MATHEMATICAL PHYSICS

P. E. KRASNUSHKIN, S. P. LOMNEV, A. G. TRAGOV

AN EXACT METHOD FOR CALCULATING A PERIODIC CELLULAR WAVEGUIDE

(Presented by Academician I. M. Vinogradov on 18 VI 1964)

The calculation of a periodic cellular waveguide is reduced to determining, from the given geometrical dimensions of the cell and the frequency \(\omega\), the wave numbers \(\psi_l\) and the forms
\(\left\|\begin{matrix}F_1^{(l)}\\ F_2^{(l)}\end{matrix}\right\|\)
of the normal waves traveling along the chain of cells \(j=1,2,\ldots\)

\[ \left\|\begin{matrix} F_1^{(l)}(q_1)\\ F_2^{(l)}(q_2) \end{matrix}\right\| \exp \left[i(\omega t-\psi_l j)\right], \tag{1} \]

where \(F_1^{(l)}(q_1)\) and \(F_2^{(l)}(q_2)\) are distribution functions of the tangential components of the field \(E_\tau\) and \(H_\tau\), specified in some section \(S_1\) of the cell, for example at the input aperture \(S_1=S_{\mathrm{in}}\), i.e. \(F_1=E_\tau(q_1)\), \(F_2=H_\tau(q_1)\), \(q_1\in S_1\), or else any other pair of functions \(\{E_\tau(q_1),\,H_\tau(q_2)\}\), \(\{E_\tau(q_1),\,E_\tau(q_2)\}\), \(\{H_\tau(q_1),\,H_\tau(q_2)\}\), specified in two different sections of the cell \(S_1\) and \(S_2\) \((q_1\in S_1,\ q_2\in S_2)\), and in one-to-one correspondence with \(\{E_\tau(q_1),\,H_\tau(q_1)\}\), \(q_1\in S_1\). The method proposed here for calculating the parameters of the normal waves of the first numbers \(l\) is based on the work \((^1)\). It consists in solving the eigenvalue equation \(\lambda_l=\exp(-i\psi_l)\) for the Breisig operator of the cell

\[ A\left\|\begin{matrix}F_1\\ F_2\end{matrix}\right\| = \lambda \left\|\begin{matrix}F_1\\ F_2\end{matrix}\right\|; \qquad A= \begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{pmatrix}. \tag{2} \]

With the aid of a certain system of basis functions \(\{F_{1s}\}_1^\infty\) and \(\{F_{2s}\}_1^\infty\), introduced in the sections \(S_1\) and \(S_2\), the functional operators \(A_{ik}\) are replaced by matrices \(A_{ik}\), and (2) becomes a homogeneous infinite system of algebraic equations. The condition for its consistency is the characteristic equation of the Breisig operator

\[ |A-\lambda \mathscr{E}|=0 \tag{3} \]

with roots \(\psi_l\) (the so-called dispersion equation of the waveguide). Replacing the infinite matrices \(A_{ik}\) by finite matrices \(A_{ik}^N\) of order \(N\), we find approximate values \(\psi_l^N\). To improve the convergence of \(\psi_l^N\) to \(\psi_l\) as \(N\to\infty\), electrostatic singularities of the field \(E\) on the sharp edges of the inner surface of the cell are taken into account in \(\{F_{1s}\}_1^\infty\) and \(\{F_{2s}\}_1^\infty\). We demonstrate this method on the classical example of axially symmetric waves of a circular iris-loaded waveguide with the cell shown in Fig. 1 \((^{2-6})\).

Fig. 1

We construct the matrix representation (2) directly from the boundary-value problem for Maxwell’s equations, taking as the basis system the functions

\[ E_r\left(r,\ z=\pm \frac{t}{2}\right) = \sum_{s=1}^{\infty} K_s^{(1,2)}\,\mathscr{E}_s(r), \quad 0<r<a, \tag{4} \]

which determine the electric fields in sections \(\langle 2\rangle\) and \(\langle 4\rangle\) (Fig. 1). Here \(\{\mathscr{E}_s(r)\}_{s=1}^{\infty}\) is an arbitrary complete system of functions. By virtue of the first problem

of electrodynamics (7), specifying the fields \(E_r\) in sections \(\langle 2\rangle\) and \(\langle 4\rangle\) is equivalent to specifying the fields \(E_r\) and \(H_\varphi\) in the input or output sections of the cell. Applying the method of partial regions (see, for example, \((^8)\)), we represent the wave fields (1) \(E_r\) and \(H_\varphi\) in the regions \(p=1,2,3\) as sums of normal waves traveling along the \(z\)-axis of the sleeves \(p=1,2,3\) (Fig. 1):

\[ E_r^{(p)}=\sum_{n=1}^{\infty} e_n^{(p)}(r)\left[C_{n1}^{(p)}e^{-\gamma_n^{(p)}z^{(p)}}-C_{n2}^{(p)}e^{\gamma_n^{(p)}z^{(p)}}\right]\frac{\gamma_n^{(p)}}{i\omega\varepsilon}, \tag{5} \]

\[ H_\varphi^{(p)}=\sum_{n=1}^{\infty} e_n^{(p)}(r)\left[C_{n1}^{(p)}e^{-\gamma_n^{(p)}z^{(p)}}+C_{n2}^{(p)}e^{\gamma_n^{(p)}z^{(p)}}\right], \tag{6} \]

where \(p\) is the number of the sleeve (region); \(n\) is the number of the wave in the sleeve; \(\gamma_n^{(p)}=\sqrt{(\rho_{0n}/r_p)^2-(\omega/c)^2}\) are the wave numbers of these waves; \(\rho_{0n}\) is the root of order \(n\) of the Bessel function \(J_0(\rho)\); \(z^{(1)}=z+t/2\); \(z^{(2)}=z\); \(z^{(3)}=z-t/2\); \(r_1=r_3=b\); \(r_2=a\); \(c=3\cdot10^{10}\ \text{cm/sec}\); \(\varepsilon\) is the dielectric constant. The functions \(e_n^{(p)}(r)=J_1(\rho_{0n}r/r_p)/\sqrt{\pi r_p}\,|J_1(\rho_{0n})|\) are normalized so that the scalar products \((e_n^{(p)},e_m^{(p)})=\delta_{n,m}\). By virtue of the definition of a normal wave (1) in a cellular waveguide,

\[ C_{n1}^{(3)}=C_{n1}^{(1)}e^{\gamma_n d-i\psi};\qquad C_{n2}^{(3)}=C_{n2}^{(1)}e^{-\gamma_n d-i\psi};\qquad \gamma_n=\gamma_n^{(1)}=\gamma_n^{(3)} \tag{7} \]

and we are in fact dealing with a closed volume composed of the volume \(V_{1+3}\) of sleeves 1 and 3 and the volume \(V_2\) of sleeve 2.

Let us solve the first problem of electrodynamics \((^7)\) for \(V_2\) for the prescribed fields (4). We expand (4) in the functions \(e_n^{(1)}=e_n^{(3)}\):

\[ E_r(r,z=\mp t/2)=\sum_{s=1}^{\infty}\sum_{n=1}^{\infty}K_s^{(1,2)}\beta_{ns}e_n^{(1)}, \tag{8} \]

where \(\beta_{ns}=(\mathscr{E}_s,e_n^{(1)})\) are Fourier coefficients; \(\widetilde{\mathscr{E}}_s=\mathscr{E}_s\) for \(0<r<a\) and \(\widetilde{\mathscr{E}}_s=0\) for \(a<r<b\).

Having determined \(E_r\) in the sections \(z=\mp t/2\) and \(z=\mp D/2\) from (5), and equating \(E_r\) at \(z=\mp t/2\) to expression (8), and for \(E_r\) at \(z=\mp D/2\) taking (7) into account, we obtain

\[ C_{n2}^{(1)}=\frac{i\omega\varepsilon}{2\gamma_n\operatorname{sh}\gamma_n d} \left(e^{i\psi}\sum_{s=1}^{\infty}\beta_{ns}K_s^{(2)} -e^{\mp\gamma_n d}\sum_{s=1}^{\infty}\beta_{ns}K_s^{(1)}\right). \tag{9} \]

Solving the first problem of electrodynamics for \(V_2\), excited at \(z=\mp t/2\) by the fields (4), we obtain

\[ C_{n2}^{(2)}=\frac{i\omega\varepsilon}{2\alpha_n\operatorname{sh}\alpha_n t} \left(e^{\pm\alpha_n t/2}\sum_{s=1}^{\infty}\alpha_{ns}K_s^{(1)} -e^{\mp\alpha_n t/2}\sum_{s=1}^{\infty}\alpha_{ns}K_s^{(2)}\right), \tag{10} \]

where \(\alpha_{ns}=(\mathscr{E}_s,e_n^{(2)})\), \(\alpha_n=\gamma_n^{(2)}\).

The continuity condition for the fields \(H_\varphi\) in the apertures \(0<r<a\), \(z=\mp t/2\), will be

\[ \sum_{n=1}^{\infty} e_n^{(1)}\left(C_{n1}^{(1,3)}+C_{n2}^{(1,3)}\right) = \sum_{n=1}^{\infty} e_n^{(2)}\left(C_{n1}^{(2)}e^{\pm\alpha_n t/2}+C_{n2}^{(2)}e^{\mp\alpha_n t/2}\right). \tag{11} \]

Multiplying (11) scalarly by \(\mathscr{E}_{s'}(r)\), \(s'=1,2,\ldots\), we replace (11) by a system of algebraic equations

\[ \sum_{n=1}^{\infty}\beta_{ns'}\left(C_{n1}^{(1,3)}+C_{n2}^{(1,3)}\right) = \sum_{n=1}^{\infty}\alpha_{ns'}\left(C_{n1}^{(2)}e^{\pm\alpha_n t/2} + C_{n2}^{(2)}e^{\mp\alpha_n t/2}\right), \tag{11'} \]

\[ s'=1,2,\ldots,\infty . \]

Substituting (9) and (10) into (11′) and taking (7) into account, we obtain an infinite homogeneous system of algebraic equations for determining \(K_s\). We write it in operator form, introducing the vectors \(\overline{K}^{(1)}\) and \(\overline{K}^{(2)}\).

\[ \begin{pmatrix} a-\alpha & \alpha\\ \alpha & a \end{pmatrix} \left\| \begin{matrix} \overline{K}^{(1)}\\ \overline{K}^{(2)} \end{matrix} \right\| = \begin{pmatrix} 0 & e^{i\psi}\beta\\ e^{-i\psi}\beta & 0 \end{pmatrix} \left\| \begin{matrix} \overline{K}^{(1)}\\ \overline{K}^{(2)} \end{matrix} \right\|, \tag{12} \]

where \(a\), \(\alpha\), and \(\beta\) are infinite symmetric square matrices with elements

\[ a_{ss'}= \sum_{n=1}^{\infty}\frac{\operatorname{cth}\gamma_n d}{\gamma_n}\,\beta_{ns}\beta_{ns'} + \sum_{n=1}^{\infty}\frac{\operatorname{cth}\alpha_n t}{\alpha_n}\,\alpha_{ns}\alpha_{ns'}, \tag{13} \]

\[ \alpha_{ss'}= \sum_{n=1}^{\infty}\frac{\alpha_{ns}\alpha_{ns'}}{\alpha_n\operatorname{sh}\alpha_n t}, \qquad \beta_{ss'}= \sum_{n=1}^{\infty}\frac{\beta_{ns}\beta_{ns'}}{\gamma_n\operatorname{sh}\gamma_n d}, \]

where \(\alpha_{ns}\) and \(\beta_{ns}\) are determined from (8) and (10). In the case when the electrostatic singularity of the field \(E_r\) at the edges of the apertures in the cross sections \(\langle 3\rangle\) and \(\langle 4\rangle\) is taken into account in the functions \(\mathscr{E}_s\) in the form \(\mathscr{E}_s=(r/a)^{2s-1}/\sqrt{1-(r/a)^2}\), they are expressed by the formulas

\[ \alpha_{ns} = \frac{2\sqrt{\pi}}{|J_1(\rho_{0n})|} \sum_{r=0}^{s-1}(-1)^r\,{}_{s-1}C_r\,(2r-1)!!\,(\rho_{0n})^{-(r+1)} I_{r+3/2}(\rho_{0n}); \tag{14} \]

\[ \beta_{ns} = \frac{a\cdot 2\sqrt{\pi}}{b\,|J_1(\rho_{0n})|} \sum_{r=0}^{s-1}(-1)^r\,{}_{s-1}C_r\,(2r-1)!! \left(\rho_{0n}\frac{a}{b}\right)^{-(r+1)} I_{r+3/2}\left(\rho_{0n}\frac{a}{b}\right), \]

where \({}_{s-1}C_r\) are binomial coefficients; \(I_q(x)=I_q'(x)/\sqrt{\pi x/2}\) are Bessel functions of order \(q\). Equation (12) is the required eigenvalue equation (3) for the Breizig operator (2) in operator form in the basis of functions (4) specified on the internal cross sections of the cell \(\langle 2\rangle\) and \(\langle 4\rangle\). For numerical calculations of \(\psi\) in the passband \((|\cos\psi|<1)\), using the substitution \(\overline{K}^{(1,2)}=\overline{X}\pm i\overline{Y}\), it is convenient to represent it in the form

\[ \begin{pmatrix} a-\alpha-\cos\psi\,\beta & -\sin\psi\,\beta\\ -\sin\psi\,\beta & a+\alpha+\cos\psi\,\beta \end{pmatrix} \left\| \begin{matrix} \overline{X}\\ \overline{Y} \end{matrix} \right\| =0. \tag{15} \]

Then the dispersion equation in matrix form takes the form

\[ \det \begin{pmatrix} a-\alpha-\beta\cos\psi & -\beta\sin\psi\\ -\beta\sin\psi & a+\alpha+\beta\cos\psi \end{pmatrix} =0. \tag{16} \]

For the numerical solution of (16), the matrix blocks in (16) were truncated at the \(N\)-th row and \(N\)-th column. For \(N=1\) and \(t\to 0\), (16) coincides with the equation obtained in work \((^9)\), and for \(N=1\) and finite \(t\), with the equation of work \((^{10})\).

Table 1 gives the results of calculations of $\psi_1^N$ (in radians) on the BESM-2 by equation (16) for the fundamental passband $(l=1)$ at various frequencies $\omega$, determined by the free-space wavelength $\lambda_0$. The calculation is given for a cell with parameters (see Fig. 1) $a/b=0.30000$, $b=4.30000$, $t=0.40000$, $d=1.20200$. As is seen from the table, $\psi_1^N$ ceases to change with increasing $N$ in the second decimal place for $N \geqslant 3$, which ensures an accuracy of the order of 10–20 min. We note that for $\lambda_0=10.677$ cm, for a cell with the dimensions indicated above, experiment gives $\psi_1=\pi/2$ with an accuracy of up to several tens of minutes, whereas the calculated value is $\psi_1=1.571$ (for $N=3$), and for a cell with $a/b=0.5$, $b=5.525$, $t=0.4$ and $d=0.778$, at $\lambda=11.039$ cm, $\psi_{\mathrm{expt}}=\pi/2$, while $\psi_{\mathrm{calc}}=1.577$ $(N=4)$. The experimental data were taken from work (10); since their accuracy is lower than the calculated one, this comparison does not make it possible to assess the accuracy of the calculation.

Table 1*

* The imaginary values of $\psi_1$ are marked with a dash.

Increasing the accuracy of the calculation of $\psi_1$ by increasing $N>4$ on a machine of the BESM-2 type is impossible, since as $N$ grows the matrices $a$, $\alpha$ and $\beta$ tend to become singular, the determinant in (16) tends to 0, and for an accurate determination of the root $\psi_1$ one must compute the elements $a$, $\alpha$ and $\beta$ with accuracy better than $10^{-8}$. In Table 1 they were computed with accuracy $10^{-6}$.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
18 VI 1964

REFERENCES

1. P. E. Krasnushkin, Radiotekhn. i elektronika (in press).
2. V. V. Vladimirskii, DAN, 52, No. 3, 219 (1946); ZhTF, 17, 11, 1277 (1947).
3. E. L. Chu, W. W. Hanson, J. Appl. Phys., 18, 996 (1947); 20, 280 (1949).
4. L. Brillouin, J. Appl. Phys., 19, 1023 (1948).
5. W. Walkinshow, Proc. Roy. Soc., 61, 246 (1948); J. Appl. Phys., 20, 634 (1949).
6. R. Combe, M. Feix, Nuovo Cim., 15, No. 5, 760 (1960).
7. Ya. N. Fel’d, Foundations of the Theory of Slotted Antennas, Moscow, 1948.
8. G. V. Kisun’ko, Electrodynamics of Hollow Systems, Leningrad, 1949.
9. G. N. Rapoport, ZhTF, 21, 9, 1076 (1951); 27, 9, 2105 (1957).
10. A. G. Tragov, Candidate’s dissertation, Moscow Engineering-Physics Institute, 1962.