi}. \]

Here, as in what follows, the factor \(e^{j\omega t}\) is omitted; \(I_n(\chi r)\), \(N_n(\chi r)\), \(\dot I_n(\chi r)\), \(\dot N_n(\chi r)\) are Bessel and Neumann functions and their derivatives with respect to the argument; \(n_s\) is the angular propagation constant;

\[ \chi^2=k^2-k_z^2;\qquad k_z=\frac{\pi}{b}. \]

It is convenient to regard the space of the resonators as a wedge-shaped transmission line short-circuited at the end. Applying the boundary conditions \(E_\varphi=0\) at \(r=R_1\) and \(r=R_2\), and taking into account that standing waves will be formed in the azimuthal direction, we obtain, for region II \((r<r_1)\),

\[ E_\varphi = jk\sum_{l=0}^{\infty}\chi C_l \left[\dot I_\alpha(\chi r)+P_{\mathrm{II}}\dot N_\alpha(\chi r)\right] \cos\alpha\varphi\sin k_z z; \]

\[ H_\varphi = -\frac{1}{r}\sum_{l=0}^{\infty}\alpha k_z C_l \left[I_\alpha(\chi r)+P_{\mathrm{II}}N_\alpha(\chi r)\right] \sin\alpha\varphi\cos k_z z; \tag{5} \]

\[ H_z = \sum_{l=0}^{\infty}C_l \left[I_\alpha(\chi r)+P_{\mathrm{II}}N_\alpha(\chi r)\right] \cos\alpha\varphi\sin k_z z, \]

and for region III \((r>r_2)\) one must replace \(C_l\) and \(P_{\mathrm{II}}\) by \(D_l\) and \(P_{\mathrm{III}}\), where

\[ P_{\mathrm{II}}=-\frac{\dot I_\alpha(\chi R_1)}{\dot N_\alpha(\chi R_1)}, \qquad P_{\mathrm{III}}=-\frac{\dot I_\alpha(\chi R_2)}{\dot N_\alpha(\chi R_2)}, \qquad \alpha=\frac{l\pi}{\psi}, \]

\(l=0,1,2\ldots\) is the number of half-waves in the \(\varphi\) direction.

We match the field components (4) and (5) at the boundary \(r=r_1=R_1+h_1\), and then at the boundary \(r=r_2=R_2-h_2\). As a result we obtain a system of 6 equations with 4 sequences of unknowns \(A_s\), \(B_s\), \(C_l\), and \(D_l\).

\[ \mathrm{I}\qquad \sum_{s=-\infty}^{\infty} \left[A_s\dot I_{n_s}(\chi r_1)+B_s\dot N_{n_s}(\chi r_1)\right]e^{-jn_s\varphi} = \]

\[ = -\sum_{l=0}^{\infty} C_l\left[\dot I_\alpha(\chi r_1)+P_{\mathrm{II}}\dot N_\alpha(\chi r_1)\right]\cos\alpha\varphi, \]

\[ \mathrm{II}\qquad j\sum_{s=-\infty}^{\infty} n_s\left[A_s I_{n_s}(\chi r_1)+B_s N_{n_s}(\chi r_1)\right]e^{-jn_s\varphi} = \]

\[ = \sum_{l=1}^{\infty} \alpha C_l\left[I_\alpha(\chi r_1)+P_{\mathrm{II}}N_\alpha(\chi r_1)\right]\sin\alpha\varphi, \]

\[ \mathrm{III}\qquad \sum_{s=-\infty}^{\infty} \left[A_s I_{n_s}(\chi r_1)+B_s N_{n_s}(\chi r_1)\right]e^{-jn_s\varphi} = \tag{6} \]

\[ = \sum_{l=0}^{\infty} C_l\left[I_\alpha(\chi r_1)+P_{\mathrm{II}}N_\alpha(\chi r_1)\right]\cos\alpha\varphi, \]

IV.
\[ \sum_{s=-\infty}^{\infty} \left[A_s I_{n_s}(\chi r_2)+B_s N_{n_s}(\chi r_2)\right]e^{-j n_s\varphi} = \sum_{l=0}^{\infty}D_l\left[I_\alpha(\chi r_2)+P_{\mathrm{III}}N_\alpha(\chi r_2)\right]\cos \alpha\varphi . \]

V.
\[ j\sum_{s=-\infty}^{\infty} n_s\left[A_s I_{n_s}(\chi r_2)+B_s N_{n_s}(\chi r_2)\right]e^{-j n_s\varphi} = \sum_{l=1}^{\infty}\alpha D_l\left[I_\alpha(\chi r_2)+P_{\mathrm{III}}N_\alpha(\chi r_2)\right]\sin \alpha\varphi . \]

VI.
\[ \sum_{s=-\infty}^{\infty} \left[A_s I_{n_s}(\chi r_2)+B_s N_{n_s}(\chi r_2)\right]e^{-j n_s\varphi} = \sum_{l=0}^{\infty}C_l\left[I_\alpha(\chi r_2)+P_{\mathrm{III}}(\chi r_2)\right]\cos \alpha\varphi . \]

We shall make use of the orthogonality property of the functions \(\sin \alpha\varphi\) and \(\cos \alpha\varphi\) within one period of the structure. To this end, we multiply equations II and V of system (6) by \(\sin \dfrac{q\pi}{\psi}\varphi\), and I, III, IV, and VI by \(\cos \dfrac{q\pi}{\psi}\varphi\), and integrate I and IV from \(0\) to \(\Psi\), and the remaining equations from \(0\) to \(\psi\). Introducing the notation

\[ f_l=I_\alpha(\chi r_1)+P_{\mathrm{II}}N_\alpha(\chi r_1), \]

\[ f'_l=I'_\alpha(\chi r_1)+P_{\mathrm{II}}N'_\alpha(\chi r_1), \]

\[ F_l=I_\alpha(\chi r_2)+P_{\mathrm{III}}N_\alpha(\chi r_2), \]

\[ F'_l=I'_\alpha(\chi r_2)+P_{\mathrm{III}}N'_\alpha(\chi r_2) \tag{7} \]

\[ Q_{sl}=j\left[ \frac{1-e^{-j(n_s+\alpha)\psi}}{(n_s+\alpha)} + \frac{1-e^{-j(n_s-\alpha)\psi}}{(n_s-\alpha)} \right], \]

\[ M_{sl}= \left[ \frac{1-e^{-j(n_s+\alpha)\psi}}{(n_s+\alpha)} - \frac{1-e^{-j(n_s-\alpha)\psi}}{(n_s-\alpha)} \right], \]

\[ E_{sl}=j\left[ \frac{1-e^{-j(n_s+\alpha)\psi}}{(n_s+\alpha)} - \frac{1-e^{-j(n_s-\alpha)\psi}}{(n_s-\alpha)} \right], \]

we obtain from (6, II)

I.
\[ \sum_{s=-\infty}^{\infty} \left[A_s I'_{n_s}(\chi r_1)+B_s N'_{n_s}(\chi r_1)\right]Q_{sl}=f'_l; \]

II.
\[ j\sum_{s=-\infty}^{\infty} n_s\left[A_s I_{n_s}(\chi r_1)+B_s N_{n_s}(\chi r_1)\right]S_{sl}=l f_l,\quad l\ne 0, \]

III.
\[ \sum_{s=-\infty}^{\infty}\left[A_s I_{n_s}(\chi r_1)+B_s N_{n_s}(\chi r_1)\right]M_{sl}=\psi f_l, \tag{8} \]

IV.
\[ \sum_{s=-\infty}^{\infty}\left[A_s \dot I_{n_s}(\chi r_2)+B_s \dot N_{n_s}(\chi r_2)\right]Q_{sl}=F'_l, \]

V.
\[ j\sum_{s=-\infty}^{\infty}\left[A_s I_{n_s}(\chi r_2)+B_s N_{n_s}(\chi r_2)\right]S_{sl}=lF_l,\quad l\ne 0, \]

VI.
\[ \sum_{s=-\infty}^{\infty}\left[A_s I_{n_s}(\chi r_2)+B_s N_{n_s}(\chi r_2)\right]M_{sl}=\psi F_l. \]

However, it turns out that equation II of system (8) is a consequence of equation III (this also applies to equations V and VI), and, since sequence III is more complete with respect to \(l\), system (8) is identical to the system

\[ \sum_{s=-\infty}^{\infty}\left[A_s \dot I_{n_s}(\chi r_1)+B_s \dot N_{n_s}(\chi r_1)\right]Q_{sl}=C_l f'_l, \]

\[ \sum_{s=-\infty}^{\infty}\left[A_s I_{n_s}(\chi r_1)+B_s N_{n_s}(\chi r_1)\right]M_{sl}=\psi C_l f_l, \tag{9} \]

\[ \sum_{s=-\infty}^{\infty}\left[A_s \dot I_{n_s}(\chi r_2)+B'_s \dot N_{n_s}(\chi r_2)\right]Q_{sl}=D_l F'_l, \]

\[ \sum_{s=-\infty}^{\infty}\left[A_s I_{n_s}(\chi r_2)+B_s N_{n_s}(\chi r_2)\right]M_{sl}=\psi D_l F_l. \]

Thus, the apparent ambiguity of the solution is absent, since system (9) contains four sequences of unknowns \(A_s, B_s, C_l, D_l\) and four sequences of equations. The system of sequences (9) is easily reduced to two infinite systems of linear homogeneous equations with respect to the amplitude coefficients

\[ \sum_{s=-\infty}^{\infty} A_s a_{sl}+B_s b_{sl}=0,\qquad \sum_{s=-\infty}^{\infty} A_s c_{sl}+B_s d_{sl}=0, \tag{10} \]

where the following notation has been introduced:

\[ a_{sl}=f_l \dot I_{n_s}(\chi r_1)Q_{sl}-f'_l I_{n_s}(\chi r_1)M_{sl}, \]

\[ c_{sl}=F_l \dot I_{n_s}(\chi r_2)Q_{sl}-F'_l I_{n_s}(\chi r_2)M_{sl}, \]

and for \(b_{sl}\) and \(d_{sl}\), \(I_n\) and \(\dot I_n\) are to be replaced by \(N_n\) and \(\dot N_n\).

A nonzero solution of system (10) will exist in the case when the determinant of the system is equal to zero. Consequently, the dispersion equation of the waveguide under consideration, characterizing the relation between the radius of curvature, the geometrical parameters of the waveguide, and the wavelength, is written in the form of the infinite determinant

\[ \left| \begin{array}{cccccccc} \cdots & a_{-10} & b_{-10} & a_{00} & b_{00} & a_{10} & b_{10} & \cdots \\ \cdots & c_{-10} & d_{-10} & c_{00} & d_{00} & c_{10} & d_{10} & \cdots \\ \cdots & a_{-11} & b_{-11} & a_{01} & b_{01} & a_{11} & b_{11} & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \right|=0 \]

It is of interest to dwell in more detail on the analysis of the solution in the first approximation, which corresponds to the determinant with zero indices \((s=0,\ l=0)\)

\[ \left| \begin{array}{cc} a_{00} & b_{00}\\ c_{00} & d_{00} \end{array} \right|=0 \]

or, in expanded form,

\[ \frac{ T_1 I_{n_0}[\chi(R-a)]-\dot I_{n_0}[\chi(R\mp a)] }{ T_1 N_{n_0}[\chi(R\mp a)]-\dot N_{n_0}[\chi(R\mp a)] } = \]

\[ = \frac{ T_2 I_{n_0}[\chi(R-a)]-\dot I_{n_0}[\chi(R-a)] }{ T_2 N_{n_0}[\chi(R-a)]-\dot N_{n_0}[\chi(R-a)] }, \tag{11} \]

where

\[ T_{1,2}=\frac{t_1-t_2}{t_3-t_4} \]

and

\[ t_1=I_0[\chi(R\mp a)]\dot N_0[\chi(R\mp a\pm h_{2,1})], \]

\[ t_2=\dot I_0[\chi(R\mp a\pm h_{2,1})]\dot N_0[\chi(R\mp a)], \]

\[ t_3=I_0[\chi(R\mp a)]\dot N_0[\chi(R\mp a\mp h_{2,1})], \]

\[ t_4=\dot I_0[\chi(R\mp a\mp h_{2,1})]N_0[\chi(R\mp a)]. \]

The quantity characterizing the slowing of the electromagnetic wave enters into the index of the cylindrical functions

\[ \left(\nu=\frac{c}{v_{\phi}}=\frac{n_0}{kR}\right). \]

The first approximation corresponds to the case when the distance between neighboring irises is much smaller than the wavelength in the waveguide, the iris thickness is considerably smaller than the distance between them, and in the resonators there exists only the fundamental type of oscillation, whose field does not depend on the azimuthal coordinate. The results of analysis of problems of this kind for various types of iris-loaded waveguides [3, 4] give grounds for considering that the theoretical investigation in the first approximation is sufficient.

to elucidate the basic properties of the system under consideration. The subsequent approximations, which take higher harmonics into account, can change the result only quantitatively (the magnitude of the corrections is 15–20% relative to the first approximation), without changing the qualitative picture.

Let us note that, for comparatively weak bending \((\kappa R \gg 1)\) and a symmetrically iris-loaded waveguide \((h_1=h_2)\), equation (11) becomes

\[ \frac{\operatorname{tg}\chi h\, I_{n_0}[\chi(R-a)]+\dot I_{n_0}[\chi(R-a)]} {\operatorname{tg}\chi h\, N_{n_0}[\chi(R-a)]+\dot N_{n_0}[\chi(R-a)]} = \]

\[ = \frac{\operatorname{tg}\chi h\, I_{n_0}[\chi(R+a)]-\dot I_{n_0}[\chi(R+a)]} {\operatorname{tg}\chi h\, N_{n_0}[\chi(R+a)]-\dot N_{n_0}[\chi(R+a)]}, \]

which, being quadratic with respect to \(\operatorname{tg}\chi h\), simultaneously describes the dispersion properties of in-phase and antiphase waves, the former of which are symmetric, and the latter antisymmetric, in the radial direction with respect to the center of the interaction space. This is readily verified by taking the limiting transition, for \(\nu\kappa R \ll \sqrt{\kappa R}\), to an unbent waveguide. If the waveguide is iris-loaded along one wall \((h_2=0\) or \(h_1=0)\) or asymmetrically \((h_1\ne h_2)\), then only antiphase waves of the type under consideration can exist in it.

Fig. 2. Curves illustrating the influence of bending on the properties of an iris-loaded waveguide:
1) in-phase waves;
2) antiphase waves.

Figure 2 shows the dependence of the relative iris height \(\kappa h\) on the relative bending radius for \(\vartheta_\varphi=c\), \(\kappa a=2\), and two different values of the waveguide height \(b\). The calculations performed show that a bent waveguide, iris-loaded along the bent walls, provides \(\nu=1\) at the mean radius \(R\) for geometric dimensions that make it possible to use it as the accelerating chamber of a waveguide cyclic accelerator with a guiding magnetic field.

Derivation of the dispersion equation for a waveguide iris-loaded along the end walls

A curved waveguide iris-loaded along the end walls is shown in Fig. 3. The initial assumptions will be the same as above. An essential difference in the present case from the preceding problem is the fact that in a waveguide loaded along the end walls, only hybrid waves can exist, since the boundary conditions at the boundary between the interaction region and the resonator regions are not satisfied by a wave of a single type. The field in the resonator may be considered as the sum of the fields of an “electric” and a “magnetic” wave:

Fig. 3. Curved waveguide iris-loaded along the end walls.

\[ E_r=-\frac{1}{\kappa_c^2}\left(k'_z\frac{\partial E_z}{\partial r} +j\,\frac{\kappa}{r}\frac{\partial H_z}{\partial \varphi}\right); \]

\[ E_\varphi=\frac{1}{\kappa_c^2}\left(-\frac{k'_z}{r}\frac{\partial E_z}{\partial \varphi} +j\kappa\frac{\partial H_z}{\partial r}\right); \]

\[ H_r=\frac{1}{\kappa_c^2}\left(-k'_z\frac{\partial H_z}{\partial r} +j\,\frac{\kappa}{r}\frac{\partial E_z}{\partial \varphi}\right); \]

\[ H_\varphi=-\frac{1}{\kappa_c^2}\left(\frac{k'_z}{r}\frac{\partial H_z}{\partial \varphi} +j\kappa\frac{\partial E_z}{\partial r}\right), \]

where \(\kappa_c^2=\kappa^2-k_z^{\prime 2}\); \(k'_z\) is the propagation constant of the waves along the \(z\)-axis. \(k'_z\) may be real or imaginary depending on the ratio \(\kappa_c^2/\kappa^2\), which, in turn, is determined by the wavelength and the distance between the diaphragms. The longitudinal field components \(E_z\) and \(H_z\) are determined by the relations:

\[ E_z= \left[ \sum_{l_1=1,3,5,\ldots}^{\infty} B_l f_l\cos\alpha\varphi - \sum_{l_1=2,4,6,\ldots}^{\infty} B_l f_l\sin\alpha\varphi \right]\operatorname{ch} k'_{zl}(z+h), \]

\[ H_z=-B'_0F'_l\sin k'_{z0}(z+h) - \left[ \sum_{l_1=1,3,5,\ldots}^{\infty} B'_lF'_l\sin\alpha\varphi - \sum_{l_1=2,4,6,\ldots}^{\infty} B'_lF'_l\cos\alpha\varphi \right]\operatorname{sh} k'_{zl}(z+h). \]

At the boundary (resonator space—interaction space) the transverse field components will be equal to

\[ E_r=j\sum_{l_1=1,3,5,\ldots}^{\infty} \left(A_l\,\kappa'_{zl} f'_l \kappa'_{cl} - A'_l\,\frac{\kappa\alpha}{r}F_l\right)\cos\alpha\varphi+\cdots \]

\[ {}+j\sum_{l_2=2,4,6\ldots}^{\infty} \left(A_l\kappa'_{zl} f'_l\kappa_{cl}-A'_l\frac{\kappa a}{r}F_l\right)\sin a\varphi; \]

\[ E_\varphi = jA'_0\kappa\kappa_{c0}F'_0 +j\sum_{l_1=1,3,5\ldots}^{\infty} \left(A_l\kappa'_{zl}\frac{a}{r}f_l-A'_l\kappa\kappa_{cl}F'_l\right)\sin a\varphi \]
\[ {}+j\sum_{l_2=2,4,6\ldots}^{\infty} \left(A_l\kappa'_{zl}\frac{a}{r}f_l+A'_l\kappa\kappa_{cl}F'_l\right)\cos a\varphi; \]

\[ H_r=-A'_0\kappa'_{z0}\kappa_{c0}F'_0\operatorname{ctg}\kappa'_{z0}h +\sum_{l_1=1,3,5\ldots}^{\infty} \left(-A_l\frac{\kappa a}{r}f_l-A'_l\kappa'_{zl}\kappa_{cl}F'_l\right) \]
\[ {}\times \operatorname{cth}\kappa'_{zl}h\,\sin a\varphi -\sum_{l_2=2,4,6\ldots}^{\infty} \left(-A_l\frac{\kappa a}{r}f_l-A'_l\kappa'_{zl}\kappa_{cl}F'_l\right) \operatorname{cth}\kappa'_{zl}h\cdot\cos a\varphi; \]

\[ H_\varphi= \sum_{l_1=1,3,5\ldots}^{\infty} \left(A'_l\frac{\kappa'_{zl}a}{r}F_l-A_l\kappa\kappa_{cl}f'_l\right) \operatorname{cth}\kappa'_{zl}h\cdot\cos a\varphi \]
\[ {}+\sum_{l_2=2,4,6\ldots}^{\infty} \left(A'_l\frac{\kappa'_{zl}a}{r}F_l-A_l\kappa\kappa_{cl}f'_l\right) \operatorname{cth}\kappa'_{zl}h\cdot\sin a\varphi, \]

where

\[ A'_0=B'_0\sin\kappa'_{z0}h,\qquad A'_l=B'_l\operatorname{sh}\kappa'_{zl}h,\qquad A_l=B_l\operatorname{sh}\kappa'_{zl}h. \]

\(f_l,\ F_l,\ F'_l\) and \(f'_l\) coincide with their values from [7], with \(x\) replaced by \(\kappa\) and \(r_1,\ r_2\) by \(r\).

If the groove width \(\psi\ll\lambda_\varphi\), then one may assume that only the fundamental wave \(H_{01}\) propagates in the resonator, and the field at the resonator boundary has two components:

\[ E_\varphi=jA'_0\kappa\kappa_{c0}F'_0,\qquad H_r=-A'_0\kappa'_{z0}\kappa_{c0}F'_0\operatorname{ctg}\kappa'_{z0}h. \tag{13} \]

The field in the interaction space can also be represented as the sum of two waves, an “electric” and a “magnetic” one. To obtain a hybrid wave, two potential functions are introduced, \(\Pi_E(r,\varphi,z)\) and \(\Pi_H(r,\varphi,z)\), which are the components of the electric and magnetic Hertz vectors along the \(z\)-axis. Both these functions satisfy the wave equation in the interaction space of the waveguide. The field equations for the interaction space are written in the form

\[ E_r=\sum_{s=-\infty}^{\infty} e^{-jn_s\varphi} \left(D'_s\kappa\frac{n_s}{r}F_{ns}-D_s\kappa_{zs}\chi_s f'_{ns}\right) \operatorname{ctg}\kappa_{zs}(g-z); \]

\[ E_\varphi=j\sum_{s=-\infty}^{\infty} e^{-jn_s\varphi} \left(-D_s\kappa_{zs}\frac{n_s}{r}f_{ns}-D'_s\kappa\chi_sF'_{ns}\right) \operatorname{ctg}\kappa_{zs}(g-z); \]

\[ H_r=\sum_{s=-\infty}^{\infty} e^{-j n_s\varphi} \left(-D'_s\varkappa_{zs}\varkappa_s F'_{ns} -D_s\varkappa\,\frac{n_s}{r}\,f_{ns}\right); \]

\[ H_\varphi=j\sum_{s=-\infty}^{\infty} e^{-j n_s\varphi} \left(-D_s\varkappa\varkappa_s f'_{ns} +D'_s\varkappa_{zs}\,\frac{n_s}{r}\,F_{ns}\right). \tag{14} \]

Here

\[ D=C\sin zk_z,\qquad D'=C'\sin k_z z; \]

\[ f_n=I_n(\varkappa r)+P N_n(\varkappa r); \]

\[ f'_n=I'_n(\varkappa r)+P N'_n(\varkappa r); \]

\[ F_n=I_n(\varkappa r)+P' N_n(\varkappa r); \]

\[ F'_n=I'_n(\varkappa r)+P' N'_n(\varkappa r); \]

\[ P=-\,\frac{I_n(\varkappa r_2)}{N_n(\varkappa r_2)},\qquad P'=-\,\frac{I'_n(\varkappa r_2)}{N'_n(\varkappa r_2)} . \]

\(\varkappa_z=\sqrt{\varkappa^2-k_z^2}\) is the transverse wave number; \(k_z\), the propagation constant of the waves along the \(z\)-axis, may be imaginary if \(\varkappa_z^2<\varkappa^2\). In this case the trigonometric functions that determine the character of the field variation along the \(z\)-axis become hyperbolic. For appreciable slowdowns \(k_z\) is always imaginary.

Matching the tangential components from (12) and (14) at the boundary of the regions leads to the equations:

\[ \left\{ \begin{aligned} &\sum_{s=-\infty}^{\infty} \left(D'_s\varkappa\,\frac{n_s}{r}\,F_{ns} -D_s\varkappa_{zs}\varkappa_s f'_{ns}\right) \operatorname{cth}\varkappa_{zs}g\,B_{sl_1} \\ &\qquad =-A_{l_1}k'_{zl} f'_l k_{cl}\,\frac{\psi}{2} +A'_{l_1}\frac{\varkappa}{r}\,\frac{l\pi}{2}\,f_l; \\[1.2em] &\sum_{s=-\infty}^{\infty} \left(D'_s\varkappa\,\frac{n_s}{r}\,F_{ns} -D_s\varkappa_{zs}\varkappa_s f'_{ns}\right) \operatorname{cth}\varkappa_{zs}g\,C_{sl_2} \\ &\qquad =+A_{l_2}k'_{zl} f'_l k_{cl}\,\frac{\psi}{2} -A'_{l_2}\frac{\varkappa}{r}\,\frac{l\pi}{2}\,F_l . \end{aligned} \right. \tag{15} \]

\[ \left\{ \begin{aligned} &\sum_{s=-\infty}^{\infty} \left(D_s\varkappa_{zl}\,\frac{n_s}{r}\,f_{ns} +D'_s\varkappa\varkappa_s F'_{ns}\right) \operatorname{cth}\varkappa_{zs}g\,C'_{s0} \\ &\qquad =A'_0\varkappa k_{c0}F'_0\psi; \\[1.2em] &\sum_{s=-\infty}^{\infty} \left(D_s\varkappa_{zs}\,\frac{n_s}{r}\,f_{ns} +D'_s\varkappa\varkappa_s F'_{ns}\right) \operatorname{cth}\varkappa_{zs}g\,B'_{sl_1}= \end{aligned} \right. \tag{16} \]

\[ \left\{ \begin{aligned} &=-A_{l_1}\,\frac{\kappa'_{zl}}{r}\,\frac{l\pi}{2}\,f_l +A'_{l_1}\kappa\kappa_{cl}\,\frac{\psi}{2}\,F'_l;\\[4pt] &\sum_{s=-\infty}^{\infty} \left(D_s\kappa_{zs}\,\frac{n_s}{r}\,Y_{ns} +D'_s\kappa\chi_s F'_{ns}\right)\operatorname{cth}\kappa_{zs}g\,C'_{sl_2} =\\[4pt] &=A_{l_2}\,\frac{\kappa'_{zl}}{r}\,\frac{l\pi}{2}\,f_l +A'_{l_2}\kappa\kappa_{cl}\,\frac{\psi}{2}\,F'_l . \end{aligned} \right. \]

\[ \left\{ \begin{aligned} &\sum_{s=-\infty}^{\infty} \left(-D_s\kappa\chi_s f'_{ns} +D'_s\,\frac{\kappa_{zs}n_s}{r}\,F_{ns}\right)C'_{sl_2} =\\[4pt] &=\left(-A_{l_2}\kappa\kappa_{cl}f'_l\,\frac{\psi}{2} +A'_{l_2}\,\frac{\kappa'_{zl}}{r}\,\frac{l\pi}{2}\,F_l\right) \operatorname{cth}\kappa'_{zl}h;\\[6pt] &\sum_{s=-\infty}^{\infty} \left(-D_s\kappa\chi_s f'_{ns} +D'_s\,\frac{\kappa_{zs}n_s}{r}\,F_{ns}\right)B'_{s1} =\\[4pt] &=-\left(A_{l_1}\kappa\kappa_{c1}f'_l\,\frac{\psi}{2} +A'_{l_1}\,\frac{\kappa'_{zl}}{r}\,\frac{l\pi}{2}\,F_l\right) \operatorname{cth}\kappa'_{zl}h . \end{aligned} \right. \tag{17} \]

\[ \left\{ \begin{aligned} &\sum_{s=-\infty}^{\infty} \left(D_s\,\frac{\kappa n_s}{r}\,f_{ns} +D'_s\chi_s\kappa_{zs}F'_{ns}\right)C_{s0} =\\[4pt] &=\psi A'_0\kappa'_{z0}\kappa_{c0}F'_0\operatorname{ctg}\kappa'_{z0}h;\\[6pt] &\sum_{s=-\infty}^{\infty} \left(D_s\,\frac{\kappa n_s}{r}\,f_{ns} +D'_s\kappa_{zs}\chi_s F'_{ns}\right)C_{2l} =\\[4pt] &=\left(A_{l_2}\,\frac{\kappa}{r}\,\frac{l\pi}{2}\,f_l +A'_{l_2}\kappa'_{zl}\kappa_{cl}F'_l\,\frac{\psi}{2}\right) \operatorname{cth}\kappa'_{zl}h;\\[6pt] &\sum_{s=-\infty}^{\infty} \left(D_s\,\frac{\kappa n_s}{r}\,f_{ns} +D'_s\kappa_{zs}\chi_s F'_{ns}\right)B_{sl_1} =\\[4pt] &=\left(A_{l_1}\,\frac{\kappa}{r}\,\frac{l\pi}{2}\,f_l +A'_{l_1}\kappa_{zl}\kappa_{cl}F'_l\,\frac{\psi}{2}\right) \operatorname{cth}\kappa'_{zl}h . \end{aligned} \right. \tag{18} \]

Here \(C_{sl_2}\), \(C'_{sl_2}\), \(B_{sl_1}\), \(B'_{sl_1}\) are constant factors that depend on the geometric dimensions of the structure [4].

From equations (15, 16, 17, and 18) it follows that, in order for the condition of the vanishing of the \(E_r\) and \(H_\varphi\) components in the resonator space to be satisfied for fields with indices \(l=0\), it is necessary that these components also vanish in the interaction space.

Analyzing the character of the functions \(f'\), \(F'\) and \(f\), \(F\), one can show that the \(E_r\), \(H_\varphi\) components of the field of the hybrid wave cancel, whereas \(E_\varphi\) and

\(H_r\)—are added. On the basis of what has been said, \(E_r\) may be set equal to zero. For an unbent iris-loaded waveguide an analogous assumption is satisfied exactly. In the present case it is satisfied approximately,

\[ -D_s\kappa_{zs}\chi_s f'_{ns}+D'_s\frac{\kappa n_s}{r}F_{ns}=0, \]

whence

\[ D_s=D'_s\frac{\kappa n_s F_{ns}}{r\chi_s\kappa_{zs}f'_{ns}} . \]

Then equations (15)—(18) may be written

\[ \left\{ \begin{aligned} &\sum_{s=-\infty}^{\infty} D'_s\frac{\kappa}{r^2\chi_s f'_{ns}} \left(n_s^2 f_{ns}F_{ns}+\chi_s^2 r^2 f'_{ns}F'_{ns}\right) C'_{s0}\operatorname{cth}\kappa_{zs}g \\ &\qquad =A'_0\kappa\kappa_{c0}F'_0\,\frac{\psi}{2}; \\[6pt] &\sum_{s=-\infty}^{\infty} D'_s\frac{\kappa}{r^2\chi_s f'_{ns}} \left(n_s^2 f_{ns}F_{ns}+\chi_s^2 r^2 f'_{ns}F'_{ns}\right) C'_{s l_2}\operatorname{cth}\kappa_{zs}g \\ &\qquad = A_{l_2}\frac{\kappa'_{z l}}{r}\frac{l\pi}{2}f_l + A'_{l_2}\kappa F'_l\,\frac{\psi}{2}; \\[6pt] &\sum_{s=-\infty}^{\infty} D'_s\frac{\kappa}{r^2\chi_s f'_{ns}} \left(n_s^2 f_{ns}F_{ns}+\chi_s^2 r^2 f'_{ns}F'_{ns}\right) B'_{s l_1}\operatorname{cth}\kappa_{zs}g \\ &\qquad = A_{l_1}\frac{\kappa'_z l}{r}\frac{l\pi}{2}f_l + A^s_{l_1}\kappa\kappa_{c l}F'_l\,\frac{\psi}{2}. \tag{19} \end{aligned} \right. \]

\[ \left\{ \begin{aligned} &\sum_{s=-\infty}^{\infty} D_s\frac{1}{r^2\kappa_{zs}\chi_s f'_{ns}} \left(\kappa_s^2 n_s^2 F_{ns}f_{ns} +r^2\kappa_{zs}^2\chi_s^2 f'_{ns}F'_{ns}\right)C_{s0} \\ &\qquad = A'_0\kappa'_{z0}\kappa_{c0}F'_0\psi\,\operatorname{ctg}\kappa'_{z0}h; \\[6pt] &\sum_{s=-\infty}^{\infty} D'_s\frac{1}{r^2\kappa_{zs}\chi_s f'_{ns}} \left(\kappa^2 n_s^2 f_{ns}F_{ns} +r^2\kappa_{zs}^2\chi_s^2 f'_{ns}F'_{ns}\right)C_{s l_2} \\ &\qquad = \left( A_{l_2}\frac{\kappa}{r}\frac{l\pi}{2}f_l + A'_{l_2}\kappa'_{z l}\kappa_{c l}F_l\,\frac{\psi}{2} \right)\operatorname{cth}\kappa'_{z l}h; \\[6pt] &\sum_{s=-\infty}^{\infty} D'_s\frac{1}{r^2\kappa_{zs}\chi_s f'_{ns}} \left(\kappa^2 n_s^2 f_{ns}F_{ns} +r^2\kappa_{zs}^2\chi_s^2 f'_{ns}F'_{ns}\right)B_{s l_1} \\ &\qquad = \left( A_{l_1}\frac{\kappa}{r}\frac{l\pi}{2}f_l + A'_{l_1}\kappa'_{z l}\kappa_{c l}F_l\,\frac{\psi}{2} \right)\operatorname{cth}\kappa'_{z l}h . \tag{20} \end{aligned} \right. \]

\[ \left\{ \begin{aligned} \sum_{s=-\infty}^{\infty} D'_s \frac{n_s}{r k_{zs}} F_{ns}(k_{zs}^{2}-k^{2}) C'_{sl_2} &= \left( - A_{l_2}\varkappa k_{c1} f'_1 \frac{\psi}{2} - A'_{l_2}\frac{\varkappa_{zl}}{2}\frac{l\pi}{2}F_1 \right)\operatorname{cth}\varkappa'_{zl}h,\\ \sum_{s=-\infty}^{\infty} D'_s \frac{n_s}{r k_{zs}} F_{ns}(k_{zs}^{2}-k^{2}) B'_{sl_1} &= \left( - A_{l_1}\varkappa k_{c1} f'_1 \frac{\psi}{2} + A'_{l_1}\frac{\varkappa'_{zl}}{r}\frac{l\pi}{2}F_1 \right)\operatorname{cth}\varkappa'_{zl}h . \end{aligned} \right. \tag{21} \]

Using equations (19, 20, and 21), one can compose an infinite system of equations with respect to \(D'_s\), eliminating the unknowns \(A_0, A_l, A_{l_2}, A'_{l_1}, A'_{l_2}\). The dispersion equation can be written in the form of an infinite determinant, from which solutions may be obtained in any approximation. In view of the great complexity of the solution obtained, and also taking into account that the derivation is approximate in character, it is expedient to restrict ourselves to the solution in the first approximation, which, after a number of transformations, is written as follows:

\[ \frac{ n^2 f_n F_n + \dfrac{x_n^2}{4}(c+1)^2 F'_n f'_n }{ \dfrac{n^2 f_n F_n}{ 1-\left[\dfrac{A\lambda}{(r_2-r_1)2\pi}\right]^2 } + \dfrac{x_n^2}{4}(c+1)^2 F'_n f'_n } = \frac{ \sqrt{\,1-\left[\dfrac{A\lambda}{(r_2-r_1)2\pi}\right]^2\,} \cdot \operatorname{tg}\sqrt{\,1-\left[\dfrac{c^E_1}{(r_2-r_1)2\pi}\right]^2\,}\,\Theta }{ \sqrt{\,1-\left[\dfrac{c^E_1}{(r_2-r_1)2\pi}\right]^2\,} \cdot \operatorname{cth}\dfrac{g}{h}\Theta \sqrt{\,1-\left[\dfrac{A\lambda}{(r_2-r_1)2\pi}\right]^2\,} } \tag{22} \]

Here \(\Theta=\varkappa h\) is the electrical depth of the groove,

\[ A=\varkappa(r_2-r_1),\qquad c^E_1=\frac{x_{01}(r_2-r_1)}{r_1},\qquad \frac{r_2+r_1}{2}=R,\qquad c=\frac{r_2}{r_1}. \]

Equation (22) makes it possible to match the fields at the mean point of the waveguide. Calculations show that the geometrical dimensions of a waveguide iris-loaded along the end walls turn out to be very close to the dimensions of the waveguide considered in the preceding paragraph.

REFERENCES

1. Vorob'ev A. A., Izv. vuzov MVO SSSR Elektromekhanika, 1, 1958.
2. Kisun'ko G. V., Electrodynamics of Hollow Systems, VKAS Publ., L., 1949.
3. Leont'eva R. M., Radiotekhnika, vol. 12, No. 12, 1957.
4. Trunova N. V., Izv. vuzov MVO SSSR, Radiotekhnika, 1, 1958.