A rectangular radio pulse propagation in a selectively absorbing medium

A pulse with rectangular envelope propagation, in which carrier frequency is close to the medium spectral absorption line’s one frequency, is considered. It is shown that when the signal carrier frequency is shifted relative to the spectral line centre, the primary interference and the response signal can lead to the total signal significant oscillations over time.


Introduction
In [1], general relations were obtained for a pulse propagating in a selectively absorbing medium. These results' importance is associated with the fact that their application allows one to obtain answers to general questions related to the causality principle application and the light limiting the vacuum speed principle for the information transmission in physical systems (see ). In this paper, these general relationships are used to analyse a rectangular envelope signal propagation in a selectively absorbing medium.

Methods
The final formula for the signal is a high-frequency signal with the carrier frequency is the envelope at the starting point  is the spectral line full width at a level 1/2 of the maximum).

Results and discussions
To analyse a non-zero signal duration effect in time on the exciting field, we first consider a rectangular pulse propagation, in which carrier frequency coincides with the spectral line centre frequency. Let the signal time dependence at the initial point 0 = z have the unit amplitude and duration a rectangular pulse form Then series (1) can be rewritten as ( The calculations results performed for the spectral line Lorentzian form factor using formula (3) (3). For the most part, the dashed lines merge with the solid ones; therefore, the agreement between the analytical and numerical results at the parameter values l s t  / less than or of the unity order can be considered quantitative. Let us note two curious circumstances. First, the primary signal (rectangular pulse) is distinguishable against the response signal background, and their interference does not lead to the total signal oscillations in time. This is due to both signal's complex envelopes realness and their carrier frequency identity. Second, the decrease rate in the response signal against the primary signal background is almost the same as at its end.
Until now, we have identified the signal carrier frequency and the spectral line frequency. Nevertheless, the proposed estimates can be used for a signal whose carrier frequency does not coincide with the line frequency, as long as the carrier shift is small compared to its frequency (but not necessarily small compared to the signal bandwidth). In this case, it is sufficient to simply include an additional phase factor in the expression for the signal complex envelope.
Therefore, the calculations next series (figure 2-4) was carried out to study the signal carrier frequency shift effect relative to the spectral gain line centre frequency. Let the signal time dependence at the initial point is the packet carrier frequency shift relative to the spectral line centre. Now series (1) can be rewritten as The calculations results were carried out using formula (6) with the parameters' values

Conclusion
The main conclusion that can be drawn from considering figures 2-4, is concluded that in the signal carrier frequency a shift case relative to the spectral line centre, the primary signal and the response signal interference can lead to the total signal significant oscillations over time. It can be verified that this is indeed interference by observing that temporal oscillations only occur in the area in which the primary signal and the response signal overlap (no oscillations are observed at the primary signal end). In addition, the oscillations' frequency corresponds to the primary signal and the response signal beat's time   , and their amplitude decreases under the response signal amplitude. From this viewpoint, it is not surprising that the oscillations' amplitude does not exceed either the primary signal amplitude or the response signal amplitude.