Abstract
Full Text
UDC 621.373.413
PHYSICS
A. T. FIALKOVSKII
ON THE THEORY OF OPEN RESONATORS FORMED BY PARALLEL DISKS
(Presented by Academician V. A. Fok on 16 X 1965)
1. We consider resonators consisting of two disks placed opposite one another (Fig. 1), with \(kl \gg 1\), where \(k=\omega/c\) is the wave number. The calculation of these resonators is based on the assumption that, at their edge \((r=a)\), electromagnetic waves at frequencies close to their critical frequencies are reflected with the reflection coefficient from the open end of a plane waveguide, which for waves \(E_{0q}\) and \(H_{0q}\) can be written in the form
Fig. 1
\[ \begin{gathered} R_E=-e^{i(\beta' + i\beta_E'')s},\\ R_H=-e^{i(\beta' + i\beta_H'')s}, \end{gathered} \tag{1} \]
where \(s=\sqrt{2l/k}\,w\), and \(w\), in our case, is the radial wave number.
The dependence of \(\beta'\), \(\beta_E''\), \(\beta_H''\) on the index \(q\) is given in Fig. 2, constructed on the basis of (1). Taking into account the difference between \(\beta_E''\) and \(\beta_H''\), which is significant for small values of \(q\), leads to qualitatively new results.
2. We introduce the functions \(f_E(r)\) and \(f_H(r)\), proportional respectively to the electric and magnetic azimuthal components of the field:
\[ \begin{gathered} f_E(r)=C J_{m-1}(wr)+D J_{m+1}(wr),\\ f_H(r)=C J_{m-1}(wr)-D J_{m+1}(wr), \end{gathered} \tag{2} \]
where \(C\) and \(D\) are arbitrary constants, \(J_m(x)\) is the Bessel function, and \(m\) is the azimuthal index.
Using expressions (1), we write the impedance boundary conditions at the edge of the resonator:
\[ \begin{gathered} f_E(r)+\frac{y_E}{w}\frac{df_E(r)}{dr}=0,\\ f_H(r)+\frac{y_H}{w}\frac{df_H(r)}{dr}=0, \end{gathered} \tag{3} \]
where (for \(|s|\ll 1\))
\[ y_E=\frac{\beta'+i\beta_E''}{2}s,\qquad y_H=\frac{\beta'+i\beta_H''}{2}s. \tag{4} \]
Substitution of (4) into (2) gives, in the first approximation in \(s\), for \(m=1,2,\ldots\)
\[ s=\frac{2\nu_{m-1,n}}{M+\beta'+i\beta''} \quad \text{or} \quad s=\frac{2\nu_{m+1,n}}{M+\beta'+i\beta''}, \tag{5} \]
where
\[ M=\sqrt{2k/l}\,a=2ka/\sqrt{\pi q};\qquad \beta''=(\beta_E''+\beta_H'')/2, \]
\(v_{m,n}\) is the \(n\)-th zero of the Bessel function \(J_m(x)\). Expressions (5) give the natural frequencies of the resonator
\[ \omega=\frac{\pi c}{l}\left(\frac{q}{2}+p\right), \qquad p=\frac{s^2}{4\pi}=p'-ip'' . \tag{6} \]
The ratio \(D/C\), according to the order of the formulas in (5), is determined from
\[ \frac{D}{C} = -i\,\frac{wa}{2m}\, \frac{\beta_E''-\beta_H''}{4}\,s \quad \text{or} \quad \frac{C}{D} = i\,\frac{wa}{2m}\, \frac{\beta_E''-\beta_H''}{4}\,s . \tag{7} \]
For \(m=0\), the natural frequencies are determined by the formulas
\[ s=\frac{2v_{1,n}}{M+\beta'+i\beta_E''} \quad \text{or} \quad s=\frac{2v_{1,n}}{M+\beta'+i\beta_H''}. \tag{8} \]
- For \(m=1,2,\ldots\), the natural oscillations are characterized by three indices \(m,n,q\) and also by that Bessel function (\(J_{m-1}\) or \(J_{m+1}\)) whose roots give the natural frequencies and which, in the first approximation, determines the field of the natural oscillation.
Fig. 2
Therefore we shall denote these oscillations by four indices: \((m,\,m-1,\,n,\,q)\); \((m,\,m+1,\,n,\,q)\).
The derived relations are valid for \(|s|\ll 1\). This means that for \(q\sim 1\) one must have \(ka\gg 1,\ a\gg l\); for \(q\gg 1\) it may also be that \(l\gg a\). Thus, no restrictions are imposed on the ratio of the diameter of the disks to the distance between them (for each such ratio there exists its own spectrum of high-\(Q\) oscillations). As \(q\to\infty\), the parameters \(\beta'\), \(\beta_E''\), \(\beta_H''\) coincide with \(\beta=0.824\), and our formulas pass into the formulas of paper (2).
- When the disks have small transparency, a plane wave incident on the resonator along the \(z\) axis excites in the resonator an electromagnetic field containing only the oscillations \((1,0,n,q)\):
\[ E_x=\sum_{n,q} A_{nq}J_0\!\left(\frac{v_{0n}r}{a}\right) \left[e^{iv_q z}-(-1)^q e^{-iv_q z}\right], \tag{9} \]
\[ H_y=\sum_{n,q} B_{nq}J_0\!\left(\frac{v_{0n}r}{a}\right) \left[e^{iv_q z}+(-1)^q e^{-iv_q z}\right]. \]
and the components \(E_z\) and \(H_z\), which we do not write out. Here
\(kl=v_q l+\pi p,\quad v_q l=\pi(q/2+\rho)\), where the parameter \(\rho\) is related to the complex reflection coefficient of the disks \(R\) by the relation
\[ \rho=\frac{i}{2\pi}\ln R=\rho'-i\rho''. \]
The coefficients \(A_{nq}\) and \(B_{nq}\) are determined by the formulas
\[ \begin{aligned} A_{nq}&=e^{-i(k-v_q)l}\, \frac{iTE_0}{klv_{0n}J_1(v_{0n})}\, \frac{\omega_{nq}\omega}{\omega^2-\omega_{nq}^2},\\ B_{nq}&=e^{-i(k-v_q)l}\, \frac{iTE_0}{klv_{0n}J_1(v_{0n})}\, \frac{\omega^2}{\omega^2-\omega_{nq}^2}, \end{aligned} \tag{10} \]
where \(T\) is the transmission coefficient of the disks; the natural frequencies
\(\omega_{nq}=\omega_{nq}'-i\omega_{nq}''\) are complex; \(\omega\) is the excitation frequency. The resonant values of the amplitudes (10), when diffraction losses predominate \((p''\gg\rho'')\), are inversely proportional to \(v_{0n}^{5/2}\) and \(q^{3/2}\), i.e., with increasing indices \(n\) and \(q\) they decrease rather rapidly. When losses in the disks predominate, the resonant values (10) are inversely proportional to \(v_{0n}^{1/2}\) and do not depend on \(q\).
I express my gratitude to L. A. Vainshtein for formulating the problem and for his guidance.
Institute of Physical Problems named after S. I. Vavilov
Academy of Sciences of the USSR
Received
28 IX 1965
REFERENCES
- L. A. Vainshtein, Diffraction of Electromagnetic and Acoustic Waves at the Open End of a Waveguide, Moscow, 1953.
- L. A. Vainshtein, ZhETF, 44, No. 3, 1050 (1963).