Classical Cherenkov radiation is a celebrated physics phenomenon of electromagnetic (EM) radiation stimulated by an electric charge moving with constant velocity in a three-dimensional 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 one-dimensional non-neutral 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 high-frequency limit of our generalized Lagrangian theory. Finally, we derive for the first time expressions identifying low- and high-frequency 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 one-dimensional (1D) and three-dimensional (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 plasma-like medium, or a polarizable medium similar to a dielectric. We use a Lagrangian field theory generalization8,9 of Pierce's TWT theory4 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 high-frequency 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 high-frequency cutoffs for amplification. These cutoff frequencies depend on two significant TWT parameters: (i) the ratio χ = w v ° of the phase velocity of the relevant mode of the slow wave structure (SWS) w and the velocity of the electron flow v °; 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 γ = 2 χ 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 ° is not necessary for amplification. In other words, even when w is larger than v °, the TWT modes associated with amplification always have phase velocities that are below the electron flow velocity v ° 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 plasma-like 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.

Our interest in Cherenkov radiation here is largely motivated by its intimate relation with amplification in TWTs. This relation and its importance was recognized by a number of researchers [Refs. 10, 11, 12 (Sec. 1.1–1.2, 4.4, 4.8–4.9, 7.3, 7.6; Chap. 8), and 13 (Sec. 8.2)]. The broadly treated phenomenon of Cherenkov radiation is a fascinating subject with a rich history going back to Heaviside and Sommerfeld [Refs. 14, 15 (Chap. 1 and 2), 16, 17, and 18 (Chap. 2)]. According to Frank and Tamm,19 Tamm,16,20 and Ginzburg,17 the Vavilov–Cherenkov effect in a somewhat narrow sense of the term is essentially radiation of electromagnetic (EM) waves with a continuous spectrum and specific angular distribution by an electric charge moving with constant velocity v that exceeds the phase speed of light c ( ω ) in the surrounding medium under consideration, that is
v > c ( ω ) = c / n ( ω ) ,
(1)
where n ( ω ) is the index of refraction for light at frequency ω in the medium and c is the velocity of light in vacuum. The inequality (1) plays an important role in the Cherenkov effect and we name it the velocity inequality. This radiation is very directional, and waves of a given frequency ω are emitted at a specific angle Θ to the direction of motion of the system so that
cos Θ = c ( ω ) v .
(2)
Since for real-valued Θ, the value of | cos Θ | cannot exceed 1, Eq. (2) readily implies inequality (1). Or in other words, if the velocity inequality does not hold, then (2) cannot hold either for any real Θ. The radiation has a clear-cut front, which forms a cone with the angle of opening π 2 Θ and the moving charge at its apex. This cone is analogous to the Mach cone that characterizes a shock wave generated by the motion of a supersonic source in air or other media. The velocity of the shock wave or sound plays the role of the phase velocity of light c ( ω ). Note that the emitted EM waves must carry energy, and, consequently, energy conservation demands the particle kinetic energy reduction. In other words, the Cherenkov radiation assumes energy transfer from the moving charge to EM waves.

The TWT is a vacuum electron device with a pencil-like (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 model4 (Ref. 21, Sec. I). We generalized Pierce's theory as a 1D TWT field theory in Refs. 8 and 9 adding it to space-charge 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 well-established features of Cherenkov radiation are special to 3D space, such as the Mach cone and the corresponding angle Θ defined by Eq. (2). Second, what exactly is the dielectric/polarizable medium and c ( ω ) 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 ( ω ). Third, the inequality (1), that is, v > c ( ω ), 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 ( ω ) 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.

In all of our considerations, frequency ω is assumed to be real. Let us consider a TWT eigenmode f ω ( z , t ) of frequency ω, which is always of the form
f ω ( z , t ) = a ω exp { i ( ω t k z ) } , k = k ( ω ) ,
(3)
where a ω is a complex-valued constant and k ( ω ) is a complex-valued wavenumber, and its dependence on ω is determined by the dispersion relation. If k ( ω ) is real, then f ω ( z , t ) is oscillatory, that is, harmonic in space and time. In the case where the imaginary part of k is not equal to zero [( ( k ) 0], f ω ( z , t ) exponentially grows or decays if z ± , and this is a convective instability, in other words, amplification.
Let us now introduce a complex-valued phase velocity
u = u ( ω ) = ω k ( ω )
(4)
and note that since ω is real, the real part of k ( { k }) is given by { k } = ω { 1 u }, implying
{ k } = ω { u } , { u } | u | 2 { u } .
(5)
We call { u }, defined by the second equality in Eq. (5), the “pseudo-real part” of the complex number u (see Ref. 9, Chap. 57). It turns out that { u } is of physical significance and can be identified with the wave's energy velocity u en (Ref. 9, Chap. 57). As to the representation of the Cherenkov effect in the case of the TWT, we make the following identifications: (i) v is the velocity of stationary electron flow in the TWT and (ii) the phase velocity c ( ω ) is { u ( ω ) }, defined by the second equation in Eq. (5), that is,
c ( ω ) { u ( ω ) } .
(6)
To tie quantities v and { u ( ω ) } to features of the Cherenkov effect, we proceed as follows. The common way to express amplification in the TWT quantitatively is by relating it to the so-called convective instability. The convective instability, in turn, is manifested through exponential growth in space of TWT eigenmodes f ω ( z , t ) defined by Eq. (3). Then, we can naturally quantify the rate of exponential growth of the eigenmode f ω ( z , t ) by ( k ( ω ) ). Consequently, f ω ( z , t ) is a convective instability mode if and only if
( k ( ω ) ) 0 or equivalently , ( u ( ω ) ) = ω ( k ( ω ) ) | k ( ω ) | 2 0 ,
(7)
where ( k ( ω ) ) is the imaginary part of k ( ω ) and ( u ( ω ) ) is the imaginary part of u ( ω ). Finally, our analysis shows that the following mathematical implication always holds:
( u ( ω ) ) 0 implies v > c ( ω )
(8)
or, in other words, if there is amplification, then the velocity inequality (1) that is critical to the Cherenkov effect also holds. Let us compare this fact with another implication related to the Cherenkov effect, namely, Eq. (2) implies the velocity inequality (1). This comparison suggests that the relations in Eq. (7) can be viewed as the TWT equivalent of the critical Cherenkov effect [Eq. (2)]. In other words, we may view amplification as a representation of the Cherenkov effect in a 1D field theory of a TWT. This mathematically precise characterization of the Cherenkov effect in the 1D field theory of a TWT can be strengthened further using the following physically transparent argument. The electron beam in the TWT is a flow of electrons moving with nearly constant velocity v. These electrons are moving close to the SWS, which can be viewed as a polarizable medium similar to a dielectric. This situation is known to lead to generation of EM radiation accompanied by electrons losing kinetic energy, and it is one of the well-known manifestations of the Cherenkov effect. The synchronism and electron bunching boost the amplification, and, consequently, the Cherenkov effect.

The TWT field theory we use here is a generalization of the experimentally well-tested Pierce's theory4 (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.

The TWT field theory is a linear theory constructed based on Hamilton's principle of least action. The states (configurations) of the electron beam are conceived as perturbations of the stationary flow of electrons represented by charge q = q ( z , t ) with current i ( z , t ) = t q ( z , t ). The states of the TL are charge Q = Q ( z , t ) with current I ( z , t ) = t Q ( z , t ). Then, the TWT Lagrangian L TB is defined as follows (Ref. 9, Chaps. 4 and 24):
L TB = L Tb + L B , L Tb = L 2 ( t Q ) 2 1 2 C ( z Q + b z q ) 2 , L B = 1 2 β ( t q + v ° z q ) 2 2 π σ B q 2 ,
(9)
where L B represents the Lagrangian for the electron beam and L Tb represents the Lagrangian for the TL, including the interaction with the electron beam. The parameter β in the electron beam Lagrangian L B is defined by
β = σ B 4 π R sc 2 ω p 2 = e 2 m R sc 2 σ B n ° , ω p 2 = 4 π n ° e 2 m ,
(10)
where e is the electron charge with e > 0, m is the electron mass, ω p is the electron beam plasma frequency, σ B is the area of the cross section of the electron beam, the constant R sc is the so-called plasma frequency reduction factor that accounts phenomenologically for the finite dimensions of the electron beam cylinder as well as geometric features of the SWS, and n ° is the number density of electrons. Note that the parameter v ° in the electron beam Lagrangian L B is the velocity of the stationary electron flow, and the expression t q + v ° z q is the so-called convective derivative to be expected for the “Eulerian point of view” associated with the field theory. The term 2 π σ B q 2 in the Lagrangian L B represents space-charge effects, particularly electron-to-electron repulsion (Ref. 9, Chaps. 4 and 24).
Parameters of the TL Lagrangian L Tb are as follows: (i) C > 0 and L > 0 are, respectively, its shunt capacitance and inductance per unit length; (ii) 0 < b 1 is a phenomenological parameter that couples the electron beam and the TL. Note that the electron beam-TL interaction enters the Lagrangian L Tb through the term 1 2 C ( z Q + b z q ) 2 that has shunt capacitance C as a factor associated physically with the spatial gap between the electron beam surface and the wall of the SWS. Consequently, the shunt charges z Q and z q enter the interaction term on equal footing except that the coupling factor b effectively reduces the inductive input of the electron beam current, the coupling (see Ref. 9, Chap. 3 for more details). Then, the corresponding Euler–Lagrange (EL) equations are
L t 2 Q 1 C z 2 ( Q + b q ) = 0 ,
(11)
1 β ( t + v ° z ) 2 q + 4 π σ B q b C z 2 ( Q + b q ) = 0.
(12)
To analyze the EL equations, we introduce TWT eigenmodes of the form
Q ( z , t ) = Q ̂ ( k , ω ) e i ( ω t k z ) q ( z , t ) = q ̂ ( k , ω ) e i ( ω t k z ) ,
(13)
and apply to Eqs. (11) and (12) the Fourier transform in time t and for spatial variable z. That and under the assumption that ω is fixed yield
M u ω X = 0 , X = [ Q ̂ q ̂ ] , u = ω k
(14)
for a generalized eigenvalue u, which is the phase velocity, and the corresponding eigenvector X, where M u ω is a 2 × 2 matrix of the form
M u ω = [ 1 u 2 1 w 2 b u 2 b u 2 [ 1 u 2 + 1 γ ( ω rp 2 ω 2 ( u v ° ) 2 u 2 ) ] b 2 ] ,
(15)
and the TWT principal parameter γ is defined by
γ = b 2 C β = b 2 C σ B 4 π ω rp 2 , ω rp = R sc ω p ,
(16)
where ω rp is the so-called reduced plasma frequency.
The problem of finding the generalized eigenvalue u is reduced to the characteristic equation det { M u ω } = 0, which, after elementary algebraic transformations, turns into the following characteristic equation:
D ( u , γ ) = γ w 2 u 2 + ( u v ° ) 2 u 2 = 1 ω 2 , ω = ω ω rp , u = ω k
(17)
and we refer to the function D ( u , γ ) as the characteristic function. Note that the characteristic equation in Eq. (17) encodes all the information about the dispersion relation in the form of a frequency-dependent phase velocity u ( ω ). Indeed, if we find u ( ω ), we can readily recover k ( ω ) = u ( ω ) ω, which is the dispersion relation. We refer to solutions u of the characteristic equation (17) as characteristic velocities.
A convenient dimensionless form of the characteristic equation (17) is
D ( u ˇ , γ ˇ ) = γ ˇ χ 2 u ˇ 2 + ( u ˇ 1 ) 2 u ˇ 2 = 1 ω 2 , ω = ω ω rp , γ ˇ = γ v ° 2 , u ˇ = u v ° = ω k v ° , χ = w v ° .
(18)
Not to clutter notation, we will use the same symbols for dimensionless version of the parameters and variables as for most of the time, the dimensionless nature of equations is evident.
Pierce's theory emerges from our TWT field theory as its high-frequency approximation, namely (Ref. 9, Chap. 4, 9, and 62),
D ( u ) = γ ˇ χ 2 u ˇ 2 + ( u ˇ 1 ) 2 u ˇ 2 = 0.
(19)
(It should be noted that this is the first time Pierce's theory has been derived from first principles in a rigorous, self-consistent manner.) The relation between our TWT principal parameter γ ˇ and Pierce's gain parameter C P is given by
γ ˇ = γ v ° 2 = 2 w v ° C P 3 = 2 χ C P 3 .
(20)

Our TWT field theory reveals for the first time well-defined low- and high-frequency 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 χ < 1, there exists a critical value γ Pcr > 0 of the parameter χ such that

  1. for 0 < γ < γ Pcr, all solutions u to Eq. (19) are real-valued, and there is no amplification;

  2. for γ > γ Pcr, there are exactly two different real-valued solutions u to Eq. (19) and exactly two different complex-valued solutions that are complex conjugate so that there is amplification.

Consequently, for the case when χ < 1 amplification is possible if and only if γ > γ Pcr and, if that is the case, it occurs for all frequencies.

The critical value γ Pcr and its corresponding critical value u Pcr are intimately related to Pierce's theory, and they satisfy the following relations:
u Pcr ( χ ) = χ 2 3 , γ Pcr ( χ ) = ( u 2 χ 2 ) ( u 1 ) 2 u 2 | u = u Pcr = ( 1 χ 2 3 ) 3 = ( 1 u Pcr ( χ ) ) 3 .
(21)
For the case when χ = w v ° < 1, there exist two fundamental phase velocities u ( γ , χ ), which are stationary points of the characteristic function D ( u ), satisfying the relations
u D ( u ) = 0 , D ( u ) > 0 , u = u ( γ , χ ) ,
(22)
0 < u ( γ , χ ) < χ ; χ < u + ( γ , χ ) < 1 ;
(23)
that is, u are extremum points and D ( u ) are the corresponding extreme values of the characteristic function D ( u ). Then, having u and Eq. (22) in mind, we introduce the following two frequencies:
Ω ( γ , χ ) = 1 D ( u ( γ , χ ) ) .
(24)
We refer to frequencies Ω ( γ , χ ) and Ω + ( γ , χ ) as the low- and high-frequency cutoffs for the instability, respectively (see Fig. 2). The names are justified by the fact that, if u is a characteristic velocity, then the following implication holds:
{ u } 0 Ω ( γ , χ ) < ω < Ω + ( γ , χ ) .

Figure 1 shows the fragments of the characteristic function D ( u ) with the extremum points. Figure 1(a) shows the case when γ = 0.0002 < γ Pcr 0.000 312 1 when both cutoff frequencies Ω ( γ , χ ) are finite, whereas in the case when γ = 0.002 > γ Pcr 0.000 312 1, we have Ω + ( γ , χ ) = + with the corresponding D ( u + ( γ , χ ) ) = 0 as one can see in Fig. 1(b).

FIG. 1.

Plots of fragments of the characteristic function D ( u ) with integrated instability branches for χ = 0.9 and γ = 0.002 > γ Pcr 0.000 312 1 (horizontal axis— { u }, vertical axis—D): (a) for γ = 0.0002 < γ Pcr 0.000 312 1; (b) for γ = 0.002 > γ Pcr 0.000 312 1. Solid (brown) curves represent instability branches with { u } 0, dash-dotted (blue) curves represent oscillatory branches with { u } = 0, and vertical dashed straight lines represent the asymptotes of D ( u ). The instability nodes (points of transition from stability to instability) are represented by solid (brown) square dots.

FIG. 1.

Plots of fragments of the characteristic function D ( u ) with integrated instability branches for χ = 0.9 and γ = 0.002 > γ Pcr 0.000 312 1 (horizontal axis— { u }, vertical axis—D): (a) for γ = 0.0002 < γ Pcr 0.000 312 1; (b) for γ = 0.002 > γ Pcr 0.000 312 1. Solid (brown) curves represent instability branches with { u } 0, dash-dotted (blue) curves represent oscillatory branches with { u } = 0, and vertical dashed straight lines represent the asymptotes of D ( u ). The instability nodes (points of transition from stability to instability) are represented by solid (brown) square dots.

Close modal
The dimensionless characteristic equation (18) allows to completely analyze the set of non-real characteristic velocities associated with the TWT instability and amplification. The low- and the high-frequency cutoffs' behavior as a function of γ depends significantly on whether χ < 1 or χ > 1. A summary of our analysis of the low- and the high-frequency cutoffs Ω ( γ , χ ) and Ω + ( γ , χ ) is as follows. For the case when χ = w v ° < 1, the cutoff frequencies Ω ( γ , χ ) and Ω + ( γ , χ ) satisfy the following inequalities:
Ω ( γ , χ ) < χ 1 χ < Ω + ( γ , χ ) , γ > 0 , χ < 1
(25)
and the following limit identities:
lim γ 0 Ω ( γ , χ ) = lim γ 0 Ω + ( γ , χ ) = χ 1 χ , χ < 1 ,
(26)
lim γ γ Pcr ( χ ) Ω + ( γ , χ ) = + , lim γ Ω ( γ , χ ) = 0 , χ < 1.
(27)
For the case χ = w v ° > 1, the high-frequency cutoff Ω + ( γ , χ ) is infinite for any γ, that is
Ω + ( γ , χ ) = + , χ > 1
(28)
and the following limit relations hold for the low-frequency cutoff Ω ( γ , χ ):
lim γ 0 Ω ( γ , χ ) = , lim γ Ω ( γ , χ ) = 0 , χ > 1.
(29)
Figure 2 shows the low- and high-frequency cutoffs for χ = w v ° = 0.9 illustrating, in particular, inequalities (26) and the limit relations (26) by means of the dashed (green) horizontal straight line. Figure 3 shows the fragments of the dispersion-instability graph (see the definition below), indicating the low- and the high-frequency cutoffs Ω ( γ , χ ) and Ω + ( γ , χ ), respectively.
FIG. 2.

Plot of the low-frequency cutoff Ω ( γ , χ ) (solid red curve) and the high-frequency cutoff Ω + ( γ , χ ) (dashed blue curve) for χ = w v ° = 0.9 < 1 and for 0.1 γ Pcr ( 0.9 ) < γ < 0.9 γ Pcr ( 0.9 ). Note that the high-frequency Ω + ( γ , χ ) approaches infinity as γ approaches the critical value γ Pcr ( χ ), that is lim γ γ Pcr ( χ ) Ω + ( γ , χ ) = + . The horizontal dashed (green) straight line identifies the lim γ 0 Ω ( γ , χ ) = lim γ 0 Ω + ( γ , χ ) = χ 1 χ = 9. Hence, the plot illustrates, in particular, the inequalities in Eq. (25) and the limit relations in Eq. (26).

FIG. 2.

Plot of the low-frequency cutoff Ω ( γ , χ ) (solid red curve) and the high-frequency cutoff Ω + ( γ , χ ) (dashed blue curve) for χ = w v ° = 0.9 < 1 and for 0.1 γ Pcr ( 0.9 ) < γ < 0.9 γ Pcr ( 0.9 ). Note that the high-frequency Ω + ( γ , χ ) approaches infinity as γ approaches the critical value γ Pcr ( χ ), that is lim γ γ Pcr ( χ ) Ω + ( γ , χ ) = + . The horizontal dashed (green) straight line identifies the lim γ 0 Ω ( γ , χ ) = lim γ 0 Ω + ( γ , χ ) = χ 1 χ = 9. Hence, the plot illustrates, in particular, the inequalities in Eq. (25) and the limit relations in Eq. (26).

Close modal
FIG. 3.

Fragments of the dispersion-instability graph (vertical axis—ω, horizontal axis— { k }) for χ = 0.9 , γ = 0.0002 < γ Pcr 0.000 312 1: (a) for positive phase velocities about the low-frequency cutoff; (b) for positive phase velocities about the high-frequency cutoff. Solid (brown) curves represent unstable branches with { k } 0, dash-dotted (blue) curves represent oscillatory branches with { k } = 0, and dashed straight lines represent the dispersion relations of the cold TL (blue) and cold electron beam (red). The instability nodes (points of transition from stability to instability) are represented by solid (brown) boxes. The frequencies of the instability nodes for { k } > 0 are the low- (a) and high-frequency (b) cutoffs.

FIG. 3.

Fragments of the dispersion-instability graph (vertical axis—ω, horizontal axis— { k }) for χ = 0.9 , γ = 0.0002 < γ Pcr 0.000 312 1: (a) for positive phase velocities about the low-frequency cutoff; (b) for positive phase velocities about the high-frequency cutoff. Solid (brown) curves represent unstable branches with { k } 0, dash-dotted (blue) curves represent oscillatory branches with { k } = 0, and dashed straight lines represent the dispersion relations of the cold TL (blue) and cold electron beam (red). The instability nodes (points of transition from stability to instability) are represented by solid (brown) boxes. The frequencies of the instability nodes for { k } > 0 are the low- (a) and high-frequency (b) cutoffs.

Close modal

It is useful to integrate the information about the TWT instability into the dispersion relations using the concept of a dispersion-instability graph that we developed in Ref. 9, Chap. 7. Recall that conventional dispersion relations are defined as the relations between the real-valued frequency ω and the real-valued wavenumber k associated with the relevant eigenmodes. In the case of the convective instability, frequency ω is real and wavenumber k is complex-valued. To represent the corresponding modes geometrically as points in the real ω k plane, we proceed as follows. In this case, we parameterize every mode of the TWT system uniquely by the pair ( k ( ω ) , ω ). In view of the importance to us of the mode instability, that is, when { k ( ω ) } 0, we partition all the system modes represented by pairs ( ω , k ( ω ) ) into two distinct classes—oscillatory modes and unstable ones—based on whether the wavenumber k ( ω ) is real- or complex-valued with { k ( ω ) } 0. We refer to a mode (eigenmode) of the system as an oscillatory mode if its wavenumber k ( ω ) is real-valued. We associate with such an oscillatory mode point ( ω , k ( ω ) ) in the ω 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 ( ω ) is complex-valued with a nonzero imaginary part, that is, { k ( ω ) } 0. We associate with such an unstable mode point ( ω , ( { k ( ω ) } ) in the ω 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 ( ω , k ( ω ) ) and ( ω , { k ( ω ) } ) in the ω k plane. We name this set the dispersion-instability graph. To distinguish graphically points ( ω , k ( ω ) ) associated with oscillatory modes when k ( ω ) is real-valued from points ( ω , { k ( ω ) } ) associated with unstable modes when k ( ω ) is complex-valued with { k ( ω ) } 0, we show points { k ( ω ) } = 0 in blue, whereas points with { k ( ω ) } 0 are shown in brown. We remind once again that every point ( ω , { k ( ω ) } ) with { k ( ω ) } 0 represents exactly two complex conjugate convectively unstable modes associated with ± { k ( ω ) }.

Finally, the low- and high-frequency cutoffs that we have identified for amplification in TWTs have recently been verified in particle-in-cell (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 high-frequency limit of our generalized Lagrangian theory; (2) we derive expressions for the first time that identify both low- and high-frequency 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. FA9550-20-1-0409 administered through the University of New Mexico.

The authors have no conflicts to disclose.

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).

The authors confirm that the data supporting the findings of this study are available within the article.

In this  Appendix, we provide a precise mathematical argument, proving that the velocity of any TWT eigenmode associated with amplification is strictly below the velocity of the electron flow, which is unity in dimensionless units. Suppose u is a complex-valued characteristic velocity satisfying the characteristic equation (18) and relations
{ u } > 0 , { u } 0.
(A1)
We claim then that u also satisfies the inequalities
0 < { u } < { u } | u | 2 { u } < 1.
(A2)
We call { u } in Eq. (A2) the pseudo-real part of the complex number u (see Ref. 9, Chap. 57). This turns out to be of physical significance and can be identified with the wave's energy velocity u en (Ref. 9, Chap. 57). Note first that inequality { u } < { u } always holds as long as { u } > 0. It remains to show that { u } < 1. Note also that { D ( u ˇ , γ ˇ ) } = 0 since the right-hand-side 1 ω 2 of the characteristic Eq. (18) is real-valued. After tedious but elementary algebraic transformations, equation { D ( u ˇ , γ ˇ ) } = 0 can be transformed into
( 1 { u } ) [ ( { u } + χ ) 2 + ( { u } ) 2 ] × [ ( { u } χ ) 2 + ( { u } ) 2 ] = γ ˇ | u | 4 .
(A3)
Since γ ˇ and all terms in Eq. (A3) except for 1 { u } are a priori positive, the desired inequality { u } < 1 must hold, and that completes the argument.

The physical significance of inequalities (A2) is that they assure that if (i) the characteristic velocity u corresponds to an unstable eigenmode and, consequently, { u } 0 and (ii) its real phase velocity { u } is positive, then { u } < { u } < 1, manifesting that the eigenmode velocity is always below the velocity of the electron flow, which is unity in dimensionless units.

1.
M. V.
Kuzelev
and
A. A.
Rukhadze
, “
Stimulated radiation from high-current relativistic electron beams
,”
Sov. Phys. Usp.
30
,
507
524
(
1987
).
2.
M. V.
Kuzelev
and
A. A.
Rukhadze
, “
Mechanisms of spontaneous and stimulated emission of relativistic electron beams
,” in
Problems of Theoretical Physics and Astrophysics (70th Anniversary of V.L. Ginzburg)
(
Nauka
,
Moscow
,
1989
), pp.
70
92
.
3.
M. V.
Kuzelev
and
A. A.
Rukhadze
, “
Spontaneous and stimulated emission induced by an electron, electron bunch, and electron beam in a plasma
,”
Phys. Usp.
51
,
989
1018
(
2008
).
4.
J. R.
Pierce
,
Traveling-Wave Tubes
(
D. van Nostrand
,
New York
,
1950
).
5.
T. M.
Antonsen
, Jr.
and
B.
Levush
, “
Traveling-wave tube devices with nonlinear dielectric elements
,”
IEEE Trans. Plasma Sci.
26
,
774
(
1998
).
6.
Z.
Duan
,
Y.
Gong
,
Y.
Wei
, and
W.
Wang
, “
Impact of attenuator models on computed traveling wave tube performances
,”
Phys. Plasmas
14
,
093103
(
2007
).
7.
D.
Chernin
,
T. M.
Antonsen
, Jr.
,
B.
Levush
, and
D. R.
Whaley
, “
A three-dimensional multifrequency large signal model for helix traveling wave tubes
,”
IEEE Trans. Electron Dev.
48
,
3
11
(
2001
).
8.
A.
Figotin
and
G.
Reyes
, “
Multi-transmission-line-beam interactive system
,”
J. Math. Phys.
54
,
111901
(
2013
).
9.
A.
Figotin
,
An Analytic Theory of Multi-Stream Electron Beams in Traveling Wave Tubes
(
World Scientific
,
Singapore
,
2020
).
10.
I. N.
Kartashov
,
M. V.
Kuzelev
,
A. A.
Rukhadze
, and
N.
Sepehri Javan
, “
Collective Cherenkov effect and anomalous Doppler effect in a bounded spatial region
,”
Tech. Phys.
50
,
298
307
(
2005
).
11.
S. H.
Gold
and
G. S.
Nusinovich
, “
Review of high-power microwave source research
,”
Rev. Sci. Instrum.
68
,
3945
3974
(
1997
).
12.
J.
Benford
,
J. A.
Swegle
, and
E.
Schamiloglu
,
High Power Microwaves
, 3rd ed. (
CRC Press
,
Boca Raton, FL
,
2016
).
13.
S.
Tsimring
,
Electron Beams and Microwave Vacuum Electronics
(
John Wiley & Sons
,
New York
,
2007
).
14.
E.
Fermi
, “
The ionization loss of energy in gases and in condensed materials
,”
Phys. Rev.
57
,
485
493
(
1940
).
15.
G.
Afanasiev
,
Vavilov-Cherenkov and Synchrotron Radiation
(
Springer
,
Berlin
,
2004
).
16.
I.
Tamm
, “
General characteristics of radiation emitted by systems moving with super-light velocities with some applications to plasma physics
,” in
Nobel Lecture
(
Springer
,
1958
).
17.
V.
Ginzburg
, “
Radiation from uniformly moving sources (Vavilov-Cherenkov effect, transition radiation, and some other phenomena)
,”
Acoust. Phys.
54
,
11
23
(
2005
).
18.
V. A.
Buts
,
A. N.
Levedev
, and
V. I.
Kurilko
,
The Theory of Coherent Radiation by Intense Electron Beams
(
Springer
,
Berlin
,
2006
).
19.
I.
Frank
and
I.
Tamm
, “
Coherent radiation of fast electrons in a medium
,”
Dokl. Akad. Nauk
14
,
107
113
(
1937
); available at http://web.ihep.su/owa/dbserv/hw.part2?s_c=TAMM+1937.
20.
I.
Tamm
, “
Radiation emitted by uniformly moving electrons
,”
J. Phys. USSR
1
,
439
455
(
1939
); available at https://link.springer.com/chapter/10.1007/978-3-642-74626-0_3.
21.
J. R.
Pierce
, “
Waves in electron streams and circuits
,”
Bell Syst. Tech. J.
30
,
626
651
(
1951
).
22.
R.
Barker
,
J.
Booske
,
N.
Luhmann
, and
G.
Nusinovich
,
Modern Microwave and Millimeter-Wave Power Electronics
(
John Wiley & Sons
,
New York
,
2005
).
23.
L.
Schächter
,
Beam-Wave Interaction in Periodic and Quasi-Periodic Structures
, 2nd ed. (
Springer
,
Berlin
,
2011
).
24.
A. S.
Gilmour
,
Principles of Klystrons, Traveling Wave Tubes, Magnetrons, Cross-Field Amplifiers, and Gyrotrons
(
Artech House
,
Norwood, MA
,
2011
).
25.
D. H.
Simon
,
P.
Wong
,
D.
Chernin
,
Y. Y.
Lau
,
B.
Hoff
,
P.
Zhang
,
C. F.
Dong
, and
R. M.
Gilgenbach
, “
On the evaluation of Pierce parameters C and Q in a traveling wave tube
,”
Phys. Plasmas
24
,
033114
(
2017
).
26.
K.
Rouhi
,
R.
Marosi
,
T.
Mealy
,
A.
Figotin
, and
F.
Capolino
, “
Parametric modeling of serpentine waveguide traveling wave tubes
,” arXiv:2308.03786v1 (
2023
).
27.
K. N.
Islam
and
E.
Schamiloglu
, “
Multiple electron beam generation with different energies and comparable currents from a single cathode potential for high power traveling wave tubes (TWTs)
,”
J. Appl. Phys.
131
,
044901
(
2022
).
28.
K. N.
Islam
,
L. D.
Ludeking
,
A. D.
Andreev
,
S.
Portillo
,
A. M.
Elfrgani
, and
E.
Schamiloglu
, “
Modeling and simulation of relativistic multiple electron beam generation with different energies from a single-cathode potential for high-power microwave sources
,”
IEEE Trans. Electron Dev.
69
,
1380
1388
(
2022
).
29.
E.
Schamiloglu
and
A.
Figotin
, “
Fundamental limitations of amplification in traveling wave tubes
,” (in preparation).