Coupled-cavity traveling wave tube (CCTWT) is a high power microwave vacuum electronic device used to amplify radio frequency signals. CCTWTs have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. Microwave-generating interactions in CCTWTs take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Operational features of a CCTWT, particularly the amplification mechanism, are similar to those of a multicavity klystron. We advance here a Lagrangian field theory of CCTWTs with the space being represented by one-dimensional continuum. The theory integrates into it the space-charge effects, including the so-called debunching (electron-to-electron repulsion). The corresponding Euler–Lagrange field equations are ordinary differential equations with coefficients varying periodically in the space. Utilizing the system periodicity, we develop instrumental features of the Floquet theory, including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include, in particular, dispersion relations and the frequency dependent gain foundational to the RF signal amplification. Serpentine (folded, corrugated) traveling wave tubes are very similar to CCTWTs, and our theory applies to them also.

We start with general principles of microwave radiation generation and amplification of RF signals (see Ref. 1, Chap. 4): “ANY generating or amplifying device converts d.c. energy into high-frequency electric field energy, and this conversion is affected by means of an electron beam. All energy exchanges between the electron beam and the alternating electric field are a result of acceleration or retardation of electrons. The kinetic energy of electrons is converted into electromagnetic energy, and vice versa. Therefore, although the mechanisms of various devices are different, in each of them, power is transferred from the constant voltage source to the alternating electromagnetic field. This is brought about in the oscillatory system by means of a density-modulated electron beam in which electrons are accelerated in the constant electric field and retarded in the alternating electric field. Density modulation of the electron beam makes it possible to retard a greater number of electrons than are accelerated by the same alternating field, thus producing the transfer of energy.”

The last sentence in the above quote underlines the critical role played by the density modulation of the electron beam (known also as “electron bunching”) in the energy transfer from the electron beam to the electromagnetic (EM) radiation.

A coupled-cavity traveling wave tube (CCTWT) shown schematically in Fig. 1 is the primary subject we pursue here. The CCTWT is a special type of traveling wave tube (TWT) that utilizes the coupled-cavity structure (CCS)8 as a slow-wave structure (SWS) (see Ref. 2, Chap. 15) (see Ref. 3, Sec. 4). The CCS commonly is a periodic linear chain of several tens of cavities coupled by coupling holes or slots and a beam tunnel (see Ref. 4, 8.7.5). The cavities can be similar to those in klystrons. As to the physical implementation, cavities are often constructed of sections of a slow-wave structure that are made resonant by suitable terminations. The quality factor of each cavity is required to be sufficiently high so that the RF field distribution in each separate cavity is substantially unaffected by the interaction with the beam.5 

FIG. 1.

Schematic representation of a coupled-cavity traveling wave tube (CCTWT) composed of a periodic array of coupled cavities (often of toroidal shape) interacting with the pencil-like electron beam. The interaction causes electron bunching and consequent amplification of the RF signal.

FIG. 1.

Schematic representation of a coupled-cavity traveling wave tube (CCTWT) composed of a periodic array of coupled cavities (often of toroidal shape) interacting with the pencil-like electron beam. The interaction causes electron bunching and consequent amplification of the RF signal.

Close modal

By its very design, the CCS is mechanically and thermally more robust than a helix, which is often used as the SWS, allowing for much greater average power, especially in short-wave bands of microwave range6,7 (see Ref. 4, Secs. 8.3, 8.7.5). Serpentine (folded, corrugated) TWTs are based on the corresponding waveguides that permit for electron interaction below the velocity of light (see Ref. 2, Secs. 15.1, 15.2). They are very similar to CCTWTs, and the results of our studies also apply to serpentine (folded, corrugated) TWTs.

A distinct and important feature of a CCTWT is that the interaction between the e-beam and the coupled-cavity structure (CCS), particularly the electron velocity modulation by a high-frequency EM field, is limited mostly to EM cavities positioned periodically along the e-beam axis. The cavity properties that are significant for an effective interaction with the e-beam are as follows (see Ref. 1, Sec. 2): “In order to be used in an electron tube, a cavity resonator must have a region with a relatively strong high-frequency field, which is polarized along the direction of electron flow. This region should, in the majority of cases, be so small that the electron transit time is less than the period of change of the field. Hollow toroidal resonators satisfy these conditions. Toroidal resonators consist of cylinders with a very prominent “bulge” in the middle.

The region in the above quote is commonly referred to as the cavity gap or just gap, and it is there the electron velocity is modulated, leading to electron bunching and consequent RF signal amplification. If v̊ is the stationary (dc) velocity of the electron flow and lg is the length of the cavity gap, then the mentioned condition of smallness of lg and the electron transit time τg=lgv̊ can be written as
lg<2πv̊ω.
(1.1)
It is a common assumption for one-dimensional models for charge-waves that the e-beam is not dense in the sense that the operational frequency ω satisfies the following (see Ref. 4, p. 277):
ωpω,
(1.2)
where ωp is the relevant plasma frequency. Then, combining inequalities (1.2) and (1.1), we obtain the following upper bound on the gap length:
lg<2πv̊ωλp=2πv̊ωp.
(1.3)
Idealized theories including the one we advance here assume that the narrow cavity gaps are just of zero width corresponding to the zero transit time of the electron (see Ref, 1, II.5) (see Ref. 9, III.3). That is, we make the following simplifying assumption:
lg=0,τg=lgv̊=0.
(1.4)

The primary subject of our studies here is the construction of one-dimensional Lagrangian field theory of a coupled-cavity traveling wave tube (CCTWT), a schematic sketch of which is shown in Fig. 1 [compare it with Fig. 11 with a schematic sketch of a multicavity klystron (MCK)]. This theory integrates into it (i) our one-dimensional Lagrangian field theory for TWTs introduced and studied in Ref. 10, Chap. 4, 24 and reviewed in Sec. II; (ii) one-dimensional Lagrangian field theory for the multicavity klystron we developed in Ref. 11 and reviewed in Sec. VIII. The theory takes into account the space-charge effects, and it applies also to serpentine (folded, corrugated) traveling wave tubes.

This paper is organized as follows. In Sec. II, we review concisely the one-dimensional Lagrangian field theory for TWTs introduced and studied in Ref. 10, Chap. 4, 24. In Sec. III, we construct the Lagrangian of the CCTWT, derive the corresponding Euler–Lagrange equations, and introduce CCTWT constitutive subsystems: coupled cavity structure and the e-beam. In Sec. IV, we use the Floquet theory to study solutions to Euler–Lagrange equations. In particular, we construct the monodromy matrix. In Sec. V, we analyze Floquet multiplies, which are solutions to characteristic equations. In Sec. VI, we construct dispersion relations and study their properties. In Sec. VII, we derive expressions for the frequency dependent gain associated with CCTWT eigenmodes. In Sec. VIII, we review concisely the one-dimensional Lagrangian field theory for multicavity klystrons developed in Ref. 11 that allows us to see some of its features in the CCTWT. In Sec. IX, we study the couple-cavity structure when it is not coupled to the e-beam. That allows us to see some of its features in properties of the CCTWT. In  Appendixes A–I, we review for the reader’s convenience a number of mathematical and physical subjects relevant to the analysis of the CCTWT. The kinetic and field points of view on the gap interaction are considered in Sec. X. The Lagrangian variational framework of our analytical theory is developed in Sec. XI. In Sec. XII, we consider special polynomials of the fourth degree and their root degeneracies that are useful for our studies of CCTWT exceptional points of degeneracy. In  Appendixes A–I, we review some mathematical and physical subject relevant to our studies.

While quoting monographs, we identify the relevant sections as follows: Reference [X,Y] refers to Section/Chapter “Y” of monograph (article) “X”, whereas [X, p. Y] refers to page “Y” of monograph (article) “X”. For instance, Ref. 12, VI.3 refers to monograph,12 Section VI.3; Ref. 12, p. 131 refers to page 131 of monograph.12 

When constructing the CCTWT Lagrangian, we use the elements of the analytic theory of TWTs developed in Refs. 10 and 13. The purpose of this section is to introduce those elements and the relevant variables of the analytic model of TWT. TWT converts the energy of the electron beam (e-beam) into the EM energy of the amplified RF signal. A schematic sketch of typical TWT is shown in Fig. 2. To facilitate the energy conversion and signal amplification, the e-beam is enclosed in the so-called slow wave structure (SWS), which supports waves that are slow enough to effectively interact with the electron flow. As a result of this interaction, the kinetic energy of electrons is converted into the EM energy stored in the field2,4 (see Ref. 14, Sec. 2.2, Ref. 15, Chap. 4). Consequently, the key operational principle of a TWT is a positive feedback interaction between the slow-wave structure and the flow of electrons. The physical mechanism of the radiation generation and its amplification is electron bunching caused by the acceleration and deceleration of electrons along the e-beam (see quotes in Sec. I).

FIG. 2.

The upper picture is a schematic representation of a traveling wave tube. The lower picture shows an RF perturbation in the form of a space-charge wave, which is amplified exponentially as it propagates through the traveling wave tube.

FIG. 2.

The upper picture is a schematic representation of a traveling wave tube. The lower picture shows an RF perturbation in the form of a space-charge wave, which is amplified exponentially as it propagates through the traveling wave tube.

Close modal

A typical TWT consists of a vacuum tube containing the e-beam that passes down the middle of an SWS, such as an RF circuit. It operates as follows. The left end of the RF circuit is fed with a low-powered RF signal to be amplified. The SWS electromagnetic field acts upon the e-beam, causing electron bunching and the formation of the so-called space-space-charge wave. In turn, the electromagnetic field generated by space-charge wave induces more current back into the RF circuit with a consequent enhancement of electron bunching. As a result, the EM field is amplified as the RF signal passes down the structure until a saturation regime is reached and a large RF signal is collected at the output. The role of the SWS is to provide slow-wave modes to match up with the velocity of electrons in the e-beam. This velocity is usually a small fraction of the speed of light. Importantly, synchronism is required for effective in-phase interaction between the SWS and the e-beam with an optimal extraction of the kinetic energy of electrons. A typical simple SWS is the helix, which reduces the speed of propagation according to its pitch. The TWT is designed so that the RF signal travels along the tube at nearly the same speed as electrons in the e-beam to facilitate effective coupling. Technical details on the designs and operation of TWTs can be found in Ref. 2 (see Ref. 14, Chap. 4).4,16 As for a rich and interesting history of traveling wave tubes, we refer the reader to Ref. 3 and the references therein.

An effective mathematical model for a TWT interacting with the e-beam was introduced by Pierce (Ref. 17, I).16 The Pierce model is one-dimensional; it accounts for wave amplification, energy extraction from the e-beam, and its conversion into microwave radiation in the TWT2,13 (Ref. 14, Chap. 4) (Ref. 15, Sec. 4).4 This model captures remarkably well significant features of the wave amplification and the beam-wave energy transfer and is still used for basic design estimates. In our paper,18 we have constructed a Lagrangian field theory by generalizing and extending the Pierce theory to the case of a possibly inhomogeneous MTL coupled to the e-beam. This work was extended to an analytic theory of multi-stream electron beams in traveling wave tubes in Ref. 10. We concisely review here this theory. According to the simplest version of the theory, an ideal TWT is represented by a single-stream electron beam (e-beam) interacting with a single transmission line (TL) just as in the Pierce model (Ref. 17, Sec. I). The main parameter describing the single-stream e-beam is e-beam intensity,
β=σB4πRsc2ωp2=e2mRsc2σBn̊,ωp2=4πn̊e2m,
(2.1)
where −e is the electron charge with e > 0, m is the electron mass, ωp is the e-beam plasma frequency, σB is the area of the cross section of the e-beam, s v̊>0 is the stationary velocity of electrons in the e-beam, and n̊ is the density of the number of electrons. The constant Rsc is the plasma frequency reduction factor that accounts phenomenologically for finite dimensions of the e-beam cylinder and geometric features of the slow-wave structure19 (Ref. 2, Sec. 9.2 and Ref. 14, Sec. 3.3.3). The frequency
ωrp=Rscωp
(2.2)
is known as the reduced plasma frequency (Ref. 2, 9.2).

Assume the Gaussian system of units of physical dimensions of a complete set of e-beam parameters, as in Tables I and II.

TABLE I.

Natural units relevant to the e-beam.

FrequencyPlasma frequencyωp=4πn̊e2m
Velocity e-beam velocity v̊ 
Wavenumber  kq=ωrpv̊=Rscωpv̊ 
Length Wavelength for kq λrp=2πv̊ωrp,ωrp=Rscωp 
Time Wave time period τ̊=2πωp 
FrequencyPlasma frequencyωp=4πn̊e2m
Velocity e-beam velocity v̊ 
Wavenumber  kq=ωrpv̊=Rscωpv̊ 
Length Wavelength for kq λrp=2πv̊ωrp,ωrp=Rscωp 
Time Wave time period τ̊=2πωp 
TABLE II.

Physical dimensions of e-beam parameters. Abbreviations: dimensionless: dim-less; p/u: per unit.

icurrentchargetime
q Charge charge 
n̊ Number of electrons p/u of volume 1length3 
λrp=2πv̊ωrp,ωrp=Rscωp The electron plasma wavelength length 
gB=σB4λrp The e-beam spatial scale length 
β=σB4πRsc2ωp2=e2mRsc2σBn̊ e-beam intensity length2time2 
β=βv̊2=πσBλrp2=4πgBλrp Dimensionless e-beam intensity dim-less 
icurrentchargetime
q Charge charge 
n̊ Number of electrons p/u of volume 1length3 
λrp=2πv̊ωrp,ωrp=Rscωp The electron plasma wavelength length 
gB=σB4λrp The e-beam spatial scale length 
β=σB4πRsc2ωp2=e2mRsc2σBn̊ e-beam intensity length2time2 
β=βv̊2=πσBλrp2=4πgBλrp Dimensionless e-beam intensity dim-less 
We would like to point to an important spatial scale related to the e-beam, namely,
λrp=2πv̊Rscωp,ωrp=Rscωp,
(2.3)
which is the distance passed by an electron for the time period 2πωrp associated with the plasma oscillations at the reduced plasma frequency ωrp. This scale is well known in the theory of klystrons and is referred to as the electron plasma wavelength (Ref. 2, 9.2). Another spatial scale related to the e-beam that arises in our analysis later on is
gB=σB4λrp,
(2.4)
and we will refer to it as e-beam spatial scale. Using these spatial scales, we obtain the following representation for the dimensionless form β′ of the e-beam intensity:
β=βv̊2=πσBλrp2=4πgBλrp.
(2.5)
As for the single transmission line, its shunt capacitance per unit of length is a real number C > 0 and its inductance per unit of length is another real number L > 0. The coupling constant 0 < b ≤ 1 is also a number; see Ref. 10, Chap. 3 for more details. The TL single characteristic velocity w and the single TL principal coefficient θ are defined by
w=1CL,θ=b2C.
(2.6)
Following Refs. 10 and 20, we assume that
0<v̊<w.
(2.7)
Following the developments in Ref. 10, we introduce the TWT principal parameter γ̄=θβ. This parameter in view of Eqs. (2.1) and (2.6) can be represented as follows:
γ=θβ=b2CσB4πRsc2ωp2=b2Ce2mRsc2σBn̊,θ=b2C,β=e2mRsc2σBn̊.
(2.8)
The TWT Lagrangian LTB in the simplest case of a single transmission line and one stream e-beam is of the following form (Ref. 10, Chap. 4, 24):
LQ,q=LTbQ,q+LBq,
(2.9)
LTb=L2tQ212CzQ+bzq2,LB=12βtq+v̊zq22πσBq2,
where b > 0 is a coupling coefficient, and
Q=Q,zQ,tQ,Q=Qz,t,q=q,zq,tq,q=qz,t,
(2.10)
where qz,t and Qz,t are charges associated, respectively, with the e-beam and the TL. The charges are defined as time integrals of the corresponding e-beam currents i(z, t) and TL current I(z, t), that is,
q(z,t)=tq(z,t)dt,.Q(z,t)=tI(z,t)dt.
(2.11)
Note that the term 2πσBq2 in the Lagrangian LB defined in Eq. (2.9) represents space-charge effects, including the so-called debunching (electron-to-electron repulsion). The corresponding Euler–Lagrange equations are represented by the following system of second-order differential equations:
Lt2QzC1zQ+bzq=0,
(2.12)
1βt+v̊z2q+4πσBqbzC1zQ+bzq=0,
(2.13)
where v̊ is the stationary velocity of electrons in the e-beam, σB is the area of the cross section of the e-beam, and β is the e-beam intensity defined by Eq. (2.8).
Following the field theory constructions in Refs. 10 and 21, we consider the total electron density N=n̊+n and V=v̊+v, where n̊ and v̊ are, respectively, densities of the electron number and the electron velocity of the stationary dc electron flow, and n=nz,t and v=vz,t are, respectively, position and time dependent ac densities of the electron number and the electron velocity of the space-charge wave field q=qz,t and the electric field E=Ez,t. To comply with the linear theory approximation, we assume n and v to be relatively small,
nn̊,vv̊.
(2.14)
We also consider the ac current density field j = j(z, t) that satisfies
j=j(z,t)=en̊v+v̊n,ten+zj=0.
(2.15)
Then, the following relations between charge density field q=qz,t and fields n=nz,t and v=vz,t hold (Ref. 10, 22.2):
zq=σBen,tq=σBj=J,
(2.16)
where J is the e-beam current. The first term of Eq. (2.16) readily implies that
n=1σBezq.
(2.17)
The relations between charge variable q = q(z, t) defined in Sec. II A, the velocity v = v(z, t), and the current associated with it Jv = Jv(z, t) are as follows (Ref. 10, 22.2):
eσBn̊v=Dtq,Jv=eσBn̊v=Dtq,Dt=t+v̊z,
(2.18)
where Dt is the so-called material time derivative. The second equation in (2.18) evidently implies that current Jv is exactly Dtq, whereas the first equation in (2.18) yields the following representations of the velocity v:
v=1eσBn̊Dtq=JveσBn̊.
(2.19)
The electric field E = E(z, t) associated with the space-charge wave satisfies the Poisson equation (Ref. 10, Sec. 22.2),
zE=4πσBzq=4πen.
(2.20)
Under the additional natural assumption that “if there is no charges there is no electric field,” that is, q̄=0 must imply E = 0, the above equation yields
E=4πσBq.̄
(2.21)
If we introduce the e-beam voltage Vb=Vbz,t=zE, then the Poisson equation [Eq. (2.20)] can be recast as
z2Vb=4πen.
(2.22)

When integrating into the mathematical model significant features of the CCTWT, we make a number of simplifying assumptions. In particular, we use the following basic assumptions of one-dimensional model of space-charge waves in velocity-modulated beams (Ref. 4, Sec. 7.6.1): (i) all quantities of interest only depend on a single space variable z; (ii) the electric field has only an z-component; (iii) there are no transverse velocities of electrons; (iv) ac values are small compared with dc values; (v) electrons have a constant dc velocity, which is much smaller than the speed of light; and (vi) electron beams are nondense. The list of preliminary assumptions of our ideal model for the CCTWT is as follows.

Assumption 1.

(ideal model of the e-beam and the TL interaction).

  • E-beam is represented by a cylinder of an infinitesimally small radius having as its axis the z axis (see Fig. 1).

  • Coupled-cavity structure (CCS) is represented mathematically by a periodic array of adjacent segments of a transmission line (TL) of length a > 0 connected by cavities at points aℓ, Z by cavities.

  • Every cavity carries shunt capacitance c0. The e-beam interacts with the CCS exclusively through the cavities located at a discrete set of equidistant points, that is, the lattice
    aZ:Z=,2,1,0,1,2,,
    (3.1)
    where a > 0, and we refer to this parameter as the CCS period or just period. The cavity width lg and the corresponding transit time τg are assumed to be zero; see Eq. (1.4) and comments above it.

The CCTWT state is described by charges q=qz,t and Q=Qz,t for, respectively, the e-beam and the TL defined as the time integrals of the corresponding currents,
Q=Qz,t=tIz,tdt,q=qz,t=tiz,tdt.
(3.2)
Since according to the formulated above assumptions the interaction should occur only at the discrete set aZ (lattice) of points embedded into one-dimensional continuum of real numbers R, some degree of singularity of functions Qz,t and qz,t is expected. As the analysis shows, it is appropriate to impose the following jump and continuity conditions on charge functions Qz,t and qz,t.

Assumption 2.
(jump-continuity of charge functions).
  • Functions Qz,t and qz,t and their time derivatives tjQz,t and tjqz,t for j = 1, 2 are continuous for all real t and z.

  • Derivatives tjQz,t, tjQz,t, zjqz,t, and zjqz,t for j = 1, 2 and the mixed derivatives ztQz,t=tzQz,t, ztqz,t=tzqz,t exist and continuous for all real t and z except for the interaction points in the lattice aZ.

  • Let for a function Fz and a real number b symbols Fb0 and Fb+0 stand for its left and right limit at b assuming their existence, that is,
    Fb±0=limzb±0Fz.
    (3.3)
Let us also denote by Fb the jump of function Fz at b, that is,
Fb=Fb+0Fb0.
(3.4)
The following right and left limits exist:
zjQa±0,t,zjqa±0,t,j=1,2,Z,
(3.5)
and these limits are continuously differentiable functions of t. The values zQa±0,t and zqa±0,t can be different, and consequently, the jumps zQa,t and zqa,t can be nonzero.

Remark 1
(physical significance of jumps). Although according to assumption 1 we neglect the widths of EM cavities, their interaction with the electron flow is represented through jumps zqa,t, which are of the direct physical significance. Indeed, the field interpretation of kinetic properties of the electron flow in Sec. X B, namely, Eq. (10.3), implies that
zqa,t=σBena,t.
(3.6)
Equations (3.6) shows that jump zqa,t up to a multiplicative constant 1σBe represents jump na,t=zqa,tσBe in the number of electron density, manifesting the electron bunching that occurs in the EM cavity centered at aℓ. In view of Eq. (10.4), we also have va,t=v̊zqa,teσBn̊, manifesting the ac electron velocity modulation in the EM cavity centered at aℓ.

The physical dimensions of quantities related to the cavities and the TL are summarized, respectively, in Tables III and IV.

TABLE III.

Physical dimensions of cavity related quantities. Abbreviations: dimensionless: dim-less.

ICurrentchargetime
Q Charge charge 
c0 Cavity capacitance length 
l0 Cavity inductance time2length 
b Coupling parameter dim-less 
ICurrentchargetime
Q Charge charge 
c0 Cavity capacitance length 
l0 Cavity inductance time2length 
b Coupling parameter dim-less 
TABLE IV.

Physical dimensions of the TL related quantities. Abbreviations: dimensionless: dim-less; p/u: per unit.

ICurrentchargetime
Q Charge charge 
C Shunt capacitance p/u of length dim-less 
L Series inductance p/u of length time2length2 
ICurrentchargetime
Q Charge charge 
C Shunt capacitance p/u of length dim-less 
L Series inductance p/u of length time2length2 
To simplify expressions, we use the following notations:
Q=Q,zQ,tQ,Q=Qz,t,q=q,zq,tq,q=qz,t,
(3.7)
x=Qq,x=Qq,zQzq,tQtq.
(3.8)
The CCTWT Lagrangian L is defined as the sum of its three components: (i) LT is the TL Lagrangian, (ii) LB is the e-beam Lagrangian; and (iii) LTB represent the TL and e-beam interaction Lagrangian. That is,
Lx=LTQ+LBq+LTBx,
(3.9)
where we used notations (3.7) and (3.8). The expressions for LT and LB are similar to the TWT Lagrangian components in Eqs. (2.9) and (2.10), namely,
LTQ=LTbb=0=L2tQ212CzQ2,LBq=12βtq+v̊zq22πσBq2,
(3.10)
where LTb is defined by Eq. (2.9) and the interaction Lagrangian LTB is defined by
LTBx==δzal02tQa212c0Qa+bqa2.
(3.11)
Parameters C and L are, respectively, distributed shunt capacitance and inductance of the TL, σB is the area of the cross section, and β is the e-beam intensity defined in Sec. II. Lagrangian LB in Eq. (2.10) represents the e-beam, and the term 2πσBq2 models the space-charge effects, including the so-called debunching (electron-to-electron repulsion). Lagrangian LTb in Eq. (2.9) integrates into it the interactions between the TL and the e-beam, whereas for the CCTWT, the Lagrangian LT corresponds to the decoupled TL. This is why we set LT to be LTb for b = 0. Note also that (i) expression (3.11) for the interaction Lagrangian LTB is limited by design to the interaction points aℓ as indicated by delta functions δza, (ii) the factors before delta functions δza are expressions similar to density LTb in Eq. (2.9) adapted to lattice aZ of discrete interaction points aℓ, and (iii) capacitance c0 is of the most significance for the interaction between the TL and the e-beam, and we refer to it as the cavity capacitance. It follows from Eqs. (3.9), (3.10), and (3.11) that L is a periodic Lagrangian of the period a.
As we derive in Sec. XI, the Euler–Lagrange (EL) equations for points z outside the lattice aZ are
Lt2QC1z2Q=0,1βt+v̊z2q+4πσBq=0,za,Z,
(3.12)
or equivalently,
z2Q1w2t2Q=0,1vt+z2q+4πβσBv̊2q=0,za,Z.
(3.13)
The EL equations at the interaction points aℓ are
Qa=0,qa=0,
(3.14)
zQa=C0t2ω02+1Qa+bqa,zqa=bβ0v̊2Qa+bqa,
(3.15)
where we make use of parameters
C0=Cc0,ω0=1l0c0,β0=βc0,
(3.16)
and jumps Qa, qa, zQa, and zqa are defined by Eq. (3.4). We refer to β0 as nodal e-beam interaction parameter. Note that Eq. (3.14) is just an acknowledgment of the continuity of charges Qz,t and qz,t at the interaction points in consistency with assumption 2. Equations (3.14) and (3.15) can be viewed as boundary conditions at interaction points that are complementary to the differential equations [Eqs. (3.12) and (3.13)].
The Euler–Lagrange differential equations [Eqs. (3.12) and (3.13)] together with the boundary conditions [(3.14) and (3.15)] form the complete set of equation describing the CCTWT evolution. Boundary conditions [(3.14) and (3.15)] can be recast into the following matrix form:
zxa=PTBxa,x=Qq,PTB=C0t2ω02+1C0bbβ0v̊2b2β0v̊2,
(3.17)
where parameters C0 and β0 are defined by Eq. (3.16). Hence, the complete set of the boundary conditions at interaction points aℓ can be concisely written as
x+a=xa,zx+a=zxa+PTBx.
(3.18)
Consequently,
Xa+0=SbXa0,Sb=I0PTBI,X=xzx,x=Qq,
(3.19)
where I is the 2 × 2 identity matrix, and in view of Eq. (3.17), we have
Sb=I0PTBI=10000100C0t2ω02+1C0b10bβ0v̊2b2β0v̊201,C0=Cc0,β0=βc0.
(3.20)

The natural units relevant to the e-beam in CCTWT are shown in Table V.

TABLE V.

Natural units relevant to the e-beam in CCTWT.

Velocitye-beam velocityv̊
Length Period a 
Length Wavelength λ̊=1k̊=v̊ωp 
Frequency Period frequency ωa=v̊a 
Frequency Plasma frequency ωp=4πn̊e2m 
Time Time of passing the period a 1ωa = av̊ 
Time Plasma oscillation time period τ̊=1ωp 
Velocitye-beam velocityv̊
Length Period a 
Length Wavelength λ̊=1k̊=v̊ωp 
Frequency Period frequency ωa=v̊a 
Frequency Plasma frequency ωp=4πn̊e2m 
Time Time of passing the period a 1ωa = av̊ 
Time Plasma oscillation time period τ̊=1ωp 
Other variables that arise in our analysis are
λrp=2πv̊ωrp,ωrp=Rscωp,χ=wv̊=1v̊CL,
(3.21)
fB=4πβσBv̊2=ωrpv̊=2πλrp,C0=Cc0,β0=βc0,
where ωrp and λrp are, respectively, the reduced plasma frequency and the electron plasma wavelength.
The dimensionless variables of importance are
z=za,z=az,t=v̊at,t=av̊t,ω=ωωa=a2πv̊ω,ω0=ω0ωa=a2πv̊ω0,
(3.22)
L=v̊2L,C=C,β=βv̊2,σB=σBa2,c0=c0a,l0=v̊2al0,
(3.23)
C0=Cc0=aCc0=aC0,β0=βc0=aβc0v̊2,fB=afB=4πβσB=aωrpv̊=2πωrpωa=2πaλrp.
(3.24)
Note also that since w = χv,
LC=v̊2w2=1χ2,aδz=δz,z=az.
(3.25)
For the convenience of the reader, we collected in Table VI all significant parameters associated with CCTWT.
TABLE VI.

The CCTWT significant parameters. Abbreviations: dimensionless: dim-less; p/u: per unit, par.: parameter. For the sake of simplicity of the notation, we often omit “prime” super-index, indicating that the dimensionless version of the relevant parameter is involved when it is clear from the context.

aThe MCK periodlength
v̊ The e-beam stationary velocity lengthtime 
ωa=2πv̊a The period frequency 1time 
ωp=4πn̊e2m The plasma frequency 1time 
λrp=2πv̊ωrp,ωrp=Rscωp The electron plasma wavelength length 
gB=σB4λrp The e-beam spatial scale length 
f=af=2πωrpωa=2πaλrp Normalized period in units of λrp2π dim-less 
n̊ The number of electrons p/u of volume 1length3 
c0, l0 The cavity capacitance, inductance length,time2length 
ω0=1l0c0 The cavity resonant frequency 1time 
β=σBRsc2ωp24π=e2Rsc2σBn̊m=πσBv̊2λrp2 The e-beam intensity length2time2 
β=βv̊2=πσBλrp2=4πgBλrp Dim-less e-beam intensity dim-less 
β0=βc0=aβc0v̊2 The first interaction par. dim-less 
Bω=B0ω2ω2ω02,B0=b2β0 The second interaction par. dim-less 
K0=B02f=b2β02f=b2σB4λrpc0=b2gBc0 The gain coefficient dim-less 
Kω=Bω2f=K0ω2ω2ω02 The gain parameter dim-less 
C0=Cc0=aCc0=aC0 The capacitance parameter dim-less 
aThe MCK periodlength
v̊ The e-beam stationary velocity lengthtime 
ωa=2πv̊a The period frequency 1time 
ωp=4πn̊e2m The plasma frequency 1time 
λrp=2πv̊ωrp,ωrp=Rscωp The electron plasma wavelength length 
gB=σB4λrp The e-beam spatial scale length 
f=af=2πωrpωa=2πaλrp Normalized period in units of λrp2π dim-less 
n̊ The number of electrons p/u of volume 1length3 
c0, l0 The cavity capacitance, inductance length,time2length 
ω0=1l0c0 The cavity resonant frequency 1time 
β=σBRsc2ωp24π=e2Rsc2σBn̊m=πσBv̊2λrp2 The e-beam intensity length2time2 
β=βv̊2=πσBλrp2=4πgBλrp Dim-less e-beam intensity dim-less 
β0=βc0=aβc0v̊2 The first interaction par. dim-less 
Bω=B0ω2ω2ω02,B0=b2β0 The second interaction par. dim-less 
K0=B02f=b2β02f=b2σB4λrpc0=b2gBc0 The gain coefficient dim-less 
Kω=Bω2f=K0ω2ω2ω02 The gain parameter dim-less 
C0=Cc0=aCc0=aC0 The capacitance parameter dim-less 
The component Lagrangians represented in dimensionless variables are as follows:
LT=L2tQ212CzQ2,LB=12βtq+zq22πσBq2,Z,
(3.26)
LTB=12c0=δzQ+bq2.
(3.27)
The corresponding EL equations are
t2Q1χ2z2Q=0,t+z2q+fB2q=0,z,fB=4πβσB=Rscωpωa,
(3.28)
zQ=C0t2ω02+1Q+bq,zq=bβ0Q+bq,Z.
(3.29)
To simplify notations, we will omit the prime symbol identifying the dimensionless variables in equations but rather simply will acknowledge their dimensionless form. Hence, we will use from now on the following dimensionless form of the EL equations [Eqs. (3.28) and (3.24)]:
t2Q1χ2z2Q=0,t+z2q+f2q=0,z,Z,f=Rscωpωa,
(3.30)
zQ=C0t2ω02+1Q+bq,zq=bβ0Q+bq,Z.
(3.31)
Equations (3.30) and (3.31) are linear partial differential equations in time and space variables. Their analysis is simplified considerably if we recast them as equations in frequency and space variable. With that in mind, we apply the Fourier transform in t (see  Appendix A) to Eqs. (3.30) and (3.31) and obtain the following equations:
z2Q̌+ω2χ2Q̌=0,ziω2q̌+f2q̌=0,z,
(3.32)
zQ=C0ω02ω2ω02Q+bq,zq=bβ0Q+bq,Z,
(3.33)
where Q̌ and q̌ are the time Fourier transform of the corresponding quantities. Equations (3.32) and (3.33) are ordinary differential equations (ODEs) in space variable z with frequency dependent coefficients.
To apply the constructions of the Floquet theory reviewed in  Appendix F, we recast the system of Eqs. (3.32) and (3.33) yet another time into the following manifestly spatially periodic vector ODE:
z2x+A1zx+A0x+=δzPTBx=0,x=Q̌q̌,
(3.34)
where 2 × 2 matrices Aj and PTB are defined by
A1=A1ω=0002iω,A0=A0ω=ω2χ200f2ω2,
(3.35)
PTB=PTBω=C0ω02ω2ω02C0bbβ0b2β0.
(3.36)
Equations (3.34)(3.36) are evidently the second-order vector ODE with spatially periodic frequency dependent singular matrix potential =δzPTBω. These equations becomes the object of our studies below.
According to  Appendix E, the second-order differential equation [Eqs. (3.34)–(3.36)] is equivalent to the first-order spatially periodic differential equation of the following form:
zX=AzX,Az=Az,ω=0IA0PzA1,X=xzx,
(3.37)
Pz=Pz,ω==δzPTBω,
where frequency dependent matrices A0, A1, and PTB satisfy Eqs. (3.35) and (3.36).
Using results of Subsection  2 of  Appendix G, we find that the spatially periodic vector ODE (3.37) is Hamiltonian with the following choice of nonsingular Hermitian matrix G=Gω:
G=G*=00i002ωC0β00iC0β0i0000iC0β000,detG=C02β02.
(3.38)
Indeed, it is an elementary exercise to verify that for each value of z matrix Az is G-skew-Hermitian, that is,
GAz+A*zG=0.
(3.39)
Then, if Φz is the matrizant of the Hamiltonian equation [Eq. (3.37)], then according to results of  Appendix G, Φz is the G-unitary matrix satisfying
Φ*zGΦz=G.
(3.40)
Consequently, its spectrum σΦz is invariant with respect to the inversion transformation ζ1ζ̄, that is, it is symmetric with respect to the unit circle,
ζσΦz1ζ̄σΦz.
(3.41)

It is instructive to take a view on the CCTWT system as a composition of its integral components, which are the coupled cavity structure (CCS) and the electron beam (e-beam). It comes as no surprise that special features of the CCS and the e-beam are manifested in fundamental properties of the CCTWT justifying their thorough analysis. This section provides the initial steps of the analysis, whereas more detailed studies of the CCS features are pursued in Sec. IX.

One way to identify the CCS and the e-beam components of the CCTWT is to use its analysis carried out in Secs. III AIII C, setting there the coupling coefficient b to be zero. With that in mind, we consider the monodromy matrix T defined by Eqs. (4.16)–(4.18) and set there b = 0. To separate variables relevant to the CCS and the e-beam, we use permutation matrix P23 defined by Eq. (4.10) and transform Tb=0 as follows:
P23Tb=0P231=TC00TB,
(3.42)
where TC and TB are 2 × 2 matrices defined by
TC=cosωχχωsinωχ1ω2ω02C0cosωχωχsinωχ1ω2ω02C0χωsinωχ+cosωχ,
(3.43)
TB=eiωcosfisinffωsinfeiωfω2f2sinffcosf+isinffωeiω.
(3.44)
Evidently, TC defined by Eq. (3.43) is the CCS monodromy matrix and TB defined by Eq. (3.44) is the e-beam monodromy matrix.
It follows from Eq. (3.43) for matrix TC that the corresponding characteristic equation detTCsI=0 for the Floquet multipliers s (see  Appendix F) is
detTCsI=s2+2WCωs+1=0,s=expik,
(3.45)
WCω=C02ωω021ωχsinωχcosω.
The real-valued function WCω in the second equation in (3.45) plays an important role in the analysis of the CCS, and its plot is depicted in Fig. 16(b). We refer to WCω as the CCS instability parameter for as we will find that it completely determines if the Floquet multipliers satisfy the instability criterion s>1.
It also follows from Eq. (3.44) for matrix TB that the corresponding characteristic equation detTBsI=0 for the Floquet multipliers s is
detTBsI=s22cosfeiωs+e2iω=0,s=expik.
(3.46)
In view of Eqs. (3.42), (3.45), and (3.46), the following factorization holds for the characteristic function of the monodromy matrix Tb=0 of the decoupled system:
detTb=0sI=s2+2WCωs+1s22cosfeiωs+e2iω.
(3.47)

The dimensionless form of the EL equations [Eqs. (3.34)–(3.36)] and their solutions can be analyzed by applying the Floquet theory reviewed in  Appendix F. To use the Floquet theory, we recast first the second-order vector ODE as the first-order vector ODE following our review on the subject in  Appendix E.

We begin with introducing expressions for (i) the characteristic scalar polynomial ATs associated with the first equation in (3.32) for the TL and (ii) the characteristic scalar polynomial ABs associated with the second equation in (3.32) for the e-beam,
ATs=ω2χ2+s2,AB=s22iωs+f2ω2.
(4.1)
Note that in the accordance with the general theory of differential equations (see  Appendixes C– E), the spectral parameter s in expressions for the characteristic polynomials ATs and ABs represents symbolically the differential operator z.
The 2 × 2 companion matrix CT of the scalar characteristic polynomial ATs (see  Appendixes C– E) is
CT=01ω2χ20=ZTiωχ00iωχZT1,ZT=iχωiχω11,x=QzQ,
(4.2)
where the columns of matrix ZT are eigenvectors of the companion matrix CT with the corresponding eigenvalues being the relevant entries of the diagonal matrix in Eq. (4.2). The expression of vector x in Eq. (4.2) clarifies the meaning of entries of relevant matrices. Consequently, the exponent expzCT that is the fundamental matrix solution to the first-order ODE associated with the first equation in (3.32) satisfies
expzCT=cosωzχχωsinωzχωχsinωχcoszωχ=ZTexpiωzχ00iωzχZT1.
(4.3)
The 2 × 2 companion matrix CB of the scalar characteristic polynomial ABs is given by
CB=01ω2f22iω=ZBiωf00iω+fZB1,ZB=iωfiω+f11,
(4.4)
x=qzq,
where the columns of matrix ZB are eigenvectors of the companion matrix CB with the corresponding eigenvalues being the relevant entries of the diagonal matrix in Eq. (4.4). The expression of vector x in Eq. (4.4) clarifies the meaning of entries of relevant matrices. Consequently, the exponent expzCB that is the fundamental matrix solution to the first-order ODE associated with the second equation in (3.32) satisfies
expzCB=1fexpiωzfcosfziωsinfzsinfzω2f2sinfzfcosfz+iωsinfz
(4.5)
=ZBexpiωfz00iω+fzZB1.
The 2 × 2 matrix characteristic polynomial ATBs of non-interacting TL and the e-beam is the following diagonal matrix polynomial:
ATBs=ATs00ABs=ω2χ2+s200s22iωs+f2ω2.
(4.6)
The 4 × 4 companion matrix of the matrix polynomial ATBs4×4 is (see  Appendixes C– E)
CTB=00100001ω2χ20000ω2f202iω,X=QqzQzq,
(4.7)
where vector X clarifies the meaning of the entries of matrix CTB.
The exponential expzCTB of the component matrix CTB is the fundamental matrix solution to the first-order ODE associated with the system of Eq. (3.32), and it satisfies
expzCTB=cosωzχ0χωsinωzχ00eiωzcosfziωfsinfz01feiωsinfzωχsinωzχ0cosωzχ001feiωzω2f2sinzf0eiωzcosfziωfsinfz.
(4.8)
In particular, for z = 1, which is the period in dimensionless variables, we get
expCTB=cosωχ0χωsinωχ00eiωcosfiωfsinf01feiωsinfωχsinωχ0cosωχ001feiωω2f2sinf0eiωcosfiωfsinf.
(4.9)
Using 4 × 4 matrix P23 that permutes the second and third coordinates, that is,
P23=1000001001000001=P231,
(4.10)
we get the following representation of expzCTB in terms and expzCT and expzCB:
expzCTB=P23expzCT00expzCBP231.
(4.11)
One can also verify the correctness of identity (4.11) by tedious straightforward evaluation.
The fundamental matrix solution expzCTB represented by Eq. (4.11) provides the solution of the relevant first-order vector ODE strictly inside the period 0,1. The complete solution on the interval 0,1 that includes boundary 1 has to account for the boundary jump conditions represented by Eq. (3.31). These boundary conditions are equivalent to
X+0=SbX0,Sb=I0PTBI,X=xzx,x=Q̌q̌,
(4.12)
where the boundary matrix Sb satisfies
Sb=I0PTBI=100001001ω2ω02C0C0b10bβ0b2β001.
(4.13)
The Floquet theory (see  Appendix F) when applied to the first-order ODE equivalent to the EL equations [Eqs. (3.34)–(3.36)] yields the following equations for the fundamental 4 × 4 matrix solution Φz:
Φz=expzCTBΦ+0,<z<+1,
(4.14)
Φ0+0=I,Φ+0=SbΦ0,Z,
where expzCTB is defined by Eq. (4.8). In particular, according to the Floquet theorem (Theorem 22), the monodromy matrix is T=Φ1+0, and in view of Eqs. (4.13) and (4.14), we have
T=Φ1+0=SbexpCTB=T11T12T21T22,
(4.15)
where expCTB is defined by Eq. (4.9) and 2 × 2 matrix blocks of T are as follows:
T11=cosωχ00cosfisinffωeiω,T12=χωsinωχ00sinfeiωf,
(4.16)
T21=1ω2ω02C0cosωχωχsinωχC0bcosfisinffωeiωβ0bcosωχβ0b2isinffωcosf+ω2f2sinffeiω,
(4.17)
T22=1ω2ω02C0χωsinωχ+cosωχC0bsinfeiωfβ0bχωsinωχisinffω+cosfβ0b2sinffeiω,
(4.18)
and the involved above dimensionless constants satisfy (see Sec. III B)
χ=wv̊=1v̊CL,C0=aCc0,β0=aβc0v̊2,f=aRscωpv̊=Rscωpωa.
(4.19)
An important special and simpler case of the monodromy matrix T defined by Eqs. (4.16)–(4.18) is when the following equations hold:
χ=1,b=1.
(4.20)
The condition χ = 1 according to Eq. (3.21) is equivalent to the equality of the phase velocity w associated with TL and the e-beam stationary flow velocity v̊. Equation b = 1 signifies the maximal coupling between the TL and e-beam at interaction points. In this case, the monodromy matrix T turns into
T1=T11T12T21T22,χ=1,b=1,
(4.21)
where
T11=cosω00cosfisinffωeiω,T12=sinωω00sinfeiωf,
(4.22)
T21=1ω2ω02C0cosωωsinωC0cosfisinffωeiωβ0cosωβ0isinffωcosf+ω2f2sinffeiω,
(4.23)
T22=1ω2ω02C0sinωω+cosωC0sinfeiωfβ0sinωωisinffω+cosfβ0sinffeiω,
(4.24)
where dimensionless constants C0, β0, and f satisfy Eq. (4.19).
We turn now to the four Floquet multipliers s, which are the eigenvalues of the CCTWT monodromy matrix T defined by Eqs. (4.15)–(4.18). Consequently, s are solutions to characteristic equation detTsI=0, which represents the following polynomial equation of order 4 (see Fig. 3):
S4+c3S3+c2S2+c̄3S+1=0,s=expik,S=seiω2=expikω2,
(5.1)
where k is the wavenumber that can be real or complex-valued and coefficients c3 and c2 satisfy
c3=2ei2ωbf+ei2ωC0χωsinωχω2ω0212cosωχ,
(5.2)
c2=2bfC0χωsinωχω2ω022cosωχ+2cosfC0χωsinωχ+2cosω,
(5.3)
where
bf=K0sinfcosf,K0=b2β02f=b2gBc0,gB=σB4λrp.
(5.4)
Note that the quantity bf in Eq. (5.4) arises also in the theory of the MCK reviewed in Sec. VIII [see Eq. (8.21)]. Note also that coefficient c2 defined by Eq. (5.3) is manifestly real. The utility of representing the Floquet multipliers s in the form s=Seiω2 is explained by the fact that Eq. (5.1) for S possesses a manifest symmetry: if S is a solution to Eq. (5.1), then 1S̄ is its solution as well. The forth-order polynomials carrying this special symmetry are considered in Sec. XII.

In what follows, to simplify analytical evaluations, we make the following assumption.

Assumption 3
(exact synchronism). To assure efficient cavity coupling, we assume the so-called exact synchronism condition, that is, χ = 1, meaning that TL velocity w exactly equals the e-beam stationary velocity v̊, namely, w=v̊. It is also convenient to choose frequency units so that ω0 = 1. Combining these two conditions, we assume
χ=1,ω0=1.
(5.5)

The CCTWT evolution is governed by spatially periodic ODEs [(3.34)–(3.37)], implying that the dispersion relations as the relations between frequency ω and wavenumber k are constructed based on the Floquet theory reviewed in  Appendix F. Specifically, in view of the relation s=expik between the Floquet multiplier s and the wave number k (see  Appendix F and Remark 25), the characteristic equations [Eqs. (5.1)–(5.4)] can be viewed as an expression of the dispersion relations between the frequency ω and the wavenumber k, and we will refer to it as the CCTWT dispersion relations or just the dispersion relations.

Under simplifying exact synchronism assumption 3, that is, χ = 1 and ω0 = 1, the dispersion relations described by Eqs. (5.1)–(5.4) turn into
S4+c3S3+c2S2+c̄3S+1=0,s=expik,S=expikω2,
(6.1)
c3=2ei2ωbf+ei2ωC0ω21ωsinω2cosω,
(6.2)
c2=2bfC0ωsinω2cosω+2cosfC0sinωω+2cosω,
(6.3)
where
bf=K0sinfcosf,K0=b2β02f=b2gBc0,gB=σB4λrp.
(6.4)
Yet another form of dispersion relations (6.1), under the same exact synchronism assumption χ = 1 and ω0 = 1, is its high-frequency form, namely,
Dω,k=D0ω,k+D1ω,kω+D2ω,kω2=0,
(6.5)
where
D0ω,k=C0sinωbf+cosωk,
(6.6)
D1ω,k=2coskcosωbf+cos2kωcosk2ω+cosωcosk,
(6.7)
D2ω,k=cosfcoskωsinωC0,
(6.8)
where bf is defined by Eq. (6.4). We refer to function Dω,k in Eq. (6.5) as the CCTWT dispersion function.

There exists a remarkability in its simplicity relations between the CCTWT dispersion function Dω,k and the dispersion functions DCω,k and DKω,k for, respectively, the CCS and the MCK systems. These relations can be verified by tedious but elementary algebraic evaluations, and they are subjects of the following theorem.

FIG. 3.

The plots of the four complex eigenvalues (the Floquet multipliers) s=Seiω2 that solve the characteristic equation [Eq. (5.1)] for the monodromy matrix T1 defined by Eqs. (4.21)–(4.24) in case when χ = 1, ω0 = 1, f = 1, and C0 = 4: (a) K0 = 4, ω = 0.28; (b) K0 = 4, ω = 0.29; and (c) K0 = 3.9, ω = 0.29. The horizontal and vertical axes represent, respectively, Rs and Is. The eigenvalues are shown by (blue) solid dots.

FIG. 3.

The plots of the four complex eigenvalues (the Floquet multipliers) s=Seiω2 that solve the characteristic equation [Eq. (5.1)] for the monodromy matrix T1 defined by Eqs. (4.21)–(4.24) in case when χ = 1, ω0 = 1, f = 1, and C0 = 4: (a) K0 = 4, ω = 0.28; (b) K0 = 4, ω = 0.29; and (c) K0 = 3.9, ω = 0.29. The horizontal and vertical axes represent, respectively, Rs and Is. The eigenvalues are shown by (blue) solid dots.

Close modal

Theorem 2
(dispersion function factorization). Let us assume that χ = ω0 = 1. Let the CCTWT, the CCS, and the MCK dispersion functions Dω,k, DCω,k, and DKω,k be defined by, respectively, Eqs. (6.1), (9.17), and (8.36). Then, the following identity holds:
Dω,kDCω,kDKω,k=K0ω2cosωcoskω21C0sinω1sinfω,
(6.9)
K0=b2β02f=b2gBc0,gB=σB4λrp.
(6.10)
In the case of the high-frequency approximation, the following identity holds:
D0ω,k=DC0ω,kDK0ω,k=C0sinωbf+cosωk.
(6.11)
The dispersion function identities [(6.9) and (6.11)] signify a very particular way the CCS and the MCK subsystems are coupled and integrated into the CCTWT system. The right-hand side of identity (6.9) can be naturally viewed as a measure of coupling between the CCS and the MCK subsystems.

Remark 3

(graphical confirmation of the dispersion factorization). The statements of Theorem 2 are well illustrated in Figs. 4(f), 5(f), 6(f), and 7 when compared with Fig. 16 for the CCS and Fig. 15 for the MCK. One can confidently identify in the CCTWT dispersion-instability graphs the patterns of the dispersion-instability graphs of its integral components—the CCS and the MCK.

FIG. 4.

The dispersion-instability graphs for the CCTWT as the gain coefficient K0 varies. In all plots, the horizontal and vertical axes represent, respectively, Rk and ω. Each of the plots shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones Rk3π,3π for χ = 1, ω0 = 1, f = 1, and C0 = 1: (a) K0 = 2, fcr ≅ 0.927 292 180, and fmax ≅ 2.034; (b) K0 = 2.25, fcr ≅ 0.837, and fmax ≅ 1.99; (c) K0 = 2.3, fcr ≅ 0.820, and fmax ≅ 1.981; (d) K0 = 2.5, fcr ≅ 0.761, and fmax ≅ 1.951; (e) K0 = 2.7, fcr ≅ 0.709, and fmax ≅ 1.926; and (f) K0 = 5, fcr ≅ 0.395, and fmax ≅ 1.768 (see Theorem 2). When Ik±ω=0, which is the case of oscillatory modes, and Rkω=kω, the corresponding branches are shown as (blue) solid curves. When Ik±ω0, that is, there is an instability, and Rk+ω=Rkω, then the corresponding branches overlap, they are shown as bold solid curves in brown color, and each point of these branches represents exactly two modes with complex-conjugate wave numbers k±.

FIG. 4.

The dispersion-instability graphs for the CCTWT as the gain coefficient K0 varies. In all plots, the horizontal and vertical axes represent, respectively, Rk and ω. Each of the plots shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones Rk3π,3π for χ = 1, ω0 = 1, f = 1, and C0 = 1: (a) K0 = 2, fcr ≅ 0.927 292 180, and fmax ≅ 2.034; (b) K0 = 2.25, fcr ≅ 0.837, and fmax ≅ 1.99; (c) K0 = 2.3, fcr ≅ 0.820, and fmax ≅ 1.981; (d) K0 = 2.5, fcr ≅ 0.761, and fmax ≅ 1.951; (e) K0 = 2.7, fcr ≅ 0.709, and fmax ≅ 1.926; and (f) K0 = 5, fcr ≅ 0.395, and fmax ≅ 1.768 (see Theorem 2). When Ik±ω=0, which is the case of oscillatory modes, and Rkω=kω, the corresponding branches are shown as (blue) solid curves. When Ik±ω0, that is, there is an instability, and Rk+ω=Rkω, then the corresponding branches overlap, they are shown as bold solid curves in brown color, and each point of these branches represents exactly two modes with complex-conjugate wave numbers k±.

Close modal
FIG. 5.

The dispersion-instability graphs for the CCTWT as the capacitance parameter C0 varies. In all plots, the horizontal and vertical axes represent, respectively, Rk and ω. Each of the plots shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones Rk3π,3π for χ = 1, ω0 = 1, f = 1, and K0 = 2, fcr ≅ 0.927 292 180, and fmax ≅ 2.034: (a) C0 = 0.7, (b) C0 = 0.3, (c) C0 = 0.15, (d) C0 = 0.1, (e) C0 = 0.05, and (f) C0 = 0.03 (see Theorem 2). When Ik±ω=0, that is, the case of oscillatory modes, and Rkω=kω, the corresponding branches are shown as (blue) solid curves. When Ik±ω0, that is, there is an instability, and Rk+ω=Rkω, then the corresponding branches overlap, they are shown as bold solid curves in brown color, and each point of these branches represents exactly two modes with complex-conjugate wave numbers k±.

FIG. 5.

The dispersion-instability graphs for the CCTWT as the capacitance parameter C0 varies. In all plots, the horizontal and vertical axes represent, respectively, Rk and ω. Each of the plots shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones Rk3π,3π for χ = 1, ω0 = 1, f = 1, and K0 = 2, fcr ≅ 0.927 292 180, and fmax ≅ 2.034: (a) C0 = 0.7, (b) C0 = 0.3, (c) C0 = 0.15, (d) C0 = 0.1, (e) C0 = 0.05, and (f) C0 = 0.03 (see Theorem 2). When Ik±ω=0, that is, the case of oscillatory modes, and Rkω=kω, the corresponding branches are shown as (blue) solid curves. When Ik±ω0, that is, there is an instability, and Rk+ω=Rkω, then the corresponding branches overlap, they are shown as bold solid curves in brown color, and each point of these branches represents exactly two modes with complex-conjugate wave numbers k±.

Close modal
FIG. 6.

The dispersion-instability graphs for the CCTWT as the normalized period f varies. In all plots, the horizontal and vertical axes represent, respectively, Rk and ω. Each of the plots shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones Rk3π,3π for χ = 1, ω0 = 1, C0 = 1, K0 = 2, fcr ≅ 0.927 292 180, and fmax ≅ 2.034: (a) f = 0.2, (b) f = 0.8, (c) f = 1.1, (d) f = 1.3, (e) f = 1.7, and (f) f = 2 (see Theorem 2). When Ik±ω=0, that is, the case of oscillatory modes, and Rkω=kω, the corresponding branches are shown as (blue) solid curves. When Ik±ω0, that is, there is an instability, and Rk+ω=Rkω, then the corresponding branches overlap, they are shown as bold solid curves in brown color, and each point of these branches represents exactly two modes with complex-conjugate wave numbers k±.

FIG. 6.

The dispersion-instability graphs for the CCTWT as the normalized period f varies. In all plots, the horizontal and vertical axes represent, respectively, Rk and ω. Each of the plots shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones Rk3π,3π for χ = 1, ω0 = 1, C0 = 1, K0 = 2, fcr ≅ 0.927 292 180, and fmax ≅ 2.034: (a) f = 0.2, (b) f = 0.8, (c) f = 1.1, (d) f = 1.3, (e) f = 1.7, and (f) f = 2 (see Theorem 2). When Ik±ω=0, that is, the case of oscillatory modes, and Rkω=kω, the corresponding branches are shown as (blue) solid curves. When Ik±ω0, that is, there is an instability, and Rk+ω=Rkω, then the corresponding branches overlap, they are shown as bold solid curves in brown color, and each point of these branches represents exactly two modes with complex-conjugate wave numbers k±.

Close modal
FIG. 7.

The dispersion-instability graphs for the CCTWT as the normalized period f varies (see Theorem 2 and Remark 3). The horizontal and vertical axes represent, respectively, Rk and ω. The plot shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones Rk3π,3π for χ = 1, ω0 = 1, C0 = 1, f ≅ 1.77, K0 = K0T = 4.95 (typical value) and, consequently, fcr ≅ 0.397, fmax ≅ 1.770. When Ik±ω=0, which is the case of oscillatory modes, and Rkω=kω, the corresponding branches are shown as (blue) solid curves. When Ik±ω0, that is, there is an instability, and Rk+ω=Rkω, then the corresponding branches overlap, they are shown as bold solid curves in brown color, and each point of these branches represents exactly two modes with complex-conjugate wave numbers k± (see Theorem 2 and Remark 3).

FIG. 7.

The dispersion-instability graphs for the CCTWT as the normalized period f varies (see Theorem 2 and Remark 3). The horizontal and vertical axes represent, respectively, Rk and ω. The plot shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones Rk3π,3π for χ = 1, ω0 = 1, C0 = 1, f ≅ 1.77, K0 = K0T = 4.95 (typical value) and, consequently, fcr ≅ 0.397, fmax ≅ 1.770. When Ik±ω=0, which is the case of oscillatory modes, and Rkω=kω, the corresponding branches are shown as (blue) solid curves. When Ik±ω0, that is, there is an instability, and Rk+ω=Rkω, then the corresponding branches overlap, they are shown as bold solid curves in brown color, and each point of these branches represents exactly two modes with complex-conjugate wave numbers k± (see Theorem 2 and Remark 3).

Close modal
Let us consider now the conventional dispersion relation, assuming that k and ω must be real numbers. Then, dividing Eq. (6.1) by S2 and carrying elementary transformations, we arrive at the following trigonometric form of the conventional dispersion relation:
cos2kω+c3coskω2+α=c22,
(6.12)
c3=c3expiα,S=expikω2,k,ωR,
where c3=c3ω, c2=c2ω, and α=argc3ω are frequency dependent parameters satisfying Eqs. (6.2)–(6.4).

As to the graphical representation of the dispersion relation, recall that conventional dispersion relations are defined as the relations between real-valued frequency ω and real-valued wavenumber k associated with the relevant eigenmodes. In the case of interest, k can be complex-valued, and to represent all system modes geometrically, we follow Ref. 10, Chap. 7. First, we parameterize every mode of the system uniquely by the pair kω,ω, where ω is its frequency and kω is its wavenumber. If kω is degenerate, it is counted a number of times according to its multiplicity. In view of the importance to us of the mode instability, that is, when Ikω0, we partition all the system modes represented by pairs kω,ω into two distinct classes—oscillatory modes and unstable ones—based on whether the wavenumber kω is real- or complex-valued with Ikω0. We refer to a mode (eigenmode) of the system as an oscillatory mode if its wavenumber kω is real-valued. We associate with such an oscillatory mode point kω,ω in the -plane, with k being the horizontal axis and ω being the vertical one. Similarly, we refer to a mode (eigenmode) of the system as a (convective) unstable mode if its wavenumber k is complex-valued with a nonzero imaginary part, that is, Ikω0. We associate with such an unstable mode point Rkω,ω in the -plane. Since we consider here only convective unstable modes, we refer to them shortly as unstable modes. Note that every point Rkω,ω is, in fact, associated with two complex conjugate system modes with ±Ikω.

Based on the above discussion, we represent the set of all oscillatory and unstable modes of the system geometrically by the set of the corresponding modal points kω,ω and Rkω,ω in the -plane. We name this set the dispersion-instability graph. To distinguish graphically points kω,ω associated oscillatory modes when kω is real-valued from points Rkω,ω associated unstable modes when kω is complex-valued with Ikω0, we show points Ikω=0 in blue color, whereas points with Ikω0 are shown in brown color. We remind once again that every point ω,Rkω with Ikω0 represents exactly two complex conjugate unstable modes associated with ±Ikω.

When Ik±ω0 and Rk+ω=Rkω, consequently, the corresponding branches overlap with each point on the segments representing two modes with complex-conjugate wave numbers k±. These branches represent exponential growth or decay in space modes and are shown in plot (c) in brown color.

We generated three sets of dispersion-instability graphs for the CCTWT shown in Figs. 46 to demonstrate their dependence on the gain coefficient K0, the capacitance parameter C0, and the normalized period f as they vary in indicated ranges. Figures 4(f), 5(f), 6(f), and 7 when compared with Fig. 16 for the CCS and Fig. 15 for the MCK clearly indicate that the CCTWT dispersion-instability graph is composed of dispersion-instability graphs of its integral components—the CCS and the MCK. The latter is important since the CCS and the MCK are significantly simpler systems compared to the original CCTWT.

Jordan eigenvector degeneracy, which is a degeneracy of the system evolution matrix when not only some eigenvalues coincide but also the corresponding eigenvectors coincide, is sometimes referred to as exceptional point of degeneracy (EPD) (Ref. 22, Sec. II.1). Our prior studies of traveling wave tubes (TWTs) in Ref. 10, Chap. 4, 7, 13, 14, 54, 55 demonstrate that TWTs always have EPDs. A particularly important class of applications of EPDs is sensing.23–27 For applications of EPDs for traveling wave tubes, see Refs. 28–32.

Applying the results of Appendix 12, particularly using the system of Eq. (12.4), we obtain the following trigonometric form of equations for exceptional points of degeneracy (EPDs):
cos2kω+c3coskω2+α=c22,2sin2kω+c3sinkω2+α=0,
(6.13)
c3=c3expiαS=expikω2,k,ωR.
Note that the first equation in (6.13) is identical to the trigonometric form (6.12) of CCTWT dispersion relations.

Figure 8 shows examples of the dispersion-instability graphs with EPDs as points, which are the points of the transition to instability. In particular, Fig. 8(c) when compared with Fig. 16 for the CCS and Fig. 15 for the MCK indicates convincingly that the components of the CCTWT dispersion-instability graph can be attributed to the dispersion-instability graphs of its integral components—the CCS and the MCK (see Theorem 2 and Remark 3).

FIG. 8.

The dispersion-instability graphs for the CCTWT showing the degeneracy (transition to instability) points as (green) diamond dots for C0 = 1, K0 = 2 and, consequently, fcr ≅ 0.927, fmax ≅ 1.770: (a) f = 0.9 < fcr, (b) f = 1.1 > fcr, and (c) f = 1.5 > fcr (see Theorem 2 and Remark 3). In all plots, the horizontal and vertical axes represent, respectively, Rk and ω. Solid (grin) diamond dots identify points of the transition from the instability to the stability, which are also EPD points.

FIG. 8.

The dispersion-instability graphs for the CCTWT showing the degeneracy (transition to instability) points as (green) diamond dots for C0 = 1, K0 = 2 and, consequently, fcr ≅ 0.927, fmax ≅ 1.770: (a) f = 0.9 < fcr, (b) f = 1.1 > fcr, and (c) f = 1.5 > fcr (see Theorem 2 and Remark 3). In all plots, the horizontal and vertical axes represent, respectively, Rk and ω. Solid (grin) diamond dots identify points of the transition from the instability to the stability, which are also EPD points.

Close modal
Based on the prior analysis, we introduce the CCTWT gain G in dB per one period as the rate of the exponential growth of the CCTWT eigenmodes associated with the Floquet multipliers s, which are the solutions to the characteristic equations [Eqs. (5.1)–(5.4)], namely,
G=Gf,ω,K0=20logsf,ω,K0.
(7.1)
Consequently, to analyze gain expression (7.1), we have to turn to the indicated characteristic equations.
In turns out that gain expression (7.1) can be significantly simplified under exact synchronism assumption 3, that is, χ = 1 and ω0 = 1. Under these conditions, the CCTWT characteristic equations [Eqs. (5.1)–(5.4)] can be recast into the following form:
P=Pf,ω,K0=s4+p3s3+p2s2+p1s+1=0,
(7.2)
where the coefficients of the CCTWT characteristic polynomial P have the following expressions:
p3=2eiωbf+C0ω21ωsinω2cosω,
(7.3)
p2=eiω2bfC0ωsinω2cosω+2cosfC0sinωω+2cosω,
(7.4)
p1=e2iωC0ω21ωsinω2cosω+2eiωbf=e2iωp̄3,
(7.5)
where
bf=K0sinfcosf,K0=b2β02f=b2gBc0,gB=σB4λrp.
(7.6)
According to Eq. (9.13), the CCS characteristic polynomial PC has the following expression:
PCω,s=s2+C0ω1ωsinω2cosωs+1=0.
(7.7)
The MCK characteristic equation [Eq. (8.26)] can be recast as follows:
PKω,s=s2+2eiω2K0sinfω2ω21cosfs+e2iω=0,
(7.8)
and we refer to PK as the MCK characteristic polynomial.
Using definitions (7.2), (7.7), and (7.8) for, respectively, the characteristic polynomials Pf,ω,K0, PCω,s, and PKω,s, one can identify their leading terms P0, PC0ω,s, and PK0 as ω → ∞, which are
P0ω,s=C0sinωss2+2bfeiωs+e2iω,
(7.9)
PC0ω,s=C0sinωs,PK0ω,s=s2+2bfeiωs+e2iω,
(7.10)
where bf is defined by Eq. (7.6).

Just as in the case of the dispersion relations that we analyzed in Sec. VI, there are simple relations between the CCTWT characteristic function Pω,s and the characteristic polynomials PCω,s and PKω,s for, respectively, the CCS and MCK systems. These relations can be verified by tedious but elementary algebraic evaluations, and they are subjects of the following theorem that relates the characteristic polynomials for CCTTX, CCS, and MCK systems.

Theorem 4
(characteristic polynomial factorization). Let us assume that χ = ω0 = 1. Let the CCTWT, the CCS, and the MCK dispersion functions Pω,s, PCω,s, and PKω,s be defined by, respectively, Eqs. (7.2), (7.7), and (7.8). Then, the following identity holds:
Pω,sPCω,sPKω,s=2K0sinfeiωsωs22cosωs+1ω21,
(7.11)
K0=b2β02f=b2gBc0,gB=σB4λrp.
(7.12)
In the case of the high-frequency approximation, the following identity holds:
P0ω,s=PC0ω,sPK0ω,s=C0sinωss2+2bfeiωs+e2iω.
(7.13)
Identities (7.11) and (7.13) represent a particular way the CCS and the MCK subsystems are coupled and integrated into the CCTWT system. The right-hand side of the identity (7.11) can be naturally viewed as a measure of coupling between the CCS and the MCK subsystems

Remark 5

(graphical confirmation of the characteristic polynomial factorization). The statements of Theorem 4 are well illustrated in Figs. 9 and 10 when compared with Fig. 17 for the CCS and Fig. 12 for the MCK. One can confidently recognize in components of the graph of the gain CCTWT the patents of the graphs for the gain of the CCS and the MCK.

FIG. 9.

Plots of gain G per one period in dB as a function of frequency ω defined by Eq. (7.1) for ω0 = 1, K0 = 2.5 and, consequently, fcr ≅ 0.761 012 7542, fmax ≅ 1.951 302 704, Gmax = 14.307 667 94 (see Sec. VIII B for the definition of the MCK quantities fcr, fmax, and Gmax): (a) f = 1.2 > fcr; (b) f = 1.95 ≈ fmax. In all plots, the horizontal and vertical axes represent, respectively, frequency ω and gain G in dB. The (brown) solid curves represent gain G as a function of frequency ω; the (blue) dashed horizontal line G = Gmax represents the maximal Gmax value of G in the high frequency limit (see Sec. VIII B); and the (green) dashed horizontal line represents the value of G in the high frequency limit for given value of f (see Sec. VIII B). The envelope of the local maxima of the gain for large values of frequency ω behaves as 20logC0ω (see captions of Fig. 17), and it is shown as a (blue) dashed curve.

FIG. 9.

Plots of gain G per one period in dB as a function of frequency ω defined by Eq. (7.1) for ω0 = 1, K0 = 2.5 and, consequently, fcr ≅ 0.761 012 7542, fmax ≅ 1.951 302 704, Gmax = 14.307 667 94 (see Sec. VIII B for the definition of the MCK quantities fcr, fmax, and Gmax): (a) f = 1.2 > fcr; (b) f = 1.95 ≈ fmax. In all plots, the horizontal and vertical axes represent, respectively, frequency ω and gain G in dB. The (brown) solid curves represent gain G as a function of frequency ω; the (blue) dashed horizontal line G = Gmax represents the maximal Gmax value of G in the high frequency limit (see Sec. VIII B); and the (green) dashed horizontal line represents the value of G in the high frequency limit for given value of f (see Sec. VIII B). The envelope of the local maxima of the gain for large values of frequency ω behaves as 20logC0ω (see captions of Fig. 17), and it is shown as a (blue) dashed curve.

Close modal
FIG. 10.

Plot of gain G per one period in dB as a function of frequency ω defined by Eq. (9.24) for ω0 = 1, K0 = K0T = 4.95 and, consequently, fcr ≅ 0.398 674 6100, fmax ≅ 1.770 133 632, Gmax = 20 (see Sec. VIII B for the definition of the MCK quantities fcr, fmax, and Gmax), and f = 1.2 > fcr. The horizontal and vertical axes represent, respectively, frequency ω and gain G in dB. The (brown) solid curves represent gain G as a function of frequency ω; the (blue) dashed horizontal line G = Gmax represents the maximal Gmax value of G in the high frequency limit (see Sec. VIII B); and the (green) dashed horizontal line represents the value of G in the high frequency limit for f = fcr (see Sec. VIII B). The envelope of the local maxima of the gain for large values of frequency ω behaves as 20logC0ω (see captions of Fig. 17), and it is shown as a (blue) dashed curve.

FIG. 10.

Plot of gain G per one period in dB as a function of frequency ω defined by Eq. (9.24) for ω0 = 1, K0 = K0T = 4.95 and, consequently, fcr ≅ 0.398 674 6100, fmax ≅ 1.770 133 632, Gmax = 20 (see Sec. VIII B for the definition of the MCK quantities fcr, fmax, and Gmax), and f = 1.2 > fcr. The horizontal and vertical axes represent, respectively, frequency ω and gain G in dB. The (brown) solid curves represent gain G as a function of frequency ω; the (blue) dashed horizontal line G = Gmax represents the maximal Gmax value of G in the high frequency limit (see Sec. VIII B); and the (green) dashed horizontal line represents the value of G in the high frequency limit for f = fcr (see Sec. VIII B). The envelope of the local maxima of the gain for large values of frequency ω behaves as 20logC0ω (see captions of Fig. 17), and it is shown as a (blue) dashed curve.

Close modal

Remark 6

(amplification in stopbands). E-beam interactions in periodic slow-wave structures were studied by Solntsev in Ref. 33. Under the condition of exact synchronism as in our assumption 3, the amplification was observed in stopbands, known also as spectral gaps in the system (oscillatory) spectrum. Our theory accounts for this general spectral phenomenon too as indicated by growing in magnitude “bumps” in Figs. 9 and 10. One can also see similar bumps in Fig. 17 for the CCS.

It is instructive to relate and compare the frequency dependent gain G per one period in dB for CCTWT defined by Eqs. (7.1), (5.1)–(5.4) with its expressions by Eq. (8.19) for the MCK gain in Sec. VIII B and Eq. (9.24) for the CCS gain GC=GCω,C0.

Usage of cavity resonators in the klystron was a revolutionary idea of Hansen and the Varians (Ref. 4, Sec. 7.1). In the pursuit of higher power and efficiency, the original design of Vairan klystrons evolves significantly over years featuring today multiple cavities and multiple electron beam (Ref. 4, Sec. 7.7). The advantages of klystrons are their high power and efficiency, potentially wide bandwidth, phase, and amplitude stability (Ref. 34, Sec. 9.1).

The construction of an analytic model for the multicavity klystron (MCK) in Ref. 11 utilizes elements of the analytic model of the traveling wave tube (TWT) introduced and studied in our monograph (Ref. 10, Chap. 4, 24); see Sec. II. Multicavity klystron, known also as cascade amplifier (Ref. 9, Sec. IIb), is composed of the e-beam interacting with a periodic array of electromagnetic cavities; see Fig. 11. Consequently, the MCK can be naturally viewed as a subsystem of the CCTWT that contributes to the properties of CCTWT.

FIG. 11.

A schematic representation of a multicavity klystron (MCK) that exploits constructive interaction between the pencil-like electron beam and an array of electromagnetic cavities (often of toroidal shape). The interaction causes electron bunching and consequent amplification of the RF signal.

FIG. 11.

A schematic representation of a multicavity klystron (MCK) that exploits constructive interaction between the pencil-like electron beam and an array of electromagnetic cavities (often of toroidal shape). The interaction causes electron bunching and consequent amplification of the RF signal.

Close modal
As to basic variables related to the e-beam and the klystron cavities, we refer the reader to Secs. II and III B and Tables II, III, and VI. The dimensionless form of L of the Lagrangians is as follows:
L=LB+LCB,LB=12βtq+zq22πσBq2,Z,
(8.1)
LCB==δzl02tQa212c0Qa+bqa2.
(8.2)
Just as we did before to simplify notations, we will omit the prime symbol in equations but rather will simply acknowledge their dimensionless form. The dimensionless form of EL equations for the MCK is
t+z2q+f2q=0,z,Z,f=2πRscωpωa=2πaλrp,
(8.3)
2Qa+ω02Qa+bqa=0,ω0=1l0c0,β0=βc0,
(8.4)
zqa=bβ0Qa+bqa,Z.
Note that term 2πσBq2 in the Lagrangian LB defined in Eq. (8.1) represents space-charge effects, including the so-called debunching (electron-to-electron repulsion).
The Fourier transform in t (see  Appendix A) of Eqs. (8.3) and (8.4) is
ziω2q̌+f2q̌=0,z,
(8.5)
subject to the boundary conditions at interaction points,
q̌a=0,zq̌a=Bωq̌aZ,
(8.6)
where q̌ is the time Fourier transform of q and Bω is an important parameter defined by
B=Bω=b2β0ω2ω2ω02=B0ω2ω2ω02,B0=b2β0=b2βc0.
(8.7)
We refer to it as cavity e-beam interaction parameter. The Fourier transform in time of Eq. (8.4) yields
Q̌a=ω02ω2ω02bq̌a,Z,
(8.8)
where Q̌ is the time Fourier transform of Q, and Eq. (8.8) was used to obtain the second equation in (8.6).
Boundary conditions (8.6) can be recast into the matrix form as follows:
Xa+0=SbXa0,Sb=10Bω1,X=q̌zq̌,Bω=b2β0ω2ω2ω02.
(8.9)
In order to use the standard form of the Floquet theory reviewed in  Appendix F, we recast the ordinary differential equation [Eq. (8.5)] with boundary (interface) conditions (8.6) as the following single second-order ordinary differential equation with singular, frequency dependent, periodic potential:
z2q̌2iωzq̌+f2ω2q̌Bωpzq̌=0,pz==δz,q̌=q̌z,
(8.10)
where the second interaction parameter Bω is defined by Eq. (8.7).
Analysis of Eq. (8.10) based on the Floquet theory (see  Appendix F) becomes now the primary subject of studies. The second-order ordinary differential equation [Eq. (8.10)] can in turn be recast into the following matrix ordinary differential equation:
zX=AKzX,AKz=AKz,ω=01ω2f2+Bωpz2iω,X=qzq,
(8.11)
Bω=b2β0ω2ω2ω02=2fK0ω2ω2ω02,pz==δz.
Note that normalized period f=2πaλrp and the MCK gain coefficient K0=b2gBc0 play particularly significant roles for MCK properties.
One can verify by straightforward evaluation that Eq. (8.10) has the Hamiltonian structure (see  Appendix G) with the following selection for the metric matrix:
GK=GK*=2ωii0,GK1=0ii2ω,detGK=1.
(8.12)
The eigenvalues are eigenvectors of metric matrix GK that are as follows:
ω+ω2+1,iω+ω2+11,ωω2+1,iω+ω2+11.
(8.13)
Using expressions (8.11) and (8.12) for, respectively, matrices Az and G, one can readily verify that Az is G-skew-Hermitian matrix, that is,
GKAKz+AK*zGK=0,
(8.14)
and that according to  Appendix G implies that system (8.11) is Hamiltonian. Consequently, according to  Appendix G, the matrizant ΦKz of the Hamiltonian system (8.11) is GK-unitary and its spectrum σΦKz is symmetric with respect to the unit circle, that is,
ΦK*zGKΦKz=G,ζσΦKz1ζ̄σΦKz.
(8.15)
The MCK monodromy matrix TK (see  Appendix F) is as follows:
TK=eiωcosfiωsincfsincfsincfω2+2iωcosf+bf2bfcosf+cos2f+1sincfiωsincfcosf2bf,
(8.16)
where
bf=Bω2sincfcosf,sincf=sinff,Bω=b2β0ω2ω2ω02.
(8.17)
We assume that the MCK normalized period f=2πaλrp, an important parameter that affects the instability, satisfies the following inequalities.

Assumption 4
(smaller MCK period). The MCK normalized period f satisfies the following bounds:
0<f=2πaλrp<π.
(8.18)

The MCK gain is defined by the following expression:
G=Gf,ω,K0=20logbf+bf21ifbf>1,0ifbf1,
(8.19)
bf=bfω=Kωsinfcosf,Kω=K0ω2ω2ω02,K0=b2β02f=b2gBc0.
Note that the following high-frequency decomposition holds for the instability parameter bf:
bfω=bf+K0ω02ω2ω02,
(8.20)
where
bf=limωbfω=K0sinfcosf,K0=b2gBc0.
(8.21)
It turns out that the high-frequency limit bf of instability parameter bf defined by Eq. (8.21) plays a significant role in the analysis of the MCK instability and its gain. In particular, there exists a unique value fcr on interval 0,π of the normalized period f such that
bfcr=1,0<fcr<π,
(8.22)
and we refer to it as the critical value and the following representation holds:
fcr=2arctan1K0, where K0=b2gBc0,gB=σB4λrp,arctan*<π2.
(8.23)
The significance of the critical value fcr is that for fcr < f < π, any ω > ω0 is an instability frequency. One can see that Fig. 12 shows the frequency dependence of the gain G and its asymptotic behavior as ω → +∞.
FIG. 12.

Plots of gain G as a function of frequency ω defined by Eq. (8.19) for ω0 = 1 and (a) K0 = 2, f = 2 > fcr with fcr ≅ 0.927 295 2180, fmax ≅ 2.034 443 936, and Gmax ≅ 12.539 258 41; (b) K0 = 1, f = 2.4 > fcr with fcr ≅ 1.570 796 327, fmax ≅ 2.356 194 491, and Gmax ≅ 7.655 513 706. In all plots, the horizontal and vertical axes represent, respectively, the frequency ω and gain G in dB. The (brown) solid curves represent gain G as a function of frequency ω; the (blue) dashed line G = Gmax represents the maximal Gmax value of G in the high frequency limit. The (green) diamond solid dots mark the values of Ωf, which is the lower frequency boundary of the instability interval.

FIG. 12.

Plots of gain G as a function of frequency ω defined by Eq. (8.19) for ω0 = 1 and (a) K0 = 2, f = 2 > fcr with fcr ≅ 0.927 295 2180, fmax ≅ 2.034 443 936, and Gmax ≅ 12.539 258 41; (b) K0 = 1, f = 2.4 > fcr with fcr ≅ 1.570 796 327, fmax ≅ 2.356 194 491, and Gmax ≅ 7.655 513 706. In all plots, the horizontal and vertical axes represent, respectively, the frequency ω and gain G in dB. The (brown) solid curves represent gain G as a function of frequency ω; the (blue) dashed line G = Gmax represents the maximal Gmax value of G in the high frequency limit. The (green) diamond solid dots mark the values of Ωf, which is the lower frequency boundary of the instability interval.

Close modal
FIG. 13.

Plots of fmax=πarctanK0 as the (brown) solid curve and fcr=2arctan1K0 as the (blue) dashed curve. The horizontal and vertical axes represent, respectively, K0 and f.

FIG. 13.

Plots of fmax=πarctanK0 as the (brown) solid curve and fcr=2arctan1K0 as the (blue) dashed curve. The horizontal and vertical axes represent, respectively, K0 and f.

Close modal
The maximal value
Gmax=20logK0+K02+1=20lnK0+K02+1ln10
(8.24)
of gain G is attained at f = fmax that satisfies (see Fig. 13)
fmax=πarctanK0,fcr<fmax<π.
(8.25)
Let s=expik, where k is the wave number of the Floquet multiplier of the monodromy matrix TK defined by Eqs. (8.16) and (8.17) (see  Appendix F and Remark 25). Then, the two Floquet multipliers s± are solutions to the characteristic equation detTKsI=0, which is11 
s=eik=eiωS:S2+2bfS+1=0,S=S±=bf±bf21,
(8.26)
readily implying
s±=eik±=eiωS±=eiωbf±bf21,bf=bfω=Kωsinfcosf,
(8.27)
Kω=K0ω2ω2ω02,K0=b2β02f=b2gBc0.
Equation (8.26) shows that parameter bf completely determines the two Floquet multipliers s± justifying its name the instability parameter. Importantly, the characteristic equation [Eq. (8.26)] can be viewed as an expression of the dispersion relations between the frequency ω and the wavenumber k, as we discuss in Sec. VI. Equations (8.26) can be readily recast as
S+2bf+S1=0,S=sexpiω=expikω,s=expik,
(8.28)
or, equivalently, as
coskω+bfω=0,bfω=Kωsinfcosf.
(8.29)
Equations (8.28) and (8.29) can be viewed as expressions of the dispersion relations between the frequency ω and the wavenumber k, and we will refer to it as the MCK dispersion relations. Dispersion relation (8.29) can be readily recast as
k±ω=ω±arccosbfω,bfω=Kωsinfcosf,
(8.30)
Kω=K0ω2ω2ω02,K0=b2β02f=b2gBc0.

Equations (8.30) in turn can be recast into an even more explicit form as stated in the following theorem.11 

Theorem 7
(MCK dispersion relations). Let s± be the MCK Floquet multipliers, that is, solutions to (8.28), and let k±ω be the corresponding complex-valued wave numbers satisfying
s±=s±ω=expik±ω.
(8.31)
Then, the following representation for k±ω holds:
k±ω=1+signbfω2π+ω+2πm±ilnbfω+bf2ω1if bf2>1,1+signbfω2π+ω+2πm±arccosbfωif bf21,mZ,
(8.32)
where 0 < f < π and
bfω=K0ω2ω2ω02sinfcosf,K0=b2β02=b2gBc0,gB=σB4λrp.
(8.33)
Requirement for Rk±ω to be in the first (main) Brillouin zone π,π effectively selects the band number m that depends on ω as follows: For any given ω > 0 and 0 < f < π, the band number mZ is determined by the requirement to satisfy the following inequalities:
π<1+signbfω2π+ω+2πmπif bf2ω>1,π<1+signbfω2π±arccosbfω+ω+2πmπif bf2ω<1.
(8.34)

Equations (8.32) for the complex-valued wave numbers k±ω represent the dispersion relations of the MCK.

Remark 8.
Note that according to expression (8.32) in Theorem 7, we have
Rk±ω=π+ω+2πm,ω0<ω<Ωf+,
(8.35)
where Ωf+ is the upper boundary of instability frequencies. Figure 15 illustrates graphically Eq. (8.35) by perfect straight lines parallel to Rk=ω in the shadowed area.

There is yet another form of the dispersion relation [(8.28) and (8.29)], which is the high-frequency form,
DKω,k=DK0ω,k+K0ω02ω2ω02=0,
(8.36)
DK0ω,k=cosωk+bf,bf=K0sinfcosf.
(8.37)
We refer to function DKω,k as the MCK dispersion function.
This form readily yields the following high-frequency approximation to the MCK dispersion relations:
cosωk+bf=0,bf=K0sinfcosf,bf1,
(8.38)
or, equivalently,
ω=k±arccosbf+2πm,bf=K0sinfcosf,bf1,mZ,
(8.39)
where inequality bf1 is necessary and sufficient for the existence of real-valued ω and k satisfying the dispersion relation.

Theorem 2 shows how the MCK dispersion function DKω,k and its high-frequency approximation DK0ω,k are integrated into the relevant dispersion functions associated with the CCTWT.

Figures 14 and 15 illustrate graphically the dispersion relations k±ω described by Eq. (8.32). The pairs of nearly straight lines above the shadowed instability zone depicted in Fig. 14 are consistent with the high-frequency approximation (8.39) to the MCK dispersion relation.

FIG. 14.

The MCK dispersion-instability plot (brown solid curves and lines) over three Brillouin zones 3π,π for K0 = 3, ω0 = 1 for which fcr ≅ 0.643 5011 088, fmax ≅ 1.892 546 882, and f = 0.5 < fcr ≅ 0.643 501 1088. The horizontal and vertical axes represent, respectively, Rk and ωω0. Two (green) solid diamond dots identify the values of Ωf and Ωf+, which are the frequency boundaries of the instability. The (brown) solid disk dots identify points of the transition from the instability to the stability, which are also EPD points. Two (brown) horizontal dashed-dotted lines ω=Ωf± identify the frequency boundaries of the instability, and the (light blue) shaded region between the lines identifies points Rk,ω of instability. The (green) dashed horizontal line ω = ω0 identifies the resonance frequency ω0. Note that the plot has a jump-discontinuity along the (green) dashed line, namely, Rk±ω jumps by π according to Eq. (8.32) as the frequency ω passes through the resonance frequency ω0 and the sign of bfω changes. The shadowed area marks points Rk,ω associated with the instability. The (blue) dashed straight lines correspond to the high frequency approximation defined by Eq. (8.39).

FIG. 14.

The MCK dispersion-instability plot (brown solid curves and lines) over three Brillouin zones 3π,π for K0 = 3, ω0 = 1 for which fcr ≅ 0.643 5011 088, fmax ≅ 1.892 546 882, and f = 0.5 < fcr ≅ 0.643 501 1088. The horizontal and vertical axes represent, respectively, Rk and ωω0. Two (green) solid diamond dots identify the values of Ωf and Ωf+, which are the frequency boundaries of the instability. The (brown) solid disk dots identify points of the transition from the instability to the stability, which are also EPD points. Two (brown) horizontal dashed-dotted lines ω=Ωf± identify the frequency boundaries of the instability, and the (light blue) shaded region between the lines identifies points Rk,ω of instability. The (green) dashed horizontal line ω = ω0 identifies the resonance frequency ω0. Note that the plot has a jump-discontinuity along the (green) dashed line, namely, Rk±ω jumps by π according to Eq. (8.32) as the frequency ω passes through the resonance frequency ω0 and the sign of bfω changes. The shadowed area marks points Rk,ω associated with the instability. The (blue) dashed straight lines correspond to the high frequency approximation defined by Eq. (8.39).

Close modal
FIG. 15.

The MCK dispersion-instability plot (solid brown curves and lines) over three Brillouin zones 3π,π for K0 = 1, ω0 = 1 for which fcr ≅ 1.570 796 327, fmax ≅ 1.892 546 882, and f = 1.569 ≅ fcr ≅ 1.570 796 327. The horizontal and vertical axes represent, respectively, Rk and ωω0. Two (green) solid diamond dots identify the values of Ωf and Ωf+, which are the frequency boundaries of the instability. The (brown) solid disk dots identify points of the transition from the instability to the stability, which are also EPD points. Two (brown) horizontal dashed-dotted lines ω=Ωf± identify the frequency boundaries of the instability, and the (light blue) shaded region between the lines identifies points Rk,ω of instability. The (green) dashed horizontal line ω = ω0 identifies the resonance frequency ω0. Note that the plot has a jump-discontinuity along the (green) dashed line, namely, Rk±ω jumps by π according to Eq. (8.32) as the frequency ω passes through the resonance frequency ω0 and the sign of bfω changes. The shadowed area marks points Rk,ω associated with the instability.

FIG. 15.

The MCK dispersion-instability plot (solid brown curves and lines) over three Brillouin zones 3π,π for K0 = 1, ω0 = 1 for which fcr ≅ 1.570 796 327, fmax ≅ 1.892 546 882, and f = 1.569 ≅ fcr ≅ 1.570 796 327. The horizontal and vertical axes represent, respectively, Rk and ωω0. Two (green) solid diamond dots identify the values of Ωf and Ωf+, which are the frequency boundaries of the instability. The (brown) solid disk dots identify points of the transition from the instability to the stability, which are also EPD points. Two (brown) horizontal dashed-dotted lines ω=Ωf± identify the frequency boundaries of the instability, and the (light blue) shaded region between the lines identifies points Rk,ω of instability. The (green) dashed horizontal line ω = ω0 identifies the resonance frequency ω0. Note that the plot has a jump-discontinuity along the (green) dashed line, namely, Rk±ω jumps by π according to Eq. (8.32) as the frequency ω passes through the resonance frequency ω0 and the sign of bfω changes. The shadowed area marks points Rk,ω associated with the instability.

Close modal
Interestingly, there is an empirical formula due to Tsimring that shows the dependence of the maximum power gain GTN on the number N of cavities in the klystron (see Ref. 4, Sec. 7.7.1, Ref. 35, Sec. 7.2.6, and Ref. 36, Chap. 16),
GTN=15+20N2dB.
(8.40)
Realistically achievable maximum amplification values, though, are smaller and are of the order of 50–70 dB. The main limiting factors are noise and self-excitation of the klystron because of parasitic feedback between cavities.

We introduce and study here basic properties of the coupled cavity structure (CCS). Since CCS is naturally an integral part of CCTWT, the knowledge of its properties would allow us to find out its contribution to the properties of CCTWT. For particular designs of coupled cavities and the way they interact with TWTs, see Ref. 34, Sec. 9.1, 9.3.3.

The Lagrangian LCQ of the CCS system can be readily obtained from the Lagrangian L of CCTWT defined by Eqs. (3.9), (3.10), and (3.11) by assuming b = 0 and omitting component LB, that is,
LCQ=L2tQ212CzQ212c0=δzaQa2.
(9.1)
Then, the corresponding EL equations [Eqs. (3.12) and (3.14)] are reduced to
Lt2QC1z2Q=0,zQa=C0t2ω02+1Qa,C0=Cc0,aZ,
(9.2)
where jumps Qa and zQa are defined by Eq. (3.4), and consequently,
Xa+0=SbXa0,Sb=10C0t2ω02+11,X=QzQ.
(9.3)
Using the same set of dimensionless variables as in Sec. III B and omitting the prime symbol for notation simplicity, we obtain the following dimensionless form of the EL equation [Eq. (9.2)]:
t2Q1χ2z2Q=0,z,zQ=C0t2ω02+1Q,Z.
(9.4)
The Fourier transform in t (see  Appendix A) of Eq. (9.4) yields
z2Q̌+ω2χ2Q̌=0,z,zQ=C01ω2ω02Q,Z,
(9.5)
where Q̌ and q̌ are the time Fourier transform of the corresponding quantities.
An alternative form of the system of Eq. (9.5) is the following second-order vector ODE with the periodic singular potential:
z2Q̌+ω2χ2Q̌+C01ω2ω02=δzQ̌=0.
(9.6)
According to  Appendix E, second-order differential equation [Eq. (9.6)] is equivalent to the first-order differential equation of the form
zX=ACzX,ACz=01ω2χ2pz0,X=Q̌zQ̌,
(9.7)
pz=C01ω2ω02=δz.
Using results of Subsection  2 of  Appendix G, we find that system (9.7) is Hamiltonian for the following choice of nonsingular Hermitian matrix G:
GC=GC*=0ii0,detGC=1.
(9.8)
In particular, it is an elementary exercise to verify that for each value of z matrix ACz is GC-skew-Hermitian, that is,
GCACz+AC*zGC=0.
(9.9)
Then, if Φz is the matrizant of the Hamiltonian equation[Eq. (9.7)], then according to results of  Appendix G, ΦCz is GC-unitary matrix,
ΦC*zGCΦCz=GC,
(9.10)
and consequently, its spectrum σΦCz is invariant with respect to the inversion transformation ζ1ζ̄, that is, it symmetric with respect to the unit circle,
ζσΦCz1ζ̄σΦCz.
(9.11)
To simplify analytic evaluations, we assume as before that assumption 3 holds.
By simplifying assumption 3 (χ = ω0 = 1), the monodromy matrix TC defined by Eq. (3.43) takes the form
TC=cosωω1sinω1ω2C0cosωωsinωcosωC0ωω1sinω.
(9.12)
The corresponding characteristic equation [Eq. (3.45)] turns into
PCω,s=s2+2WCωs+1=0,WCω=C02ω1ωsinωcosω,
(9.13)
s=expik,
where quadratic polynomial PC is referred to as the CCS characteristic polynomial and quantity WCω is the CCS instability parameter, which is depicted in Fig. 16(b).
The two Floquet multipliers s±, which are the eigenvalues of the monodromy matrix TC defined by Eq. (9.12) and, consequently, are the solutions to its characteristic equation [Eq. (9.13)], can be represented as follows:
s±=eik±=WC±WC21,WC=WCω=C02ω1ωsinωcosω.
(9.14)
Equation (9.14) implies that instability parameter WCω there completely determines the two Floquet multipliers s± justifying its name.
Importantly, the characteristic equation [Eq. (9.13)] can be viewed as an expression of the dispersion relations between the frequency ω and the wavenumber k. To obtain an explicit form of the dispersion relations for the CCS by simplifying assumption 3 (χ = ω0 = 1), we divide the characteristic equation [Eq. (9.13)] by 2s and substitute s=expik obtaining the following equations:
cosk+WCω=0,WCω=C02ω1ωsinωcosω,
(9.15)
where WCω is the principle CCS function [see Fig. 16(b)]. Alternatively, the dispersion-instability relations (9.14) can be represented in the form
k±=k±ω=ilnWC±WC21,
(9.16)
WC=WCω=C02ω1ωsinωcosω.
Dividing Eq. (9.15) by ω, we obtain the following high-frequency form of the dispersion relations for the MCK:
DCω,k=DC0ω,k+2coskcosωωC0sinωω2=0,
(9.17)
DC0ω,k=C0sinω.
We refer to function DCω,k as the CCS dispersion function.
As to the e-beam transforming characteristic equation [Eq. (3.46)] for the e-beam, the same way we obtain the following explicit form of the dispersion relations for the e-beam:
coskωcosf=0,ω=k±f.
(9.18)
Expression (9.15) for the CCS dispersion relation readily implies that its EPD frequencies are solutions to the following CCS EPD equation:
WCω=C02ω1ωsinωcosω=±1,
(9.19)
where WCω is the principle CCS function defined by the second equation in (9.13) and its plot is depicted in Fig. 16(b). Straightforward evaluations show that WCω satisfies the following equations:
WCπn=1n+1,ωWCπn=1n,n=1,2,.
(9.20)
Consequently, πn for positive integers n are CCS EPD points. Remaining sets of EPD frequencies ξm, m ≥ 1 are found by solving the CCS EPD equation [Eq. (9.19)].
FIG. 16.

The CCS for C0 = 1. (a) Dispersion-instability graph: horizontal axis is Rk and the vertical axis is ω. (b) The plot of the instability parameter WCω defined by the second equation in (9.13). The horizontal axis is Rk, and the vertical axis is W.

FIG. 16.

The CCS for C0 = 1. (a) Dispersion-instability graph: horizontal axis is Rk and the vertical axis is ω. (b) The plot of the instability parameter WCω defined by the second equation in (9.13). The horizontal axis is Rk, and the vertical axis is W.

Close modal
Instability (oscillatory spectrum) bands (intervals) are
0,ξ0,ξm,πm,m1:0<ξ0<ξ1<π,πm1<ξm<πm,m2,
(9.21)
where numbers ξm satisfy also the following equations:
WCξ0=1,WCωξ1=1,WCξm=1m1,m2.
(9.22)
Stability (oscillatory) spectrum bands (intervals) are
ξ0,ξ1,πm,ξm+1,m1:0<ξ0<ξ1<π,πm1<ξm<πm,m2.
(9.23)

Based on the prior analysis, we introduce the CCS gain GC in dB per one period as a the rate of the exponential growth of the CCS eigenmodes associated with Floquet multipliers s± defined by Eq. (9.16). More precisely, the definition is as follows.

Definition 9
(CCS gain per one period). Let s± be the CCS Floquet multipliers defined by Eq. (9.16). Then, corresponding to them, gain GC in dB per one period is defined by
GC=GCω,C0=20logs+=20logWC+WC21ifWC>1,0ifWC1,
(9.24)
WC=WCω=C02ω1ωsinωcosω.

Figure 17 shows the frequency dependence of the gain GC per one period. Growing in magnitude “bumps” in Fig. 17 indicates the presence of gain/amplification inside of stopbands, known also as spectral gaps in the system (oscillatory) spectrum, of the CCS; see Remark 6.

FIG. 17.

The plot of the CCS gain GCω,C0 per one period for C0 = 0.5. The horizontal and vertical axes represent, respectively, the frequency ω and gain G in dB. The instability frequencies ω are identified by condition GCω,C0>0. The envelope of the local maxima of the gain GCω,C0 behaves asymptotically for large values of frequency ω as 20logC0ω as it follows from Eq. (9.24). It is shown as a (blue) dashed curve.

FIG. 17.

The plot of the CCS gain GCω,C0 per one period for C0 = 0.5. The horizontal and vertical axes represent, respectively, the frequency ω and gain G in dB. The instability frequencies ω are identified by condition GCω,C0>0. The envelope of the local maxima of the gain GCω,C0 behaves asymptotically for large values of frequency ω as 20logC0ω as it follows from Eq. (9.24). It is shown as a (blue) dashed curve.

Close modal
The monodromy matrix TC defined by Eq. (9.12) and its Jordan form at ω = πn are as follows:
TC=1n01π2n2C01n1n=ZC1n101nZC1,ω=πn,
where matrix ZC is
ZC=011π2n2C01n0,
and columns of the matrix ZC are the Jordan basis of the monodromy matrix TC.
The monodromy matrix expression at EPDs is as follows:
TC=cosωω1sinωωsinωcosω1cosω+12cosω=ZC1101ZC1,WCω=1,ωπn,
where
ZC=cosω11ωsinωcosω1cosω+10,
and
TC=cosωω1sinωωcosω+12sinω2cosω=ZC1101ZC1,WCω=1,ωπn,
where
ZC=cosω+11ωcosω+12sinω0.

We compare here some of the features of our field theory with the relevant features of the kinematic/ballistic theory of the CCTWT operation. Before going into technical details, we would like to point out that from the outset, our Lagrangian field theory takes into account the space-charge forces, that is, the electron-to-electron repulsion, whereas the standard hydrokinetic analysis completely neglects them.

We briefly review here some points of the kinetic/ballistic theory. Kinematic analysis of the CCTWT operation involves (i) the electron velocity modulation in gaps of the klystron cavities, (ii) consequent electron bunching, (iii) the energy exchange between the e-beam to the EM field, and (iv) the energy transfer from the e-beam to the EM field under proper conditions and consequent RF signal amplification. The listed subjects were thoroughly studied by many scholars; see, for instance, Refs. 5, 21, and 37 (see Ref. 2, Chap. 15, Ref. 35, Sec. 7.2, Ref. 4, Secs. 6.1–6.3; 7.1–7.7, and Ref. 1, Chap. II] and the references therein. When presenting relevant to us conclusions of the studies, we mostly follow the hydrokinetic (ballistic) approach that utilizes the Eulerian (spatial) and the Lagrangian (material) descriptions (points of view) as in Ref. 4, Sec. 7.1–7.7 and Ref. 1, Chap. II. As to general aspects of the hydrokinetic approach in continua, which includes, in particular, the Eulerian and the Lagrangian descriptions, we refer the reader to Ref. 38, Secs. I.4–I.8, Ref. 39, Secs. 3.1–3.2, and Ref. 40, Sec. 1.7.

Our field theory assumes that the cavity width lg and the corresponding transit time τg are zeros; see Eq. (1.4) and assumption 1. Consequently, the most sophisticated developments of the kinetic theory dealing with cavity gaps of finite lengths are outside the scope of our studies. In our simpler case, when lg = 0 and τg = 0 following Ref. 1, Sec. II.5, we suppose that Ů is the constant accelerating voltage, so the stationary dc electron flow velocity v̊=2emŮ, where m and −e is, respectively, the electron mass and its charge. Suppose also that U1sinωt is the gap voltage. Then, based on the elementary energy conservation law, one gets
mv22=mv̊22+U1sinωt,
(10.1)
where v is the modulated velocity. Solving Eq. (10.1) for v and assuming “small signal” approximation, we obtain
vv̊1+ξ2sinωt,ξ=U1Ů1.
(10.2)
Then, following Ref. 1, Secs. II.6 and II.7, we suppose that the velocity-modulated in the cavity electron beam as described by Eq. (10.2) enters the field-free drift space beyond the gap. Then, we follow Ref. 1, Secs. II.6 and II.7: “Whilst passing through the drift space, some electrons overtake other, slower, electrons which entered the drift space earlier, and the initial distribution of charge in the beam is changed. If the drift space is long enough the initial velocity modulation can lead to substantial density modulation of the electron beam.”

In other words, according the above scenario, electron bunching takes place. More precisely, the velocity-modulated, uniformly dense electron beam becomes a density-modulated beam with nearly constant dc velocity v̊.

According the CCTWT design, all the interactions between the electron flow and the EM field occur in cavity gaps. In what follows, we use notations and results from Sec. II B. Let us consider first the action of the cavity ac EM field on the electron flow. The cavity ac EM field acts upon the e-beam by accelerating and decelerating its electrons and effectively modulating their velocities by the relatively small compared to v̊ electron velocity field v=vz,t. Therefore, as to this part of the interaction, we may view the electron density to be essentially constant n̊, whereas its ac velocity field v=vz,t is modulated by ac EM field. Consider now the action of the e-beam on the cavity ac EM field. The space charge acts upon the cavity ac EM field essentially quasi-electrostatically through relatively small ac electron number density field n=nz,t. Hence, for this part of the interaction, we may view the electron flow to be of nearly constant velocity v̊ perturbed by relatively small ac electron number density n=nz,t. Following the results of Sec. III, let us take a look at the variation of the ac electron velocity v=vz,t and ac electron number density n=nz,t in the vicinity of centers aℓ of the cavity gaps.

Note first that the action of the ac cavity EM field on the e-beam is manifested directly through a variation of the electron velocity v=vz,t in a vicinity of the gap center aℓ. The action of the e-beam on the cavity EM field is produced by the electron number density n=nz,t. As to the quantitative assessment of the variations, note that Eq. (2.17) implies that the electron velocity v=vz,t and number density n=nz,t have the following jumps na and va at the interaction points aℓ:
na,t=zqa,tσBe,va,t=v̊zqa,teσBn̊=vn̊a,tn̊,
(10.3)
readily implying
va,tv̊=na,tn̊.
(10.4)
Equations (2.15) and (10.4) in turn yield
ja,t=en̊v,t+v̊na,t=0,
(10.5)
signifying that the e-beam current density j is continuous in z at the interaction points aℓ. In view of the Poisson equation [Eq. (2.20)] and the first equation in (2.15), the following representation holds for the zEa at the interaction points aℓ:
zEa,t=4πσBzqa,t=4πena,t.
(10.6)

Note that according to Eq. (10.4), the jumps in the velocity the number density are in antiphase.

An insightful comparative analysis of “electron-wave theory” and the kinetic/ballistic theory of bunching is provided in Ref. 1, Sec. II.15.

A description of the mechanism of phase focusing as a phenomenon of oscillating space-charge waves is only a mathematical description of a process, the essence of which is as follows: The initial velocity modulation gives rise to periodic concentration and dispersion of electron space charge. The amount of bunching and the associated alternating current increase through the bunching region, provided that there are no repulsive space-charge forces affecting this process. Space-charge forces oppose the initial velocity modulation and cause additional retardation and acceleration of the electrons. Thus, the law of conservation of energy is obeyed. On the other hand, the ballistic theory is fundamentally contradictory to this.

In fact, the ballistic theory of bunching assumes that the alternating velocity acquired by the electrons in the modulator remains constant along the whole path. However, the potential energy necessarily increases after electron bunching, and so the total energy of the electron beam constantly varies, and this conflicts with the law of conservation of energy. Despite this contradiction, the ballistic theory is a good enough approximation for many of the cases met with in practice. In this case, both ballistic and electron-wave theories lead to identical results.

In agreement with the above quotation, our field theory of the space-charge wave can be viewed as effective mathematical descriptions of the underlying physical complexity involving the electron velocity and the electron number densities.

As to the energy conservation, unlike the kinetic theory, our Lagrangian field theory surely provides for that. The field theory under some conditions agrees at least with some points of the kinetic/ballistic theory as we discuss below.

The hydrokinetic point of view on our simplifying assumption that the cavity width lg and the corresponding transit time τg are zeros, see Eq. (1.4) and assumption 1, is as follows (Ref. 1, Sec. II.5):

“Let us assume further that the transit time of electrons between grids 1 and 2 is infinitesimally small, which means a physically small transit time compared with the period of oscillation of the high-frequency field. If the transit time is negligible, electrons can be considered to move through a constant (momentarily) alternating field, i.e., virtually in a static field. The electrons acquire or lose an amount of energy equal to the product of the electron charge and the momentary value of the voltage. Therefore electrons entering the space between the grids at different moments in time, with equal velocities, pass out of this space at different velocities which are determined by the momentary value of the alternating voltage. The electron beam is thus velocity modulated and has a uniform density of space charge.”

The direct link between our field theory and the hydrokinetic theory is provided by the e-beam Lagrangian LB defined by Eqs. (2.9) and (3.10),
LBq=12βDtq22πσBq2,Dt=t+v̊z.
(10.7)
Indeed, its first kinetic term 12βDtq2 involves the material time derivative, which represents an important concept of “particle” in the hydrokinetic theory. The second term 2πσBq2 in the e-beam Lagrangian accounts for the electron-to-electron repulsion, a phenomenon neglected by the standard ballistic analysis of the electron bunching.

Another link between the field and the kinetic theories comes from our analysis in Sec. X B. In view of Eqs. (10.3) and (10.4), jumps zqa,t that are explicitly allowed by the field theory represent jumps na,t and va,t related to the kinetic properties of the electron flow; see Remark 1. Namely, jump na,t manifests the electron bunching, jump va,t manifests the ac electron velocity modulation, and Eq. (10.4) relates the two of them.

We construct here the Lagrangian variational framework for our model of CCTWT. According to assumption 1, the model integrates into it quantities associated with continuum of real numbers, on one hand, and features associated with discrete points, on the other hand. The continuum features are represented by Lagrangian densities LT and LB in Eq. (3.10), whereas discrete features are represented by Lagrangian LTB in Eq. (3.11) with energies concentrated in a set of discrete points aZ. One possibility for constructing the desired Lagrangian variational framework is to apply the general approach developed in Ref. 41 when the “rigidity” condition holds. Another possibility is to directly construct the Lagrangian variational framework using some ideas from Ref. 41, and that is what we actually pursue here.

Following the standard procedures of the least action principle (Ref. 12, Sec. II.3, Ref. 42, Chap. 3, Ref. 43, Chap. 7, Ref. 44, Sec. 8.6), we start with setting up the action integral S based on the Lagrangian L defined by Eqs. (3.9)–(3.11). Using notations (3.7) and (3.8), we define the action integral S as follows:
Sx=t0t1dtz1z2Lxdz=STQ+SBq+STBx,t0<t1,z0<z1,
(11.1)
where
STQ=t0t1dtz1z2LTQdz=t0t1dtz1z2L2tQ212CzQ2dz,
(11.2)
SBq=t0t1dtz1z2LBqdz=t0t1dtz1z212βtq+v̊zq22πσBq2dz,
(11.3)
STBQ=t0t1dtz1z2LTBQ,qdz
(11.4)
=z1<a<z2t0t1dtl02tQa212c0Qa+bqa2.
To make expressions of the action integrals less cluttered, we suppress notationally their dependence on intervals z0,z1 and t0,t1 that can be chosen arbitrarily. We consider then variation δS of action S, assuming that variations δQ and δq of charges q=qz,t and Q=Qz,t vanish outside intervals z0,z1 and t0,t1, that is,
δQz,t=δqz,t=0,z,tz0,z1×t0,t1,
(11.5)
implying, in particular, that δQ and δq vanish on the boundary of the rectangle z0,z1×t0,t1, that is,
δQz,t=δqz,t=0if z=z0,z1or if t=t0,t1.
(11.6)
We refer to variations δQ and δq satisfying Eq. (11.5) and hence (11.5) for a rectangle z0,z1×t0,t1 as admissible.
Following the least action principle, we introduce the functional differential δS of the action by the following formula [Ref. 43, 7(35)]:
δS=limε0Sx+εδxSxε.
(11.7)
Then, the system configurations x=xz,t that actually can occur must satisfy
δS=limε0Sx+εδxSxε=0 for all admissible variations.
(11.8)
Let us choose now any z outside lattice aZ. Then, there always exist a sufficiently small ξ > 0 and an integer 0 such that
a0<z0=zξ<z<z1=z+ξ<a0+1.
(11.9)
If we apply now the variational principle (11.9) for all admissible variations δQ and δq such that space interval z0,z1 is compliant with inequalities (11.8), we readily find that
δS=δST+δSB=0,
(11.10)
where ST and SB are defined by expressions (11.2) and (11.3). Using Eq. (11.6) and carrying out in the standard way the integration by parts transformations, we arrive at
δST=t0t1dtz1z2Lt2QC1z2QδQdz,
(11.11)
δSB=t0t1dtz1z21βt+v̊z2q+4πσBqδqdz.
(11.12)
Combining Eqs. (11.10)–(11.12), we arrive at the following EL equations:
Lt2QC1z2Q=0,1βt+v̊z2q+4πσBq=0,za,Z.
(11.13)
Consider now the case when z = aℓ0 for an integer 0 and select space interval z0,z1 as follows:
a01<z0=a012<z=a0<z1=a0+12<a0+1.
(11.14)
Note that in this case, all actions ST, ST, and STB contribute to the variation δS. In particular, as a consequence of the presence of delta functions δza in the expression of the Lagrangian LTB defined by Eq. (3.11), the space derivatives zQ and zq can have jumps at z = aℓ0 as was already acknowledged by assumption 2. Based on this circumstance, we proceed as follows: (i) we split the integral with respect to the space variable z into two integrals,
z0z1=a012a0+a0a0+12;
(11.15)
(ii) we carry out the integration by parts for each of the two integrals in the right-hand side of Eq. (11.15); and (iii) we use already established the EL equation [Eq. (11.13)] to simplify the integral expressions. When that is all done, we arrive at the following equation:
δST=t0t11CzQa0,tδQa0,tdt,δSB=t0t1v̊2βzqa0,tδqa0,tdt,
(11.16)
where jumps zQa and zqa are defined by Eq. (3.4), and
δSTB=t0t1l0t2Qa+1c0Qa0,t+bqa0,tδQa0,tdt
(11.17)
t0t1bc0Qa0,t+bqa0,tδqa0,tdt.
Using the variational principle (11.8), that is,
δST+δSB+δSTB=0,
(11.18)
and the fact that variations δQa0,t and δqa0,t can be chosen arbitrarily, we arrive at the following equations:
1CzQa0,t=l0t2Qa+1c0Qa0,t+bqa0,t,
(11.19)
v̊2βzqa0,t=bc0Qa0+bqa0,t,
where jumps zQa and zqa are defined by Eq. (3.4). Equation (11.19) can be readily recast into the following boundary conditions:
zQa=C0t2ω02+1Qa+bqa,zqa=bβ0v̊2Qa+bqa,
(11.20)
where
C0=Cc0,ω0=1l0c0,β0=βc0.
(11.21)
We remind also that as a consequence of continuity of Q and q, we also have
Qa0,t=0,qa0,t=0.
(11.22)
Hence, Eqs. (11.20) and (11.22) can be viewed as the EL equations at point aℓ0.
Equation (11.20) at an interaction point aℓ0 is perfectly consistent with boundary conditions (2.12) of the general treatment in Ref. 41, which are
LD1ψD(b1,t)+LBψB(b1,t)0LB0ψB(b1,t)=0,
(11.23)
LD1ψD(b2,t)+LBψB(b2,t)0LB0ψB(b2,t)=0,
where (i) b1 = aℓ0 − 0 and b2 = aℓ0 + 0; (ii) LD corresponds to LT+LB; (iii) LB corresponds to LTB; (iv) fields ψD correspond to charges Q and q; and (v) boundary fields ψB correspond to Qa0,t and qa0,t. We remind the reader that boundary conditions (2.12) in Ref. 41 are an implementation of the “rigidity” requirement, which is appropriate for Lagrangian LTB defined by Eq. (3.11). In fact, the signs of the terms containing LD in Eq. (11.23) are altered compared to original Eq. (2.12) in Ref. 41 to correct an unfortunate typo there.

Thus, Eqs. (11.20) and (11.22) form a complete set of EL equations.

The complex plane transformation z1z̄ is known as the unit (circle) inversion (Ref. 45, III.13), and if a set is invariant under the transformation, we refer to it as inversion symmetric set. Let us consider the general form of polynomial Eq. (5.1) of order 4,
S4+aS3+bS2+āS+1=0,aC,bR.
(12.1)
If S is a solution to Eq. (12.1), which is a degenerate one, then the following equation must also hold:
SS4+aS3+bS2+āS+1=4S3+3aS2+2bS+ā=0.
(12.2)
Subtracting from two times Eq. (12.1) S times Eq. (12.2) and dividing the result by S2, we obtain
2S2+2S2aS+āS2=0.
(12.3)
If a solution S to the system of Eqs. (12.1) and (12.3) lies on the unit circle, that is, S=1, then S1=S̄ and the system is equivalent to the following system of equations:
RS2+aS+b2=0,I2S2+aS=0,S=eiφ,φR.
(12.4)
A trigonometric version of the system equation [Eq. (12.4)] is
cos2φ+acosφ+α=b2,2sin2φ+asinφ+α=0,
(12.5)
a=aexpiα,S=expiφ,φ,αR.

This research was supported by AFOSR MURI under Grant No. FA9550-20-1-0409 administered through the University of New Mexico. The author is grateful to E. Schamiloglu for sharing his deep and vast knowledge of high power microwave devices and inspiring discussions.

The author has no conflicts to disclose.

Alexander Figotin: Writing – original draft (lead).

The data that support the findings of this study are available within the article.

C

set of complex number

Cn

set of n dimensional column vectors with complex complex-valued entries

Cn×m

set of n × m matrices with complex-valued entries

Dω,k

CCTWT dispersion function

DCω,k

CCS dispersion function

DKω,k

MCK dispersion function

detA

the determinant of matrix A

diagA1,A2,,Ar

block diagonal matrix with indicated blocks

dimW

dimension of the vector space W

EL

the Euler–Lagrange (equations)

Iνν×ν

identity matrix

kerA

kernel of matrix A, which is the vector space of vector x such that Ax = 0

MT

matrix transposed to matrix M

ODE

ordinary differential equation

s̄

complex-conjugate to complex number s

σA

spectrum of matrix A

Rn×m

set of n × m matrices with real-valued entries

χAs=detsIνA

characteristic polynomial of a ν × ν matrix A

Our preferred form of Fourier transforms as in Ref. 46, Secs. 7.2 and 7.5 and Ref. 20, Sec. 20.2 is as follows:
ft=f̂ωeiωtdω,f̂ω=12πfteiωtdt,
(A1)
fz,t=f̂k,ωeiωtkzdkdω,
(A2)
f̂k,ω=12π2fz,teiωtkzdzdt.
This preference was motivated by the fact that the so-defined Fourier transform of the convolution of two functions has its simplest form. Namely, the convolution f*g of two functions f and g is defined by Ref. 46, Secs. 7.2 and 7.5,
f*gt=g*ft=fttgtdt,
(A3)
f*gz,t=g*fz,t=fzz,ttgz,tdzdt.
(A4)
Then, its Fourier transform as defined by Eqs. (A1) and (A2) satisfies the following properties:
f*ĝω=f̂ωĝω,
(A5)
f*ĝk,ω=f̂k,ωĝk,ω.
(A6)

We provide here a very concise review of Jordan canonical forms following mostly Ref. 47, Sec. III.4 and Ref. 48, Secs. 3.1 and 3.2. As to a demonstration of how Jordan block arises in the case of a single nth order differential equation, we refer to Ref. 49, Sec. 25.4.

Let A be an n × n matrix and λ be its eigenvalue, and let rλ be the least integer k such that NAλIk=NAλIk+1, where NC is a null space of a matrix C. Then, we refer to Mλ=NAλIrλ as the generalized eigenspace of matrix A corresponding to eigenvalue λ. Then, the following statements hold (Ref. 47, Sec. III.4).

Proposition 10
(generalized eigenspaces). Let A be an n × n matrix and λ1, …, λp be its distinct eigenvalues. Then, generalized eigenspaces Mλ1,,Mλp are linearly independent, invariant under the matrix A and
Cn=Mλ1Mλp.
(B1)
Consequently, any vector x0 in Cn can be represented uniquely as
x0=j=1px0,j,x0,jMλj,
(B2)
and
expAtx0=j=1peλjtpjt,
(B3)
where column-vector polynomials pjt satisfy
pjt=k=0rλj1AλjIktkk!x0,j,x0,jMλj,1jp.
(B4)

For a complex number λ, a Jordan block Jrλ of size r ≥ 1 is an r × r upper triangular matrix of the form
Jrλ=λIr+Kr=λ1000λ1000λ1000λ,J1λ=λ,J2λ=λ10λ,
(B5)
Kr=Jr0=0100001000010000.
(B6)
The special Jordan block Kr=Jr0 defined by Eq. (B6) is an nilpotent matrix that satisfies the following identities:
Kr2=0010000001000000,,Krr1=0001000000000000,Krr=0.
(B7)
A general Jordan n × n matrix J is defined as a direct sum of Jordan blocks, that is,
J=Jn1λ10000Jn2λ20000Jnq1λnq10000Jnqλnq,n1+n2+nq=n,
(B8)
where λj need not be distinct. Any square matrix A is similar to a Jordan matrix as in Eq. (B8), which is called Jordan canonical form of A. Namely, the following statement holds (Ref. 48, 3.1).

Proposition 11
(Jordan canonical form). Let A be an n × n matrix. Then, there exists a non-singular n × n matrix Q such that the following block-diagonal representation holds:
Q1AQ=J,
(B9)
where J is the Jordan matrix defined by Eq. (B8) and λj, 1 ≤ jq, are not necessarily different eigenvalues of matrix A. Representation (B9) is known as the Jordan canonical form of matrix A, and matrices Jj are called Jordan blocks. The columns of the n × n matrix Q constitute the Jordan basis, providing for the Jordan canonical form (B9) of matrix A.

A function fJrs of a Jordan block Jrs is represented by the following equation (Ref. 50, Sec. 7.9, Ref. 51, Sec. 10.5):
fJrs=fsfs2fs2r1fsr1!0fsfsr2fsr2!00fsfs000fs.
(B10)
Note that any function fJrs of the Jordan block Jrs is evidently an upper triangular Toeplitz matrix.
There are two particular cases of formula (B10), which can also be derived straightforwardly using Eq. (B7),
expKrt=k=0r1tkk!Krk=1tt22!tr1r1!01ttr2r2!001t0001,
(B11)
Jrs1=k=0r1sk1Krk=1s1s21s31r1sr01s1s21r2sr1001s1s20001s.
(B12)
The companion matrix Ca for the monic polynomial
as=sν+1kνaνksνk,
(C1)
where coefficients ak are complex numbers, is defined by Ref. 51, Sec. 5.2,
Ca=0100001000001a0a1aν2aν1.
(C2)
Note that
detCa=1νa0.
(C3)

An eigenvalue is called cyclic (nonderogatory) if its geometric multiplicity is 1. A square matrix is called cyclic (nonderogatory) if all its eigenvalues are cyclic (Ref. 51, Sec. 5.5). The following statement provides different equivalent descriptions of a cyclic matrix (Ref. 51, Sec. 5.5).

Proposition 12
(criteria for a matrix to be cyclic). Let ACn×n be an n × n matrix with complex-valued entries. Let s