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MEASUREMENT OF DIELECTRICS AT MILLIMETER WAVES
Knowledge of the frequency dependences of the electrical characteristics ($\varepsilon$ and $\operatorname{tg}\delta$) of dielectrics over the entire frequency range, from the very lowest up to the optical, is extremely important both for theoretical explanation of the processes occurring during the interaction of a substance (dielectric)
with a field (electric), and for practical application of dielectrics. The microwave technique, which has developed widely mainly in recent years, has made it possible to develop a number of methods based on the use of volume resonators and measuring—waveguide and coaxial—lines, and making it possible to measure the electrical properties of dielectrics at very high frequencies, up to frequencies of the order of \(3\cdot 10^{10}\) cps, which corresponds to wavelengths of \(1\) cm.
At present, a very timely task is the development of methods of measurement and of carrying out the corresponding investigations at still shorter wavelengths. For understandable reasons, at such wavelengths of electromagnetic waves the use of resonant methods, as well as methods based on the use of measuring lines, becomes difficult or simply impossible. Therefore the problem of investigating dielectrics in the millimeter-wave region is now being solved in the direction of developing methods analogous to optical ones.
In 1953 a number of papers were published in which methods, developed by various authors, for investigating solid dielectrics in the wavelength range shorter than \(1\) cm are described and the first results are reported. The content of some of these papers, which are of considerable interest, is set forth in the present note.
In the paper by Goulshaw\(^1\), an abstract of which was published in our journal\(^2\), a microwave Fabry–Perot interferometer is described in detail, by means of which measurements were made on a number of dielectrics at \(\lambda = 8\) mm.
At this same wavelength, in another paper\(^3\) a study is presented of the electrical properties of several dielectrics by using a method based on measuring the reflection of electromagnetic waves from plates of the specimen, whose dimensions are considerably larger than \(\lambda\). By varying the angle of incidence, the author determines the Brewster angle \(\theta_B\), at which the reflection from the specimen is minimal. Then the dielectric permittivity was determined (under the condition \(\operatorname{tg}\delta \ll 1\), \(\mu = 1\))* as \(\varepsilon = \operatorname{tg}^2 \theta_B\).
To determine \(\operatorname{tg}\delta\), the ratio was measured of the energy \((W)\) transmitted through the specimen when incident at the Brewster angle to the energy transmitted \((W_0)\) without the specimen.
If this ratio is denoted by
\[ \frac{W}{W_0} = \frac{1}{r^2}, \]
then, neglecting terms containing \(\operatorname{tg}\delta\) in the second degree and higher, the angle of dielectric losses can be determined from the formula
\[ \operatorname{tg}\delta = \frac{\lambda \ln r}{\pi d \sqrt{\varepsilon + 1}}, \tag{1} \]
where \(\lambda\) is the wavelength, and \(d\) is the thickness of the specimen.
By the method described, in paper\(^3\) measurements were carried out on a number of dielectrics (glass, polystyrene, and others). The results, for example, for polystyrene are: \(\varepsilon = 2.52\), \(\operatorname{tg}\delta = 0.0092\) (at \(\lambda = 8\) mm).
In paper\(^4\) it proved possible to carry out measurements of \(\varepsilon\) and \(\operatorname{tg}\delta\) for several solid dielectrics at still shorter wavelengths, in the range from \(4.2\) to \(1.8\) mm. Harmonics of a 3131 magnetron (the fundamental wavelength \(\lambda = 12.5\) mm) were used as the source of electromagnetic waves. The dispersion necessary for separating the required harmonic was provided by a slotted diffraction grating similar to that described in\(^5\). The detector of the oscillations that passed through the specimen (or arrived at the same place in the absence of the specimen) was a bolometer placed in a specially
* In paper\(^3\) a generalization of the calculation formulas is also given, making it possible to carry out measurements not only of \(\varepsilon\) and \(\operatorname{tg}\delta\), but also of the magnetic properties of materials (\(\mu\) and \(\operatorname{tg}\delta_\mu\)) by this method at millimeter wavelengths.
an adapted section of waveguide, which could be tuned to the frequency of the selected harmonic.
The experimental procedure consisted in measuring the complex transmission coefficient \(t=|t|e^{i\varphi}\), where \(|t|\)—the amplitude of the transmission coefficient—was determined in the usual way as the square root of the ratio
\[ \frac{W}{W_0}, \]
where \(W\) is the radiation density recorded by a bolometer at a certain point behind the specimen, and \(W_0\) is the same quantity before the specimen was removed. To determine the phase \((\varphi)\) of the transmission coefficient, the interference pattern of electromagnetic waves that had passed through the specimen was recorded; at the same place (in the same plane), in the absence of the specimen, \(\varphi\) was determined as the ratio, multiplied by \(360^\circ\), of the shift of the minima caused by placing the specimen to the distance between the minima.
Fig. 1.
Next, using the experimentally found quantities \(|t|\) and \(\varphi\), the complex refractive index \(n=\eta+jk\) was determined with the aid of the following formula:
\[ \frac{1}{t}=\frac{e^{j\gamma d}}{4n}\left[(n+1)^2 e^{-j\gamma_1 d}-(n-1)^2 e^{j\gamma_1 d}\right], \tag{2} \]
where \(\gamma=\dfrac{2\pi}{\lambda}\) (\(\lambda\) is the wavelength of the working harmonic), \(d\) is the thickness of the dielectric plate, \(\gamma_1=\dfrac{2\pi}{\lambda}n=\gamma n\), \(n=\eta+jk\), \(\eta\) is the refractive index, and \(k\) is the attenuation coefficient.
Formula (2) is obtained by considering the expressions for the electric and magnetic field vectors in the dielectric plate and in air and by using the boundary conditions at the front and back faces of the specimen. The determination of \(\eta\) and \(k\) from formula (2) was carried out by a numerical method, using successive approximations, after first transforming the formula accordingly; as the first approximation it is assumed that \(k=0\), i.e. that \(\eta\) is a real number.
FROM CURRENT LITERATURE
The results obtained by the authors4 ($\eta$ and $k$), as well as the corresponding values of $\varepsilon$ and $\operatorname{tg}\delta$ calculated by us from these data, are given in the table (the last line of the table gives, for comparison, the refractive-index values $n_D$ of these materials for visible light).
Table
| $\lambda$ in mm | Polystyrene $\eta$ | Polystyrene $k$ | Polystyrene $\varepsilon$ | Polystyrene $\operatorname{tg}\delta$ | Plexiglas $\eta$ | Plexiglas $k$ | Plexiglas $\varepsilon$ | Plexiglas $\operatorname{tg}\delta$ |
|---|---|---|---|---|---|---|---|---|
| 4.18 | 1.594 | 0.001 | 2.54 | 0.0012 | 1.604 | 0.007 | 2.57 | 0.0087 |
| 3.13 | 1.594 | 0.002 | 2.54 | 0.0025 | 1.603 | 0.005 | 2.57 | 0.0062 |
| 2.51 | 1.593 | 0.001 | 2.54 | 0.0012 | 1.608 | 0.006 | 2.59 | 0.0075 |
| 1.79 | 1.593 | 0.002 | 2.54 | 0.0025 | 1.600 | 0.007 | 2.56 | 0.0087 |
| 1.551 | 1.502 |
| $\lambda$ in mm | Paraffin $\eta$ | Paraffin $k$ | Paraffin $\varepsilon$ | Paraffin $\operatorname{tg}\delta$ |
|---|---|---|---|---|
| 4.18 | 1.482 | 0.002 | 2.20 | 0.0027 |
| 3.13 | 1.483 | 0.001 | 2.20 | 0.0013 |
| 2.51 | 1.480 | 0.002 | 2.19 | 0.0027 |
| 1.79 | 1.483 | 0.001 | 2.20 | 0.0013 |
| 1.439 |
The results of these works show that, as was to be expected from general theoretical considerations, the dielectric constant of the dielectrics studied here, whose polarization is due to electronic polarizability, does not depend on frequency up to wavelengths of the order of 1 mm; the tangent of the dielectric losses, however, is somewhat increased in comparison with longer waves.
Fig. 2.
The values of $\eta$ for millimeter waves pre—
exceed the values of \(n_D\), which indicates the presence in these materials of dispersion somewhere in the intermediate region (at \(\lambda < 1\) mm).
In considering the question of studying the electrical characteristics of dielectrics at wavelengths shorter than centimeter waves, mention should be made of the work of Meier\(^6\), in which measurements were made of some dielectrics with a very high \(\varepsilon\) (of the order of several thousand) at wavelengths from 0.15 to 0.6 mm. The method used in this work is already typical for infrared spectroscopy (a high-pressure mercury lamp served as the source of electromagnetic waves). Studying several dielectrics prepared on the basis of various titanates (including barium titanate), the author found (Figs. 1, 2) a very strong decrease in \(\varepsilon\) at these wavelengths (\(\lambda = 0.15\text{--}0.6\) mm), which confirms the results obtained earlier\(^7\) for barium titanate at longer wavelengths (\(\lambda = 3.2\text{--}1.25\) cm).
V. Sarafanov
CITED LITERATURE
- Gulshaw, Proc. Phys. Soc. 66, 597 (1953).
- UFN 52, 173 (1954).
- Talpey, L’onde électrique 319, 561 (1953).
- Books, Greig et al., J. Opt. Soc. Am. 43, 1191 (1953).
- Greig, Ferguson, J. Opt. Soc. Am. 40, 504 (1950).
- Meier, Ann. d. Phys. 19, 26 (1953).
- Powles, Jackson, Proc. IEE 96, Part III, 383 (1949).