FABRY–PEROT INTERFEROMETER FOR MILLIMETER-WAVE RADIO WAVES

One of the characteristic features of the microwave region of the spectrum is that, along with radio-engineering methods and devices, instruments and methods typical of optics prove applicable here. The use of the latter is connected with the possibility of making lenses, mirrors, diaphragms, and other optical components whose dimensions are far greater than the wavelength of the radiation. However, the relation between the dimensions of the components of optical instruments and the wavelength in the case of microwaves is, in general, several orders of magnitude different from that encountered in ordinary optical instruments. As a result, diffraction phenomena acquire an especially important role; to ignore them, as is often permissible under the conditions of ordinary optics, becomes quite impossible. Hence there arise certain specific features that are of both practical and theoretical interest.

It was natural that an effort should arise to create for microwave radio waves a multi-beam interferometric device, namely a Fabry–Perot interferometer. Such an interferometer was first implemented for operation “in reflected light”[^1] and had comparatively modest performance. A considerably better interferometer, intended for operation “in transmitted light” with radiation in the millimeter range, was constructed and investigated by the author of the paper under review[^2].

The interferometer consisted of a pair of reflectors whose effective diameter was 27.5 cm. The distance between the reflectors \(t\) could be varied (by moving one of them) from a few millimeters to 2 meters. This displacement was effected by means of a micrometer screw and was measured with an accuracy of up to \(1\,\mu\). As reflectors, stacks of 8 polystyrene disks of thickness

\[ \frac{\lambda'}{4} \]

(\(\lambda'\) is the wavelength of the radiation in polystyrene) were used, separated by air spacers of thickness

\[ \frac{\lambda}{4} \]

(\(\lambda\) is the wavelength of the radiation in air). Practically each of the disks had a rim whose thickness was equal to

\[ \frac{\lambda' + \lambda}{4}; \]

the disks were placed tightly one upon another by their rims and fastened in a common mount. This composite reflector, an immediate analogue of the multilayer dielectric reflectors that have come into wide use in optics in recent years[^3], was employed on the basis of the following considerations.

As is known, in order to obtain high resolving power and high intensity of the fringes it is necessary to use reflectors with reflection coefficients \(R\) as large as possible and with absorption of radiation as small as possible. (In particular, the resolving power of an interferometer is determined by the value \(Qn\), where \(n\) is the order of interference and \(Q\) is the quantity reciprocal to the half-width of the first-order fringe, expressed in fractions of the distance between fringes:

\[ Q=\frac{\pi}{\varphi_1};\quad \cos \varphi_1=\frac{2R-(1-R)^2}{2R}. \]

) Multilayer dielectric reflectors best satisfy this requirement.

Taking for polystyrene (at \(\lambda=1.25\ \text{cm}\)) \(\varepsilon=2.56\) and \(\operatorname{tg}\delta=0.001\) (\(\delta\) is the loss angle), the author found the following values of the amplitude reflection coefficient \(r_m\) from \(m\) quarter-wave layers of polystyrene separated by quarter-wave layers of air (Table 1).

Table 1

Thus, \(r_m\) rapidly increases with increasing \(m\) and at \(m=8\) almost reaches its limiting value. Accordingly, the energy reflection coefficient \(R=r^2\) of a fifteen-layer reflector (eight layers of polystyrene and 7 air gaps) is \(R_{m=8}=0.9954\) and \(Q=675\). (Even without taking account of such weak dielectric losses, substantially different values are obtained: \(R_{m=8}=0.9978\) and \(Q=1350\).) Since for fused quartz \(\varepsilon=3.83\) and \(\operatorname{tg}\delta=0.0001\), with it considerably larger values of \(Q\) can be obtained. Of course, no single-layer reflectors, including metallic ones, can provide such favorable values of \(R\) and, correspondingly, \(Q\).

The essential elements of an interferometric setup are the emitter and the radiation receiver. As the radiation source a klystron was used. The wavelength was equal to \(8\ \text{mm}\) and was stabilized by a cavity resonator. The degree of monochromaticity of the radiation ensured the “visibility” of the fringes when the reflectors of the interferometer were separated by a distance \(t\) of the order of several meters, which made it possible to investigate the properties of the interferometer itself. From the \(H_{10}\)-type generator the wave was transmitted along a rectangular waveguide ending in a radiating horn (with aperture \(15\ \text{cm}\times15\ \text{cm}\) or \(8.75\ \text{cm}\times8.75\ \text{cm}\)), closed by a short-focus (\(F=15\ \text{cm};\ 50\ \text{cm}\)) polystyrene lens (Fig. 1). To avoid reflections, the perimeter of the horn aperture was beveled at an angle of \(45^\circ\). Under these conditions the emitted wave was uniform in phase; as for the amplitude, it was the same in the vertical plane (vector \(E\)), while in the horizontal plane it varied according to a sinusoidal law. As a result, in addition to the plane wave propagating in the direction \(\vartheta=0\), there arose a set of likewise plane diffracted waves diverging in a fan in the horizontal plane. In addition, diffraction occurred at the aperture of the horn device.

The receiving device also consisted of a horn with a lens, passing into a rectangular waveguide in which a crystal detector of millimeter waves was placed. At the output, the rectified current was measured by a galvanometer. The measurements consisted in gradually changing \(t\) (by means of a micrometer screw) and determining the dependence of the readings

of the galvanometer on \(t\). At the same time, control measurements of \(\lambda\) were made with the aid of a strip wavemeter, with an accuracy of up to several units in \(10^{-5}\).

Allowance for the diffraction structure of the irradiating beam, as well as for diffraction in the interferometer itself, was made by a number of authors\(^{4,5}\). In broad outline the results of this analysis are as follows. The radiation field may be

Fig. 1.

regarded as a superposition of an undiffracted and a number of diffracted plane waves, having different directions (different \(\vartheta\)) and, correspondingly, passing through the interferometer to different degrees. The angular half-width of the transmitted diffraction spectrum is determined by the relation

\[ \cos \vartheta_{\left(\frac{1}{2} I_{\max}\right)} = 1 - \frac{1}{2nQ}. \]

In addition, if the distance between the transmitter and the interferometer or the receiver is large, then waves diffracted at large angles will not enter the receiver. Further, the intensity of the radiation passing through the interferometer is weakened owing to the cutting off or attenuation of diffracted waves. At the same time, the presence of diffracted waves at small interference orders may lead to a displacement of the maximum. However, at large interference orders the position of the maximum corresponds to \(\vartheta = 0\). Finally, as \(t\) increases, the effective value of \(Q\), and consequently the sharpness of the fringes, should decrease. All these phenomena were observed experimentally.

The measurements were made at various distances from the transmitter to the front reflector of the interferometer \(a\) and from the rear reflector of the interferometer to the receiver \(b\), namely \(9.5 \leq a \leq 115\ \text{cm}\) and \(150\ \text{cm} \leq b \leq 300\ \text{cm}\). Measurements of the wavelength (according to the wavemeter, \(\lambda = 8.3299\ \text{mm}\)), carried out with the aid of the interferometer by measuring \(\Delta t\), corresponding to a shift by 120 fringes, gave the following values (Table II).

Table II

Thus, for large \(n\) the accuracy of the measurements corresponds to \(10^{-5}\) and decreases as \(n\) decreases, owing to a systematic displacement

bands, having an evidently diffractional character, as is, in particular, evidenced by the dependence of these quantities on the choice of \(a\) and \(b\).

Fig. 2.
Vertical axis: Intensity; horizontal axis: Reflector displacement \(R2\) (in cm).

Fig. 2 shows the character of the observed dependence of the output-current strength on \(t\) and illustrates the sharpness of the observed interference pattern. From Fig. 3 it is seen that at \(t \sim 2\ \text{m}\) the interference bands remain sufficiently sharp and the selectivity factor \(\dfrac{Qn}{2}\) is still very large, but at the same time \(Q\) and \(Qn\) decrease with increasing \(t\) (and consequently,

Fig. 3.
Vertical axis: Intensity; horizontal axis: Reflector position \(R2\) (in cm).
For maximum “1”: \(t = 14\ \text{cm}\).

Annotations in the figure:

1. \(Q = 695\), \(Qn = 123000\)
2. \(Q = 309\), \(Qn = 91000\)
3. \(Q = 163\), \(Qn = 70000\)

and \(n\)). Fig. 4 shows 10 successive maxima, detected at the initial value \(t = 3\ \text{cm}\). The small secondary maxima (satellites) evidently have a diffractional origin — they are located on the side of larger values of \(t\) (i.e. \(\vartheta > 0\)) and move away from the principal maximum as \(n\) increases.

The dependence of the intensity maximum on \(t\) is shown in Fig. 5 (curve 1 corresponds to a horn \(15\ \text{cm} \times 15\ \text{cm}\), curve 2 to a horn \(8.75\ \text{cm} \times 8.75\ \text{cm}\)). It is seen from the figure that, owing to diffraction phenomena,

the effective value of \(Q\) first increases and then decreases, and the character of this dependence is different for different apertures of the emitter. Removing the lenses from the horns of the emitter and receiver entailed a decrease in the intensity of the maxima (by \(60\) db) and a simultaneous increase in the amplitude and number of diffraction maxima, as a consequence of the broadening of the diffraction spectrum of the radiation being analyzed. The diffraction maxima disappeared at \(t \gtrsim 43\) cm.

Fig. 4.

Similarly to a band resonator, a Fabry–Perot interferometer may be used to determine dielectric properties and, in particular, small dielectric losses in plates. It should be borne in mind, however, that, owing to interference phenomena, the immediate result of the measurements depends on the position of the layer of the substance under investigation in the interferometer.

The introduction of a polystyrene plate \(0.865\) inch thick caused, according to the author’s evidence, a displacement of the band corresponding to \(\Delta t = 1.314\) cm. Since

\[ \Delta t = d\left(\sqrt{\varepsilon}-1\right), \]

the value \(\varepsilon = 2.55\) was thereby found, in agreement with the data of other measurements. The same value \(\varepsilon = 2.64\) was found for Plexiglas.

Fig. 5.

Further, in the case of a resonator (if ohmic losses are neglected)

\[ \frac{1}{Q_{\lambda}} = B \tan \delta, \]

where

\[ Q_{\lambda} = \frac{Q_n}{2}, \]

and \(B\) is the fraction of the total energy stored in the dielectric and depending on the relative volume of the dielectric and on the ratio \(A\) of the field amplitudes in the dielectric and outside it. When the dielectric is moved, \(A\) changes periodically and, correspondingly, so does \(Q\) (the latter by a factor \(\varepsilon\) when the layer is moved from a minimum to a maximum of the standing wave).

Thus, for example, if for \(n = 158\) in the absence of a dielectric \(Q = 416\), then upon insertion of a layer of polystyrene the bands broadened substantially.

and (depending on the position of the layer) \(Q\) varied within the range from 124 to 273. For Perspex, respectively, \(11 \ll Q \ll 35\) was found, while the line intensity decreased by 26 dB. The following values of \(\operatorname{tg}\delta\) were correspondingly obtained:

for polystyrene—0.0006 (according to other data, 0.0008 at \(\lambda = 1.25\) cm),

for Perspex—0.009 (according to other data, 0.012 at \(\lambda = 1.25\) cm).

Thus, the use of the interferometer makes it possible to measure \(\varepsilon\) with an accuracy of \(\sim 1\%\) and gives satisfactory results in determining \(\operatorname{tg}\delta\).

It should be noted that in the present case values \(Q_\lambda \sim 60\,000\) were achieved, which is comparable with the values of \(Q_\lambda\) for the best cavity resonators. In other words, the interference method of filtering radiation and measuring its wavelength can already compete with other methods. In particular, wavelength measurements with an accuracy of the order of \(10^{-5}\) are readily carried out. The use of fused quartz for making reflectors should substantially improve the results and make it possible to apply this method even to such precision measurements as, for example, measuring the speed of light in vacuum.

G. R.

CITED LITERATURE

1. M. Sachs, J. O. Artman and E. Richter, Columbia Univ. Radiation Labor. Rep. 1 June 1952.
2. W. Culshaw, Proc. Phys. Soc. 66, No. 403, B, 597 (1953).
3. See, for example, G. V. Rozenberg, UFN 47, Nos. 1 and 2, 3 (1952).
4. F. A. Korolev, Trudy FIAN, 2, No. 1, 3 (1940).
5. H. G. Booker and P. C. Clemmow, J. Inst. Electr. Eng., Part III, 97, 11 (1950), et al.