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.

## I. INTRODUCTION

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}

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 range^{6,7} (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.

*cavity gap*or just

*gap*, and it is there the electron velocity is modulated, leading to electron bunching and consequent RF signal amplification. If $v\u030a$ is the stationary (dc) velocity of the electron flow and

*l*

_{g}is the length of the cavity gap, then the mentioned condition of smallness of

*l*

_{g}and the electron transit time $\tau g=lgv\u030a$ can be written as

*ω*satisfies the following (see Ref. 4, p. 277):

*ω*

_{p}is the relevant plasma frequency. Then, combining inequalities (1.2) and (1.1), we obtain the following upper bound on the gap length:

*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:

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}

## II. SKETCH OF THE ANALYTIC MODEL OF THE TRAVELING WAVE TUBE

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 field^{2,4} (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).

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.

^{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 TWT

^{2,13}(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,

*e*is the electron charge with

*e*> 0,

*m*is the electron mass,

*ω*

_{p}is the e-beam plasma frequency,

*σ*

_{B}is the area of the cross section of the e-beam, s $v\u030a>0$ is the stationary velocity of electrons in the e-beam, and $n\u030a$ is the density of the number of electrons. The constant

*R*

_{sc}is the plasma frequency reduction factor that accounts phenomenologically for finite dimensions of the e-beam cylinder and geometric features of the slow-wave structure

^{19}(Ref. 2, Sec. 9.2 and Ref. 14, Sec. 3.3.3). The frequency

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

Frequency . | Plasma frequency . | $\omega p=4\pi n\u030ae2m$ . |
---|---|---|

Velocity | e-beam velocity | $v\u030a$ |

Wavenumber | $kq=\omega rpv\u030a=Rsc\omega pv\u030a$ | |

Length | Wavelength for k_{q} | $\lambda rp=2\pi v\u030a\omega rp,\omega rp=Rsc\omega p$ |

Time | Wave time period | $\tau \u030a=2\pi \omega p$ |

Frequency . | Plasma frequency . | $\omega p=4\pi n\u030ae2m$ . |
---|---|---|

Velocity | e-beam velocity | $v\u030a$ |

Wavenumber | $kq=\omega rpv\u030a=Rsc\omega pv\u030a$ | |

Length | Wavelength for k_{q} | $\lambda rp=2\pi v\u030a\omega rp,\omega rp=Rsc\omega p$ |

Time | Wave time period | $\tau \u030a=2\pi \omega p$ |

i
. | current . | $chargetime$ . |
---|---|---|

q | Charge | $charge$ |

$n\u030a$ | Number of electrons p/u of volume | $1length3$ |

$\lambda rp=2\pi v\u030a\omega rp,\omega rp=Rsc\omega p$ | The electron plasma wavelength | $length$ |

$gB=\sigma B4\lambda rp$ | The e-beam spatial scale | $length$ |

$\beta =\sigma B4\pi Rsc2\omega p2=e2mRsc2\sigma Bn\u030a$ | e-beam intensity | $length2time2$ |

$\beta \u2032=\beta v\u030a2=\pi \sigma B\lambda rp2=4\pi gB\lambda rp$ | Dimensionless e-beam intensity | $dim-less$ |

i
. | current . | $chargetime$ . |
---|---|---|

q | Charge | $charge$ |

$n\u030a$ | Number of electrons p/u of volume | $1length3$ |

$\lambda rp=2\pi v\u030a\omega rp,\omega rp=Rsc\omega p$ | The electron plasma wavelength | $length$ |

$gB=\sigma B4\lambda rp$ | The e-beam spatial scale | $length$ |

$\beta =\sigma B4\pi Rsc2\omega p2=e2mRsc2\sigma Bn\u030a$ | e-beam intensity | $length2time2$ |

$\beta \u2032=\beta v\u030a2=\pi \sigma B\lambda rp2=4\pi gB\lambda rp$ | Dimensionless e-beam intensity | $dim-less$ |

*ω*

_{rp}. 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

*e-beam spatial scale*. Using these spatial scales, we obtain the following representation for the dimensionless form

*β*′ of the e-beam intensity:

*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

### A. TWT Lagrangian and evolution equations

*TWT principal parameter*$\gamma \u0304=\theta \beta $. This parameter in view of Eqs. (2.1) and (2.6) can be represented as follows:

*b*> 0 is a coupling coefficient, and

*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,

*σ*

_{B}is the area of the cross section of the e-beam, and

*β*is the e-beam intensity defined by Eq. (2.8).

### B. Space-charge wave velocity and electron density fields

*n*and

*v*to be relatively small,

*j*=

*j*(

*z*,

*t*) that satisfies

*J*is the e-beam current. The first term of Eq. (2.16) readily implies that

*q*=

*q*(

*z*,

*t*) defined in Sec. II A, the velocity

*v*=

*v*(

*z*,

*t*), and the current associated with it

*J*

_{v}=

*J*

_{v}(

*z*,

*t*) are as follows (Ref. 10, 22.2):

*D*

_{t}is the so-called

*material time derivative*. The second equation in (2.18) evidently implies that current

*J*

_{v}is exactly

*D*

_{t}

*q*, whereas the first equation in (2.18) yields the following representations of the velocity

*v*:

*E*=

*E*(

*z*,

*t*) associated with the space-charge wave satisfies the Poisson equation (Ref. 10, Sec. 22.2),

*E*= 0, the above equation yields

## III. ANALYTIC MODEL OF COUPLED-CAVITY TRAVELING WAVE TUBE

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.

*(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**,*$\u2113\u2208Z$*by**cavities.**Every cavity carries shunt capacitance**c*_{0}*. The e-beam interacts with the CCS exclusively through the cavities located at a discrete set of equidistant points, that is, the lattice*(3.1)$aZ:Z=\u2026,\u22122,\u22121,0,1,2,\u2026,$*where**a*> 0,*and we refer to this parameter as the CCS period or just period. The cavity width**l*_{g}*and the corresponding transit time**τ*_{g}*are assumed to be zero; see**Eq. (1.4)**and comments above**it.*

*jump and continuity conditions*on charge functions $Qz,t$ and $qz,t$.

*(jump-continuity of charge functions).*

*Functions*$Qz,t$*and*$qz,t$*and their time derivatives*$\u2202tjQz,t$*and*$\u2202tjqz,t$*for**j*= 1, 2*are continuous for all real**t**and**z**.**Derivatives*$\u2202tjQz,t$*,*$\u2202tjQz,t$*,*$\u2202zjqz,t$*, and*$\u2202zjqz,t$*for**j*= 1, 2*and the mixed derivatives*$\u2202z\u2202tQz,t=\u2202t\u2202zQz,t$*,*$\u2202z\u2202tqz,t=\u2202t\u2202zqz,t$*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*$Fb\u22120$*and*$Fb+0$*stand for its left and right limit at**b**assuming their existence, that is,*(3.3)$Fb\xb10=limz\u2192b\xb10Fz.$

*Let us also denote by*$Fb$

*the jump of function*$Fz$

*at*

*b*

*, that is,*

*The following right and left limits exist:*

*and these limits are continuously differentiable functions of*

*t*

*. The values*$\u2202zQa\u2113\xb10,t$

*and*$\u2202zqa\u2113\xb10,t$

*can be different, and consequently, the jumps*$\u2202zQa\u2113,t$

*and*$\u2202zqa\u2113,t$

*can be nonzero.*

*(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 $\u2202zqa\u2113,t$, 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

*manifesting the electron bunching*that occurs in the EM cavity centered at

*aℓ*. In view of Eq. (10.4), we also have $va\u2113,t=\u2212v\u030a\u2202zqa\u2113,te\sigma Bn\u030a$, 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.

I
. | Current . | $chargetime$ . |
---|---|---|

Q | Charge | $charge$ |

c_{0} | Cavity capacitance | $length$ |

l_{0} | Cavity inductance | $time2length$ |

b | Coupling parameter | $dim-less$ |

I
. | Current . | $chargetime$ . |
---|---|---|

Q | Charge | $charge$ |

c_{0} | Cavity capacitance | $length$ |

l_{0} | Cavity inductance | $time2length$ |

b | Coupling parameter | $dim-less$ |

### A. CCTWT Lagrangian and the Euler–Lagrange equations

*C*and

*L*are, respectively, distributed shunt capacitance and inductance of the TL,

*σ*

_{B}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 $\u22122\pi \sigma Bq2$ 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 $\delta z\u2212a\u2113$, (ii) the factors before delta functions $\delta z\u2212a\u2113$ are expressions similar to density $LTb$ in Eq. (2.9) adapted to lattice $aZ$ of discrete interaction points

*aℓ*, and (iii) capacitance

*c*

_{0}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*

*.*

*z*outside the lattice $aZ$ are

*aℓ*are

*β*

_{0}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:

*C*

_{0}and

*β*

_{0}are defined by Eq. (3.16). Hence, the complete set of the boundary conditions at interaction points

*aℓ*can be concisely written as

### B. Natural units and dimensionless parameters

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

Velocity . | e-beam velocity . | $v\u030a$ . |
---|---|---|

Length | Period | a |

Length | Wavelength | $\lambda \u030a=1k\u030a=v\u030a\omega p$ |

Frequency | Period frequency | $\omega a=v\u030aa$ |

Frequency | Plasma frequency | $\omega p=4\pi n\u030ae2m$ |

Time | Time of passing the period a | $1\omega a$ = $av\u030a$ |

Time | Plasma oscillation time period | $\tau \u030a=1\omega p$ |

Velocity . | e-beam velocity . | $v\u030a$ . |
---|---|---|

Length | Period | a |

Length | Wavelength | $\lambda \u030a=1k\u030a=v\u030a\omega p$ |

Frequency | Period frequency | $\omega a=v\u030aa$ |

Frequency | Plasma frequency | $\omega p=4\pi n\u030ae2m$ |

Time | Time of passing the period a | $1\omega a$ = $av\u030a$ |

Time | Plasma oscillation time period | $\tau \u030a=1\omega p$ |

*ω*

_{rp}and

*λ*

_{rp}are, respectively, the

*reduced plasma frequency*and

*the electron plasma*

*wavelength*.

*w*=

*χv*,

a
. | The MCK period . | $length$ . |
---|---|---|

$v\u030a$ | The e-beam stationary velocity | $lengthtime$ |

$\omega a=2\pi v\u030aa$ | The period frequency | $1time$ |

$\omega p=4\pi n\u030ae2m$ | The plasma frequency | $1time$ |

$\lambda rp=2\pi v\u030a\omega rp,\omega rp=Rsc\omega p$ | The electron plasma wavelength | $length$ |

$gB=\sigma B4\lambda rp$ | The e-beam spatial scale | $length$ |

$f\u2032=af=2\pi \omega rp\omega a=2\pi a\lambda rp$ | Normalized period in units of $\lambda rp2\pi $ | $dim-less$ |

$n\u030a$ | The number of electrons p/u of volume | $1length3$ |

c_{0}, l_{0} | The cavity capacitance, inductance | $length,time2length$ |

$\omega 0=1l0c0$ | The cavity resonant frequency | $1time$ |

$\beta =\sigma BRsc2\omega p24\pi =e2Rsc2\sigma Bn\u030am=\pi \sigma Bv\u030a2\lambda rp2$ | The e-beam intensity | $length2time2$ |

$\beta \u2032=\beta v\u030a2=\pi \sigma B\lambda rp2=4\pi gB\lambda rp$ | Dim-less e-beam intensity | $dim-less$ |

$\beta 0\u2032=\beta \u2032c0\u2032=a\beta c0v\u030a2$ | The first interaction par. | $dim-less$ |

$B\omega =B0\omega 2\omega 2\u2212\omega 02,B0=b2\beta 0\u2032$ | The second interaction par. | $dim-less$ |

$K0=B02f=b2\beta 0\u20322f=b2\sigma B4\lambda rpc0=b2gBc0$ | The gain coefficient | $dim-less$ |

$K\omega =B\omega 2f=K0\omega 2\omega 2\u2212\omega 02$ | The gain parameter | $dim-less$ |

$C0\u2032=C\u2032c0\u2032=aCc0=aC0$ | The capacitance parameter | $dim-less$ |

a
. | The MCK period . | $length$ . |
---|---|---|

$v\u030a$ | The e-beam stationary velocity | $lengthtime$ |

$\omega a=2\pi v\u030aa$ | The period frequency | $1time$ |

$\omega p=4\pi n\u030ae2m$ | The plasma frequency | $1time$ |

$\lambda rp=2\pi v\u030a\omega rp,\omega rp=Rsc\omega p$ | The electron plasma wavelength | $length$ |

$gB=\sigma B4\lambda rp$ | The e-beam spatial scale | $length$ |

$f\u2032=af=2\pi \omega rp\omega a=2\pi a\lambda rp$ | Normalized period in units of $\lambda rp2\pi $ | $dim-less$ |

$n\u030a$ | The number of electrons p/u of volume | $1length3$ |

c_{0}, l_{0} | The cavity capacitance, inductance | $length,time2length$ |

$\omega 0=1l0c0$ | The cavity resonant frequency | $1time$ |

$\beta =\sigma BRsc2\omega p24\pi =e2Rsc2\sigma Bn\u030am=\pi \sigma Bv\u030a2\lambda rp2$ | The e-beam intensity | $length2time2$ |

$\beta \u2032=\beta v\u030a2=\pi \sigma B\lambda rp2=4\pi gB\lambda rp$ | Dim-less e-beam intensity | $dim-less$ |

$\beta 0\u2032=\beta \u2032c0\u2032=a\beta c0v\u030a2$ | The first interaction par. | $dim-less$ |

$B\omega =B0\omega 2\omega 2\u2212\omega 02,B0=b2\beta 0\u2032$ | The second interaction par. | $dim-less$ |

$K0=B02f=b2\beta 0\u20322f=b2\sigma B4\lambda rpc0=b2gBc0$ | The gain coefficient | $dim-less$ |

$K\omega =B\omega 2f=K0\omega 2\omega 2\u2212\omega 02$ | The gain parameter | $dim-less$ |

$C0\u2032=C\u2032c0\u2032=aCc0=aC0$ | The capacitance parameter | $dim-less$ |

### C. Euler–Lagrange equations in dimensionless variables

*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)]:*

*t*(see Appendix A) to Eqs. (3.30) and (3.31) and obtain the following equations:

*z*with frequency dependent coefficients.

*A*

_{j}and

*P*

_{TB}are defined by

*Equations (3.34)*–

*(3.36)*

*are evidently the second-order vector ODE with spatially periodic frequency dependent singular matrix potential*$\u2211\u2113=\u2212\u221e\u221e\delta z\u2212\u2113PTB\omega $

*. These equations becomes the object of our studies*

*below.*

*A*

_{0},

*A*

_{1}, and

*P*

_{TB}satisfy Eqs. (3.35) and (3.36).

*z*matrix $Az$ is

*G*-skew-Hermitian, that is,

*G*-unitary matrix satisfying

### D. CCTWT subsystems: The coupled cavity structure and the e-beam

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.

*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

*P*

_{23}defined by Eq. (4.10) and transform $Tb=0$ as follows:

*s*(see Appendix F) is

*CCS instability parameter*for as we will find that it completely determines if the Floquet multipliers satisfy the instability criterion $s>1$.

*s*is

## IV. SOLUTIONS TO THE COUPLED-CAVITY TWT EQUATIONS

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.

### A. Solutions to the Euler–Lagrange equation inside the period

*s*in expressions for the characteristic polynomials $ATs$ and $ABs$ represents symbolically the differential operator

*∂*

_{z}.

*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

*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

*non-interacting*

*TL*and the e-beam is the following diagonal matrix polynomial:

*X*clarifies the meaning of the entries of matrix $CTB$.

*z*= 1, which is the period in dimensionless variables, we get

*P*

_{23}that permutes the second and third coordinates, that is,

### B. The boundary conditions and the monodromy matrix in dimensionless variables

*boundary matrix*$Sb$ satisfies

*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 $v\u030a$. 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

*C*

_{0},

*β*

_{0}, and

*f*satisfy Eq. (4.19).

## V. CHARACTERISTIC EQUATION, THE FLOQUET MULTIPLIERS, AND THE DISPERSION RELATION

*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*$detT\u2212sI=0$, which represents the following polynomial equation of order 4 (see Fig. 3):

*k*is the wavenumber that can be real or complex-valued and coefficients

*c*

_{3}and

*c*

_{2}satisfy

*c*

_{2}defined by Eq. (5.3) is manifestly real. The utility of representing the Floquet multipliers

*s*in the form $s=Sei\omega 2$ 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 $1S\u0304$ 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.

*(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*$v\u030a$

*, namely,*$w=v\u030a$

*. It is also convenient to choose frequency units so that*

*ω*

_{0}= 1

*. Combining these two conditions, we assume*

## VI. DISPERSION RELATIONS

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.*

*χ*= 1 and

*ω*

_{0}= 1, the dispersion relations described by Eqs. (5.1)–(5.4) turn into

*χ*= 1 and

*ω*

_{0}= 1, is its high-frequency form, namely,

*the CCTWT dispersion*

*function*.

There exists a remarkability in its simplicity relations between the CCTWT dispersion function $D\omega ,k$ and the dispersion functions $DC\omega ,k$ and $DK\omega ,k$ 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.

**.**

*Let us assume that*

*χ*=

*ω*

_{0}= 1

*. Let the CCTWT, the CCS, and the MCK dispersion functions*$D\omega ,k$

*,*$DC\omega ,k$,

*and*$DK\omega ,k$

*be defined by, respectively, Eqs. (6.1), (9.17), and (8.36). Then, the following identity holds:*

*In the case of the high-frequency approximation, the following identity holds:*

*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.*

*(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.

*k*and

*ω*must be real numbers. Then, dividing Eq. (6.1) by

*S*

^{2}and carrying elementary transformations, we arrive at the following trigonometric form of the conventional dispersion relation:

### A. Graphical representation of the dispersion relations

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 $k\omega ,\omega $, where *ω* is its frequency and $k\omega $ is its wavenumber. If $k\omega $ 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 $Ik\omega \u22600$, we partition all the system modes represented by pairs $k\omega ,\omega $ into two distinct classes—oscillatory modes and unstable ones—based on whether the wavenumber $k\omega $ is real- or complex-valued with $Ik\omega \u22600$. We refer to a mode (eigenmode) of the system as an *oscillatory mode* if its wavenumber $k\omega $ is real-valued. We associate with such an oscillatory mode point $k\omega ,\omega $ in the *kω*-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, $Ik\omega \u22600$. We associate with such an unstable mode point $Rk\omega ,\omega $ in the *kω*-plane. Since we consider here only *convective unstable modes*, we refer to them shortly as *unstable modes*. Note that every point $Rk\omega ,\omega $ is, in fact, associated with two complex conjugate system modes with $\xb1Ik\omega $.

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 $k\omega ,\omega $ and $Rk\omega ,\omega $ in the *kω*-plane. We name this set the *dispersion-instability graph*. To distinguish graphically points $k\omega ,\omega $ associated oscillatory modes when $k\omega $ is real-valued from points $Rk\omega ,\omega $ associated unstable modes when $k\omega $ is complex-valued with $Ik\omega \u22600$, we show points $Ik\omega =0$ in blue color, whereas points with $Ik\omega \u22600$ are shown in brown color. We remind once again that every point $\omega ,Rk\omega $ with $Ik\omega \u22600$ represents exactly two complex conjugate unstable modes associated with $\xb1Ik\omega $.

When $Ik\xb1\omega \u22600$ and $Rk+\omega =Rk\u2212\omega $, 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. 4–6 to demonstrate their dependence on the gain coefficient *K*_{0}, the capacitance parameter *C*_{0}, 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.

### B. Exceptional points of degeneracy

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.

*exceptional points of degeneracy (EPDs)*:

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).

## VII. GAIN EXPRESSION IN TERMS OF THE FLOQUET MULTIPLIERS

*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,

*χ*= 1 and

*ω*

_{0}= 1. Under these conditions, the CCTWT characteristic equations [Eqs. (5.1)–(5.4)] can be recast into the following form:

*CCTWT characteristic polynomial*

*P*have the following expressions:

*P*

_{C}has the following expression:

*P*

_{K}as the

*MCK characteristic*

*polynomial*.

*ω*→ ∞, which are

Just as in the case of the dispersion relations that we analyzed in Sec. VI, there are simple relations between the CCTWT characteristic function $P\omega ,s$ and the characteristic polynomials $PC\omega ,s$ and $PK\omega ,s$ 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.

*(characteristic polynomial factorization)*

**.**

*Let us assume that*

*χ*=

*ω*

_{0}= 1

*. Let the CCTWT, the CCS, and the MCK dispersion functions*$P\omega ,s$

*,*$PC\omega ,s$,

*and*$PK\omega ,s$

*be defined by, respectively,*Eqs. (7.2), (7.7), and (7.8)

*. Then, the following identity holds:*

*In the case of the high-frequency approximation, the following identity holds:*

*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*

*(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.

*(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.

## VIII. SKETCH OF THE MULTICAVITY KLYSTRON ANALYTICAL MODEL

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.

### A. The Euler–Lagrange equations in dimensionless variables

*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*(see Appendix A) of Eqs. (8.3) and (8.4) is

*q*and $B\omega $ is an important parameter defined by

*cavity e-beam interaction parameter*. The Fourier transform in time of Eq. (8.4) yields

*Q*, and Eq. (8.8) was used to obtain the second equation in (8.6).

*G*

_{K}that are as follows:

*G*, one can readily verify that $Az$ is

*G*-skew-Hermitian matrix, that is,

*G*

_{K}-unitary and its spectrum $\sigma \Phi Kz$ is symmetric with respect to the unit circle, that is,

### B. The monodromy matrix, the dispersion-instability relations, and the gain

*(smaller MCK period). The MCK normalized period*

*f*

*satisfies the following bounds:*

*b*

_{f}:

*b*

_{f}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

*f*

_{cr}on interval $0,\pi $ of the normalized period

*f*such that

*critical value*and the following representation holds:

*f*

_{cr}is that for

*f*

_{cr}<

*f*<

*π*, any

*ω*>

*ω*

_{0}is an instability frequency. One can see that Fig. 12 shows the frequency dependence of the gain

*G*and its asymptotic behavior as

*ω*→ +∞.

*G*is attained at

*f*=

*f*

_{max}that satisfies (see Fig. 13)

*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 $detTK\u2212sI=0$, which is

^{11}

*parameter*

*b*

_{f}

*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

*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

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

**.**

*Let*

*s*

_{±}

*be the MCK Floquet multipliers, that is, solutions to (8.28), and let*$k\xb1\omega $

*be the corresponding complex-valued wave numbers satisfying*

*Then, the following representation for*$k\xb1\omega $

*holds:*

*f*<

*π*and

*Requirement for*$Rk\xb1\omega $

*to be in the first (main) Brillouin zone*$\u2212\pi ,\pi $

*effectively selects the band number*

*m*

*that depends on*

*ω*

*as follows: For any given*

*ω*> 0

*and*0 <

*f*<

*π*,

*the band number*$m\u2208Z$

*is determined by the requirement to satisfy the following inequalities:*

*Equations (8.32)* *for the complex-valued wave numbers* $k\xb1\omega $ *represent the dispersion relations of the* *MCK.*

*MCK dispersion*

*function*.

*ω*and

*k*satisfying the dispersion relation.

Theorem 2 shows how the MCK dispersion function $DK\omega ,k$ and its high-frequency approximation $DK0\omega ,k$ are integrated into the relevant dispersion functions associated with the CCTWT.

Figures 14 and 15 illustrate graphically the dispersion relations $k\xb1\omega $ 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.

*N*of cavities in the klystron (see Ref. 4, Sec. 7.7.1, Ref. 35, Sec. 7.2.6, and Ref. 36, Chap. 16),

## IX. COUPLED CAVITY STRUCTURE

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.

*b*= 0 and omitting component $LB$, that is,

*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)]:

*t*(see Appendix A) of Eq. (9.4) yields

*G*:

*z*matrix $ACz$ is

*G*

_{C}-skew-Hermitian, that is,

*G*

_{C}-unitary matrix,

### A. Monodromy matrix and the dispersion-instability relations

*χ*=

*ω*

_{0}= 1), the monodromy matrix $TC$ defined by Eq. (3.43) takes the form

*P*

_{C}is referred to as the

*CCS characteristic polynomial*and quantity $WC\omega $ is the CCS instability parameter, which is depicted in Fig. 16(b).

*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:

*instability parameter*$WC\omega $

*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*=

*ω*

_{0}= 1), we divide the characteristic equation [Eq. (9.13)] by 2

*s*and substitute $s=expik$ obtaining the following equations:

*ω*, we obtain the following

*high-frequency form of the dispersion relations*for the MCK:

*CCS dispersion*

*function*.

*dispersion relations for the e-beam:*

*CCS EPD equation:*

*πn*for positive integers

*n*are CCS EPD points. Remaining sets of EPD frequencies

*ξ*

_{m},

*m*≥ 1 are found by solving the CCS EPD equation [Eq. (9.19)].

*ξ*

_{m}satisfy also the following equations:

Based on the prior analysis, we introduce the CCS gain *G*_{C} 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.

*(CCS gain per one period)*

**.**Let

*s*

_{±}be the CCS Floquet multipliers defined by Eq. (9.16). Then, corresponding to them, gain

*G*

_{C}in dB per one period is defined by

### B. Exceptional points of degeneracy

*ω*=

*πn*are as follows:

## X. THE KINETIC AND FIELD POINTS OF VIEW ON THE GAP INTERACTION

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.

### A. Some points from the kinetic theory

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.

*l*

_{g}and the corresponding transit time

*τ*

_{g}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

*l*

_{g}= 0 and

*τ*

_{g}= 0 following Ref. 1, Sec. II.5, we suppose that $U\u030a$ is the constant accelerating voltage, so the stationary dc electron flow velocity $v\u030a=2emU\u030a$, where

*m*and −

*e*is, respectively, the electron mass and its charge. Suppose also that $U1\u2061sin\omega t$ is the gap voltage. Then, based on the elementary energy conservation law, one gets

*v*is the modulated velocity. Solving Eq. (10.1) for

*v*and assuming “small signal” approximation, we obtain

*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 $v\u030a$.

### B. Field theory point of view on the kinetic properties of the electron flow

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 $v\u030a$ electron velocity field $v=vz,t$. Therefore, as to this part of the interaction, we may view the electron density to be essentially constant $n\u030a$, 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 $v\u030a$ 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.

*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 $na\u2113$ and $va\u2113$ at the interaction points

*aℓ*:

*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 $\u2202zEa\u2113$ at the interaction points

*aℓ*:

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

### C. Relation between the kinetic and the field points of view on the gap interaction

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 *l*_{g} and the corresponding transit time *τ*_{g} 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.”

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 $\u2202zqa\u2113,t$ that are explicitly allowed by the field theory represent jumps $na\u2113,t$ and $va\u2113,t$ related to the kinetic properties of the electron flow; see Remark 1. Namely, jump $na\u2113,t$ manifests the electron bunching, jump $va\u2113,t$ manifests the ac electron velocity modulation, and Eq. (10.4) relates the two of them.

## XI. LAGRANGIAN VARIATIONAL FRAMEWORK

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.

*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:

*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,

*δQ*and

*δq*vanish on the boundary of the rectangle $z0,z1\xd7t0,t1$, that is,

*δQ*and

*δq*satisfying Eq. (11.5) and hence (11.5) for a rectangle $z0,z1\xd7t0,t1$ as

*admissible*.

*δS*of the action by the following formula [Ref. 43, 7(35)]:

*z*outside lattice $aZ$. Then, there always exist a sufficiently small

*ξ*> 0 and an integer

*ℓ*

_{0}such that

*δQ*and

*δq*such that space interval $z0,z1$ is compliant with inequalities (11.8), we readily find that

*S*

_{T}and

*S*

_{B}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

*z*=

*aℓ*

_{0}for an integer

*ℓ*

_{0}and select space interval $z0,z1$ as follows:

*S*

_{T},

*S*

_{T}, and

*S*

_{TB}contribute to the variation

*δS*. In particular, as a consequence of the presence of delta functions $\delta z\u2212a\u2113$ in the expression of the Lagrangian $LTB$ defined by Eq. (3.11), the space derivatives

*∂*

_{z}

*Q*and

*∂*

_{z}

*q*can have jumps at

*z*=

*aℓ*

_{0}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,

*Q*and

*q*, we also have

*aℓ*

_{0}.

*aℓ*

_{0}is perfectly consistent with boundary conditions (2.12) of the general treatment in Ref. 41, which are

*b*

_{1}=

*aℓ*

_{0}− 0 and

*b*

_{2}=

*aℓ*

_{0}+ 0; (ii)

*L*

_{D}corresponds to $LT+LB$; (iii)

*L*

_{B}corresponds to $LTB$; (iv) fields $\psi D\u2113$ correspond to charges

*Q*and

*q*; and (v) boundary fields $\psi B\u2113$ correspond to $Qa\u21130,t$ and $qa\u21130,t$. 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

*L*

_{D}in Eq. (11.23) are altered compared to original Eq. (2.12) in Ref. 41 to correct an unfortunate typo there.

## XII. ROOT DEGENERACY FOR A SPECIAL POLYNOMIAL OF THE FOURTH DEGREE

*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,

*S*is a solution to Eq. (12.1), which is a degenerate one, then the following equation must also hold:

*S*times Eq. (12.2) and dividing the result by

*S*

^{2}, we obtain

*S*to the system of Eqs. (12.1) and (12.3) lies on the unit circle, that is, $S=1$, then $S\u22121=S\u0304$ and the system is equivalent to the following system of equations:

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Alexander Figotin**: Writing – original draft (lead).

## DATA AVAILABILITY

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

## NOMENCLATURE

- $C$
set of complex number

- $Cn$
set of

*n*dimensional column vectors with complex complex-valued entries- $Cn\xd7m$
set of

*n*×*m*matrices with complex-valued entries- $D\omega ,k$
CCTWT dispersion function

- $DC\omega ,k$
CCS dispersion function

- $DK\omega ,k$
MCK dispersion function

- $detA$
the determinant of matrix

*A*- $diagA1,A2,\u2026,Ar$
block diagonal matrix with indicated blocks

- $dimW$
dimension of the vector space

*W*- EL
the Euler–Lagrange (equations)

- $I\nu \nu \xd7\nu $
identity matrix

- $kerA$
kernel of matrix

*A*, which is the vector space of vector*x*such that*Ax*= 0*M*^{T}matrix transposed to matrix

*M*- ODE
ordinary differential equation

- $s\u0304$
complex-conjugate to complex number

*s*- $\sigma A$
spectrum of matrix

*A*- $Rn\xd7m$
set of

*n*×*m*matrices with real-valued entries- $\chi As=detsI\nu \u2212A$
characteristic polynomial of a

*ν*×*ν*matrix*A*

### APPENDIX A: FOURIER TRANSFORM

*f**

*g*of two functions

*f*and

*g*is defined by Ref. 46, Secs. 7.2 and 7.5,

### APPENDIX B: JORDAN CANONICAL FORM

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 *n*th 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 $r\lambda $ be the least integer *k* such that $NA\u2212\lambda Ik=NA\u2212\lambda Ik+1$, where $NC$ is a null space of a matrix *C*. Then, we refer to $M\lambda =NA\u2212\lambda Ir\lambda $ as the *generalized eigenspace* of matrix *A* corresponding to eigenvalue *λ*. Then, the following statements hold (Ref. 47, Sec. III.4).

*(generalized eigenspaces)*

**.**

*Let*

*A*

*be an*

*n*×

*n*

*matrix and*

*λ*

_{1}, …,

*λ*

_{p}

*be its distinct eigenvalues. Then, generalized eigenspaces*$M\lambda 1,\u2026,M\lambda p$

*are linearly independent, invariant under the matrix*

*A*

*and*

*Consequently, any vector*

*x*

_{0}

*in*$Cn$

*can be represented uniquely as*

*and*

*where column-vector polynomials*$pjt$

*satisfy*

*λ*, a Jordan block $Jr\lambda $ of size

*r*≥ 1 is an

*r*×

*r*upper triangular matrix of the form

*n*×

*n*matrix

*J*is defined as a direct sum of Jordan blocks, that is,

*λ*

_{j}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).

*(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:*

*where*

*J*

*is the Jordan matrix defined by Eq. (B8) and*

*λ*

_{j}

*,*1 ≤

*j*≤

*q*,

*are not necessarily different eigenvalues of matrix*

*A*

*. Representation (B9) is known as the Jordan canonical form of matrix*

*A*

*, and matrices*

*J*

_{j}

*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*

*.*

### APPENDIX C: COMPANION MATRIX AND CYCLICITY CONDITION

*a*

_{k}are complex numbers, is defined by Ref. 51, Sec. 5.2,

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).

*(criteria for a matrix to be cyclic)*

**.**

*Let*$A\u2208Cn\xd7n$

*be an*

*n*×

*n*

*matrix with complex-valued entries. Let*$s$