Classical Cherenkov radiation is a celebrated physics phenomenon of electromagnetic (EM) radiation stimulated by an electric charge moving with constant velocity in a threedimensional dielectric medium. Cherenkov radiation has a wide spectrum and a particular distribution in space similar to the Mach cone created by a supersonic source. It is also characterized by the energy transfer from the charge's kinetic energy to the EM radiation. In the case of an electron beam passing through the middle of an EM waveguide, the radiation is manifested as collective Cherenkov radiation. In this case, the electron beam can be viewed as a onedimensional nonneutral plasma, whereas the waveguide can be viewed as a slow wave structure. This collective radiation occurs, in particular, in traveling wave tubes (TWTs), and it features the energy transfer from the electron beam to the EM radiation in the waveguide. Based on a first principles Lagrangian field theory, we develop a convincing argument that the collective Cherenkov effect in TWTs is, in fact, a convective instability, that is, amplification. We also recover Pierce's theory as a highfrequency limit of our generalized Lagrangian theory. Finally, we derive for the first time expressions identifying low and highfrequency cutoffs for amplification in TWTs.
The collective Cherenkov effect is one of the fundamental mechanisms for stimulated emission of radiation from electron beams propagating in media with slow waves,^{1–3} such as in a traveling wave tube (TWT).^{4} It is well known that the mechanism of signal amplification in TWTs is based on the Cherenkov radiation effect occurring in dielectric media. Though some features of Cherenkov radiation depend on details of the dielectric environment, there is one feature that stands out as universal. This universal feature is manifested as a higher speed of the electron flow compare to the characteristic velocity in the dielectric medium.
Researchers have developed onedimensional (1D) and threedimensional (3D) theories for TWTs, where the electrons are considered as a collection of charged particles.^{5–7} In contrast, in this article, these electrons can be viewed as a plasma or plasmalike medium, or a polarizable medium similar to a dielectric. We use a Lagrangian field theory generalization^{8,9} of Pierce's TWT theory^{4} to establish as its mathematical implication that the velocity of the electron flow is always above the phase velocity of any TWT mode associated with amplification. (It should be noted that Pierce's theory is recovered from our general theory as a highfrequency limit.) Remarkably, this statement holds for any conceivable values of TWT parameters, implying that the primary condition for Cherenkov radiation is always fulfilled in our theory. The theory also yields, for the first time, explicit formulas describing the low and highfrequency cutoffs for amplification. These cutoff frequencies depend on two significant TWT parameters: (i) the ratio $ \chi = w v \xb0$ of the phase velocity of the relevant mode of the slow wave structure (SWS) w and the velocity of the electron flow $ v \xb0$; and (ii) a single parameter γ that integrates into the intensity of the electron flow and the strength of its interaction with the SWS. It turns out that $ \gamma = 2 \chi C P 3$, where $ C P$ is the Pierce gain parameter.^{4} Interestingly, our analysis shows that the commonly made assumption requiring the characteristic velocity w to be below the velocity of the electron flow $ v \xb0$ is not necessary for amplification. In other words, even when w is larger than $ v \xb0$, the TWT modes associated with amplification always have phase velocities that are below the electron flow velocity $ v \xb0$ in conformity with the primary Cherenkov radiation requirement.
Important theoretical studies of the Cherenkov effect in TWTs conducted in Ref. 10 resulted in the following significant conclusions: (i) in the electrodynamics of plasmas and plasmalike media, the collective Cherenkov effect can be classified as related to wave–wave interactions in which the energy of one of the interacting waves is negative; (ii) the collective Cherenkov effect can be treated as one of numerous electron beam instabilities; (iii) the fundamental role played by plasma collective effects for Cherenkov radiation remains virtually untouched in the case of the Cherenkov effect; (iv) the methods and terminology of the general theory of instabilities developed in plasma physics can be successfully applied to study the collective Cherenkov effect. The TWT field theory and the results we obtained here are consistent with these conclusions.
The TWT is a vacuum electron device with a pencillike (or annular) electron beam propagating on its axis. Therefore, it is quite natural to view the TWT spatially as a 1D continuum, and it is exactly what Pierce did in his model^{4} (Ref. 21, Sec. I). We generalized Pierce's theory as a 1D TWT field theory in Refs. 8 and 9 adding it to spacecharge effects. One of the goals we pursue here is to find the relation between the Cherenkov effect and amplification in TWTs as it is applicable to a 1D field theory. There are several challenges in the pursuit of this goal. First, many wellestablished features of Cherenkov radiation are special to 3D space, such as the Mach cone and the corresponding angle $\Theta $ defined by Eq. (2). Second, what exactly is the dielectric/polarizable medium and $ c \u2032 ( \omega )$ in the case of a 1D TWT field theory? To answer this, we need to expand the dielectric point of view of the TWT, and, in particular, we have to identify an analog of the phase velocity of light $ c \u2032 ( \omega )$. Third, the inequality (1), that is, $ v > c \u2032 ( \omega )$, can be viewed as a key property of the Cherenkov effect. In addition to that, it also selects the frequencies for which the Cherenkov effect can occur. The key question is whether this inequality is applicable to a 1D theory, provided the velocity $ c \u2032 ( \omega )$ is identified, and if that is the case, what can we say about frequencies for which it holds? Assume now that a satisfactory 1D version of the Cherenkov effect is somehow constructed. We want to answer our main question: what is the relation between the 1D version of the Cherenkov effect and TWT amplification? In a nutshell, our approach to establishing this relationship is as follows.
The TWT field theory we use here is a generalization of the experimentally welltested Pierce's theory^{4} (Ref. 21, Sec. I). It was introduced and studied in Ref. 9, Chap. 4. We remind the reader that the celebrated Pierce theory is a 1D theory of TWTs that accounts for the signal amplification and the energy transfer from the electron flow to microwave radiation [Refs. 13, 22 (Chap. 4), 23 (Chap. 4), and 24]. Pierce's theory assumes: (i) an idealized linear representation of the electron beam as a dynamic system; (ii) a lossless transmission line (TL) representing the relevant eigenmode of the SWS that interacts with the electron beam; (iii) the TL is spatially homogeneous with uniformly distributed shunt capacitance and serial inductance. Further modifications of Pierce's theory can be found in Ref. 25.
Our TWT field theory reveals for the first time welldefined low and highfrequency cutoffs for amplification in TWTs. The TWT field theory we present is constructed based on the principle of least action. Therefore, energy conservation and energy transfer from the electron beam to the EM radiation (represented by the state of the TL) are exact, and one may view the amplification frequency limits as fundamental.
An analysis of the characteristic equation (19) shows that when $ \chi < 1$, there exists a critical value $ \gamma Pcr > 0$ of the parameter χ such that

for $ 0 < \gamma < \gamma Pcr$, all solutions u to Eq. (19) are realvalued, and there is no amplification;

for $ \gamma > \gamma Pcr$, there are exactly two different realvalued solutions u to Eq. (19) and exactly two different complexvalued solutions that are complex conjugate so that there is amplification.
Consequently, for the case when $ \chi < 1$ amplification is possible if and only if $ \gamma > \gamma Pcr$ and, if that is the case, it occurs for all frequencies.
Figure 1 shows the fragments of the characteristic function $ D ( u )$ with the extremum points. Figure 1(a) shows the case when $ \gamma = 0.0002 < \gamma Pcr \u2245 0.000 \u2009 312 \u2009 1$ when both cutoff frequencies $ \Omega \u2213 ( \gamma , \chi )$ are finite, whereas in the case when $ \gamma = 0.002 > \gamma Pcr \u2245 0.000 \u2009 312 \u2009 1$, we have $ \Omega + ( \gamma , \chi ) = + \u221e$ with the corresponding $ D ( u + ( \gamma , \chi ) ) = 0$ as one can see in Fig. 1(b).
It is useful to integrate the information about the TWT instability into the dispersion relations using the concept of a dispersioninstability graph that we developed in Ref. 9, Chap. 7. Recall that conventional dispersion relations are defined as the relations between the realvalued frequency ω and the realvalued wavenumber k associated with the relevant eigenmodes. In the case of the convective instability, frequency ω is real and wavenumber k is complexvalued. To represent the corresponding modes geometrically as points in the real $ \omega \u2212 k$ plane, we proceed as follows. In this case, we parameterize every mode of the TWT system uniquely by the pair $ ( k ( \omega ) , \omega )$. In view of the importance to us of the mode instability, that is, when $ \u2111 { k ( \omega )} \u2260 0$, we partition all the system modes represented by pairs $ ( \omega , k ( \omega ) )$ into two distinct classes—oscillatory modes and unstable ones—based on whether the wavenumber $ k ( \omega )$ is real or complexvalued with $ \u2111 { k ( \omega )} \u2260 0$. We refer to a mode (eigenmode) of the system as an oscillatory mode if its wavenumber $ k ( \omega )$ is realvalued. We associate with such an oscillatory mode point $ ( \omega , k ( \omega ) )$ in the $ \omega \u2212 k$ plane with ω being the vertical axis and k being the horizontal one. Similarly, we refer to a mode (eigenmode) of the system as a convectively unstable mode if its wavenumber $ k = k ( \omega )$ is complexvalued with a nonzero imaginary part, that is, $ \u2111 { k ( \omega )} \u2260 0$. We associate with such an unstable mode point $ ( \omega , ( \u211c { k ( \omega )} )$ in the $ \omega \u2212 k$ plane.
Based on the aforementioned considerations, we represent the set of all oscillatory and convectively unstable modes of the system geometrically by the set of the corresponding modal points $ ( \omega , k ( \omega ) )$ and $ ( \omega , \u211c { k ( \omega )} )$ in the $ \omega \u2212 k$ plane. We name this set the dispersioninstability graph. To distinguish graphically points $ ( \omega , k ( \omega ) )$ associated with oscillatory modes when $ k ( \omega )$ is realvalued from points $ ( \omega , \u211c { k ( \omega )} )$ associated with unstable modes when $ k ( \omega )$ is complexvalued with $ \u2111 { k ( \omega )} \u2260 0$, we show points $ \u2111 { k ( \omega )} = 0$ in blue, whereas points with $ \u2111 { k ( \omega )} \u2260 0$ are shown in brown. We remind once again that every point $ ( \omega , \u211c { k ( \omega )} )$ with $ \u2111 { k ( \omega )} \u2260 0$ represents exactly two complex conjugate convectively unstable modes associated with $ \xb1 \u2111 { k ( \omega )}$.
Finally, the low and highfrequency cutoffs that we have identified for amplification in TWTs have recently been verified in particleincell (PIC) simulations.^{26} Plans for an experimental campaign to validate the theoretical and PIC simulation results are under way.^{27,28}
In conclusion, we present results from a Lagrangian field theory generalization of Pierce's TWT theory that convincingly shows that the Cherenkov effect in TWTs is a convective instability leading to amplification. The novelty of our approach is that (1) we recover Pierce's theory as a highfrequency limit of our generalized Lagrangian theory; (2) we derive expressions for the first time that identify both low and highfrequency cutoffs for amplification in TWTs (which have been verified in PIC simulations). These results can be tested in experiment and will prove valuable in designing future TWT experiments and in explaining experimental observations where TWT amplifiers transition from amplification to oscillation, as we will describe in our next publication.^{29}
This research was supported by AFOSR MURI under Grant No. FA95502010409 administered through the University of New Mexico.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Edl Schamiloglu: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Alexander Figotin: Conceptualization (equal); Formal analysis (lead); Funding acquisition (supporting); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The authors confirm that the data supporting the findings of this study are available within the article.
APPENDIX: MATHEMATICAL ARGUMENT FOR THE PRINCIPLE STATEMENT ON THE VELOCITIES
The physical significance of inequalities (A2) is that they assure that if (i) the characteristic velocity u corresponds to an unstable eigenmode and, consequently, $ \u2111 { u} \u2260 0$ and (ii) its real phase velocity $ \u211c { u}$ is positive, then $ \u211c { u} < \u211c \u2323 { u} < 1$, manifesting that the eigenmode velocity is always below the velocity of the electron flow, which is unity in dimensionless units.