Calculation of angular-frequency characteristics of conical radiation based on the Doppler effect
Abstract Generated abstract
The paper addresses determination of the angular frequency characteristic of coherent sources whose radiation pattern consists of narrow hollow cones, where emission frequency is linked to cone aperture angle. It proposes using heterodyne reception with a Doppler shifted copy of the radiation produced by reflection from a moving, rotating plane reflector. The authors derive relations for the Doppler frequency increment as a function of reflector speed, distance, cone angle, incidence geometry, and position along the elliptical reflection contour. The resulting coefficient functions, including plotted dependences for different cone angles, are presented as tools for calculating or measuring angular frequency characteristics of conical radiation.
Full Text
Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov
Vol. 228 1974
CALCULATION OF THE ANGULAR-FREQUENCY CHARACTERISTICS OF CONICAL RADIATION BASED ON THE DOPPLER EFFECT
M. S. Aleinikov, E. P. Koltsov, V. I. Lukovnikov
(Presented by the scientific and technical seminar of the Department of Electrical Equipment, AEMF)
Known sources of coherent radiation are those whose directional diagram is a series of hollow cones with aperture angles \(\theta\) relative to the axis on the order of \(1^\circ\)–\(5^\circ\) [1, 2]. Here the angle \(\theta\) is related to the frequency of the radiation propagating in the given cone; that is, between the spatial and frequency characteristics of the radiation source there exists a definite relationship, which we call here the angular-frequency characteristic.
In studying new radiation sources of this type, it becomes necessary to determine their angular-frequency characteristic.
Since existing radiation receivers do not respond to the instantaneous value of the carrier frequency in the optical and IR ranges [3], the heterodyne method of reception is used in solving the reception problem. The role of the heterodyne is then played by the source of the analyzed radiation; the role of the signal is played by the radiation of the source shifted relative to the original by a frequency \(\Delta \nu\).
A shift \(\Delta \nu\) can be produced, for example, by means of the Doppler effect, using reflection of the radiation under study from a moving reflector.
The present work is devoted to finding and analyzing the dependence \(\Delta \nu(\theta)\) for various reflector parameters, which ultimately determines the angular-frequency characteristic of the emitter.
As is known, the Doppler phenomenon in optics obeys the law
\[ \nu=\nu_0(1+VC^{-1}\cos\beta)^{-1}\cdot(1-V^2C^{-2}), \tag{1} \]
where
\(\nu\) and \(\nu_0\) are the observed optical frequencies for moving and stationary source and receiver, respectively;
\(V\) is the absolute velocity of the relative motion of the source;
\(\beta\) is the angle between the line of observation and the direction of motion of the source.
The addition due to the Doppler effect \(\Delta \nu=\nu_0-\nu\) is determined by the coefficient \(K_D=\Delta\nu\cdot\nu_0^{-1}\), which, for \(V\cdot C^{-1}\ll1\), as is true for most of the cases analyzed, will be determined by the expression, according to (1),
\[ K_D=(1+\xi^{-1})^{-1}, \tag{2} \]
where
\[ \xi = V C^{-1}\cos \beta . \]
For a plane reflector rotating at a speed \(n\), the magnitude of the linear velocity at each point of the elliptical boundary of reflection of the conical radiation surface is
\[ V=\frac{1}{30}\pi n a b\left(b^2\cos^2\varphi+a^2\sin^2\varphi\right)^{-0.5}, \tag{3} \]
where
\[ a=L\sin\theta\,\sin^{-1}(\gamma+\theta) \]
and
\[ b=L\tan\theta \]
are the semiaxes of the ellipse, \(\varphi\) is the angular coordinate of the given point of the ellipse, \(2\theta\) is the angle of the radiation cone at the apex, \(\gamma\) is the angle between the direction of radiation and the plane reflector, and \(L\) is the distance between the reflector (Fig. 1) and the emitter.
Fig. 1. For the derivation of the calculation relations
Using the laws of geometrical optics and taking into account that the direction of the velocity \(V\) at each point of the reflection ellipse is perpendicular to the “radius” \(\rho\) of the ellipse drawn to the given point, the expression for \(\cos\beta\) can be obtained in the following form (for small \(\theta\) and \(\gamma\approx45^\circ\)):
\[ \cos\beta= \frac{ L^2\sin^2\theta \pm L\cos\theta\sqrt{\rho^2-L^2\sin^2\theta} }{ \rho\cot\varphi\left(L\cos\theta\pm\sqrt{\rho^2-L^2\sin^2\theta}\right) }. \tag{4} \]
Taking \(C=3\cdot10^8\ \text{m/sec}\), after multiplying relations (3) and (4) and carrying out transformations, we find that
\[ \xi=3.49\cdot10^{-10}\cdot n\cdot L\cdot K_{\varphi}, \tag{5} \]
where
\[ K_{\varphi}=\sin\theta\cdot\tan\varphi\cdot\tan(\theta-\delta), \tag{6} \]
with
\[ \delta=\operatorname{arctg} \left[ \frac{ \sin^2\varphi\,\sin^2\theta+\cos^2\varphi\,\cos^2(\gamma-\theta) }{ \sin^2\varphi\,\cos^2\theta+\cos^2\varphi\,\sin^2(\gamma+\theta) } \right]^{0.5}. \tag{7} \]
In the formula the dimensions are \([n]=\text{rev./min.}\), \([L]=\text{m}\); and the coefficient \(K_{\varphi}\) shows how the Doppler increment changes under variations of \(\theta\) and \(\gamma\)
Fig. 2. Dependences \(K_\varphi(\varphi)\) for different \(\theta\)
over the entire reflection contour \(0 \le \varphi \le 2\pi\) of the conical radiation surface.
Figure 2 gives the dependences \(K_\varphi(\varphi)\) for different \(\theta\), which can be used in calculating or measuring the angular-frequency characteristics of conical radiation on the basis of the Doppler effect.
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