Analytic theory of coupled-cavity traveling wave tubes

Figotin, Alexander

doi:10.1063/5.0102701

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

FIG. 1.

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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.

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.

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

\overset{̊}{v}

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

τ_{g} = \frac{l_{g}}{\overset{̊}{v}}

can be written as

l_{g} < \frac{2 π \overset{̊}{v}}{ω} .

(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):

ω_{p} ≪ ω,

(1.2)

where ω_p is the relevant plasma frequency. Then, combining inequalities () and (), we obtain the following upper bound on the gap length:

l_{g} < \frac{2 π \overset{̊}{v}}{ω} ≪ λ_{p} = \frac{2 π \overset{̊}{v}}{ω_{p}} .

(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:

l_{g} = 0, τ_{g} = \frac{l_{g}}{\overset{̊}{v}} = 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

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).

FIG. 2.

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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.

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 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,

β = \frac{σ_{B}}{4 π} R_{sc}^{2} ω_{p}^{2} = \frac{e^{2}}{m} R_{sc}^{2} σ_{B} \overset{̊}{n}, ω_{p}^{2} = \frac{4 π \overset{̊}{n} e^{2}}{m},

(2.1)

where −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

\overset{̊}{v} > 0

is the stationary velocity of electrons in the e-beam, and

\overset{̊}{n}

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 structure¹⁹ (Ref. 2, Sec. 9.2 and Ref. 14, Sec. 3.3.3). The frequency

ω_{rp} = R_{sc} ω_{p}

(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.

Frequency	Plasma frequency	$ω_{p} = \sqrt{\frac{4 π \overset{̊}{n} e^{2}}{m}}$
Velocity	e-beam velocity	$\overset{̊}{v}$
Wavenumber		$k_{q} = \frac{ω_{rp}}{\overset{̊}{v}} = \frac{R_{sc} ω_{p}}{\overset{̊}{v}}$
Length	Wavelength for k_q	$λ_{rp} = \frac{2 π \overset{̊}{v}}{ω_{rp}}, ω_{rp} = R_{sc} ω_{p}$
Time	Wave time period	$\overset{̊}{τ} = \frac{2 π}{ω_{p}}$

Frequency	Plasma frequency	$ω_{p} = \sqrt{\frac{4 π \overset{̊}{n} e^{2}}{m}}$
Velocity	e-beam velocity	$\overset{̊}{v}$
Wavenumber		$k_{q} = \frac{ω_{rp}}{\overset{̊}{v}} = \frac{R_{sc} ω_{p}}{\overset{̊}{v}}$
Length	Wavelength for k_q	$λ_{rp} = \frac{2 π \overset{̊}{v}}{ω_{rp}}, ω_{rp} = R_{sc} ω_{p}$
Time	Wave time period	$\overset{̊}{τ} = \frac{2 π}{ω_{p}}$

TABLE II.

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

i	current	$\frac{[charge]}{[time]}$
q	Charge	$[charge]$
$\overset{̊}{n}$	Number of electrons p/u of volume	$\frac{[1]}{{[length]}^{3}}$
$λ_{rp} = \frac{2 π \overset{̊}{v}}{ω_{rp}}, ω_{rp} = R_{sc} ω_{p}$	The electron plasma wavelength	$[length]$
$g_{B} = \frac{σ_{B}}{4 λ_{rp}}$	The e-beam spatial scale	$[length]$
$β = \frac{σ_{B}}{4 π} R_{sc}^{2} ω_{p}^{2} = \frac{e^{2}}{m} R_{sc}^{2} σ_{B} \overset{̊}{n}$	e-beam intensity	$\frac{{[length]}^{2}}{{[time]}^{2}}$
$β^{'} = \frac{β}{{\overset{̊}{v}}^{2}} = \frac{π σ_{B}}{λ_{rp}^{2}} = \frac{4 π g_{B}}{λ_{rp}}$	Dimensionless e-beam intensity	$[\dim - less]$

i	current	$\frac{[charge]}{[time]}$
q	Charge	$[charge]$
$\overset{̊}{n}$	Number of electrons p/u of volume	$\frac{[1]}{{[length]}^{3}}$
$λ_{rp} = \frac{2 π \overset{̊}{v}}{ω_{rp}}, ω_{rp} = R_{sc} ω_{p}$	The electron plasma wavelength	$[length]$
$g_{B} = \frac{σ_{B}}{4 λ_{rp}}$	The e-beam spatial scale	$[length]$
$β = \frac{σ_{B}}{4 π} R_{sc}^{2} ω_{p}^{2} = \frac{e^{2}}{m} R_{sc}^{2} σ_{B} \overset{̊}{n}$	e-beam intensity	$\frac{{[length]}^{2}}{{[time]}^{2}}$
$β^{'} = \frac{β}{{\overset{̊}{v}}^{2}} = \frac{π σ_{B}}{λ_{rp}^{2}} = \frac{4 π g_{B}}{λ_{rp}}$	Dimensionless e-beam intensity	$[\dim - less]$

We would like to point to an important spatial scale related to the e-beam, namely,

λ_{rp} = \frac{2 π \overset{̊}{v}}{R_{sc} ω_{p}}, ω_{rp} = R_{sc} ω_{p},

(2.3)

which is the distance passed by an electron for the time period

\frac{2 π}{ω_{rp}}

associated with the plasma oscillations at the reduced plasma frequency 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

g_{B} = \frac{σ_{B}}{4 λ_{rp}},

(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^{'} = \frac{β}{{\overset{̊}{v}}^{2}} = \frac{π σ_{B}}{λ_{rp}^{2}} = \frac{4 π g_{B}}{λ_{rp}} .

(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 = \frac{1}{\sqrt{C L}}, θ = \frac{b^{2}}{C} .

(2.6)

Following Refs. 10 and 20, we assume that

0 < \overset{̊}{v} < w .

(2.7)

A. TWT Lagrangian and evolution equations

Following the developments in Ref. 10, we introduce the TWT principal parameter

\bar{γ} = θ β

⁠. This parameter in view of Eqs. () and () can be represented as follows:

γ = θ β = \frac{b^{2}}{C} \frac{σ_{B}}{4 π} R_{sc}^{2} ω_{p}^{2} = \frac{b^{2}}{C} \frac{e^{2}}{m} R_{sc}^{2} σ_{B} \overset{̊}{n}, θ = \frac{b^{2}}{C}, β = \frac{e^{2}}{m} R_{sc}^{2} σ_{B} \overset{̊}{n} .

(2.8)

The TWT Lagrangian

L_{TB}

in the simplest case of a single transmission line and one stream e-beam is of the following form (Ref. 10, Chap. 4, 24):

L (\{Q\}, \{q\}) = L_{Tb} (\{Q\}, \{q\}) + L_{B} (\{q\}),

(2.9)

L_{Tb} = \frac{L}{2} {(\partial_{t} Q)}^{2} - \frac{1}{2 C} {(\partial_{z} Q + b \partial_{z} q)}^{2}, L_{B} = \frac{1}{2 β} {(\partial_{t} q + \overset{̊}{v} \partial_{z} q)}^{2} - \frac{2 π}{σ_{B}} q^{2},

where b > 0 is a coupling coefficient, and

\{Q\} = Q, \partial_{z} Q, \partial_{t} Q, Q = Q (z, t), \{q\} = q, \partial_{z} q, \partial_{t} q, q = q (z, t),

(2.10)

where

q (z, t)

and

Q (z, 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) = \int^{t} q (z, t^{'}) d t^{'}, . Q (z, t) = \int^{t} I (z, t^{'}) d t^{'} .

(2.11)

Note that the term

- \frac{2 π}{σ_{B}} q^{2}

in the Lagrangian

L_{B}

defined in Eq. () 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:

L \partial_{t}^{2} Q - \partial_{z} [C^{- 1} (\partial_{z} Q + b \partial_{z} q)] = 0,

(2.12)

\frac{1}{β} {(\partial_{t} + \overset{̊}{v} \partial_{z})}^{2} q + \frac{4 π}{σ_{B}} q - b \partial_{z} [C^{- 1} (\partial_{z} Q + b \partial_{z} q)] = 0,

(2.13)

where

\overset{̊}{v}

is the stationary velocity of electrons in the e-beam, σ_B is the area of the cross section of the e-beam, and β is the e-beam intensity defined by Eq. ().

B. Space-charge wave velocity and electron density fields

Following the field theory constructions in Refs. 10 and 21, we consider the total electron density

N = \overset{̊}{n} + n

and

V = \overset{̊}{v} + v

⁠, where

\overset{̊}{n}

and

\overset{̊}{v}

are, respectively, densities of the electron number and the electron velocity of the stationary dc electron flow, and

n = n (z, t)

and

v = v (z, t)

are, respectively, position and time dependent ac densities of the electron number and the electron velocity of the space-charge wave field

q = q (z, t)

and the electric field

E = E (z, t)

⁠. To comply with the linear theory approximation, we assume n and v to be relatively small,

|n| ≪ \overset{̊}{n}, |v| ≪ \overset{̊}{v} .

(2.14)

We also consider the ac current density field j = j(z, t) that satisfies

j = j (z, t) = - e (\overset{̊}{n} v + \overset{̊}{v} n), \partial_{t} (- e n) + \partial_{z} j = 0 .

(2.15)

Then, the following relations between charge density field

q = q (z, t)

and fields

n = n (z, t)

and

v = v (z, t)

hold (Ref. 10, 22.2):

\partial_{z} q = σ_{B} e n, \partial_{t} q = σ_{B} j = J,

(2.16)

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

n = \frac{1}{σ_{B} e} \partial_{z} q .

(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 J_v = J_v(z, t) are as follows (Ref. 10, 22.2):

e σ_{B} \overset{̊}{n} v = - D_{t} q, J_{v} = - e σ_{B} \overset{̊}{n} v = D_{t} q, D_{t} = \partial_{t} + \overset{̊}{v} \partial_{z},

(2.18)

where D_t is the so-called material time derivative. The second equation in () evidently implies that current J_v is exactly D_tq, whereas the first equation in () yields the following representations of the velocity v:

v = - \frac{1}{e σ_{B} \overset{̊}{n}} D_{t} q = - \frac{J_{v}}{e σ_{B} \overset{̊}{n}} .

(2.19)

The electric field E = E(z, t) associated with the space-charge wave satisfies the Poisson equation (Ref. 10, Sec. 22.2),

\partial_{z} E = - \frac{4 π}{σ_{B}} \partial_{z} q = - 4 π e n .

(2.20)

Under the additional natural assumption that “if there is no charges there is no electric field,” that is,

\bar{q} = 0

must imply E = 0, the above equation yields

E = - \frac{4 π}{σ_{B}} \bar{q .}

(2.21)

If we introduce the e-beam voltage

V_{b} = V_{b} (z, t) = - \partial_{z} E

⁠, then the Poisson equation [Eq. ()] can be recast as

\partial_{z}^{2} V_{b} = 4 π e n .

(2.22)

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.

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, $ℓ \in Z$ 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
$a Z : Z = \{\dots, - 2, - 1, 0, 1, 2, \dots\},$
(3.1)
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.

The CCTWT state is described by charges

q = q (z, t)

and

Q = Q (z, t)

for, respectively, the e-beam and the TL defined as the time integrals of the corresponding currents,

Q = Q (z, t) = \int^{t} I (z, t^{'}) d t^{'}, q = q (z, t) = \int^{t} i (z, t^{'}) d t^{'} .

(3.2)

Since according to the formulated above assumptions the interaction should occur only at the discrete set

a Z

(lattice) of points embedded into one-dimensional continuum of real numbers

R

⁠, some degree of singularity of functions

Q (z, t)

and

q (z, t)

is expected. As the analysis shows, it is appropriate to impose the following jump and continuity conditions on charge functions

Q (z, t)

and

q (z, t)

⁠.

Assumption 2.

(jump-continuity of charge functions).

Functions $Q (z, t)$ and $q (z, t)$ and their time derivatives $\partial_{t}^{j} Q (z, t)$ and $\partial_{t}^{j} q (z, t)$ for j = 1, 2 are continuous for all real t and z.
Derivatives $\partial_{t}^{j} Q (z, t)$ , $\partial_{t}^{j} Q (z, t)$ , $\partial_{z}^{j} q (z, t)$ , and $\partial_{z}^{j} q (z, t)$ for j = 1, 2 and the mixed derivatives $\partial_{z} \partial_{t} Q (z, t) = \partial_{t} \partial_{z} Q (z, t)$ , $\partial_{z} \partial_{t} q (z, t) = \partial_{t} \partial_{z} q (z, t)$ exist and continuous for all real t and z except for the interaction points in the lattice $a Z$ .
Let for a function $F (z)$ and a real number b symbols $F (b - 0)$ and $F (b + 0)$ stand for its left and right limit at b assuming their existence, that is,
$F (b \pm 0) = \lim_{z \to b \pm 0} F (z) .$
(3.3)

Let us also denote by

[F] (b)

the jump of function

F (z)

at b, that is,

[F] (b) = F (b + 0) - F (b - 0) .

(3.4)

The following right and left limits exist:

\partial_{z}^{j} Q (a ℓ \pm 0, t), \partial_{z}^{j} q (a ℓ \pm 0, t), j = 1, 2, ℓ \in Z,

(3.5)

and these limits are continuously differentiable functions of t. The values

\partial_{z} Q (a ℓ \pm 0, t)

and

\partial_{z} q (a ℓ \pm 0, t)

can be different, and consequently, the jumps

[\partial_{z} Q] (a ℓ, t)

and

[\partial_{z} q] (a ℓ, 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

[\partial_{z} q] (a ℓ, 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. (), implies that

[\partial_{z} q] (a ℓ, t) = σ_{B} e [n] (a ℓ, t) .

(3.6)

Equations () shows that jump

[\partial_{z} q] (a ℓ, t)

up to a multiplicative constant

\frac{1}{σ_{B} e}

represents jump

[n] (a ℓ, t) = \frac{[\partial_{z} q] (a ℓ, t)}{σ_{B} e}

in the number of electron density, manifesting the electron bunching that occurs in the EM cavity centered at aℓ. In view of Eq. (), we also have

[v] (a ℓ, t) = - \frac{\overset{̊}{v} [\partial_{z} q] (a ℓ, t)}{e σ_{B} \overset{̊}{n}}

⁠, 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.

I	Current	$\frac{[charge]}{[time]}$
Q	Charge	$[charge]$
c₀	Cavity capacitance	$[length]$
l₀	Cavity inductance	$\frac{{[time]}^{2}}{[length]}$
b	Coupling parameter	$[\dim - less]$

TABLE IV.

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

I	Current	$\frac{[charge]}{[time]}$
Q	Charge	$[charge]$
C	Shunt capacitance p/u of length	$[\dim - less]$
L	Series inductance p/u of length	$\frac{{[time]}^{2}}{{[length]}^{2}}$

A. CCTWT Lagrangian and the Euler–Lagrange equations

To simplify expressions, we use the following notations:

\{Q\} = Q, \partial_{z} Q, \partial_{t} Q, Q = Q (z, t), \{q\} = q, \partial_{z} q, \partial_{t} q, q = q (z, t),

(3.7)

x = [\begin{matrix} Q \\ q \end{matrix}], \{x\} = [\begin{matrix} Q \\ q \end{matrix}], [\begin{matrix} \partial_{z} Q \\ \partial_{z} q \end{matrix}], [\begin{matrix} \partial_{t} Q \\ \partial_{t} q \end{matrix}] .

(3.8)

The CCTWT Lagrangian

L

is defined as the sum of its three components: (i)

L_{T}

is the TL Lagrangian, (ii)

L_{B}

is the e-beam Lagrangian; and (iii)

L_{TB}

represent the TL and e-beam interaction Lagrangian. That is,

L (\{x\}) = L_{T} (\{Q\}) + L_{B} (\{q\}) + L_{TB} (x),

(3.9)

where we used notations () and (). The expressions for

L_{T}

and

L_{B}

are similar to the TWT Lagrangian components in Eqs. () and (), namely,

L_{T} (\{Q\}) = {L_{Tb}|}_{b = 0} = \frac{L}{2} {(\partial_{t} Q)}^{2} - \frac{1}{2 C} {(\partial_{z} Q)}^{2}, L_{B} (\{q\}) = \frac{1}{2 β} {(\partial_{t} q + \overset{̊}{v} \partial_{z} q)}^{2} - \frac{2 π}{σ_{B}} q^{2},

(3.10)

where

L_{Tb}

is defined by Eq. () and the interaction Lagrangian

L_{TB}

is defined by

L_{TB} (x) = \sum_{ℓ = - \infty}^{\infty} δ (z - a ℓ) \{\frac{l_{0}}{2} {(\partial_{t} Q (a ℓ))}^{2} - \frac{1}{2 c_{0}} {[Q (a ℓ) + b q (a ℓ)]}^{2}\} .

(3.11)

Parameters 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

L_{B}

in Eq. () represents the e-beam, and the term

- \frac{2 π}{σ_{B}} q^{2}

models the space-charge effects, including the so-called debunching (electron-to-electron repulsion). Lagrangian

L_{Tb}

in Eq. () integrates into it the interactions between the TL and the e-beam, whereas for the CCTWT, the Lagrangian

L_{T}

corresponds to the decoupled TL. This is why we set

L_{T}

to be

L_{Tb}

for b = 0. Note also that (i) expression () for the interaction Lagrangian

L_{TB}

is limited by design to the interaction points aℓ as indicated by delta functions

δ (z - a ℓ)

⁠, (ii) the factors before delta functions

δ (z - a ℓ)

are expressions similar to density

L_{Tb}

in Eq. () adapted to lattice

a Z

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

a Z

are

L \partial_{t}^{2} Q - C^{- 1} \partial_{z}^{2} Q = 0, \frac{1}{β} {(\partial_{t} + \overset{̊}{v} \partial_{z})}^{2} q + \frac{4 π}{σ_{B}} q = 0, z \neq a ℓ, ℓ \in Z,

(3.12)

or equivalently,

\partial_{z}^{2} Q - \frac{1}{w^{2}} \partial_{t}^{2} Q = 0, {(\frac{1}{v} \partial_{t} + \partial_{z})}^{2} q + \frac{4 π β}{σ_{B} {\overset{̊}{v}}^{2}} q = 0, z \neq a ℓ, ℓ \in Z .

(3.13)

The EL equations at the interaction points aℓ are

[Q] (a ℓ) = 0, [q] (a ℓ) = 0,

(3.14)

[\partial_{z} Q] (a ℓ) = C_{0} [(\frac{\partial_{t}^{2}}{ω_{0}^{2}} + 1) Q (a ℓ) + b q (a ℓ)], [\partial_{z} q] (a ℓ) = - \frac{b β_{0}}{{\overset{̊}{v}}^{2}} [Q (a ℓ) + b q (a ℓ)],

(3.15)

where we make use of parameters

C_{0} = \frac{C}{c_{0}}, ω_{0} = \frac{1}{\sqrt{l_{0} c_{0}}}, β_{0} = \frac{β}{c_{0}},

(3.16)

and jumps

[Q] (a ℓ)

⁠,

[q] (a ℓ)

⁠,

[\partial_{z} Q] (a ℓ)

⁠, and

[\partial_{z} q] (a ℓ)

are defined by Eq. (). We refer to β₀ as nodal e-beam interaction parameter. Note that Eq. () is just an acknowledgment of the continuity of charges

Q (z, t)

and

q (z, t)

at the interaction points in consistency with assumption 2. Equations () and () can be viewed as boundary conditions at interaction points that are complementary to the differential equations [Eqs. () and ()].

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:

[\partial_{z} x] (a ℓ) = P_{TB} x (a ℓ), x = [\begin{matrix} Q \\ q \end{matrix}], P_{TB} = [\begin{matrix} C_{0} (\frac{\partial_{t}^{2}}{ω_{0}^{2}} + 1) & C_{0} b \\ - \frac{b β_{0}}{{\overset{̊}{v}}^{2}} & - \frac{b^{2} β_{0}}{{\overset{̊}{v}}^{2}} \end{matrix}],

(3.17)

where parameters C₀ and β₀ are defined by Eq. (). Hence, the complete set of the boundary conditions at interaction points aℓ can be concisely written as

x (+ a ℓ) = x (- a ℓ), \partial_{z} x (+ a ℓ) = \partial_{z} x (- a ℓ) + P_{TB} x .

(3.18)

Consequently,

X (a ℓ + 0) = S_{b} X (a ℓ - 0), S_{b} = [\begin{matrix} I & 0 \\ P_{TB} & I \end{matrix}], X = [\begin{matrix} x \\ \partial_{z} x \end{matrix}], x = [\begin{matrix} Q \\ q \end{matrix}],

(3.19)

where

I

is the 2 × 2 identity matrix, and in view of Eq. (), we have

S_{b} = [\begin{matrix} I & 0 \\ P_{TB} & I \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ C_{0} (\frac{\partial_{t}^{2}}{ω_{0}^{2}} + 1) & C_{0} b & 1 & 0 \\ - \frac{b β_{0}}{{\overset{̊}{v}}^{2}} & - \frac{b^{2} β_{0}}{{\overset{̊}{v}}^{2}} & 0 & 1 \end{matrix}], C_{0} = \frac{C}{c_{0}}, β_{0} = \frac{β}{c_{0}} .

(3.20)

B. Natural units and dimensionless parameters

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.

Velocity	e-beam velocity	$\overset{̊}{v}$
Length	Period	a
Length	Wavelength	$\overset{̊}{λ} = \frac{1}{\overset{̊}{k}} = \frac{\overset{̊}{v}}{ω_{p}}$
Frequency	Period frequency	$ω_{a} = \frac{\overset{̊}{v}}{a}$
Frequency	Plasma frequency	$ω_{p} = \sqrt{\frac{4 π \overset{̊}{n} e^{2}}{m}}$
Time	Time of passing the period a	$\frac{1}{ω_{a}}$ = $\frac{a}{\overset{̊}{v}}$
Time	Plasma oscillation time period	$\overset{̊}{τ} = \frac{1}{ω_{p}}$

Velocity	e-beam velocity	$\overset{̊}{v}$
Length	Period	a
Length	Wavelength	$\overset{̊}{λ} = \frac{1}{\overset{̊}{k}} = \frac{\overset{̊}{v}}{ω_{p}}$
Frequency	Period frequency	$ω_{a} = \frac{\overset{̊}{v}}{a}$
Frequency	Plasma frequency	$ω_{p} = \sqrt{\frac{4 π \overset{̊}{n} e^{2}}{m}}$
Time	Time of passing the period a	$\frac{1}{ω_{a}}$ = $\frac{a}{\overset{̊}{v}}$
Time	Plasma oscillation time period	$\overset{̊}{τ} = \frac{1}{ω_{p}}$

Other variables that arise in our analysis are

λ_{rp} = \frac{2 π \overset{̊}{v}}{ω_{rp}}, ω_{rp} = R_{sc} ω_{p}, χ = \frac{w}{\overset{̊}{v}} = \frac{1}{\overset{̊}{v} \sqrt{C L}},

(3.21)

f_{B} = \sqrt{\frac{4 π β}{σ_{B} {\overset{̊}{v}}^{2}}} = \frac{ω_{rp}}{\overset{̊}{v}} = \frac{2 π}{λ_{rp}}, C_{0} = \frac{C}{c_{0}}, β_{0} = \frac{β}{c_{0}},

where ω_rp and λ_rp are, respectively, the reduced plasma frequency and the electron plasma wavelength.

The dimensionless variables of importance are

z^{'} = \frac{z}{a}, \partial_{z^{'}} = a \partial_{z}, t^{'} = \frac{\overset{̊}{v}}{a} t, \partial_{t^{'}} = \frac{a}{\overset{̊}{v}} \partial_{t}, ω^{'} = \frac{ω}{ω_{a}} = \frac{a}{2 π \overset{̊}{v}} ω, ω_{0}^{'} = \frac{ω_{0}}{ω_{a}} = \frac{a}{2 π \overset{̊}{v}} ω_{0},

(3.22)

L^{'} = {\overset{̊}{v}}^{2} L, C^{'} = C, β^{'} = \frac{β}{{\overset{̊}{v}}^{2}}, σ_{B}^{'} = \frac{σ_{B}}{a^{2}}, c_{0}^{'} = \frac{c_{0}}{a}, l_{0}^{'} = \frac{{\overset{̊}{v}}^{2}}{a} l_{0},

(3.23)

C_{0}^{'} = \frac{C^{'}}{c_{0}^{'}} = \frac{a C}{c_{0}} = a C_{0}, β_{0}^{'} = \frac{β^{'}}{c_{0}^{'}} = \frac{a β}{c_{0} {\overset{̊}{v}}^{2}}, f_{B}^{'} = a f_{B} = \sqrt{\frac{4 π β^{'}}{σ_{B}^{'}}} = \frac{a ω_{rp}}{\overset{̊}{v}} = \frac{2 π ω_{rp}}{ω_{a}} = \frac{2 π a}{λ_{rp}} .

(3.24)

Note also that since w = χv,

L^{'} C^{'} = \frac{{\overset{̊}{v}}^{2}}{w^{2}} = \frac{1}{χ^{2}}, a δ (z) = δ (z^{'}), z = a z^{'} .

(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.

a	The MCK period	$[length]$
$\overset{̊}{v}$	The e-beam stationary velocity	$\frac{[length]}{[time]}$
$ω_{a} = \frac{2 π \overset{̊}{v}}{a}$	The period frequency	$\frac{[1]}{[time]}$
$ω_{p} = \sqrt{\frac{4 π \overset{̊}{n} e^{2}}{m}}$	The plasma frequency	$\frac{[1]}{[time]}$
$λ_{rp} = \frac{2 π \overset{̊}{v}}{ω_{rp}}, ω_{rp} = R_{sc} ω_{p}$	The electron plasma wavelength	$[length]$
$g_{B} = \frac{σ_{B}}{4 λ_{rp}}$	The e-beam spatial scale	$[length]$
$f^{'} = a f = \frac{2 π ω_{rp}}{ω_{a}} = \frac{2 π a}{λ_{rp}}$	Normalized period in units of $\frac{λ_{rp}}{2 π}$	$[\dim - less]$
$\overset{̊}{n}$	The number of electrons p/u of volume	$\frac{[1]}{{[length]}^{3}}$
c₀, l₀	The cavity capacitance, inductance	$[length], \frac{{[time]}^{2}}{[length]}$
$ω_{0} = \frac{1}{\sqrt{l_{0} c_{0}}}$	The cavity resonant frequency	$\frac{[1]}{[time]}$
$β = \frac{σ_{B} R_{sc}^{2} ω_{p}^{2}}{4 π} = \frac{e^{2} R_{sc}^{2} σ_{B} \overset{̊}{n}}{m} = \frac{π σ_{B} {\overset{̊}{v}}^{2}}{λ_{rp}^{2}}$	The e-beam intensity	$\frac{{[length]}^{2}}{{[time]}^{2}}$
$β^{'} = \frac{β}{{\overset{̊}{v}}^{2}} = \frac{π σ_{B}}{λ_{rp}^{2}} = \frac{4 π g_{B}}{λ_{rp}}$	Dim-less e-beam intensity	$[\dim - less]$
$β_{0}^{'} = \frac{β^{'}}{c_{0}^{'}} = \frac{a β}{c_{0} {\overset{̊}{v}}^{2}}$	The first interaction par.	$[\dim - less]$
$B (ω) = B_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}, B_{0} = b^{2} β_{0}^{'}$	The second interaction par.	$[\dim - less]$
$K_{0} = \frac{B_{0}}{2 f} = \frac{b^{2} β_{0}^{'}}{2 f} = \frac{b^{2} σ_{B}}{4 λ_{rp} c_{0}} = \frac{b^{2} g_{B}}{c_{0}}$	The gain coefficient	$[\dim - less]$
$K (ω) = \frac{B (ω)}{2 f} = K_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}$	The gain parameter	$[\dim - less]$
$C_{0}^{'} = \frac{C^{'}}{c_{0}^{'}} = \frac{a C}{c_{0}} = a C_{0}$	The capacitance parameter	$[\dim - less]$

a	The MCK period	$[length]$
$\overset{̊}{v}$	The e-beam stationary velocity	$\frac{[length]}{[time]}$
$ω_{a} = \frac{2 π \overset{̊}{v}}{a}$	The period frequency	$\frac{[1]}{[time]}$
$ω_{p} = \sqrt{\frac{4 π \overset{̊}{n} e^{2}}{m}}$	The plasma frequency	$\frac{[1]}{[time]}$
$λ_{rp} = \frac{2 π \overset{̊}{v}}{ω_{rp}}, ω_{rp} = R_{sc} ω_{p}$	The electron plasma wavelength	$[length]$
$g_{B} = \frac{σ_{B}}{4 λ_{rp}}$	The e-beam spatial scale	$[length]$
$f^{'} = a f = \frac{2 π ω_{rp}}{ω_{a}} = \frac{2 π a}{λ_{rp}}$	Normalized period in units of $\frac{λ_{rp}}{2 π}$	$[\dim - less]$
$\overset{̊}{n}$	The number of electrons p/u of volume	$\frac{[1]}{{[length]}^{3}}$
c₀, l₀	The cavity capacitance, inductance	$[length], \frac{{[time]}^{2}}{[length]}$
$ω_{0} = \frac{1}{\sqrt{l_{0} c_{0}}}$	The cavity resonant frequency	$\frac{[1]}{[time]}$
$β = \frac{σ_{B} R_{sc}^{2} ω_{p}^{2}}{4 π} = \frac{e^{2} R_{sc}^{2} σ_{B} \overset{̊}{n}}{m} = \frac{π σ_{B} {\overset{̊}{v}}^{2}}{λ_{rp}^{2}}$	The e-beam intensity	$\frac{{[length]}^{2}}{{[time]}^{2}}$
$β^{'} = \frac{β}{{\overset{̊}{v}}^{2}} = \frac{π σ_{B}}{λ_{rp}^{2}} = \frac{4 π g_{B}}{λ_{rp}}$	Dim-less e-beam intensity	$[\dim - less]$
$β_{0}^{'} = \frac{β^{'}}{c_{0}^{'}} = \frac{a β}{c_{0} {\overset{̊}{v}}^{2}}$	The first interaction par.	$[\dim - less]$
$B (ω) = B_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}, B_{0} = b^{2} β_{0}^{'}$	The second interaction par.	$[\dim - less]$
$K_{0} = \frac{B_{0}}{2 f} = \frac{b^{2} β_{0}^{'}}{2 f} = \frac{b^{2} σ_{B}}{4 λ_{rp} c_{0}} = \frac{b^{2} g_{B}}{c_{0}}$	The gain coefficient	$[\dim - less]$
$K (ω) = \frac{B (ω)}{2 f} = K_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}$	The gain parameter	$[\dim - less]$
$C_{0}^{'} = \frac{C^{'}}{c_{0}^{'}} = \frac{a C}{c_{0}} = a C_{0}$	The capacitance parameter	$[\dim - less]$

C. Euler–Lagrange equations in dimensionless variables

The component Lagrangians represented in dimensionless variables are as follows:

L_{T}^{'} = \frac{L^{'}}{2} {(\partial_{t^{'}} Q)}^{2} - \frac{1}{2 C^{'}} {(\partial_{z^{'}} Q)}^{2}, L_{B}^{'} = \frac{1}{2 β^{'}} {(\partial_{t^{'}} q + \partial_{z^{'}} q)}^{2} - \frac{2 π}{σ_{B}^{'}} q^{2}, ℓ \in Z,

(3.26)

L_{TB}^{'} = - \frac{1}{2 c_{0}^{'}} \sum_{ℓ = - \infty}^{\infty} δ (z^{'} - ℓ) {[Q (ℓ) + b q (ℓ)]}^{2} .

(3.27)

The corresponding EL equations are

\partial_{t^{'}}^{2} Q - \frac{1}{χ^{2}} \partial_{z^{'}}^{2} Q = 0, {(\partial_{t^{'}} + \partial_{z^{'}})}^{2} q + f_{B}^{' 2} q = 0, z^{'} \neq ℓ, f_{B}^{'} = \sqrt{\frac{4 π β^{'}}{σ_{B}^{'}}} = \frac{R_{sc} ω_{p}}{ω_{a}},

(3.28)

[\partial_{z^{'}} Q] (ℓ) = C_{0}^{'} [(\frac{\partial_{t^{'}}^{2}}{ω_{0}^{' 2}} + 1) Q (ℓ) + b q (ℓ)], [\partial_{z^{'}} q] (ℓ) = - b β_{0}^{'} [Q (ℓ) + b q (ℓ)], ℓ \in 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)]:

\partial_{t}^{2} Q - \frac{1}{χ^{2}} \partial_{z}^{2} Q = 0, {(\partial_{t} + \partial_{z})}^{2} q + f^{2} q = 0, z \neq ℓ, ℓ \in Z, f = \frac{R_{sc} ω_{p}}{ω_{a}},

(3.30)

[\partial_{z} Q] (ℓ) = C_{0} [(\frac{\partial_{t}^{2}}{ω_{0}^{2}} + 1) Q (ℓ) + b q (ℓ)], [\partial_{z} q] (ℓ) = - b β_{0} [Q (ℓ) + b q (ℓ)], ℓ \in Z .

(3.31)

Equations () and () 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. () and () and obtain the following equations:

\partial_{z}^{2} \overset{̌}{Q} + \frac{ω^{2}}{χ^{2}} \overset{̌}{Q} = 0, {(\partial_{z} - i ω)}^{2} \overset{̌}{q} + f^{2} \overset{̌}{q} = 0, z \neq ℓ,

(3.32)

[\partial_{z} Q] (ℓ) = C_{0} [\frac{ω_{0}^{2} - ω^{2}}{ω_{0}^{2}} Q (ℓ) + b q (ℓ)], [\partial_{z} q] (ℓ) = - b β_{0} [Q (ℓ) + b q (ℓ)], ℓ \in Z,

(3.33)

where

\overset{̌}{Q}

and

\overset{̌}{q}

are the time Fourier transform of the corresponding quantities. Equations () and () 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. () and () yet another time into the following manifestly spatially periodic vector ODE:

\partial_{z}^{2} x + A_{1} \partial_{z} x + A_{0} x + \sum_{ℓ = - \infty}^{\infty} δ (z - ℓ) P_{TB} x = 0, x = [\begin{matrix} \overset{̌}{Q} \\ \overset{̌}{q} \end{matrix}],

(3.34)

where 2 × 2 matrices A_j and P_TB are defined by

A_{1} = A_{1} (ω) = [\begin{matrix} 0 & 0 \\ 0 & - 2 i ω \end{matrix}], A_{0} = A_{0} (ω) = [\begin{matrix} \frac{ω^{2}}{χ^{2}} & 0 \\ 0 & f^{2} - ω^{2} \end{matrix}],

(3.35)

P_{TB} = P_{TB} (ω) = [\begin{matrix} C_{0} \frac{ω_{0}^{2} - ω^{2}}{ω_{0}^{2}} & C_{0} b \\ - b β_{0} & - b^{2} β_{0} \end{matrix}] .

(3.36)

Equations (3.34)–(3.36) are evidently the second-order vector ODE with spatially periodic frequency dependent singular matrix potential

\sum_{ℓ = - \infty}^{\infty} δ (z - ℓ) P_{TB} (ω)

. These equations becomes the object of our studies below.

According to Appendix E, the second-order differential equation [Eqs. ()–()] is equivalent to the first-order spatially periodic differential equation of the following form:

\partial_{z} X = A (z) X, A (z) = A (z, ω) = [\begin{matrix} 0 & I \\ - A_{0} - P (z) & - A_{1} \end{matrix}], X = [\begin{matrix} x \\ \partial_{z} x \end{matrix}],

(3.37)

P (z) = P (z, ω) = \sum_{ℓ = - \infty}^{\infty} δ (z - ℓ) P_{TB} (ω),

where frequency dependent matrices A₀, A₁, and P_TB satisfy Eqs. () and ().

Using results of Subsection 2 of Appendix G, we find that the spatially periodic vector ODE () is Hamiltonian with the following choice of nonsingular Hermitian matrix

G = G (ω)

⁠:

G = G^{*} = [\begin{matrix} 0 & 0 & i & 0 \\ 0 & - \frac{2 ω C_{0}}{β_{0}} & 0 & - i \frac{C_{0}}{β_{0}} \\ - i & 0 & 0 & 0 \\ 0 & i \frac{C_{0}}{β_{0}} & 0 & 0 \end{matrix}], \det \{G\} = \frac{C_{0}^{2}}{β_{0}^{2}} .

(3.38)

Indeed, it is an elementary exercise to verify that for each value of z matrix

A (z)

is G-skew-Hermitian, that is,

G A (z) + A^{*} (z) G = 0 .

(3.39)

Then, if

Φ (z)

is the matrizant of the Hamiltonian equation [Eq. ()], then according to results of Appendix G,

Φ (z)

is the G-unitary matrix satisfying

Φ^{*} (z) G Φ (z) = G .

(3.40)

Consequently, its spectrum

σ \{Φ (z)\}

is invariant with respect to the inversion transformation

ζ \to \frac{1}{\bar{ζ}}

⁠, that is, it is symmetric with respect to the unit circle,

ζ \in σ \{Φ (z)\} \Rightarrow \frac{1}{\bar{ζ}} \in σ \{Φ (z)\} .

(3.41)

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.

One way to identify the CCS and the e-beam components of the CCTWT is to use its analysis carried out in Secs. III A–III C, setting there the coupling coefficient b to be zero. With that in mind, we consider the monodromy matrix

T

defined by Eqs. ()–() and set there b = 0. To separate variables relevant to the CCS and the e-beam, we use permutation matrix P₂₃ defined by Eq. () and transform

{T|}_{b = 0}

as follows:

P_{23} {T|}_{b = 0} P_{23}^{- 1} = [\begin{matrix} T_{C} & 0 \\ 0 & T_{B} \end{matrix}],

(3.42)

where

T_{C}

and

T_{B}

are 2 × 2 matrices defined by

T_{C} = [\begin{matrix} \cos (\frac{ω}{χ}) & \frac{χ}{ω} \sin (\frac{ω}{χ}) \\ (1 - \frac{ω^{2}}{ω_{0}^{2}}) C_{0} \cos (\frac{ω}{χ}) - \frac{ω}{χ} \sin (\frac{ω}{χ}) & (1 - \frac{ω^{2}}{ω_{0}^{2}}) \frac{C_{0} χ}{ω} \sin (\frac{ω}{χ}) + \cos (\frac{ω}{χ}) \end{matrix}],

(3.43)

T_{B} = e^{i ω} [\begin{matrix} \cos (f) - i \frac{\sin (f)}{f} ω & \frac{\sin (f) e^{i ω}}{f} \\ \frac{(ω^{2} - f^{2}) \sin (f)}{f} & (\cos (f) + i \frac{\sin (f)}{f} ω) e^{i ω} \end{matrix}] .

(3.44)

Evidently,

T_{C}

defined by Eq. () is the CCS monodromy matrix and

T_{B}

defined by Eq. () is the e-beam monodromy matrix.

It follows from Eq. () for matrix

T_{C}

that the corresponding characteristic equation

\det \{T_{C} - s I\} = 0

for the Floquet multipliers s (see Appendix F) is

\det \{T_{C} - s I\} = s^{2} + 2 W_{C} (ω) s + 1 = 0, s = \exp \{i k\},

(3.45)

W_{C} (ω) = \frac{C_{0}}{2} (\frac{ω}{ω_{0}^{2}} - \frac{1}{ω}) χ \sin (\frac{ω}{χ}) - \cos (ω) .

The real-valued function

W_{C} (ω)

in the second equation in () plays an important role in the analysis of the CCS, and its plot is depicted in Fig. 16(b). We refer to

W_{C} (ω)

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. () for matrix

T_{B}

that the corresponding characteristic equation

\det \{T_{B} - s I\} = 0

for the Floquet multipliers s is

\det \{T_{B} - s I\} = s^{2} - 2 \cos (f) e^{i ω} s + e^{2 i ω} = 0, s = \exp \{i k\} .

(3.46)

In view of Eqs. (), (), and (), the following factorization holds for the characteristic function of the monodromy matrix

{T|}_{b = 0}

of the decoupled system:

\det \{{T|}_{b = 0} - s I\} = (s^{2} + 2 W_{C} (ω) s + 1) (s^{2} - 2 \cos (f) e^{i ω} s + e^{2 i ω}) .

(3.47)

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

We begin with introducing expressions for (i) the characteristic scalar polynomial

A_{T} (s)

associated with the first equation in () for the TL and (ii) the characteristic scalar polynomial

A_{B} (s)

associated with the second equation in () for the e-beam,

A_{T} (s) = \frac{ω^{2}}{χ^{2}} + s^{2}, A_{B} = s^{2} - 2 i ω s + f^{2} - ω^{2} .

(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

A_{T} (s)

and

A_{B} (s)

represents symbolically the differential operator ∂_z.

The 2 × 2 companion matrix

C_{T}

of the scalar characteristic polynomial

A_{T} (s)

(see Appendixes C– E) is

C_{T} = [\begin{matrix} 0 & 1 \\ - \frac{ω^{2}}{χ^{2}} & 0 \end{matrix}] = Z_{T} [\begin{matrix} i \frac{ω}{χ} & 0 \\ 0 & - i \frac{ω}{χ} \end{matrix}] Z_{T}^{- 1}, Z_{T} = [\begin{matrix} - i \frac{χ}{ω} & i \frac{χ}{ω} \\ 1 & 1 \end{matrix}], x = [\begin{matrix} Q \\ \partial_{z} Q \end{matrix}],

(4.2)

where the columns of matrix

Z_{T}

are eigenvectors of the companion matrix

C_{T}

with the corresponding eigenvalues being the relevant entries of the diagonal matrix in Eq. (). The expression of vector x in Eq. () clarifies the meaning of entries of relevant matrices. Consequently, the exponent

\exp \{z C_{T}\}

that is the fundamental matrix solution to the first-order ODE associated with the first equation in () satisfies

\exp \{z C_{T}\} = [\begin{matrix} \cos (\frac{ω z}{χ}) & \frac{χ}{ω} \sin (\frac{ω z}{χ}) \\ - \frac{ω}{χ} \sin (\frac{ω}{χ}) & \cos (\frac{z ω}{χ}) \end{matrix}] = Z_{T} \exp \{[\begin{matrix} i \frac{ω z}{χ} & 0 \\ 0 & - i \frac{ω z}{χ} \end{matrix}]\} Z_{T}^{- 1} .

(4.3)

The 2 × 2 companion matrix

C_{B}

of the scalar characteristic polynomial

A_{B} (s)

is given by

C_{B} = [\begin{matrix} 0 & 1 \\ ω^{2} - f^{2} & 2 i ω \end{matrix}] = Z_{B} [\begin{matrix} i (ω - f) & 0 \\ 0 & i (ω + f) \end{matrix}] Z_{B}^{- 1}, Z_{B} = [\begin{matrix} - \frac{i}{ω - f} & - \frac{i}{ω + f} \\ 1 & 1 \end{matrix}],

(4.4)

x = [\begin{matrix} q \\ \partial_{z} q \end{matrix}],

where the columns of matrix

Z_{B}

are eigenvectors of the companion matrix

C_{B}

with the corresponding eigenvalues being the relevant entries of the diagonal matrix in Eq. (). The expression of vector x in Eq. () clarifies the meaning of entries of relevant matrices. Consequently, the exponent

\exp \{z C_{B}\}

that is the fundamental matrix solution to the first-order ODE associated with the second equation in () satisfies

\exp \{z C_{B}\} = \frac{1}{f} \exp \{i ω z\} [\begin{matrix} f \cos (f z) - i ω \sin (f z) & \sin (f z) \\ (ω^{2} - f^{2}) \sin (f z) & f \cos (f z) + i ω \sin (f z) \end{matrix}]

(4.5)

= Z_{B} \exp \{[\begin{matrix} i (ω - f) z & 0 \\ 0 & i (ω + f) z \end{matrix}]\} Z_{B}^{- 1} .

The 2 × 2 matrix characteristic polynomial

A_{TB} (s)

of non-interacting TL and the e-beam is the following diagonal matrix polynomial:

A_{TB} (s) = [\begin{matrix} A_{T} (s) & 0 \\ 0 & A_{B} (s) \end{matrix}] = [\begin{matrix} \frac{ω^{2}}{χ^{2}} + s^{2} & 0 \\ 0 & s^{2} - 2 i ω s + f^{2} - ω^{2} \end{matrix}] .

(4.6)

The 4 × 4 companion matrix of the matrix polynomial

A_{TB} (s) 4 \times 4

is (see Appendixes C– E)

C_{TB} = [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - \frac{ω^{2}}{χ^{2}} & 0 & 0 & 0 \\ 0 & ω^{2} - f^{2} & 0 & 2 i ω \end{matrix}], X = [\begin{matrix} Q \\ q \\ \partial_{z} Q \\ \partial_{z} q \end{matrix}],

(4.7)

where vector X clarifies the meaning of the entries of matrix

C_{TB}

⁠.

The exponential

\exp \{z C_{TB}\}

of the component matrix

C_{TB}

is the fundamental matrix solution to the first-order ODE associated with the system of Eq. (), and it satisfies

\exp \{z C_{TB}\} = [\begin{matrix} \cos (\frac{ω z}{χ}) & 0 & \frac{χ}{ω} \sin (\frac{ω z}{χ}) & 0 \\ 0 & e^{i ω z} [\cos (f z) - i \frac{ω}{f} \sin (f z)] & 0 & \frac{1}{f} e^{i ω} \sin (f z) \\ - \frac{ω}{χ} \sin (\frac{ω z}{χ}) & 0 & \cos (\frac{ω z}{χ}) & 0 \\ 0 & \frac{1}{f} e^{i ω z} (ω^{2} - f^{2}) \sin (z f) & 0 & e^{i ω z} [\cos (f z) - i \frac{ω}{f} \sin (f z)] \end{matrix}] .

(4.8)

In particular, for z = 1, which is the period in dimensionless variables, we get

\exp \{C_{TB}\} = [\begin{matrix} \cos (\frac{ω}{χ}) & 0 & \frac{χ}{ω} \sin (\frac{ω}{χ}) & 0 \\ 0 & e^{i ω} [\cos (f) - i \frac{ω}{f} \sin (f)] & 0 & \frac{1}{f} e^{i ω} \sin (f) \\ - \frac{ω}{χ} \sin (\frac{ω}{χ}) & 0 & \cos (\frac{ω}{χ}) & 0 \\ 0 & \frac{1}{f} e^{i ω} (ω^{2} - f^{2}) \sin (f) & 0 & e^{i ω} [\cos (f) - i \frac{ω}{f} \sin (f)] \end{matrix}] .

(4.9)

Using 4 × 4 matrix P₂₃ that permutes the second and third coordinates, that is,

P_{23} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] = P_{23}^{- 1},

(4.10)

we get the following representation of

\exp \{z C_{TB}\}

in terms and

\exp \{z C_{T}\}

and

\exp \{z C_{B}\}

⁠:

\exp \{z C_{TB}\} = P_{23} [\begin{matrix} \exp \{z C_{T}\} & 0 \\ 0 & \exp \{z C_{B}\} \end{matrix}] P_{23}^{- 1} .

(4.11)

One can also verify the correctness of identity () by tedious straightforward evaluation.

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

The fundamental matrix solution

\exp \{z C_{TB}\}

represented by Eq. () 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. (). These boundary conditions are equivalent to

X (ℓ + 0) = S_{b} X (ℓ - 0), S_{b} = [\begin{matrix} I & 0 \\ P_{TB} & I \end{matrix}], X = [\begin{matrix} x \\ \partial_{z} x \end{matrix}], x = [\begin{matrix} \overset{̌}{Q} \\ \overset{̌}{q} \end{matrix}],

(4.12)

where the boundary matrix

S_{b}

satisfies

S_{b} = [\begin{matrix} I & 0 \\ P_{TB} & I \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ (1 - \frac{ω^{2}}{ω_{0}^{2}}) C_{0} & C_{0} b & 1 & 0 \\ - b β_{0} & - b^{2} β_{0} & 0 & 1 \end{matrix}] .

(4.13)

The Floquet theory (see Appendix F) when applied to the first-order ODE equivalent to the EL equations [Eqs. ()–()] yields the following equations for the fundamental 4 × 4 matrix solution

Φ (z)

⁠:

Φ (z) = \exp \{(z - ℓ) C_{TB}\} Φ (ℓ + 0), ℓ < z < ℓ + 1,

(4.14)

Φ (0 + 0) = I, Φ (ℓ + 0) = S_{b} Φ (ℓ - 0), ℓ \in Z,

where

\exp \{z C_{TB}\}

is defined by Eq. (). In particular, according to the Floquet theorem (Theorem 22), the monodromy matrix is

T = Φ (1 + 0)

⁠, and in view of Eqs. () and (), we have

T = Φ (1 + 0) = S_{b} \exp \{C_{TB}\} = [\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix}],

(4.15)

where

\exp \{C_{TB}\}

is defined by Eq. () and 2 × 2 matrix blocks of

T

are as follows:

T_{11} = [\begin{matrix} \cos (\frac{ω}{χ}) & 0 \\ 0 & (\cos (f) - i \frac{\sin (f)}{f} ω) e^{i ω} \end{matrix}], T_{12} = [\begin{matrix} \frac{χ}{ω} \sin (\frac{ω}{χ}) & 0 \\ 0 & \frac{\sin (f) e^{i ω}}{f} \end{matrix}],

(4.16)

T_{21} = [\begin{matrix} (1 - \frac{ω^{2}}{ω_{0}^{2}}) C_{0} \cos (\frac{ω}{χ}) - \frac{ω}{χ} \sin (\frac{ω}{χ}) & C_{0} b (\cos (f) - i \frac{\sin (f)}{f} ω) e^{i ω} \\ - β_{0} b \cos (\frac{ω}{χ}) & [β_{0} b^{2} (i \frac{\sin (f)}{f} ω - \cos (f)) + \frac{(ω^{2} - f^{2}) \sin (f)}{f}] e^{i ω} \end{matrix}],

(4.17)

T_{22} = [\begin{matrix} (1 - \frac{ω^{2}}{ω_{0}^{2}}) \frac{C_{0} χ}{ω} \sin (\frac{ω}{χ}) + \cos (\frac{ω}{χ}) & \frac{C_{0} b \sin (f) e^{i ω}}{f} \\ - \frac{β_{0} b χ}{ω} \sin (\frac{ω}{χ}) & [(i \frac{\sin (f)}{f} ω + \cos (f)) - \frac{β_{0} b^{2} \sin (f)}{f}] e^{i ω} \end{matrix}],

(4.18)

and the involved above dimensionless constants satisfy (see Sec. III B)

χ = \frac{w}{\overset{̊}{v}} = \frac{1}{\overset{̊}{v} \sqrt{C L}}, C_{0} = \frac{a C}{c_{0}}, β_{0} = \frac{a β}{c_{0} {\overset{̊}{v}}^{2}}, f = \frac{a R_{sc} ω_{p}}{\overset{̊}{v}} = \frac{R_{sc} ω_{p}}{ω_{a}} .

(4.19)

An important special and simpler case of the monodromy matrix

T

defined by Eqs. ()–() is when the following equations hold:

χ = 1, b = 1 .

(4.20)

The condition χ = 1 according to Eq. () is equivalent to the equality of the phase velocity w associated with TL and the e-beam stationary flow velocity

\overset{̊}{v}

⁠. 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

T_{1} = [\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix}], χ = 1, b = 1,

(4.21)

where

T_{11} = [\begin{matrix} \cos (ω) & 0 \\ 0 & (\cos (f) - i \frac{\sin (f)}{f} ω) e^{i ω} \end{matrix}], T_{12} = [\begin{matrix} \frac{\sin (ω)}{ω} & 0 \\ 0 & \frac{\sin (f) e^{i ω}}{f} \end{matrix}],

(4.22)

T_{21} = [\begin{matrix} (1 - \frac{ω^{2}}{ω_{0}^{2}}) C_{0} \cos (ω) - ω \sin (ω) & C_{0} (\cos (f) - i \frac{\sin (f)}{f} ω) e^{i ω} \\ - β_{0} \cos (ω) & [β_{0} (i \frac{\sin (f)}{f} ω - \cos (f)) + \frac{(ω^{2} - f^{2}) \sin (f)}{f}] e^{i ω} \end{matrix}],

(4.23)

T_{22} = [\begin{matrix} (1 - \frac{ω^{2}}{ω_{0}^{2}}) \frac{C_{0} \sin (ω)}{ω} + \cos (ω) & \frac{C_{0} \sin (f) e^{i ω}}{f} \\ - \frac{β_{0} \sin (ω)}{ω} & [(i \frac{\sin (f)}{f} ω + \cos (f)) - \frac{β_{0} \sin (f)}{f}] e^{i ω} \end{matrix}],

(4.24)

where dimensionless constants C₀, β₀, and f satisfy Eq. ().

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

We turn now to the four Floquet multipliers s, which are the eigenvalues of the CCTWT monodromy matrix

T

defined by Eqs. ()–(). Consequently, s are solutions to characteristic equation

\det \{T - s I\} = 0

⁠, which represents the following polynomial equation of order 4 (see Fig. 3):

S^{4} + c_{3} S^{3} + c_{2} S^{2} + {\bar{c}}_{3} S + 1 = 0, s = \exp \{i k\}, S = s e^{- i \frac{ω}{2}} = \exp \{i (k - \frac{ω}{2})\},

(5.1)

where k is the wavenumber that can be real or complex-valued and coefficients c₃ and c₂ satisfy

c_{3} = 2 e^{\frac{i}{2} ω} b_{f}^{\infty} + e^{- \frac{i}{2} ω} [C_{0} \frac{χ}{ω} \sin (\frac{ω}{χ}) (\frac{ω^{2}}{ω_{0}^{2}} - 1) - 2 \cos (\frac{ω}{χ})],

(5.2)

c_{2} = 2 b_{f}^{\infty} [C_{0} \frac{χ}{ω} \sin (\frac{ω}{χ}) \frac{ω^{2}}{ω_{0}^{2}} - 2 \cos (\frac{ω}{χ})] + 2 \cos (f) C_{0} \frac{χ}{ω} \sin (\frac{ω}{χ}) + 2 \cos (ω),

(5.3)

where

b_{f}^{\infty} = K_{0} \sin (f) - \cos (f), K_{0} = \frac{b^{2} β_{0}}{2 f} = \frac{b^{2} g_{B}}{c_{0}}, g_{B} = \frac{σ_{B}}{4 λ_{rp}} .

(5.4)

Note that the quantity

b_{f}^{\infty}

in Eq. () arises also in the theory of the MCK reviewed in Sec. VIII [see Eq. ()]. Note also that coefficient c₂ defined by Eq. () is manifestly real. The utility of representing the Floquet multipliers s in the form

s = S e^{i \frac{ω}{2}}

is explained by the fact that Eq. () for S possesses a manifest symmetry: if S is a solution to Eq. (), then

\frac{1}{\bar{S}}

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

\overset{̊}{v}

, namely,

w = \overset{̊}{v}

. It is also convenient to choose frequency units so that ω₀ = 1. Combining these two conditions, we assume

χ = 1, ω_{0} = 1 .

(5.5)

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 $s = \exp \{i k\}$ 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. ()–() turn into

S^{4} + c_{3} S^{3} + c_{2} S^{2} + {\bar{c}}_{3} S + 1 = 0, s = \exp \{i k\}, S = \exp \{i (k - \frac{ω}{2})\},

(6.1)

c_{3} = 2 e^{\frac{i}{2} ω} b_{f}^{\infty} + e^{- \frac{i}{2} ω} [C_{0} \frac{ω^{2} - 1}{ω} \sin (ω) - 2 \cos (ω)],

(6.2)

c_{2} = 2 b_{f}^{\infty} [C_{0} ω \sin (ω) - 2 \cos (ω)] + 2 \cos (f) C_{0} \frac{\sin (ω)}{ω} + 2 \cos (ω),

(6.3)

where

b_{f}^{\infty} = K_{0} \sin (f) - \cos (f), K_{0} = \frac{b^{2} β_{0}}{2 f} = \frac{b^{2} g_{B}}{c_{0}}, g_{B} = \frac{σ_{B}}{4 λ_{rp}} .

(6.4)

Yet another form of dispersion relations (), under the same exact synchronism assumption χ = 1 and ω₀ = 1, is its high-frequency form, namely,

D (ω, k) = D^{(0)} (ω, k) + \frac{D^{(1)} (ω, k)}{ω} + \frac{D^{(2)} (ω, k)}{ω^{2}} = 0,

(6.5)

where

D^{(0)} (ω, k) = C_{0} \sin (ω) (b_{f}^{\infty} + \cos (ω - k)),

(6.6)

D^{(1)} (ω, k) = 2 (\cos (k) - \cos (ω)) b_{f}^{\infty} + \cos (2 k - ω) - \cos (k - 2 ω) + \cos (ω) - \cos (k),

(6.7)

D^{(2)} (ω, k) = (\cos (f) - \cos (k - ω)) \sin (ω) C_{0},

(6.8)

where

b_{f}^{\infty}

is defined by Eq. (). We refer to function

D (ω, k)

in Eq. () as the CCTWT dispersion function.

There exists a remarkability in its simplicity relations between the CCTWT dispersion function $D (ω, k)$ and the dispersion functions $D_{C} (ω, k)$ and $D_{K} (ω, k)$ 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.

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The plots of the four complex eigenvalues (the Floquet multipliers) $s = S e^{i \frac{ω}{2}}$ that solve the characteristic equation [Eq. (5.1)] for the monodromy matrix $T_{1}$ 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, $R \{s\}$ and $I \{s\}$ ⁠. The eigenvalues are shown by (blue) solid dots.

Theorem 2

(dispersion function factorization). Let us assume that χ = ω₀ = 1. Let the CCTWT, the CCS, and the MCK dispersion functions

D (ω, k)

,

D_{C} (ω, k)

⁠, and

D_{K} (ω, k)

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

D (ω, k) - D_{C} (ω, k) D_{K} (ω, k) = \frac{K_{0}}{ω} [\frac{2 (\cos (ω) - \cos (k))}{ω^{2} - 1} - \frac{C_{0} \sin (ω) (1 - \sin (f))}{ω}],

(6.9)

K_{0} = \frac{b^{2} β_{0}}{2 f} = \frac{b^{2} g_{B}}{c_{0}}, g_{B} = \frac{σ_{B}}{4 λ_{rp}} .

(6.10)

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

D^{(0)} (ω, k) = D_{C}^{(0)} (ω, k) D_{K}^{(0)} (ω, k) = C_{0} \sin (ω) (b_{f}^{\infty} + \cos (ω - k)) .

(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 Rk∈−3π,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±ω≠0, that is, there is an instability, and Rk+ω=Rk−ω, 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±.

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The dispersion-instability graphs for the CCTWT as the gain coefficient K₀ varies. In all plots, the horizontal and vertical axes represent, respectively, $R \{k\}$ 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 $R \{k\} \in [- 3 π, 3 π]$ for χ = 1, ω₀ = 1, f = 1, and C₀ = 1: (a) K₀ = 2, f_cr ≅ 0.927 292 180, and f_max ≅ 2.034; (b) K₀ = 2.25, f_cr ≅ 0.837, and f_max ≅ 1.99; (c) K₀ = 2.3, f_cr ≅ 0.820, and f_max ≅ 1.981; (d) K₀ = 2.5, f_cr ≅ 0.761, and f_max ≅ 1.951; (e) K₀ = 2.7, f_cr ≅ 0.709, and f_max ≅ 1.926; and (f) K₀ = 5, f_cr ≅ 0.395, and f_max ≅ 1.768 (see Theorem 2). When $I \{k_{\pm} (ω)\} = 0$ ⁠, which is the case of oscillatory modes, and $R \{k (ω)\} = k (ω)$ ⁠, the corresponding branches are shown as (blue) solid curves. When $I \{k_{\pm} (ω)\} \neq 0$ ⁠, that is, there is an instability, and $R \{k_{+} (ω)\} = R \{k_{-} (ω)\}$ ⁠, 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 Rk∈−3π,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±ω≠0, that is, there is an instability, and Rk+ω=Rk−ω, 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±.

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The dispersion-instability graphs for the CCTWT as the capacitance parameter C₀ varies. In all plots, the horizontal and vertical axes represent, respectively, $R \{k\}$ 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 $R \{k\} \in [- 3 π, 3 π]$ for χ = 1, ω₀ = 1, f = 1, and K₀ = 2, f_cr ≅ 0.927 292 180, and f_max ≅ 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 $I \{k_{\pm} (ω)\} = 0$ ⁠, that is, the case of oscillatory modes, and $R \{k (ω)\} = k (ω)$ ⁠, the corresponding branches are shown as (blue) solid curves. When $I \{k_{\pm} (ω)\} \neq 0$ ⁠, that is, there is an instability, and $R \{k_{+} (ω)\} = R \{k_{-} (ω)\}$ ⁠, 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.

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The dispersion-instability graphs for the CCTWT as the normalized period f varies. In all plots, the horizontal and vertical axes represent, respectively, $R \{k\}$ 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 $R \{k\} \in [- 3 π, 3 π]$ for χ = 1, ω₀ = 1, C₀ = 1, K₀ = 2, f_cr ≅ 0.927 292 180, and f_max ≅ 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 $I \{k_{\pm} (ω)\} = 0$ ⁠, that is, the case of oscillatory modes, and $R \{k (ω)\} = k (ω)$ ⁠, the corresponding branches are shown as (blue) solid curves. When $I \{k_{\pm} (ω)\} \neq 0$ ⁠, that is, there is an instability, and $R \{k_{+} (ω)\} = R \{k_{-} (ω)\}$ ⁠, 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. 7.

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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, $R \{k\}$ and ω. The plot shows three bands of the dispersion of the CCTWT described by Eqs. (5.1)–(5.4) over three Brillouin zones $R \{k\} \in [- 3 π, 3 π]$ for χ = 1, ω₀ = 1, C₀ = 1, f ≅ 1.77, K₀ = K_0T = 4.95 (typical value) and, consequently, f_cr ≅ 0.397, f_max ≅ 1.770. When $I \{k_{\pm} (ω)\} = 0$ ⁠, which is the case of oscillatory modes, and $R \{k (ω)\} = k (ω)$ ⁠, the corresponding branches are shown as (blue) solid curves. When $I \{k_{\pm} (ω)\} \neq 0$ ⁠, that is, there is an instability, and $R \{k_{+} (ω)\} = R \{k_{-} (ω)\}$ ⁠, 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).

Let us consider now the conventional dispersion relation, assuming that k and ω must be real numbers. Then, dividing Eq. () by S² and carrying elementary transformations, we arrive at the following trigonometric form of the conventional dispersion relation:

\cos (2 k - ω) + |c_{3}| \cos (k - \frac{ω}{2} + α) = - \frac{c_{2}}{2},

(6.12)

c_{3} = |c_{3}| \exp \{i α\}, S = \exp \{i (k - \frac{ω}{2})\}, k, ω \in R,

where

c_{3} = c_{3} (ω)

⁠,

c_{2} = c_{2} (ω)

⁠, and

α = \arg \{c_{3} (ω)\}

are frequency dependent parameters satisfying Eqs. ()–().

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 $(k (ω), ω)$ ⁠, where ω is its frequency and $k (ω)$ is its wavenumber. If $k (ω)$ 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 $I \{k (ω)\} \neq 0$ ⁠, we partition all the system modes represented by pairs $(k (ω), ω)$ into two distinct classes—oscillatory modes and unstable ones—based on whether the wavenumber $k (ω)$ is real- or complex-valued with $I \{k (ω)\} \neq 0$ ⁠. We refer to a mode (eigenmode) of the system as an oscillatory mode if its wavenumber $k (ω)$ is real-valued. We associate with such an oscillatory mode point $(k (ω), ω)$ 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, $I \{k (ω)\} \neq 0$ ⁠. We associate with such an unstable mode point $(R \{k (ω)\}, ω)$ in the kω-plane. Since we consider here only convective unstable modes, we refer to them shortly as unstable modes. Note that every point $(R \{k (ω)\}, ω)$ is, in fact, associated with two complex conjugate system modes with $\pm I \{k (ω)\}$ ⁠.

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 $(k (ω), ω)$ and $(R \{k (ω)\}, ω)$ in the kω-plane. We name this set the dispersion-instability graph. To distinguish graphically points $(k (ω), ω)$ associated oscillatory modes when $k (ω)$ is real-valued from points $(R \{k (ω)\}, ω)$ associated unstable modes when $k (ω)$ is complex-valued with $I \{k (ω)\} \neq 0$ ⁠, we show points $I \{k (ω)\} = 0$ in blue color, whereas points with $I \{k (ω)\} \neq 0$ are shown in brown color. We remind once again that every point $(ω, R \{k (ω)\})$ with $I \{k (ω)\} \neq 0$ represents exactly two complex conjugate unstable modes associated with $\pm I \{k (ω)\}$ ⁠.

When $I \{k_{\pm} (ω)\} \neq 0$ and $R \{k_{+} (ω)\} = R \{k_{-} (ω)\}$ ⁠, 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 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.

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.

Applying the results of Appendix 12, particularly using the system of Eq. (), we obtain the following trigonometric form of equations for exceptional points of degeneracy (EPDs):

\cos (2 k - ω) + |c_{3}| \cos (k - \frac{ω}{2} + α) = - \frac{c_{2}}{2}, 2 \sin (2 k - ω) + |c_{3}| \sin (k - \frac{ω}{2} + α) = 0,

(6.13)

c_{3} = |c_{3}| \exp \{i α\} S = \exp \{i (k - \frac{ω}{2})\}, k, ω \in R .

Note that the first equation in () is identical to the trigonometric form () 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.

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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, f_cr ≅ 0.927, f_max ≅ 1.770: (a) f = 0.9 < f_cr, (b) f = 1.1 > f_cr, and (c) f = 1.5 > f_cr (see Theorem 2 and Remark 3). In all plots, the horizontal and vertical axes represent, respectively, $R \{k\}$ and ω. Solid (grin) diamond dots identify points of the transition from the instability to the stability, which are also EPD points.

VII. GAIN EXPRESSION IN TERMS OF THE FLOQUET MULTIPLIERS

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. ()–()], namely,

G = G (f, ω, K_{0}) = 20 |\log (|s (f, ω, K_{0})|)| .

(7.1)

Consequently, to analyze gain expression (), we have to turn to the indicated characteristic equations.

In turns out that gain expression () can be significantly simplified under exact synchronism assumption 3, that is, χ = 1 and ω₀ = 1. Under these conditions, the CCTWT characteristic equations [Eqs. ()–()] can be recast into the following form:

P = P (f, ω, K_{0}) = s^{4} + p_{3} s^{3} + p_{2} s^{2} + p_{1} s + 1 = 0,

(7.2)

where the coefficients of the CCTWT characteristic polynomial P have the following expressions:

p_{3} = 2 e^{i ω} b_{f}^{\infty} + C_{0} \frac{ω^{2} - 1}{ω} \sin (ω) - 2 \cos (ω),

(7.3)

p_{2} = e^{i ω} \{2 b_{f}^{\infty} [C_{0} ω \sin (ω) - 2 \cos (ω)] + 2 \cos (f) C_{0} \frac{\sin (ω)}{ω} + 2 \cos (ω)\},

(7.4)

p_{1} = e^{2 i ω} [C_{0} \frac{ω^{2} - 1}{ω} \sin (ω) - 2 \cos (ω)] + 2 e^{i ω} b_{f}^{\infty} = e^{2 i ω} {\bar{p}}_{3},

(7.5)

where

b_{f}^{\infty} = K_{0} \sin (f) - \cos (f), K_{0} = \frac{b^{2} β_{0}}{2 f} = \frac{b^{2} g_{B}}{c_{0}}, g_{B} = \frac{σ_{B}}{4 λ_{rp}} .

(7.6)

According to Eq. (), the CCS characteristic polynomial P_C has the following expression:

P_{C} (ω, s) = s^{2} + [C_{0} (ω - \frac{1}{ω}) \sin (ω) - 2 \cos (ω)] s + 1 = 0 .

(7.7)

The MCK characteristic equation [Eq. ()] can be recast as follows:

P_{K} (ω, s) = s^{2} + 2 e^{i ω} (\frac{2 K_{0} \sin (f) ω^{2}}{ω^{2} - 1} - \cos (f)) s + e^{2 i ω} = 0,

(7.8)

and we refer to P_K as the MCK characteristic polynomial.

Using definitions (), (), and () for, respectively, the characteristic polynomials

P (f, ω, K_{0})

⁠,

P_{C} (ω, s)

⁠, and

P_{K} (ω, s)

⁠, one can identify their leading terms

P^{(0)}

⁠,

P_{C}^{(0)} (ω, s)

⁠, and

P_{K}^{(0)}

as ω → ∞, which are

P^{(0)} (ω, s) = C_{0} \sin (ω) s (s^{2} + 2 b_{f}^{\infty} e^{i ω} s + e^{2 i ω}),

(7.9)

P_{C}^{(0)} (ω, s) = C_{0} \sin (ω) s, P_{K}^{(0)} (ω, s) = s^{2} + 2 b_{f}^{\infty} e^{i ω} s + e^{2 i ω},

(7.10)

where

b_{f}^{\infty}

is defined by Eq. ().

Just as in the case of the dispersion relations that we analyzed in Sec. VI, there are simple relations between the CCTWT characteristic function $P (ω, s)$ and the characteristic polynomials $P_{C} (ω, s)$ and $P_{K} (ω, s)$ 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

P (ω, s)

,

P_{C} (ω, s)

⁠, and

P_{K} (ω, s)

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

P (ω, s) - P_{C} (ω, s) P_{K} (ω, s) = - \frac{2 K_{0} \sin (f) e^{i ω} s}{ω} \frac{(s^{2} - 2 \cos (ω) s + 1)}{ω^{2} - 1},

(7.11)

K_{0} = \frac{b^{2} β_{0}}{2 f} = \frac{b^{2} g_{B}}{c_{0}}, g_{B} = \frac{σ_{B}}{4 λ_{rp}} .

(7.12)

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

P^{(0)} (ω, s) = P_{C}^{(0)} (ω, s) P_{K}^{(0)} (ω, s) = C_{0} \sin (ω) s (s^{2} + 2 b_{f}^{\infty} e^{i ω} s + e^{2 i ω}) .

(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.

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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, f_cr ≅ 0.761 012 7542, f_max ≅ 1.951 302 704, G_max = 14.307 667 94 (see Sec. VIII B for the definition of the MCK quantities f_cr, f_max, and G_max): (a) f = 1.2 > f_cr; (b) f = 1.95 ≈ f_max. 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 = G_max represents the maximal G_max 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 $20 |\log (C_{0} ω)|$ (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.

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Plot of gain G per one period in dB as a function of frequency ω defined by Eq. (9.24) for ω₀ = 1, K₀ = K_0T = 4.95 and, consequently, f_cr ≅ 0.398 674 6100, f_max ≅ 1.770 133 632, G_max = 20 (see Sec. VIII B for the definition of the MCK quantities f_cr, f_max, and G_max), and f = 1.2 > f_cr. 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 = G_max represents the maximal G_max 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 = f_cr (see Sec. VIII B). The envelope of the local maxima of the gain for large values of frequency ω behaves as $20 |\log (C_{0} ω)|$ (see captions of Fig. 17), and it is shown as a (blue) dashed curve.

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 $G_{C} = G_{C} (ω, C_{0})$ ⁠.

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.

FIG. 11.

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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.

A. The Euler–Lagrange equations in dimensionless variables

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^{'} = L_{B}^{'} + L_{CB}^{'}, L_{B}^{'} = \frac{1}{2 β^{'}} {(\partial_{t^{'}} q + \partial_{z^{'}} q)}^{2} - \frac{2 π}{σ_{B}^{'}} q^{2}, ℓ \in Z,

(8.1)

L_{CB}^{'} = \sum_{ℓ = - \infty}^{\infty} δ (z^{'} - ℓ) \{\frac{l_{0}^{'}}{2} {(\partial_{t} Q (a ℓ))}^{2} - \frac{1}{2 c_{0}^{'}} {[Q (a ℓ) + b q (a ℓ)]}^{2}\} .

(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

{(\partial_{t} + \partial_{z})}^{2} q + f^{2} q = 0, z \neq ℓ, ℓ \in Z, f = \frac{2 π R_{sc} ω_{p}}{ω_{a}} = \frac{2 π a}{λ_{rp}},

(8.3)

\partial_{'}^{2} Q (a ℓ) + ω_{0}^{2} [Q (a ℓ) + b q (a ℓ)] = 0, ω_{0} = \frac{1}{\sqrt{l_{0} c_{0}}}, β_{0} = \frac{β}{c_{0}},

(8.4)

[\partial_{z} q] (a ℓ) = - b β_{0} [Q (a ℓ) + b q (a ℓ)], ℓ \in Z .

Note that term

- \frac{2 π}{σ_{B}^{'}} q^{2}

in the Lagrangian

L_{B}^{'}

defined in Eq. () represents space-charge effects, including the so-called debunching (electron-to-electron repulsion).

The Fourier transform in t (see Appendix A) of Eqs. () and () is

{(\partial_{z} - i ω)}^{2} \overset{̌}{q} + f^{2} \overset{̌}{q} = 0, z \neq ℓ,

(8.5)

subject to the boundary conditions at interaction points,

[\overset{̌}{q}] (a ℓ) = 0, [\partial_{z} \overset{̌}{q}] (a ℓ) = - B (ω) \overset{̌}{q} (a ℓ) ℓ \in Z,

(8.6)

where

\overset{̌}{q}

is the time Fourier transform of q and

B (ω)

is an important parameter defined by

B = B (ω) = \frac{b^{2} β_{0} ω^{2}}{ω^{2} - ω_{0}^{2}} = B_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}, B_{0} = b^{2} β_{0} = \frac{b^{2} β}{c_{0}} .

(8.7)

We refer to it as cavity e-beam interaction parameter. The Fourier transform in time of Eq. () yields

\overset{̌}{Q} (a ℓ) = \frac{ω_{0}^{2}}{ω^{2} - ω_{0}^{2}} b \overset{̌}{q} (a ℓ), ℓ \in Z,

(8.8)

where

\overset{̌}{Q}

is the time Fourier transform of Q, and Eq. () was used to obtain the second equation in ().

Boundary conditions () can be recast into the matrix form as follows:

X (a ℓ + 0) = S_{b} X (a ℓ - 0), S_{b} = [\begin{matrix} 1 & 0 \\ - B (ω) & 1 \end{matrix}], X = [\begin{matrix} \overset{̌}{q} \\ \partial_{z} \overset{̌}{q} \end{matrix}], B (ω) = \frac{b^{2} β_{0} ω^{2}}{ω^{2} - ω_{0}^{2}} .

(8.9)

In order to use the standard form of the Floquet theory reviewed in Appendix F, we recast the ordinary differential equation [Eq. ()] with boundary (interface) conditions () as the following single second-order ordinary differential equation with singular, frequency dependent, periodic potential:

\partial_{z}^{2} \overset{̌}{q} - 2 i ω \partial_{z} \overset{̌}{q} + (f^{2} - ω^{2}) \overset{̌}{q} - B (ω) p (z) \overset{̌}{q} = 0, p (z) = \sum_{ℓ = - \infty}^{\infty} δ (z - ℓ), \overset{̌}{q} = \overset{̌}{q} (z),

(8.10)

where the second interaction parameter

B (ω)

is defined by Eq. ().

Analysis of Eq. () based on the Floquet theory (see Appendix F) becomes now the primary subject of studies. The second-order ordinary differential equation [Eq. ()] can in turn be recast into the following matrix ordinary differential equation:

\partial_{z} X = A_{K} (z) X, A_{K} (z) = A_{K} (z, ω) = [\begin{matrix} 0 & 1 \\ ω^{2} - f^{2} + B (ω) p (z) & 2 i ω \end{matrix}], X = [\begin{matrix} q \\ \partial_{z} q \end{matrix}],

(8.11)

B (ω) = \frac{b^{2} β_{0} ω^{2}}{ω^{2} - ω_{0}^{2}} = 2 f K_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}, p (z) = \sum_{ℓ = - \infty}^{\infty} δ (z - ℓ) .

Note that normalized period

f = \frac{2 π a}{λ_{rp}}

and the MCK gain coefficient

K_{0} = \frac{b^{2} g_{B}}{c_{0}}

play particularly significant roles for MCK properties.

One can verify by straightforward evaluation that Eq. () has the Hamiltonian structure (see Appendix G) with the following selection for the metric matrix:

G_{K} = G_{K}^{*} = [\begin{matrix} 2 ω & i \\ - i & 0 \end{matrix}], G_{K}^{- 1} = [\begin{matrix} 0 & i \\ - i & - 2 ω \end{matrix}], \det [G_{K}] = - 1 .

(8.12)

The eigenvalues are eigenvectors of metric matrix G_K that are as follows:

ω + \sqrt{ω^{2} + 1}, [\begin{matrix} \frac{i}{- ω + \sqrt{ω^{2} + 1}} \\ 1 \end{matrix}], ω - \sqrt{ω^{2} + 1}, [\begin{matrix} - \frac{i}{ω + \sqrt{ω^{2} + 1}} \\ 1 \end{matrix}] .

(8.13)

Using expressions () and () for, respectively, matrices

A (z)

and G, one can readily verify that

A (z)

is G-skew-Hermitian matrix, that is,

G_{K} A_{K} (z) + A_{K}^{*} (z) G_{K} = 0,

(8.14)

and that according to Appendix G implies that system () is Hamiltonian. Consequently, according to Appendix G, the matrizant

Φ_{K} (z)

of the Hamiltonian system () is G_K-unitary and its spectrum

σ \{Φ_{K} (z)\}

is symmetric with respect to the unit circle, that is,

Φ_{K}^{*} (z) G_{K} Φ_{K} (z) = G, ζ \in σ \{Φ_{K} (z)\} \Rightarrow \frac{1}{\bar{ζ}} \in σ \{Φ_{K} (z)\} .

(8.15)

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

The MCK monodromy matrix

T_{K}

(see Appendix F) is as follows:

T_{K} = e^{i ω} [\begin{matrix} \cos (f) - i ω s i n c (f) & s i n c (f) \\ s i n c (f) ω^{2} + 2 i ω (\cos (f) + b_{f}) - \frac{2 b_{f} \cos (f) + \cos^{2} (f) + 1}{s i n c (f)} & i ω s i n c (f) - \cos (f) - 2 b_{f} \end{matrix}],

(8.16)

where

b_{f} = \frac{B (ω)}{2} s i n c (f) - \cos (f), s i n c (f) = \frac{\sin (f)}{f}, B (ω) = \frac{b^{2} β_{0} ω^{2}}{ω^{2} - ω_{0}^{2}} .

(8.17)

We assume that the MCK normalized period

f = \frac{2 π a}{λ_{rp}}

⁠, 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 = \frac{2 π a}{λ_{rp}} < π .

(8.18)

The MCK gain is defined by the following expression:

G = G (f, ω, K_{0}) = \{\begin{matrix} 20 |\log (||b_{f}| + \sqrt{b_{f}^{2} - 1}|)| & if & |b_{f}| > 1, \\ 0 & if & |b_{f}| \leq 1, \end{matrix}

(8.19)

b_{f} = b_{f} (ω) = K (ω) \sin (f) - \cos (f), K (ω) = K_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}, K_{0} = \frac{b^{2} β_{0}}{2 f} = \frac{b^{2} g_{B}}{c_{0}} .

Note that the following high-frequency decomposition holds for the instability parameter b_f:

b_{f} (ω) = b_{f}^{\infty} + K_{0} \frac{ω_{0}^{2}}{ω^{2} - ω_{0}^{2}},

(8.20)

where

b_{f}^{\infty} = \lim_{ω \to \infty} b_{f} (ω) = K_{0} \sin (f) - \cos (f), K_{0} = \frac{b^{2} g_{B}}{c_{0}} .

(8.21)

It turns out that the high-frequency limit

b_{f}^{\infty}

of instability parameter b_f defined by Eq. () 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, π)

of the normalized period f such that

b_{f_{cr}}^{\infty} = 1, 0 < f_{cr} < π,

(8.22)

and we refer to it as the critical value and the following representation holds:

f_{cr} = 2 \arctan (\frac{1}{K_{0}}), where K_{0} = \frac{b^{2} g_{B}}{c_{0}}, g_{B} = \frac{σ_{B}}{4 λ_{rp}}, |\arctan (*)| < \frac{π}{2} .

(8.23)

The significance of the critical value f_cr is that for f_cr < 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.

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Plots of gain G as a function of frequency ω defined by Eq. (8.19) for ω₀ = 1 and (a) K₀ = 2, f = 2 > f_cr with f_cr ≅ 0.927 295 2180, f_max ≅ 2.034 443 936, and G_max ≅ 12.539 258 41; (b) K₀ = 1, f = 2.4 > f_cr with f_cr ≅ 1.570 796 327, f_max ≅ 2.356 194 491, and G_max ≅ 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 = G_max represents the maximal G_max value of G in the high frequency limit. The (green) diamond solid dots mark the values of $Ω_{f}^{-}$ ⁠, which is the lower frequency boundary of the instability interval.

FIG. 13.

Plots of fmax=π−arctanK0 as the (brown) solid curve and fcr=2⁡arctan1K0 as the (blue) dashed curve. The horizontal and vertical axes represent, respectively, K0 and f.

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Plots of $f_{\max} = π - \arctan (K_{0})$ as the (brown) solid curve and $f_{cr} = 2 \arctan (\frac{1}{K_{0}})$ as the (blue) dashed curve. The horizontal and vertical axes represent, respectively, K₀ and f.

The maximal value

G_{\max} = 20 |\log (|K_{0} + \sqrt{K_{0}^{2} + 1}|)| = 20 \frac{\ln (K_{0} + \sqrt{K_{0}^{2} + 1})}{\ln (10)}

(8.24)

of gain G is attained at f = f_max that satisfies (see Fig. 13)

f_{\max} = π - \arctan (K_{0}), f_{cr} < f_{\max} < π .

(8.25)

Let

s = \exp \{i k\}

⁠, where k is the wave number of the Floquet multiplier of the monodromy matrix

T_{K}

defined by Eqs. () and () (see Appendix F and Remark 25). Then, the two Floquet multipliers s_± are solutions to the characteristic equation

\det \{T_{K} - s I\} = 0

⁠, which is¹¹

s = e^{i k} = e^{i ω} S : S^{2} + 2 b_{f} S + 1 = 0, S = S_{\pm} = - b_{f} \pm \sqrt{b_{f}^{2} - 1},

(8.26)

readily implying

s_{\pm} = e^{i k_{\pm}} = e^{i ω} S_{\pm} = e^{i ω} (- b_{f} \pm \sqrt{b_{f}^{2} - 1}), b_{f} = b_{f} (ω) = K (ω) \sin (f) - \cos (f),

(8.27)

K (ω) = K_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}, K_{0} = \frac{b^{2} β_{0}}{2 f} = \frac{b^{2} g_{B}}{c_{0}} .

Equation () shows that 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 () can be readily recast as

S + 2 b_{f} + S^{- 1} = 0, S = s \exp \{- i ω\} = \exp \{i (k - ω)\}, s = \exp \{i k\},

(8.28)

or, equivalently, as

\cos (k - ω) + b_{f} (ω) = 0, b_{f} (ω) = K (ω) \sin (f) - \cos (f) .

(8.29)

Equations () and () 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 () can be readily recast as

k_{\pm} (ω) = ω \pm \arccos (- b_{f} (ω)), b_{f} (ω) = K (ω) \sin (f) - \cos (f),

(8.30)

K (ω) = K_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}}, K_{0} = \frac{b^{2} β_{0}}{2 f} = \frac{b^{2} g_{B}}{c_{0}} .

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_{\pm} (ω)

be the corresponding complex-valued wave numbers satisfying

s_{\pm} = s_{\pm} (ω) = \exp \{i k_{\pm} (ω)\} .

(8.31)

Then, the following representation for

k_{\pm} (ω)

holds:

k_{\pm} (ω) = \{\begin{matrix} - \frac{1 + sign \{b_{f} (ω)\}}{2} π + ω + 2 π m \pm i \ln [(|b_{f} (ω)| + \sqrt{b_{f}^{2} (ω) - 1})] & if & b_{f}^{2} > 1, \\ - \frac{1 + sign \{b_{f} (ω)\}}{2} π + ω + 2 π m \pm \arccos (|b_{f} (ω)|) & if & b_{f}^{2} \leq 1, \end{matrix} m \in Z,

(8.32)

where 0 < f < π and

b_{f} (ω) = K_{0} \frac{ω^{2}}{ω^{2} - ω_{0}^{2}} \sin (f) - \cos (f), K_{0} = \frac{b^{2} β_{0}}{2} = \frac{b^{2} g_{B}}{c_{0}}, g_{B} = \frac{σ_{B}}{4 λ_{rp}} .

(8.33)

Requirement for

R \{k_{\pm} (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

m \in Z

is determined by the requirement to satisfy the following inequalities:

\begin{matrix} - π < - \frac{1 + sign \{b_{f} (ω)\}}{2} π + ω + 2 π m \leq π & if & b_{f}^{2} (ω) > 1, \\ - π < - \frac{1 + sign \{b_{f} (ω)\}}{2} π \pm \arccos (- b_{f} (ω)) + ω + 2 π m \leq π & if & b_{f}^{2} (ω) < 1 . \end{matrix}

(8.34)

Equations (8.32) for the complex-valued wave numbers $k_{\pm} (ω)$ represent the dispersion relations of the MCK.

Remark 8.

Note that according to expression () in Theorem 7, we have

R \{k_{\pm} (ω)\} = π + ω + 2 π m, ω_{0} < ω < Ω_{f}^{+},

(8.35)

where

Ω_{f}^{+}

is the upper boundary of instability frequencies. Figure 15 illustrates graphically Eq. () by perfect straight lines parallel to

R \{k\} = ω

in the shadowed area.

There is yet another form of the dispersion relation [() and ()], which is the high-frequency form,

D_{K} (ω, k) = D_{K}^{(0)} (ω, k) + \frac{K_{0} ω_{0}^{2}}{ω^{2} - ω_{0}^{2}} = 0,

(8.36)

D_{K}^{(0)} (ω, k) = \cos (ω - k) + b_{f}^{\infty}, b_{f}^{\infty} = K_{0} \sin (f) - \cos (f) .

(8.37)

We refer to function

D_{K} (ω, k)

as the MCK dispersion function.

This form readily yields the following high-frequency approximation to the MCK dispersion relations:

\cos (ω - k) + b_{f}^{\infty} = 0, b_{f}^{\infty} = K_{0} \sin (f) - \cos (f), |b_{f}^{\infty}| \leq 1,

(8.38)

or, equivalently,

ω = k \pm \arccos (- b_{f}^{\infty}) + 2 π m, b_{f}^{\infty} = K_{0} \sin (f) - \cos (f), |b_{f}^{\infty}| \leq 1, m \in Z,

(8.39)

where inequality

|b_{f}^{\infty}| \leq 1

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 $D_{K} (ω, k)$ and its high-frequency approximation $D_{K}^{(0)} (ω, k)$ are integrated into the relevant dispersion functions associated with the CCTWT.

Figures 14 and 15 illustrate graphically the dispersion relations $k_{\pm} (ω)$ 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 3−π,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).

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The MCK dispersion-instability plot (brown solid curves and lines) over three Brillouin zones $3 [- π, π]$ for K₀ = 3, ω₀ = 1 for which f_cr ≅ 0.643 5011 088, f_max ≅ 1.892 546 882, and f = 0.5 < f_cr ≅ 0.643 501 1088. The horizontal and vertical axes represent, respectively, $R \{k\}$ and $\frac{ω}{ω_{0}}$ ⁠. Two (green) solid diamond dots identify the values of $Ω_{f}^{-}$ and $Ω_{f}^{+}$ ⁠, 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 $ω = Ω_{f}^{\pm}$ identify the frequency boundaries of the instability, and the (light blue) shaded region between the lines identifies points $(R \{k\}, ω)$ 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, $R \{k_{\pm} (ω)\}$ jumps by π according to Eq. (8.32) as the frequency ω passes through the resonance frequency ω₀ and the sign of $b_{f} (ω)$ changes. The shadowed area marks points $(R \{k\}, ω)$ associated with the instability. The (blue) dashed straight lines correspond to the high frequency approximation defined by Eq. (8.39).

FIG. 15.

The MCK dispersion-instability plot (solid brown curves and lines) over three Brillouin zones 3−π,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.

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The MCK dispersion-instability plot (solid brown curves and lines) over three Brillouin zones $3 [- π, π]$ for K₀ = 1, ω₀ = 1 for which f_cr ≅ 1.570 796 327, f_max ≅ 1.892 546 882, and f = 1.569 ≅ f_cr ≅ 1.570 796 327. The horizontal and vertical axes represent, respectively, $R \{k\}$ and $\frac{ω}{ω_{0}}$ ⁠. Two (green) solid diamond dots identify the values of $Ω_{f}^{-}$ and $Ω_{f}^{+}$ ⁠, 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 $ω = Ω_{f}^{\pm}$ identify the frequency boundaries of the instability, and the (light blue) shaded region between the lines identifies points $(R \{k\}, ω)$ 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, $R \{k_{\pm} (ω)\}$ jumps by π according to Eq. (8.32) as the frequency ω passes through the resonance frequency ω₀ and the sign of $b_{f} (ω)$ changes. The shadowed area marks points $(R \{k\}, ω)$ associated with the instability.

Interestingly, there is an empirical formula due to Tsimring that shows the dependence of the maximum power gain

G_{T} (N)

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),

G_{T} (N) = 15 + 20 (N - 2) d B .

(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.

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.

The Lagrangian

L_{C} (\{Q\})

of the CCS system can be readily obtained from the Lagrangian

L

of CCTWT defined by Eqs. (), (), and () by assuming b = 0 and omitting component

L_{B}

⁠, that is,

L_{C} (\{Q\}) = \frac{L}{2} {(\partial_{t} Q)}^{2} - \frac{1}{2 C} {(\partial_{z} Q)}^{2} - \frac{1}{2 c_{0}} \sum_{ℓ = - \infty}^{\infty} δ (z - a ℓ) {[Q (a ℓ)]}^{2} .

(9.1)

Then, the corresponding EL equations [Eqs. () and ()] are reduced to

L \partial_{t}^{2} Q - C^{- 1} \partial_{z}^{2} Q = 0, [\partial_{z} Q] (a ℓ) = C_{0} (\frac{\partial_{t}^{2}}{ω_{0}^{2}} + 1) Q (a ℓ), C_{0} = \frac{C}{c_{0}}, a ℓ \in Z,

(9.2)

where jumps

[Q] (a ℓ)

and

[\partial_{z} Q] (a ℓ)

are defined by Eq. (), and consequently,

X (a ℓ + 0) = S_{b} X (a ℓ - 0), S_{b} = [\begin{matrix} 1 & 0 \\ C_{0} (\frac{\partial_{t}^{2}}{ω_{0}^{2}} + 1) & 1 \end{matrix}], X = [\begin{matrix} Q \\ \partial_{z} Q \end{matrix}] .

(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. ()]:

\partial_{t}^{2} Q - \frac{1}{χ^{2}} \partial_{z}^{2} Q = 0, z \neq ℓ, [\partial_{z} Q] (ℓ) = C_{0} (\frac{\partial_{t}^{2}}{ω_{0}^{2}} + 1) Q (ℓ), ℓ \in Z .

(9.4)

The Fourier transform in t (see Appendix A) of Eq. () yields

\partial_{z}^{2} \overset{̌}{Q} + \frac{ω^{2}}{χ^{2}} \overset{̌}{Q} = 0, z \neq ℓ, [\partial_{z} Q] (ℓ) = C_{0} (1 - \frac{ω^{2}}{ω_{0}^{2}}) Q (ℓ), ℓ \in Z,

(9.5)

where

\overset{̌}{Q}

and

\overset{̌}{q}

are the time Fourier transform of the corresponding quantities.

An alternative form of the system of Eq. () is the following second-order vector ODE with the periodic singular potential:

\partial_{z}^{2} \overset{̌}{Q} + \frac{ω^{2}}{χ^{2}} \overset{̌}{Q} + C_{0} (1 - \frac{ω^{2}}{ω_{0}^{2}}) \sum_{ℓ = - \infty}^{\infty} δ (z - ℓ) \overset{̌}{Q} = 0 .

(9.6)

According to Appendix E, second-order differential equation [Eq. ()] is equivalent to the first-order differential equation of the form

\partial_{z} X = A_{C} (z) X, A_{C} (z) = [\begin{matrix} 0 & 1 \\ - \frac{ω^{2}}{χ^{2}} - p (z) & 0 \end{matrix}], X = [\begin{matrix} \overset{̌}{Q} \\ \partial_{z} \overset{̌}{Q} \end{matrix}],

(9.7)

p (z) = C_{0} (1 - \frac{ω^{2}}{ω_{0}^{2}}) \sum_{ℓ = - \infty}^{\infty} δ (z - ℓ) .

Using results of Subsection 2 of Appendix G, we find that system () is Hamiltonian for the following choice of nonsingular Hermitian matrix G:

G_{C} = G_{C}^{*} = [\begin{matrix} 0 & i \\ - i & 0 \end{matrix}], \det \{G_{C}\} = - 1 .

(9.8)

In particular, it is an elementary exercise to verify that for each value of z matrix

A_{C} (z)

is G_C-skew-Hermitian, that is,

G_{C} A_{C} (z) + A_{C}^{*} (z) G_{C} = 0 .

(9.9)

Then, if

Φ (z)

is the matrizant of the Hamiltonian equation[Eq. ()], then according to results of Appendix G,

Φ_{C} (z)

is G_C-unitary matrix,

Φ_{C}^{*} (z) G_{C} Φ_{C} (z) = G_{C},

(9.10)

and consequently, its spectrum

σ \{Φ_{C} (z)\}

is invariant with respect to the inversion transformation

ζ \to \frac{1}{\bar{ζ}}

⁠, that is, it symmetric with respect to the unit circle,

ζ \in σ \{Φ_{C} (z)\} \Rightarrow \frac{1}{\bar{ζ}} \in σ \{Φ_{C} (z)\} .

(9.11)

To simplify analytic evaluations, we assume as before that assumption 3 holds.

A. Monodromy matrix and the dispersion-instability relations

By simplifying assumption 3 (χ = ω₀ = 1), the monodromy matrix

T_{C}

defined by Eq. () takes the form

T_{C} = [\begin{matrix} \cos (ω) & ω^{- 1} \sin (ω) \\ (1 - ω^{2}) C_{0} \cos (ω) - ω \sin (ω) & \cos (ω) - C_{0} (ω - ω^{- 1}) \sin (ω) \end{matrix}] .

(9.12)

The corresponding characteristic equation [Eq. ()] turns into

P_{C} (ω, s) = s^{2} + 2 W_{C} (ω) s + 1 = 0, W_{C} (ω) = \frac{C_{0}}{2} (ω - \frac{1}{ω}) \sin (ω) - \cos (ω),

(9.13)

s = \exp \{i k\},

where quadratic polynomial P_C is referred to as the CCS characteristic polynomial and quantity

W_{C} (ω)

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

T_{C}

defined by Eq. () and, consequently, are the solutions to its characteristic equation [Eq. ()], can be represented as follows:

s_{\pm} = e^{i k_{\pm}} = W_{C} \pm \sqrt{W_{C}^{2} - 1}, W_{C} = W_{C} (ω) = \frac{C_{0}}{2} (ω - \frac{1}{ω}) \sin (ω) - \cos (ω) .

(9.14)

Equation () implies that instability parameter

W_{C} (ω)

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. ()] by 2s and substitute

s = \exp \{i k\}

obtaining the following equations:

\cos (k) + W_{C} (ω) = 0, W_{C} (ω) = \frac{C_{0}}{2} (ω - \frac{1}{ω}) \sin (ω) - \cos (ω),

(9.15)

where

W_{C} (ω)

is the principle CCS function [see Fig. 16(b)]. Alternatively, the dispersion-instability relations () can be represented in the form

k_{\pm} = k_{\pm} (ω) = - i \ln (W_{C} \pm \sqrt{W_{C}^{2} - 1}),

(9.16)

W_{C} = W_{C} (ω) = \frac{C_{0}}{2} (ω - \frac{1}{ω}) \sin (ω) - \cos (ω) .

Dividing Eq. () by ω, we obtain the following high-frequency form of the dispersion relations for the MCK:

D_{C} (ω, k) = D_{C}^{(0)} (ω, k) + \frac{2 (\cos (k) - \cos (ω))}{ω} - \frac{C_{0} \sin (ω)}{ω^{2}} = 0,

(9.17)

D_{C}^{(0)} (ω, k) = C_{0} \sin (ω) .

We refer to function

D_{C} (ω, k)

as the CCS dispersion function.

As to the e-beam transforming characteristic equation [Eq. ()] for the e-beam, the same way we obtain the following explicit form of the dispersion relations for the e-beam:

\cos (k - ω) - \cos (f) = 0, ω = k \pm f .

(9.18)

Expression () for the CCS dispersion relation readily implies that its EPD frequencies are solutions to the following CCS EPD equation:

W_{C} (ω) = \frac{C_{0}}{2} (ω - \frac{1}{ω}) \sin (ω) - \cos (ω) = \pm 1,

(9.19)

where

W_{C} (ω)

is the principle CCS function defined by the second equation in () and its plot is depicted in Fig. 16(b). Straightforward evaluations show that

W_{C} (ω)

satisfies the following equations:

W_{C} (π n) = {(- 1)}^{n + 1}, \partial_{ω} W_{C} (π n) = {(- 1)}^{n}, n = 1, 2, \dots .

(9.20)

Consequently, 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. ()].

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.

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The CCS for C₀ = 1. (a) Dispersion-instability graph: horizontal axis is $R \{k\}$ and the vertical axis is ω. (b) The plot of the instability parameter $W_{C} (ω)$ defined by the second equation in (9.13). The horizontal axis is $R \{k\}$ ⁠, and the vertical axis is W.

Instability (oscillatory spectrum) bands (intervals) are

[0, ξ_{0}], [ξ_{m}, π m], m \geq 1 : 0 < ξ_{0} < ξ_{1} < π, π (m - 1) < ξ_{m} < π m, m \geq 2,

(9.21)

where numbers ξ_m satisfy also the following equations:

W_{C} (ξ_{0}) = - 1, W_{C} (ω) (ξ_{1}) = 1, W_{C} (ξ_{m}) = {(- 1)}^{m - 1}, m \geq 2 .

(9.22)

Stability (oscillatory) spectrum bands (intervals) are

[ξ_{0}, ξ_{1}], [π m, ξ_{m + 1}], m \geq 1 : 0 < ξ_{0} < ξ_{1} < π, π (m - 1) < ξ_{m} < π m, m \geq 2 .

(9.23)

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.

Definition 9

(CCS gain per one period). Let s_± be the CCS Floquet multipliers defined by Eq. (). Then, corresponding to them, gain G_C in dB per one period is defined by

G_{C} = G_{C} (ω, C_{0}) = \{\begin{matrix} 20 |\log (|s_{+}|)| = 20 |\log (||W_{C}| + \sqrt{W_{C}^{2} - 1}|)| & if & |W_{C}| > 1, \\ 0 & if & |W_{C}| \leq 1, \end{matrix}

(9.24)

W_{C} = W_{C} (ω) = \frac{C_{0}}{2} (ω - \frac{1}{ω}) \sin (ω) - \cos (ω) .

Figure 17 shows the frequency dependence of the gain G_C 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.

View large Download slide

The plot of the CCS gain $G_{C} (ω, C_{0})$ 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 $G_{C} (ω, C_{0}) > 0$ ⁠. The envelope of the local maxima of the gain $G_{C} (ω, C_{0})$ behaves asymptotically for large values of frequency ω as $20 |\log (C_{0} ω)|$ as it follows from Eq. (9.24). It is shown as a (blue) dashed curve.

B. Exceptional points of degeneracy

The monodromy matrix

T_{C}

defined by Eq. () and its Jordan form at ω = πn are as follows:

T_{C} = [\begin{matrix} {(- 1)}^{n} & 0 \\ (1 - π^{2} n^{2}) C_{0} {(- 1)}^{n} & {(- 1)}^{n} \end{matrix}] = Z_{C} [\begin{matrix} {(- 1)}^{n} & 1 \\ 0 & {(- 1)}^{n} \end{matrix}] Z_{C}^{- 1}, ω = π n,

where matrix

Z_{C}

is

Z_{C} = [\begin{matrix} 0 & 1 \\ (1 - π^{2} n^{2}) C_{0} {(- 1)}^{n} & 0 \end{matrix}],

and columns of the matrix

Z_{C}

are the Jordan basis of the monodromy matrix

T_{C}

⁠.

The monodromy matrix expression at EPDs is as follows:

T_{C} = [\begin{matrix} \cos (ω) & ω^{- 1} \sin (ω) \\ \frac{ω \sin (ω) (\cos (ω) - 1)}{\cos (ω) + 1} & 2 - \cos (ω) \end{matrix}] = Z_{C} [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] Z_{C}^{- 1}, W_{C} (ω) = - 1, ω \neq π n,

where

Z_{C} = [\begin{matrix} \cos (ω) - 1 & 1 \\ \frac{ω \sin (ω) (\cos (ω) - 1)}{\cos (ω) + 1} & 0 \end{matrix}],

and

T_{C} = [\begin{matrix} \cos (ω) & ω^{- 1} \sin (ω) \\ - \frac{ω {(\cos (ω) + 1)}^{2}}{\sin (ω)} & - 2 - \cos (ω) \end{matrix}] = Z_{C} [\begin{matrix} - 1 & 1 \\ 0 & - 1 \end{matrix}] Z_{C}^{- 1}, W_{C} (ω) = 1, ω \neq π n,

where

Z_{C} = [\begin{matrix} \cos (ω) + 1 & 1 \\ - \frac{ω {(\cos (ω) + 1)}^{2}}{\sin (ω)} & 0 \end{matrix}] .

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.

Our field theory assumes that the cavity width l_g and the corresponding transit time τ_g are zeros; see Eq. () 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

\overset{̊}{U}

is the constant accelerating voltage, so the stationary dc electron flow velocity

\overset{̊}{v} = \sqrt{\frac{2 e}{m} \overset{̊}{U}}

⁠, where m and −e is, respectively, the electron mass and its charge. Suppose also that

U_{1} \sin (ω t)

is the gap voltage. Then, based on the elementary energy conservation law, one gets

\frac{m v^{2}}{2} = \frac{m {\overset{̊}{v}}^{2}}{2} + U_{1} \sin (ω t),

(10.1)

where v is the modulated velocity. Solving Eq. () for v and assuming “small signal” approximation, we obtain

v ≅ \overset{̊}{v} (1 + \frac{ξ}{2} \sin (ω t)), ξ = \frac{U_{1}}{\overset{̊}{U}} ≪ 1 .

(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. () 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 $\overset{̊}{v}$ ⁠.

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 $\overset{̊}{v}$ electron velocity field $v = v (z, t)$ ⁠. Therefore, as to this part of the interaction, we may view the electron density to be essentially constant $\overset{̊}{n}$ ⁠, whereas its ac velocity field $v = v (z, t)$ 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 $n = n (z, t)$ ⁠. Hence, for this part of the interaction, we may view the electron flow to be of nearly constant velocity $\overset{̊}{v}$ perturbed by relatively small ac electron number density $n = n (z, t)$ ⁠. Following the results of Sec. III, let us take a look at the variation of the ac electron velocity $v = v (z, t)$ and ac electron number density $n = n (z, t)$ 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 = v (z, 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 = n (z, t)

⁠. As to the quantitative assessment of the variations, note that Eq. () implies that the electron velocity

v = v (z, t)

and number density

n = n (z, t)

have the following jumps

[n] (a ℓ)

and

[v] (a ℓ)

at the interaction points aℓ:

[n] (a ℓ, t) = \frac{[\partial_{z} q] (a ℓ, t)}{σ_{B} e}, [v] (a ℓ, t) = - \frac{\overset{̊}{v} [\partial_{z} q] (a ℓ, t)}{e σ_{B} \overset{̊}{n}} = - \frac{v \overset{̊}{n} (a ℓ, t)}{\overset{̊}{n}},

(10.3)

readily implying

\frac{[v] (a ℓ, t)}{\overset{̊}{v}} = - \frac{[n] (a ℓ, t)}{\overset{̊}{n}} .

(10.4)

Equations () and () in turn yield

[j] (a ℓ, t) = - e \{\overset{̊}{n} [v, t] + \overset{̊}{v} [n]\} (a ℓ, 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. ()] and the first equation in (), the following representation holds for the

[\partial_{z} E] (a ℓ)

at the interaction points aℓ:

[\partial_{z} E] (a ℓ, t) = - \frac{4 π}{σ_{B}} [\partial_{z} q] (a ℓ, t) = - 4 π e [n] (a ℓ, t) .

(10.6)

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.”

The direct link between our field theory and the hydrokinetic theory is provided by the e-beam Lagrangian

L_{B}

defined by Eqs. () and (),

L_{B} (\{q\}) = \frac{1}{2 β} {(D_{t} q)}^{2} - \frac{2 π}{σ_{B}} q^{2}, D_{t} = \partial_{t} + \overset{̊}{v} \partial_{z} .

(10.7)

Indeed, its first kinetic term

\frac{1}{2 β} {(D_{t} q)}^{2}

involves the material time derivative, which represents an important concept of “particle” in the hydrokinetic theory. The second term

- \frac{2 π}{σ_{B}} q^{2}

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 $[\partial_{z} q] (a ℓ, t)$ that are explicitly allowed by the field theory represent jumps $[n] (a ℓ, t)$ and $[v] (a ℓ, t)$ related to the kinetic properties of the electron flow; see Remark 1. Namely, jump $[n] (a ℓ, t)$ manifests the electron bunching, jump $[v] (a ℓ, t)$ 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 $L_{T}$ and $L_{B}$ in Eq. (3.10), whereas discrete features are represented by Lagrangian $L_{TB}$ in Eq. (3.11) with energies concentrated in a set of discrete points $a Z$ ⁠. 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. ()–(). Using notations () and (), we define the action integral S as follows:

S (\{x\}) = \int_{t_{0}}^{t_{1}} d t \int_{z_{1}}^{z_{2}} L (\{x\}) d z = S_{T} (\{Q\}) + S_{B} (\{q\}) + S_{TB} (x), t_{0} < t_{1}, z_{0} < z_{1},

(11.1)

where

S_{T} (\{Q\}) = \int_{t_{0}}^{t_{1}} d t \int_{z_{1}}^{z_{2}} L_{T} (\{Q\}) d z = \int_{t_{0}}^{t_{1}} d t \int_{z_{1}}^{z_{2}} [\frac{L}{2} {(\partial_{t} Q)}^{2} - \frac{1}{2 C} {(\partial_{z} Q)}^{2}] d z,

(11.2)

S_{B} (\{q\}) = \int_{t_{0}}^{t_{1}} d t \int_{z_{1}}^{z_{2}} L_{B} (\{q\}) d z = \int_{t_{0}}^{t_{1}} d t \int_{z_{1}}^{z_{2}} [\frac{1}{2 β} {(\partial_{t} q + \overset{̊}{v} \partial_{z} q)}^{2} - \frac{2 π}{σ_{B}} q^{2}] d z,

(11.3)

S_{TB} (\{Q\}) = \int_{t_{0}}^{t_{1}} d t \int_{z_{1}}^{z_{2}} L_{TB} (Q, q) d z

(11.4)

= \sum_{z_{1} < a ℓ < z_{2}} \int_{t_{0}}^{t_{1}} d t {[\frac{l_{0}}{2} {(\partial_{t} Q (a ℓ))}^{2} - \frac{1}{2 c_{0}} (Q (a ℓ) + b q (a ℓ))]}^{2} .

To make expressions of the action integrals less cluttered, we suppress notationally their dependence on intervals

(z_{0}, z_{1})

and

(t_{0}, t_{1})

that can be chosen arbitrarily. We consider then variation δS of action S, assuming that variations δQ and δq of charges

q = q (z, t)

and

Q = Q (z, t)

vanish outside intervals

(z_{0}, z_{1})

and

(t_{0}, t_{1})

⁠, that is,

δ Q (z, t) = δ q (z, t) = 0, (z, t) \notin (z_{0}, z_{1}) \times (t_{0}, t_{1}),

(11.5)

implying, in particular, that δQ and δq vanish on the boundary of the rectangle

(z_{0}, z_{1}) \times (t_{0}, t_{1})

⁠, that is,

δ Q (z, t) = δ q (z, t) = 0 if z = z_{0}, z_{1} or if t = t_{0}, t_{1} .

(11.6)

We refer to variations δQ and δq satisfying Eq. () and hence () for a rectangle

(z_{0}, z_{1}) \times (t_{0}, t_{1})

as admissible.

Following the least action principle, we introduce the functional differential δS of the action by the following formula [Ref. 43, 7(35)]:

δ S = \lim_{ε \to 0} \frac{S (\{x + ε δ x\}) - S (\{x\})}{ε} .

(11.7)

Then, the system configurations

x = x (z, t)

that actually can occur must satisfy

δ S = \lim_{ε \to 0} \frac{S (\{x + ε δ x\}) - S (\{x\})}{ε} = 0 for all admissible variations.

(11.8)

Let us choose now any z outside lattice

a Z

⁠. Then, there always exist a sufficiently small ξ > 0 and an integer ℓ₀ such that

a ℓ_{0} < z_{0} = z - ξ < z < z_{1} = z + ξ < a (ℓ_{0} + 1) .

(11.9)

If we apply now the variational principle () for all admissible variations δQ and δq such that space interval

(z_{0}, z_{1})

is compliant with inequalities (), we readily find that

δ S = δ S_{T} + δ S_{B} = 0,

(11.10)

where S_T and S_B are defined by expressions () and (). Using Eq. () and carrying out in the standard way the integration by parts transformations, we arrive at

δ S_{T} = - \int_{t_{0}}^{t_{1}} d t \int_{z_{1}}^{z_{2}} [L \partial_{t}^{2} Q - C^{- 1} \partial_{z}^{2} Q] δ Q d z,

(11.11)

δ S_{B} = - \int_{t_{0}}^{t_{1}} d t \int_{z_{1}}^{z_{2}} [\frac{1}{β} {(\partial_{t} + \overset{̊}{v} \partial_{z})}^{2} q + \frac{4 π}{σ_{B}} q] δ q d z .

(11.12)

Combining Eqs. ()–(), we arrive at the following EL equations:

L \partial_{t}^{2} Q - C^{- 1} \partial_{z}^{2} Q = 0, \frac{1}{β} {(\partial_{t} + \overset{̊}{v} \partial_{z})}^{2} q + \frac{4 π}{σ_{B}} q = 0, z \neq a ℓ, ℓ \in Z .

(11.13)

Consider now the case when z = aℓ₀ for an integer ℓ₀ and select space interval

(z_{0}, z_{1})

as follows:

a (ℓ_{0} - 1) < z_{0} = a (ℓ_{0} - \frac{1}{2}) < z = a ℓ_{0} < z_{1} = a (ℓ_{0} + \frac{1}{2}) < a (ℓ_{0} + 1) .

(11.14)

Note that in this case, all actions S_T, S_T, and S_TB contribute to the variation δS. In particular, as a consequence of the presence of delta functions

δ (z - a ℓ)

in the expression of the Lagrangian

L_{TB}

defined by Eq. (), 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,

\int_{z_{0}}^{z_{1}} = \int_{a (ℓ_{0} - \frac{1}{2})}^{a ℓ_{0}} + \int_{a ℓ_{0}}^{a (ℓ_{0} + \frac{1}{2})};

(11.15)

(ii) we carry out the integration by parts for each of the two integrals in the right-hand side of Eq. (); and (iii) we use already established the EL equation [Eq. ()] to simplify the integral expressions. When that is all done, we arrive at the following equation:

δ S_{T} = \int_{t_{0}}^{t_{1}} \frac{1}{C} [\partial_{z} Q] (a ℓ_{0}, t) δ Q (a ℓ_{0}, t) d t, δ S_{B} = - \int_{t_{0}}^{t_{1}} \frac{{\overset{̊}{v}}^{2}}{β} [\partial_{z} q] (a ℓ_{0}, t) δ q (a ℓ_{0}, t) d t,

(11.16)

where jumps

[\partial_{z} Q] (a ℓ)

and

[\partial_{z} q] (a ℓ)

are defined by Eq. (), and

δ S_{TB} = - \int_{t_{0}}^{t_{1}} \{l_{0} \partial_{t}^{2} Q (a ℓ) + \frac{1}{c_{0}} [Q (a ℓ_{0}, t) + b q (a ℓ_{0}, t)]\} δ Q (a ℓ_{0}, t) d t

(11.17)

- \int_{t_{0}}^{t_{1}} \{\frac{b}{c_{0}} [Q (a ℓ_{0}, t) + b q (a ℓ_{0}, t)]\} δ q (a ℓ_{0}, t) d t .

Using the variational principle (), that is,

δ S_{T} + δ S_{B} + δ S_{TB} = 0,

(11.18)

and the fact that variations

δ Q (a ℓ_{0}, t)

and

δ q (a ℓ_{0}, t)

can be chosen arbitrarily, we arrive at the following equations:

\frac{1}{C} [\partial_{z} Q] (a ℓ_{0}, t) = l_{0} \partial_{t}^{2} Q (a ℓ) + \frac{1}{c_{0}} [Q (a ℓ_{0}, t) + b q (a ℓ_{0}, t)],

(11.19)

\frac{{\overset{̊}{v}}^{2}}{β} [\partial_{z} q] (a ℓ_{0}, t) = - \frac{b}{c_{0}} [Q (a ℓ_{0}) + b q (a ℓ_{0}, t)],

where jumps

[\partial_{z} Q] (a ℓ)

and

[\partial_{z} q] (a ℓ)

are defined by Eq. (). Equation () can be readily recast into the following boundary conditions:

[\partial_{z} Q] (a ℓ) = C_{0} [(\frac{\partial_{t}^{2}}{ω_{0}^{2}} + 1) Q (a ℓ) + b q (a ℓ)], [\partial_{z} q] (a ℓ) = - \frac{b β_{0}}{{\overset{̊}{v}}^{2}} [Q (a ℓ) + b q (a ℓ)],

(11.20)

where

C_{0} = \frac{C}{c_{0}}, ω_{0} = \frac{1}{\sqrt{l_{0} c_{0}}}, β_{0} = \frac{β}{c_{0}} .

(11.21)

We remind also that as a consequence of continuity of Q and q, we also have

[Q] (a ℓ_{0}, t) = 0, [q] (a ℓ_{0}, t) = 0 .

(11.22)

Hence, Eqs. () and () can be viewed as the EL equations at point aℓ₀.

Equation () at an interaction point aℓ₀ is perfectly consistent with boundary conditions () of the general treatment in Ref. 41, which are

- \frac{\partial L_{D}}{\partial \partial_{1} ψ_{D}^{ℓ}} (b_{1}, t) + \frac{\partial L_{B}}{\partial ψ_{B}^{ℓ} (b_{1}, t)} - \partial_{0} (\frac{\partial L_{B}}{\partial \partial_{0} ψ_{B}^{ℓ} (b_{1}, t)}) = 0,

(11.23)

\frac{\partial L_{D}}{\partial \partial_{1} ψ_{D}^{ℓ}} (b_{2}, t) + \frac{\partial L_{B}}{\partial ψ_{B}^{ℓ} (b_{2}, t)} - \partial_{0} (\frac{\partial L_{B}}{\partial (\partial_{0} ψ_{B}^{ℓ} (b_{2}, t))}) = 0,

where (i) b₁ = aℓ₀ − 0 and b₂ = aℓ₀ + 0; (ii) L_D corresponds to

L_{T} + L_{B}

⁠; (iii) L_B corresponds to

L_{TB}

⁠; (iv) fields

ψ_{D}^{ℓ}

correspond to charges Q and q; and (v) boundary fields

ψ_{B}^{ℓ}

correspond to

Q (a ℓ_{0}, t)

and

q (a ℓ_{0}, 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

L_{TB}

defined by Eq. (). In fact, the signs of the terms containing L_D in Eq. () 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.

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

The complex plane transformation

z \to \frac{1}{\bar{z}}

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. () of order 4,

S^{4} + a S^{3} + b S^{2} + \bar{a} S + 1 = 0, a \in C, b \in R .

(12.1)

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

\partial_{S} (S^{4} + a S^{3} + b S^{2} + \bar{a} S + 1) = 4 S^{3} + 3 a S^{2} + 2 b S + \bar{a} = 0 .

(12.2)

Subtracting from two times Eq. () S times Eq. () and dividing the result by S², we obtain

- 2 S^{2} + \frac{2}{S^{2}} - a S + \frac{\bar{a}}{S^{2}} = 0 .

(12.3)

If a solution S to the system of Eqs. () and () lies on the unit circle, that is,

|S| = 1

⁠, then

S^{- 1} = \bar{S}

and the system is equivalent to the following system of equations:

R \{S^{2} + a S + \frac{b}{2}\} = 0, I \{2 S^{2} + a S\} = 0, S = e^{i φ}, φ \in R .

(12.4)

A trigonometric version of the system equation [Eq. ()] is

\cos (2 φ) + |a| \cos (φ + α) = - \frac{b}{2}, 2 \sin (2 φ) + |a| \sin (φ + α) = 0,

(12.5)

a = |a| \exp \{i α\}, S = \exp \{i φ\}, φ, α \in R .

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
$C^{n}$: set of n dimensional column vectors with complex complex-valued entries
$C^{n \times m}$: set of n × m matrices with complex-valued entries
$D (ω, k)$: CCTWT dispersion function
$D_{C} (ω, k)$: CCS dispersion function
$D_{K} (ω, k)$: MCK dispersion function
$\det \{A\}$: the determinant of matrix A
$diag (A_{1}, A_{2}, \dots, A_{r})$: block diagonal matrix with indicated blocks
$\dim (W)$: dimension of the vector space W
EL: the Euler–Lagrange (equations)
$I_{ν} ν \times ν$: identity matrix
$\ker (A)$: 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
$\bar{s}$: complex-conjugate to complex number s
$σ \{A\}$: spectrum of matrix A
$R^{n \times m}$: set of n × m matrices with real-valued entries
$χ_{A} (s) = \det \{s I_{ν} - A\}$: characteristic polynomial of a ν × ν matrix A

APPENDIX A: FOURIER TRANSFORM

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:

f (t) = \int_{- \infty}^{\infty} \hat{f} (ω) e^{- i ω t} d ω, \hat{f} (ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} f (t) e^{i ω t} d t,

(A1)

f (z, t) = \int_{- \infty}^{\infty} \hat{f} (k, ω) e^{- i (ω t - k z)} d k d ω,

(A2)

\hat{f} (k, ω) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} f (z, t) e^{i (ω t - k z)} d z d t .

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 * g] (t) = [g * f] (t) = \int_{- \infty}^{\infty} f (t - t^{'}) g (t^{'}) d t^{'},

(A3)

[f * g] (z, t) = [g * f] (z, t) = \int_{- \infty}^{\infty} f (z - z^{'}, t - t^{'}) g (z^{'}, t^{'}) d z^{'} d t^{'} .

(A4)

Then, its Fourier transform as defined by Eqs. () and () satisfies the following properties:

\hat{f * g} (ω) = \hat{f} (ω) \hat{g} (ω),

(A5)

\hat{f * g} (k, ω) = \hat{f} (k, ω) \hat{g} (k, ω) .

(A6)

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 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 $r (λ)$ be the least integer k such that $N [{(A - λ I)}^{k}] = N [{(A - λ I)}^{k + 1}]$ ⁠, where $N [C]$ is a null space of a matrix C. Then, we refer to $M_{λ} = N [{(A - λ I)}^{r (λ)}]$ 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, …, λ_p be its distinct eigenvalues. Then, generalized eigenspaces

M_{λ_{1}}, \dots, M_{λ_{p}}

are linearly independent, invariant under the matrix A and

C^{n} = M_{λ_{1}} \oplus \dots \oplus M_{λ_{p}} .

(B1)

Consequently, any vector x₀ in

C^{n}

can be represented uniquely as

x_{0} = \sum_{j = 1}^{p} x_{0, j}, x_{0, j} \in M_{λ_{j}},

(B2)

and

\exp \{A t\} x_{0} = \sum_{j = 1}^{p} e^{λ_{j} t} p_{j} (t),

(B3)

where column-vector polynomials

p_{j} (t)

satisfy

p_{j} (t) = \sum_{k = 0}^{r (λ_{j}) - 1} {(A - λ_{j} I)}^{k} \frac{t^{k}}{k!} x_{0, j}, x_{0, j} \in M_{λ_{j}}, 1 \leq j \leq p .

(B4)

For a complex number λ, a Jordan block

J_{r} (λ)

of size r ≥ 1 is an r × r upper triangular matrix of the form

J_{r} (λ) = λ I_{r} + K_{r} = [\begin{matrix} λ & 1 & \dots & 0 & 0 \\ 0 & λ & 1 & \dots & 0 \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & λ & 1 \\ 0 & 0 & \dots & 0 & λ \end{matrix}], J_{1} (λ) = [λ], J_{2} (λ) = [\begin{matrix} λ & 1 \\ 0 & λ \end{matrix}],

(B5)

K_{r} = J_{r} (0) = [\begin{matrix} 0 & 1 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 & 1 \\ 0 & 0 & \dots & 0 & 0 \end{matrix}] .

(B6)

The special Jordan block

K_{r} = J_{r} (0)

defined by Eq. () is an nilpotent matrix that satisfies the following identities:

K_{r}^{2} = [\begin{matrix} 0 & 0 & 1 & \dots & 0 \\ 0 & 0 & 0 & \dots & ⋮ \\ 0 & 0 & ⋱ & \dots & 1 \\ ⋮ & ⋮ & ⋱ & 0 & 0 \\ 0 & 0 & \dots & 0 & 0 \end{matrix}], \dots, K_{r}^{r - 1} = [\begin{matrix} 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 & 0 \\ 0 & 0 & \dots & 0 & 0 \end{matrix}], K_{r}^{r} = 0 .

(B7)

A general Jordan n × n matrix J is defined as a direct sum of Jordan blocks, that is,

J = [\begin{matrix} J_{n_{1}} (λ_{1}) & 0 & \dots & 0 & 0 \\ 0 & J_{n_{2}} (λ_{2}) & 0 & \dots & 0 \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & J_{n_{q - 1}} (λ_{n_{q} - 1}) & 0 \\ 0 & 0 & \dots & 0 & J_{n_{q}} (λ_{n_{q}}) \end{matrix}], n_{1} + n_{2} + \dots n_{q} = n,

(B8)

where λ_j need not be distinct. Any square matrix A is similar to a Jordan matrix as in Eq. (), 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:

Q^{- 1} A Q = J,

(B9)

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 () of matrix A.

A function

f (J_{r} (s))

of a Jordan block

J_{r} (s)

is represented by the following equation (Ref. 50, Sec. 7.9, Ref. 51, Sec. 10.5):

f (J_{r} (s)) = [\begin{matrix} f (s) & \partial f (s) & \frac{\partial^{2} f (s)}{2} & \dots & \frac{\partial^{r - 1} f (s)}{(r - 1)!} \\ 0 & f (s) & \partial f (s) & \dots & \frac{\partial^{r - 2} f (s)}{(r - 2)!} \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & f (s) & \partial f (s) \\ 0 & 0 & \dots & 0 & f (s) \end{matrix}] .

(B10)

Note that any function

f (J_{r} (s))

of the Jordan block

J_{r} (s)

is evidently an upper triangular Toeplitz matrix.

There are two particular cases of formula (), which can also be derived straightforwardly using Eq. (),

\exp \{K_{r} t\} = \sum_{k = 0}^{r - 1} \frac{t^{k}}{k!} K_{r}^{k} = [\begin{matrix} 1 & t & \frac{t^{2}}{2!} & \dots & \frac{t^{r - 1}}{(r - 1)!} \\ 0 & 1 & t & \dots & \frac{t^{r - 2}}{(r - 2)!} \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 1 & t \\ 0 & 0 & \dots & 0 & 1 \end{matrix}],

(B11)

{[J_{r} (s)]}^{- 1} = \sum_{k = 0}^{r - 1} s^{- k - 1} {(- K_{r})}^{k} = [\begin{matrix} \frac{1}{s} & - \frac{1}{s^{2}} & \frac{1}{s^{3}} & \dots & \frac{{(- 1)}^{r - 1}}{s^{r}} \\ 0 & \frac{1}{s} & - \frac{1}{s^{2}} & \dots & \frac{{(- 1)}^{r - 2}}{s^{r - 1}} \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & \frac{1}{s} & - \frac{1}{s^{2}} \\ 0 & 0 & \dots & 0 & \frac{1}{s} \end{matrix}] .

(B12)

APPENDIX C: COMPANION MATRIX AND CYCLICITY CONDITION

The companion matrix

C (a)

for the monic polynomial

a (s) = s^{ν} + \sum_{1 \leq k \leq ν} a_{ν - k} s^{ν - k},

(C1)

where coefficients a_k are complex numbers, is defined by Ref. 51, Sec. 5.2,

C (a) = [\begin{matrix} 0 & 1 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 \\ 0 & 0 & 0 & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 & 1 \\ - a_{0} & - a_{1} & \dots & - a_{ν - 2} & - a_{ν - 1} \end{matrix}] .

(C2)

Note that

\det \{C (a)\} = {(- 1)}^{ν} a_{0} .

(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

A \in C^{n \times n}

be an n × n matrix with complex-valued entries. Let

s p e c (A) = \{ζ_{1}, ζ_{2}, \dots, ζ_{r}\}

be the set of all distinct eigenvalues and

k_{j} = {i n d}_{A} (ζ_{j})

be the largest size of Jordan block associated with ζ_j. Then, the minimal polynomial

μ_{A} (s)

of the matrix A, which is a monic polynomial of the smallest degree such that

μ_{A} (A) = 0

, satisfies

μ_{A} (s) = \prod_{j = 1}^{r} {(s - ζ_{j})}^{k_{j}} .

(C4)

Furthermore, the following statements are equivalent:

$μ_{A} (s) = χ_{A} (s) = \det \{s I - A\}$ ⁠.
A is cyclic.
For every ζ_j, the Jordan form of A contains exactly one block associated with ζ_j.
A is similar to the companion matrix $C (χ_{A})$ ⁠.

Proposition 13

(companion matrix factorization). Let

a (s)

be a monic polynomial having degree ν and

C (a)

be its ν × ν companion matrix. Then, there exist unimodular ν × ν matrices

S_{1} (s)

and

S_{2} (s)

, that is,

\det \{S_{m}\} = \pm 1

, m = 1, 2, such that

s I_{ν} - C (a) = S_{1} (s) [\begin{matrix} I_{ν - 1} & 0_{(ν - 1) \times 1} \\ 0_{1 \times (ν - 1)} & a (s) \end{matrix}] S_{2} (s) .

(C5)

Consequently,

C (a)

is cyclic and

χ_{C (a)} (s) = μ_{C (a)} (s) = a (s) .

(C6)

The following statement summarizes important information on the Jordan form of the companion matrix and the generalized Vandermonde matrix (Ref. 51, Sec. 5.16, Ref. 52, Sec. 2.11, and Ref. 50, Sec. 7.9).

Proposition 14

(Jordan form of the companion matrix). Let

C (a)

be an n × n companion matrix of the monic polynomial

a (s)

defined by Eq. (C1). Suppose that the set of distinct roots of polynomial

a (s)

is

\{ζ_{1}, ζ_{2}, \dots, ζ_{r}\}

and

\{n_{1}, n_{2}, \dots, n_{r}\}

is the corresponding set of the root multiplicities such that

n_{1} + n_{2} + \dots + n_{r} = n .

(C7)

Then,

C (a) = R J R^{- 1},

(C8)

where

J = diag \{J_{n_{1}} (ζ_{1}), J_{n_{2}} (ζ_{2}), \dots, J_{n_{r}} (ζ_{r})\}

(C9)

is the Jordan form of companion matrix

C (a)

and n × n matrix R is the so-called generalized Vandermonde matrix defined by

R = [R_{1} | R_{2} | \dots | R_{r}],

(C10)

where R_j is the n × n_j matrix of the form

R_{j} = [\begin{matrix} 1 & 0 & \dots & 0 \\ ζ_{j} & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ζ_{j}^{n - 2} & (\binom{n - 2}{1}) ζ_{j}^{n - 3} & \dots & (\binom{n - 2}{n_{j} - 1}) ζ_{j}^{n - n_{j} - 1} \\ ζ_{j}^{n - 1} & (\binom{n - 1}{1}) ζ_{j}^{n - 2} & \dots & (\binom{n - 1}{n_{j} - 1}) ζ_{j}^{n - n_{j}} \end{matrix}] .

(C11)

As a consequence of representation (C9),

C (a)

is a cyclic matrix.

As to the structure of matrix R_j in Eq. (), if we denote by

Y (ζ_{j})

its first column, then it can be expressed as follows (Ref. 52, Sec. 2.11):

R_{j} = [Y^{(0)} | Y^{(1)} | \dots | Y^{(n_{j} - 1)}], Y^{(m)} = \frac{1}{m!} \partial_{s_{j}}^{m} Y (ζ_{j}), 0 \leq m \leq n_{j} - 1 .

(C12)

In the case when all eigenvalues of a cyclic matrix are distinct, then the generalized Vandermonde matrix turns into the standard Vandermonde matrix,

V = [\begin{matrix} 1 & 1 & \dots & 1 \\ ζ_{1} & ζ_{2} & \dots & ζ_{n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ζ_{1}^{n - 2} & ζ_{2}^{n - 2} & \dots & ζ_{n}^{n - 2} \\ ζ_{1}^{n - 1} & ζ_{2}^{n - 1} & \dots & ζ_{n}^{n - 1} \end{matrix}] .

(C13)

APPENDIX D: MATRIX POLYNOMIALS

An important incentive for considering matrix polynomials is that they are relevant to the spectral theory of the differential equations of the order higher than 1, particularly the Euler–Lagrange equations, which are the second-order differential equations in time. We provide here selected elements of the theory of matrix polynomials following mostly (Ref. 53, Secs. II.7 and II.8; Ref. 54, Chap. 9). The general matrix polynomial eigenvalue problem reads

A (s) x = 0, A (s) = \sum_{j = 0}^{ν} A_{j} s^{j}, x \neq 0,

(D1)

where s is a complex number, A_k are constant m × m matrices, and

x \in C^{m}

is an m-dimensional column-vector. We refer to problem (1) of funding complex-valued s and non-zero vector

x \in C^{m}

as the polynomial eigenvalue problem.

If a pair of a complex s and non-zero vector x solves problem (), we refer to s as an eigenvalue or as a characteristic value and to x as the corresponding value to the s eigenvector. Evidently, the characteristic values of problem () can be found from polynomial characteristic equation as follows:

\det \{A (s)\} = 0 .

(D2)

We refer to matrix polynomial

A (s)

as regular if

\det \{A (s)\}

is not identically zero. We denote by

m (s_{0})

the multiplicity (called also algebraic multiplicity) of eigenvalue s₀ as a root of polynomial

\det \{A (s)\}

⁠. In contrast, the geometric multiplicity of eigenvalue s₀ is defined as

\dim \{\ker \{A (s_{0})\}\}

⁠, where

\ker \{A\}

defined for any square matrix A stands for the subspace of solutions x to equation Ax = 0. Evidently, the geometric multiplicity of eigenvalue does not exceed its algebraic one; see Corollary 17.

It turns out that the matrix polynomial eigenvalue problem () can be always recast as the standard “linear” eigenvalue problem, namely,

(s B - A) x = 0,

(D3)

where mν × mν matrices

A

and

B

are defined by

B = [\begin{matrix} I & 0 & \dots & 0 & 0 \\ 0 & I & 0 & \dots & 0 \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & I & 0 \\ 0 & 0 & \dots & 0 & A_{ν} \end{matrix}], A = [\begin{matrix} 0 & I & \dots & 0 & 0 \\ 0 & 0 & I & \dots & 0 \\ 0 & 0 & 0 & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 & I \\ - A_{0} & - A_{1} & \dots & - A_{ν - 2} & - A_{ν - 1} \end{matrix}],

(D4)

with

I

being the m × m identity matrix. Matrix

A

⁠, particularly in the monic case, is often referred to as companion matrix. In the case of monic polynomial

A (λ)

⁠, when

A_{ν} = I

is the m × m identity matrix, matrix

B = I

is the mν × mν identity matrix. The reduction of original polynomial problem () to an equivalent linear problem () is called linearization.

The linearization is not unique, and one way to accomplish is by introducing the so-called known “companion polynomial,” which is the mν × mν matrix,

C_{A} (s) = s B - A = [\begin{matrix} s I & - I & \dots & 0 & 0 \\ 0 & s I & - I & \dots & 0 \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋮ & s I & - I \\ A_{0} & A_{1} & \dots & A_{ν - 2} & s A_{ν} + A_{ν - 1} \end{matrix}] .

(D5)

Note that in the case of EL equations, the linearization can be accomplished by the relevant Hamilton equations.

To demonstrate the equivalency between the eigenvalue problems for the mν × mν companion polynomial

C_{A} (s)

and the original m × m matrix polynomial

A (s)

⁠, we introduce two mν × mν matrix polynomials

E (s)

and

F (s)

⁠. Namely,

E (s) = [\begin{matrix} E_{1} (s) & E_{2} (s) & \dots & E_{ν - 1} (s) & I \\ - I & 0 & 0 & \dots & 0 \\ 0 & - I & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 & 0 \\ 0 & 0 & \dots & - I & 0 \end{matrix}],

(D6)

\det \{E (s)\} = 1,

where m × m matrix polynomials

E_{j} (s)

are defined by the following recursive formulas:

E_{ν} (s) = A_{ν}, E_{j - 1} (s) = A_{j - 1} + s E_{j} (s), j = ν, \dots, 2 .

(D7)

Matrix polynomial

F (s)

is defined by

F (s) = [\begin{matrix} I & 0 & \dots & 0 & 0 \\ - s I & I & 0 & \dots & 0 \\ 0 & - s I & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & I & 0 \\ 0 & 0 & \dots & - s I & I \end{matrix}], \det \{F (s)\} = 1 .

(D8)

Note that both matrix polynomials

E (s)

and

F (s)

have constant determinants, readily implying that their inverses

E^{- 1} (s)

and

F^{- 1} (s)

are also matrix polynomials. Then, it is straightforward to verify that

E (s) C_{A} (s) F^{- 1} (s) = E (s) (s B - A) F^{- 1} (s) = [\begin{matrix} A (s) & 0 & \dots & 0 & 0 \\ 0 & I & 0 & \dots & 0 \\ 0 & 0 & ⋱ & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & I & 0 \\ 0 & 0 & \dots & 0 & I \end{matrix}] .

(D9)

The identity () where matrix polynomials

E (s)

and

F (s)

have constant determinants can be viewed as the definition of equivalency between matrix polynomial

A (s)

and its companion polynomial

C_{A} (s)

⁠.

Let us take a look at the eigenvalue problem for eigenvalue s and eigenvector

x \in C^{m ν}

associated with companion polynomial

C_{A} (s)

⁠, that is,

(s B - A) x = 0, x = [\begin{matrix} x_{0} \\ x_{1} \\ x_{2} \\ ⋮ \\ x_{ν - 1} \end{matrix}] \in C^{m ν}, x_{j} \in C^{m}, 0 \leq j \leq ν - 1,

(D10)

where

(s B - A) x = [\begin{matrix} s x_{0} - x_{1} \\ s x_{1} - x_{2} \\ ⋮ \\ s x_{ν - 2} - x_{ν - 1} \\ \sum_{j = 0}^{ν - 2} A_{j} x_{j} + (s A_{ν} + A_{ν - 1}) x_{ν - 1} \end{matrix}] .

(D11)

With Eqs. () and () in mind, we introduce the following vector polynomial:

x_{s} = [\begin{matrix} x_{0} \\ s x_{0} \\ ⋮ \\ s^{ν - 2} x_{0} \\ s^{ν - 1} x_{0} \end{matrix}], x_{0} \in C^{m} .

(D12)

Not accidentally, the components of the vector

x_{s}

in its representation () are in evident relation with the derivatives

\partial_{t}^{j} (x_{0} e^{s t}) = s^{j} x_{0} e^{s t}

⁠. This is just another sign of intimate relations between the matrix polynomial theory and the theory of systems of ordinary differential equations; see Appendix E.

Theorem 15

(eigenvectors). Let

A (s)

as in (D1) be regular, which

\det \{A (s)\}

is not identically zero, and let mν × mν matrices

A

and

B

be defined by (D2). Then, the following identities hold:

(s B - A) x_{s} = [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ A (s) x_{0} \end{matrix}], x_{s} = [\begin{matrix} x_{0} \\ s x_{0} \\ ⋮ \\ s^{ν - 2} x_{0} \\ s^{ν - 1} x_{0} \end{matrix}],

(D13)

\det \{A (s)\} = \det \{s B - A\}, \det \{B\} = \det \{A_{ν}\},

(D14)

where

\det \{A (s)\} = \det \{s B - A\}

is a polynomial of the degree mν if

\det \{B\} = \det \{A_{ν}\} \neq 0

. There is one-to-one correspondence between solutions of equations

A (s) x = 0

and

(s B - A) x = 0

. Namely, a pair

s, x

solves eigenvalue problem

(s B - A) x = 0

if and only if the following equalities hold:

x = x_{s} = [\begin{matrix} x_{0} \\ s x_{0} \\ ⋮ \\ s^{ν - 2} x_{0} \\ s^{ν - 1} x_{0} \end{matrix}], A (s) x_{0} = 0, x_{0} \neq 0, \det \{A (s)\} = 0 .

(D15)

Proof.

Polynomial vector identity (D13) readily follows from Eqs. (D11) and (D12). Identities (D14) for the determinants follow straightforwardly from Eqs. (D12), (D15), and (D9). If $\det \{B\} = \det \{A_{ν}\} \neq 0$ ⁠, then the degree of the polynomial $\det \{s B - A\}$ has to be mν since $A$ and $B$ are mν × mν matrices.

Suppose that Eq. (D15) holds. Then, combining them with proven identity (D13), we get $(s B - A) x_{s} = 0$ ⁠, proving that expressions (D15) define an eigenvalue s and an eigenvector $x = x_{s}$ ⁠.

Suppose now that

(s B - A) x = 0

where

x \neq 0

⁠. Combining that with Eq. (), we obtain

x_{1} = s x_{0}, x_{2} = s x_{1} = s^{2} x_{0}, \dots, x_{ν - 1} = s^{ν - 1} x_{0},

(D16)

implying that

x = x_{s} = [\begin{matrix} x_{0} \\ s x_{0} \\ ⋮ \\ s^{ν - 2} x_{0} \\ s^{ν - 1} x_{0} \end{matrix}], x_{0} \neq 0,

(D17)

and

\sum_{j = 0}^{ν - 2} A_{j} x_{j} + (s A_{ν} + A_{ν - 1}) x_{ν - 1} = A (s) x_{0} .

(D18)

Using Eq. () and identity (), we obtain

0 = (s B - A) x = (s B - A) x_{s} = [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ A (s) x_{0} \end{matrix}] .

(D19)

Equation () readily implies

A (s) x_{0} = 0

and

\det \{A (s)\} = 0

since x₀ ≠ 0. That completes the proof.□

Remark 16

(characteristic polynomial degree). Note that according to Theorem 15, the characteristic polynomial $\det \{A (s)\}$ for the m × m matrix polynomial $A (s)$ has the degree mν, whereas in the linear case $s I - A_{0}$ for the m × m identity matrix $I$ and m × m matrix A₀, the characteristic polynomial $\det \{s I - A_{0}\}$ is of the degree m. This can be explained by observing that in the non-linear case of m × m matrix polynomial $A (s)$ ⁠, we are dealing effectively with many more m × m matrices A than just a single matrix A₀.

Another problem of our particular interest related to the theory of matrix polynomials is the degeneracy of eigenvalues and eigenvectors and, consequently, the existence of non-trivial Jordan blocks, that is, Jordan blocks of dimensions higher or equal to 2. The general theory addresses this problem by introducing the so-called “Jordan chains,” which are intimately related to the theory of system of differential equations expressed as $A (\partial_{t}) x (t) = 0$ and their solutions of the form $x (t) = p (t) e^{s t}$ ⁠, where $p (t)$ is a vector polynomial; see Appendix E and Ref. 53, Chap. I, II and Ref. 54, Chap. 9. Avoiding the details of Jordan chain developments, we simply note that an important point of Theorem 15 is that there is one-to-one correspondence between solutions of equations $A (s) x = 0$ and $(s B - A) x = 0$ ⁠, and it has the following immediate implication.

Corollary 17

(equality of the dimensions of eigenspaces). Under the conditions of Theorem 15 for any eigenvalue s₀, that is,

\det \{A (s_{0})\} = 0

, we have

\dim \{\ker \{s_{0} B - A\}\} = \dim \{\ker \{A (s_{0})\}\} .

(D20)

In other words, the geometric multiplicities of the eigenvalue s₀ associated with matrices

A (s_{0})

and

s_{0} B - A

are equal. In view of identity (D20), the following inequality holds for the (algebraic) multiplicity

m (s_{0})

⁠:

m (s_{0}) \geq \dim \{\ker \{A (s_{0})\}\} .

(D21)

The next statement shows that if the geometric multiplicity of an eigenvalue is strictly less than its algebraic one, then there exist non-trivial Jordan blocks, that is, Jordan blocks of dimensions higher or equal to 2.

Theorem 18

(non-trivial Jordan block). Assuming notations introduced in Theorem 15, let us suppose that the multiplicity

m (s_{0})

of eigenvalue s₀ satisfies

m (s_{0}) > \dim \{\ker \{A (s_{0})\}\} .

(D22)

Then, the Jordan canonical form of companion polynomial

C_{A} (s) = s B - A

has at least one nontrivial Jordan block of the dimension exceeding 2.

In particular, if

\dim \{\ker \{s_{0} B - A\}\} = \dim \{\ker \{A (s_{0})\}\} = 1

(D23)

and

m (s_{0}) \geq 2

⁠, then the Jordan canonical form of companion polynomial

C_{A} (s) = s B - A

has exactly one Jordan block associated with eigenvalue s₀ and its dimension is

m (s_{0})

.

The Proof of Theorem 18 follows straightforwardly from the definition of the Jordan canonical form and its basic properties. Note that if Eq. (D23) holds, it implies that eigenvalue 0 is cyclic (nonderogatory) for matrix $A (s_{0})$ and eigenvalue s₀ is cyclic (nonderogatory) for matrix $B^{- 1} A$ ⁠, provided that $B^{- 1}$ exists; see Appendix C.

APPENDIX E: VECTOR DIFFERENTIAL EQUATIONS AND THE JORDAN CANONICAL FORM

In this section, we relate the vector ordinary differential equations to the matrix polynomials reviewed in Appendix D following Ref. 55, Secs. 5.1 and 5.7, Ref. 53, Sec. II.8.3, Ref. 47, Sec. III.4, and Ref. 50, Sec. 7.9.

Equation

A (s) x = 0

with polynomial matrix

A (s)

defined by Eq. () corresponds to the following m-vector νth order ordinary differential:

A (\partial_{t}) x (t) = 0, where A (\partial_{t}) = \sum_{j = 0}^{ν} A_{j} \partial_{t}^{j},

(E1)

where

A_{j} = A_{j} (t)

are m × m matrices. Introducing the mν-column-vector function

Y (t) = [\begin{matrix} x (t) \\ \partial_{t} x (t) \\ ⋮ \\ \partial_{t}^{ν - 2} x (t) \\ \partial_{t}^{ν - 1} x (t) \end{matrix}]

(E2)

and under the assumption that matrix

A_{ν} (t)

is the identity matrix, the differential equation [Eq. ()] can be recast and the first order differential equation

\partial_{t} Y (t) = A Y (t),

(E3)

where

A

is the mν × mν matrix defined by

A = A (t) = [\begin{matrix} 0 & I & \dots & 0 & 0 \\ 0 & 0 & I & \dots & 0 \\ 0 & 0 & 0 & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 & I \\ - A_{0} (t) & - A_{1} (t) & \dots & - A_{ν - 2} (t) & - A_{ν - 1} (t) \end{matrix}], A_{ν} (t) = I .

(E4)

1. Constant coefficients case.

Let us consider an important special case of Eq. () when matrices A_j are m × m that do not depended on t. Then, Eq. () can be recast as

B \partial_{t} Y (t) = A Y (t),

(E5)

where

A

and

B

are mν × mν companion matrices defined by Eq. ().

In the case when A_ν is an invertible m × m matrix, Eq. () can be recast further as

\partial_{t} Y (t) = \dot{A} Y (t),

(E6)

where

\dot{A} = [\begin{matrix} 0 & I & \dots & 0 & 0 \\ 0 & 0 & I & \dots & 0 \\ 0 & 0 & 0 & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 & I \\ - {\dot{A}}_{0} & - {\dot{A}}_{1} & \dots & - {\dot{A}}_{ν - 2} & - {\dot{A}}_{ν - 1} \end{matrix}], {\dot{A}}_{j} = A_{ν}^{- 1} A_{j}, 0 \leq ν - 1 .

(E7)

Note that one can interpret Eq. () as a particular case of Eq. () where matrices A_ν and

B

are identity matrices of the respective dimensions m × m and mν × mν and that polynomial matrix

A (s)

defined by Eq. () becomes monic matrix polynomial

\dot{A} (s)

⁠, that is,

\dot{A} (s) = I s^{ν} + \sum_{j = 0}^{ν - 1} {\dot{A}}_{j} s^{j}, {\dot{A}}_{j} = A_{ν}^{- 1} A_{j}, 0 \leq ν - 1 .

(E8)

Note that in view of Eq. (), one recovers

x (t)

from

Y (t)

by using the following formula:

x (t) = P_{1} Y (t), P_{1} = [\begin{matrix} I & 0 & \dots & 0 & 0 \end{matrix}],

(E9)

where P₁ is evidently the m × mν matrix.

Observe also that (Ref. 55, Proposition 5.1.2)^10,52

{[\dot{A} (s)]}^{- 1} = P_{1} {[I s - \dot{A}]}^{- 1} R_{1}, P_{1} = [\begin{matrix} I & 0 & \dots & 0 & 0 \end{matrix}], R_{1} = [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ I \end{matrix}],

(E10)

where P₁ and R₁ evidently, respectively, m × mν and mν × m matrices.

The general form for the solution to vector differential equation [Eq. ()] is

Y (t) = \exp \{\dot{A} t\} Y_{0}, Y_{0} \in C^{m ν} .

(E11)

Then, using formulas () and () and Proposition 10, we arrive at the following statement.

Proposition 19

(solution to the vector differential equation). Let

\dot{A}

be the mν × mν companion matrix defined by Eq. (E7), ζ₁, …, ζ_p be its distinct eigenvalues, and

M_{ζ_{1}}, \dots, M_{ζ_{p}}

be the corresponding generalized eigenspaces of the corresponding dimensions

r (ζ_{j})

, 1 ≤ j ≤ p. Then, the mν column-vector solution

Y (t)

to the differential equation (E6) is of the form

Y (t) = \exp \{\dot{A} t\} Y_{0} = \sum_{j = 1}^{p} e^{ζ_{j} t} p_{j} (t), Y_{0} = \sum_{j = 1}^{p} Y_{0, j}, Y_{0, j} \in M_{ζ_{j}},

(E12)

where mν-column-vector polynomials

p_{j} (t)

satisfy

p_{j} (t) = \sum_{k = 0}^{r (ζ_{j}) - 1} \frac{t^{k}}{k!} {(\dot{A} - ζ_{j} I)}^{k} Y_{0, j}, 1 \leq j \leq p .

(E13)

Consequently, the general m-column-vector solution

x (t)

to differential equation (E1) is of the form

x (t) = \sum_{j = 1}^{p} e^{ζ_{j} t} P_{1} p_{j} (t), P_{1} = [\begin{matrix} I & 0 & \dots & 0 & 0 \end{matrix}] .

(E14)

Note that

χ_{\dot{A}} (s) = \det \{s I - \dot{A}\}

is the characteristic function of the matrix

\dot{A}

⁠. Then, using notations of Proposition 19, we obtain

χ_{\dot{A}} (s) = \prod_{j = 1}^{p} {(s - ζ_{j})}^{r (ζ_{j})} .

(E15)

Note also that for any values of complex-valued coefficients b_k, we have

{(\partial_{t} - ζ_{j})}^{r (ζ_{j})} [e^{ζ_{j} t} p_{j} (t)] = 0, p_{j} (t) = \sum_{k = 0}^{r (ζ_{j}) - 1} b_{k} t^{k},

(E16)

implying together with representation () that

χ_{\dot{A}} (\partial_{t}) [e^{ζ_{j} t} p_{j} (t)] = 0, p_{j} (t) = \sum_{k = 0}^{r (ζ_{j}) - 1} b_{k} t^{k} .

(E17)

Combining now Proposition 19 with Eq. (), we obtain the following statement.

Corollary 20

(property of a solution to the vector differential equation). Let

x (t)

be the general m-column-vector solution

x (t)

to differential equation (E1). Then,

x (t)

satisfies

χ_{\dot{A}} (\partial_{t}) x (t) = 0 .

(E18)

APPENDIX F: FLOQUET THEORY

We provide here a concise review of the Floquet theory following Ref. 56, Chap. III, Ref. 47, Sec. III.7, and Ref. 57, Sec. II.2. The primary subject of the Floquet theory is the general form of solutions to the ordinary differential equations with periodic coefficients. With that in mind, suppose that (i) z is a real valued variable, (ii)

x (z)

is an n-vector valued function of z, (iii)

A (z)

is an n × n matrix-valued ς-periodic function of z, and consider the following homogeneous linear periodic system:

\partial_{z} x (z) = A (z) x (z), A (z + ς) = A (z), ς > 0 .

(F1)

We would like to give a complete characterization of the general structure of the solutions to Eq. (). We start with the following statement, showing how to define the logarithm B of a matrix C so that C = e^B.

Lemma 21

(logarithm of a matrix). Let C be an n × n matrix with

\det \{C\} \neq 0

. Suppose that C = Z⁻¹JZ, where J is the Jordan canonical form of C as described in Proposition 11. Then, using the block representation (B8) for J, that is,

J = J = diag \{J_{n_{1}} (ζ_{1}), J_{n_{2}} (ζ_{2}), \dots, J_{n_{r}} (ζ_{r})\}, n_{1} + n_{2} + \dots n_{q} = n,

(F2)

we decompose J into its diagonal and nilpotent components,

J = diag \{λ_{1} I_{n_{1}}, λ_{2} I_{n_{2}}, \dots, λ_{q} I_{n_{q}}\} + K,

(F3)

where

D = diag \{λ_{1} I_{n_{1}}, λ_{2} I_{n_{2}}, \dots, λ_{q} I_{n_{q}}\}, K = diag \{K_{n_{1}}, K_{n_{2}}, \dots, K_{n_{q}}\},

(F4)

K_{n_{j}} = J_{n_{j}} (λ_{j}) - λ_{j} I_{n_{j}}, 1 \leq j \leq q .

Then, let

\ln (*)

be a branch of the logarithm and let

H = \ln J = diag \{\ln (λ_{1}) I_{n_{1}}, \ln (λ_{2}) I_{n_{2}}, \dots, \ln (λ_{q}) I_{n_{q}}\} + S,

(F5)

where

I_{n_{j}}

are identity matrices of identified dimensions, and

S = diag \{S_{n_{1}}, S_{n_{2}}, \dots, S_{n_{q}}\}, S_{n_{j}} = \sum_{m = 1}^{n_{j} - 1} {(- 1)}^{m - 1} \frac{1}{m λ_{j}^{m}} K_{n_{j}}^{m}, 1 \leq j \leq q .

(F6)

Then,

C = e^{B}, B = \ln C = Z^{- 1} H Z,

(F7)

where matrix H is defined by Eq. (F5).

Note that matrix S in Eqs. () and () is associated with the nilpotent part of Jordan canonical form J. The expression for

S_{n_{j}}

originates in the series

\ln (1 + s) = \sum_{m = 1}^{\infty} {(- 1)}^{m - 1} \frac{1}{m} s^{m} = s - \frac{s^{2}}{2} + \frac{s^{3}}{3} + \dots,

(F8)

and it is a finite sum since

K_{n_{j}}

is a nilpotent matrix such that

K_{n_{j}}^{m} = 0, m \geq n_{j}, 1 \leq j \leq q .

(F9)

An n × n matrix

Φ (z)

is called matrizant (matriciant) of Eq. () if it satisfies the following equation:

\partial_{z} Φ (z) = A (z) Φ (z), Φ (0) = I, A (z + ς) = A (z), ς > 0,

(F10)

where

I

is the n × n identity matrix. Matrix

Φ (z)

is also called principal fundamental matrix solution to Eq. (). Evidently,

x (z) = Φ (z) x_{0}

is the solution to Eq. () with the initial condition

x (0) = x_{0}

⁠. Using the fundamental solution

Φ (z)

⁠, we can represent any matrix solution

Ψ (z)

to Eq. () based on its initial values as follows:

\partial_{z} Ψ (z) = A (z) Ψ (z), Ψ (z) = Φ (z) Ψ (0) .

(F11)

In the case of ς-periodic matrix function

A (z)

⁠, the matrix function

Ψ (z) = Φ (z + ς)

is evidently a solution to Eq. (), and consequently,

Φ (z + ς) = Φ (z) Φ (ς) .

(F12)

It turns out that matrix

M_{ς} = Φ (ς)

called the monodromy matrix is of particular importance for the analysis of solutions to Eq. () with ς-periodic matrix function

A (z)

⁠.

The monodromy matrix is integrated into the formulation of the main statement of the Floquet theory describing the structure of solutions to Eq. (F11) for ς-periodic matrix function $A (z)$ ⁠.

Theorem 22

(Floquet). Suppose that

A (z)

is a ς-periodic continuous function of z. Let

Φ (z)

be the matrizant of Eq. (F10), and let

M_{ς} = Φ (ς)

be the corresponding monodromy matrix. Using the statement of Lemma 21, we introduce matrix Γ defined by

Γ = \frac{1}{ς} \ln M_{ς} = \frac{1}{ς} \ln Φ (ς), implying M_{ς} = Φ (ς) = e^{Γ ς} .

(F13)

Then, matrizant

Φ (z)

satisfies the following equation called Floquet representation,

Φ (z) = P (z) e^{Γ z}, P (z + ς) = P (z), P (0) = I,

(F14)

where

P (z)

is a differentiable ς-periodic matrix function of z.

Proof.

Let us define matrix

P (z)

by the following equation:

P (z) = Φ (z) e^{- Γ z} .

(F15)

Then, combining representation () for

P (z)

with Eqs. () and (), we obtain

P (z + ς) = Φ (z + ς) e^{- Γ (z + ς)} = Φ (z) Φ (ς) e^{- Γ ς} e^{- Γ z} = Φ (z) e^{- Γ z} = P (z),

(F16)

that is,

P (z)

is a differentiable ς-periodic matrix function of z. Equality

P (0) = I

readily follows from Eq. () and equality

Φ (0) = I

⁠.□

The eigenvalues of the monodromy matrix $Φ (ς) = e^{Γ ς}$ are called Floquet (characteristic) multipliers, and their logarithms (not uniquely defined) are called characteristic exponents.

Definition 23

(Floquet multipliers, characteristic exponents, and eigenmodes). Using the notation of Theorem 22, let us consider complex numbers κ, s_κ and vector y_κ satisfying the following equations:

Γ y_{κ} = κ y_{κ}, Φ (ς) y_{κ} = e^{- Γ ς} y_{κ} = s_{κ} y_{κ}, s_{κ} = e^{κ ς},

(F17)

where evidently κ and y_κ are, respectively, an eigenvalue and the corresponding eigenvector of matrix Γ. We refer to κ and s_κ, respectively, as the Floquet characteristic exponent and the Floquet (characteristic) multiplier.

Using κ and y_κ defined above, we introduce the following special solution to the original differential equation ():

ψ_{κ} (z) = p_{κ} (z) e^{κ z} = Φ (z) y_{κ} = P (z) e^{Γ z} y_{κ}, p_{κ} (z) = P (z) y_{κ},

(F18)

and we refer to it as the Floquet eigenmode. Note that

p_{κ} (z)

in Eq. () is a ς-periodic vector-function of z.

Remark 24

(Floquet eigenmodes). If $ψ_{κ} (z)$ is the Floquet eigenmode defined by (F18) and $R \{κ\} > 0$ or, equivalently, $|s_{κ}| > 1$ ⁠, then $ψ_{κ} (z)$ grows exponentially as z → +∞, and we refer to such $ψ_{κ} (z)$ as the exponentially growing Floquet eigenmode. In the case when $R \{κ\} = 0$ or equivalently $|s_{κ}| = 1$ ⁠, function $ψ_{κ} (z)$ is bounded and we refer to such $ψ_{κ} (z)$ as an oscillatory Floquet eigenmode.

Remark 25

(dispersion relations). In physical applications of the Floquet theory, ς-periodic matrix valued function

A (z)

in differential equation () depends on the frequency ω, that is,

A (z) = A (z, ω)

⁠. In this case, we also have

κ = κ (ω)

⁠. If we naturally introduce the wave number k by

k = k (ω) = - i κ (ω),

(F19)

then the relation between ω and k provided by Eq. () is called the dispersion relation.

APPENDIX G: HAMILTONIAN SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS

We follow here Ref. 56, Sec. I.8, V.1 and Ref. 57, Chap. III. We introduce first indefinite scalar product

⟨x, y⟩

on the vector space

C^{n}

associated with a nonsingular Hermitian n × n matrix G, namely,

⟨x, y⟩ = \bar{⟨y, x⟩} = x^{*} G y, G^{*} = G, \det \{G\} \neq 0, x, y \in C^{n} .

(G1)

We refer to matrix the G metric matrix. We define, then, for any n × n matrix A another matrix A^† called adjoint by using the following relations:

⟨A x, y⟩ = ⟨x, A^{†} y⟩ or equivalently A^{†} = G^{- 1} A^{*} G .

(G2)

Note that relations () readily imply that

{(A B)}^{†} = B^{†} A^{†} .

(G3)

Let G and

H (t)

be Hermitian n × n matrices, and suppose that matrix G is nonsingular. We define the Hamiltonian system of equations to be a system of the form

- i G \partial_{t} x (t) = H (t) x (t), H^{*} (t) = H (t) .

(G4)

If based on matrices G and

H (t)

we introduce the G-skew-Hermitian matrix,

A (t) = i G^{- 1} H (t),

(G5)

we can recast the Hamiltonian system () in the following equivalent form:

\partial_{t} x (t) = A (t) x (t), A^{†} (t) = - A (t) .

(G6)

It turns out that the matrizant

Φ (t)

of Eq. () with G-skew-Hermitian matrix

A (t)

is a G-unitary matrix for each value of t. Indeed, using Eq. () together with Eqs. () and (), we obtain

\partial_{t} [Φ^{†} (t) Φ (t)] = {\{\partial_{t} [Φ (t)]\}}^{†} Φ (t) + Φ^{†} (t) \partial_{t} [Φ (t)]

(G7)

= - Φ^{†} (t) A (t) Φ (t) + Φ^{†} (t) A (t) Φ (t) = 0,

implying that matrizant

Φ (t)

satisfies

Φ^{†} (t) Φ (t) = I or equivalently Φ^{*} (t) G Φ (t) = G,

(G8)

implying that

Φ (t)

is a G-unitary matrix for each value of t. Identity () implies in turn that for any two solutions

x (t)

and

y (t)

to the Hamiltonian system (), we always have

⟨x (t), y (t)⟩ = x^{*} (t) G y (t) = x^{*} (0) Φ^{*} (t) G Φ (t) y (0) = ⟨x (0), y (0)⟩,

(G9)

that is,

⟨x (t), y (t)⟩

does not depend on t.

1. Symmetry of the spectra.

G-unitary, G-skew-Hermitian, and G-Hermitian matrices have special properties described in Table VII. These properties can viewed as symmetries, and not surprisingly, they imply consequent symmetries of the spectra of the matrices. Let $σ \{A\}$ denote the spectrum of matrix A. It is a straightforward exercise to verify based on matrix properties described in Table VII that the following statements hold.

TABLE VII.

G-unitary, G-skew-Hermitian, and G-Hermitian matrices.

G-unitary	G-skew-Hermitian	G-Hermitian
$⟨A x, A y⟩ = ⟨x, y⟩$	$⟨A x, y⟩ = - ⟨x, A y⟩$	$⟨A x, y⟩ = ⟨x, A y⟩$
$A^{†} A = G^{- 1} A^{*} G A = I$ ⁠,	A^† = G⁻¹A*G = −A,	A^† = G⁻¹A*G = A,
A* = GA⁻¹G⁻¹	GA + A*G = 0	GA − A*G = 0
A*GA = G	A = iG⁻¹H, H = H*	A = G⁻¹H, H = H*

Theorem 26

(spectral symmetries). Suppose that matrix A is either G-unitary or G-skew-Hermitian or G-Hermitian. Then, the following statements hold:

If A is G-unitary, then $σ \{A\}$ is symmetric with respect to the unit circle, that is,
$ζ \in σ \{Φ\} \Rightarrow \frac{1}{\bar{ζ}} \in σ \{Φ\} .$
(G10)
If A is G-skew-Hermitian, then $σ \{A\}$ is symmetric with respect the imaginary axis, that is,
$ζ \in σ \{Φ\} \Rightarrow - \bar{ζ} \in σ \{Φ\} .$
(G11)
If A is G-Hermitian, then $σ \{A\}$ is symmetric with respect to real axis, that is,
$ζ \in σ \{Φ\} \Rightarrow \bar{ζ} \in σ \{Φ\} .$
(G12)

The following statement describes the G-orthogonality of invariant subspaces of G-unitary, G-skew-Hermitian, and G-Hermitian matrices (Ref. 56, Sec. 1.8).

Theorem 27

(eigenspaces). Suppose that matrix A is either G-unitary or G-skew-Hermitian or G-Hermitian. Then, the following statements hold. Let

Λ \subset σ \{A\}

be a subset of the spectrum

σ \{A\}

of the matrix A, and let

\tilde{Λ}

be the relevant symmetric image of Λ defined by

\tilde{Λ} = \{\begin{matrix} \{\frac{1}{\bar{ζ}} : ζ \in Λ\} & if & A is G - unitary, \\ \{- \bar{ζ} : ζ \in Λ\} & if & A is G - skew - Hermitian, \\ \{\bar{ζ} : ζ \in Λ\} & if & is G - Hermitian . \end{matrix}

Let

Λ_{1}, Λ_{2} \subset σ \{A\}

be two subsets of the spectrum

σ \{A\}

so that

{\tilde{Λ}}_{1}

and Λ₂ are separated from each other by non-intersecting contours

{\tilde{Γ}}_{1}

and Γ₂. Then, the invariant subspaces E₁ and E₁ of the matrix A corresponding to Λ₁ and Λ₂ are G-orthogonal.

The statement below describes a special property of eigenvectors of a G-unitary matrix.

Lemma 28

(isotropic eigenvector). Let A be a G-unitary matrix and ζ be its eigenvalue that does not lie on the unit circuit, that is,

|ζ| \neq 1

. Then, if x is the eigenvector corresponding to ζ, it is isotropic, that is,

⟨x, x⟩ = x^{*} G x = 0, A x = ζ x, |ζ| \neq 1 .

(G13)

Proof.

Since Ax = ζx and A is a G-unitary, we have

⟨A x, A x⟩ = ⟨ζ x, ζ x⟩ = {|ζ|}^{2} ⟨x, x⟩, ⟨A x, A x⟩ = ⟨x, x⟩ .

Combining the two equations above with

|ζ| \neq 1

⁠, we conclude that

⟨x, x⟩ = 0

⁠, which is the desired equation [Eq. ()].□

2. Special Hamiltonian systems.

With Eq. () in mind, we introduce the following system:

\partial_{t} x (t) = A (t) x (t),

(G14)

where the 4 × 4 matrix function

A (z)

is of the following form special form:

A (z) = [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ a_{1} (z) & a (z) a_{2} & i c_{1} & 0 \\ a (z) a_{3} & a_{4} (z) & 0 & i c_{2} \end{matrix}], a_{1} (z), a_{2}, a_{3}, a_{4} (z), c_{1}, c_{2}, a (z) \in R .

(G15)

The system () is Hamiltonian if we select Hermitian matrix G to be

G = [\begin{matrix} c_{1} & 0 & i & 0 \\ 0 & \frac{c_{2} a_{2}}{a_{3}} 0 & 0 & i \frac{a_{2}}{a_{3}} \\ - i & 0 & 0 & 0 \\ 0 & - i \frac{a_{2}}{a_{3}} & 0 & 0 \end{matrix}], G^{- 1} = [\begin{matrix} 0 & 0 & i & 0 \\ 0 & 0 & 0 & i \frac{a_{3}}{a_{2}} \\ - i & 0 & - c_{1} & 0 \\ 0 & - i \frac{a_{3}}{a_{2}} & 0 & - \frac{c_{2} a_{3}}{a_{2}} \end{matrix}], \det \{G\} = \frac{a_{2}^{2}}{a_{3}^{2}} .

(G16)

Indeed, it is an elementary exercise to verify that for each value of z matrix,

A (z)

is G-skew-Hermitian, that is,

G A (z) + A^{*} (z) G = 0 .

(G17)

APPENDIX H: FOLDED WAVEGUIDE TWT DISPERSION RELATIONS

Using a number of approximations, the authors of Ref. 58 arrived at the following expression of the dispersion relation similar to that of the Pierce theory:

δ k^{'} {(Δ - δ k^{'})}^{2} (1 + \frac{δ k^{'}}{2}) - C^{3} = 0, k^{'} = \frac{k}{k_{c}},

(H1)

where (i) ω_c is the cut-off frequency of the TE₁₀-mode; (ii)

k_{c} = \frac{ω_{c}}{c}

⁠, c is the velocity of light; (iii) C and Δ are, respectively, the coupling (Pierce) parameter and the detuning parameter represented by explicit formulas involving folded waveguide TWT parameters and frequency ω; and (iv) δk is defined by

k_{m} = k_{m}^{0} + δ k, k_{m}^{0} = \frac{2 π m}{a} + \frac{a + h}{a} \sqrt{\frac{ω^{2} - ω_{c}^{2}}{c^{2}}},

(H2)

where

k_{m}^{0}

is an unperturbed propagation constant. In the case C³ → 0, there are four solutions to equation (…): δk = 0 (unperturbed forward propagating wave),

2 k_{0}^{0}

(unperturbed contra propagating wave), and k = k_cΔ (degenerate e-beam mode). If the interaction with the contra propagating wave is neglected,

|δ k| ≪ 2 |k_{0}^{0}|

⁠, then we obtain from () the following third-order dispersion equation:

δ k^{'} 3 - 2 Δ δ k^{'} 2 + Δ^{2} δ k^{'} - C^{3} = 0 .

(H3)

APPENDIX I: CAPACITANCE

According to Ref. 59, Sec. I.1 and Ref. 60, Sec. 3.5.2, the following formulas hold for the capacitance of capacitors of different geometries in Gaussian system of units.

Capacitance for the parallel-plate capacitor consisting of two parallel plates of area A that are separated by distance d is

C = \frac{A}{4 π d} .

(I1)

Capacitance for the spherical capacitor consisting of two concentric spherical shells of radii r₁ < r₂ is

C = \frac{r_{1} r_{2}}{r_{2} - r_{1}}, C ≅ \frac{r_{1}^{2}}{2 (r_{2} - r_{1})} if 0 < \frac{r_{2} - r_{1}}{r_{1}} ≪ 1 .

(I2)

Capacitance of the cylindrical capacitor consisting of two coaxial cylinders of radii r₁ < r₂ and height h is

C = \frac{h}{2 \ln \{\frac{r_{2}}{r_{1}}\}}, C ≅ \frac{h r_{1}}{2 (r_{2} - r_{1})} if 0 < \frac{r_{2} - r_{1}}{r_{1}} ≪ 1 .

(I3)

Capacitances for a number of different geometric shapes are available in Ref. 61, II.3, including the capacitance of the disk of radius r,

C = \frac{2 r}{π^{2}} .

(I4)

Since often the data are available in the SI system of units rather than in Gaussian, it is useful to know that the capacitances in these two systems are related as follows (Ref. 62, App. on units, 4):

C_{SI} = 4 π ε_{0} C_{Gaussian}, ε_{0} = 8.854 187 813 \cdot 1 0^{- 12} F / m .

(I5)

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2023

Author(s)

Analytic theory of coupled-cavity traveling wave tubes

I. INTRODUCTION

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

A. TWT Lagrangian and evolution equations

B. Space-charge wave velocity and electron density fields

III. ANALYTIC MODEL OF COUPLED-CAVITY TRAVELING WAVE TUBE

A. CCTWT Lagrangian and the Euler–Lagrange equations

B. Natural units and dimensionless parameters

C. Euler–Lagrange equations in dimensionless variables

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

IV. SOLUTIONS TO THE COUPLED-CAVITY TWT EQUATIONS

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

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

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

VI. DISPERSION RELATIONS

A. Graphical representation of the dispersion relations

B. Exceptional points of degeneracy

VII. GAIN EXPRESSION IN TERMS OF THE FLOQUET MULTIPLIERS

VIII. SKETCH OF THE MULTICAVITY KLYSTRON ANALYTICAL MODEL

A. The Euler–Lagrange equations in dimensionless variables

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

IX. COUPLED CAVITY STRUCTURE

A. Monodromy matrix and the dispersion-instability relations

B. Exceptional points of degeneracy

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

A. Some points from the kinetic theory

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

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

XI. LAGRANGIAN VARIATIONAL FRAMEWORK

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

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

NOMENCLATURE

APPENDIX A: FOURIER TRANSFORM

APPENDIX B: JORDAN CANONICAL FORM

APPENDIX C: COMPANION MATRIX AND CYCLICITY CONDITION

APPENDIX D: MATRIX POLYNOMIALS

APPENDIX E: VECTOR DIFFERENTIAL EQUATIONS AND THE JORDAN CANONICAL FORM

1. Constant coefficients case.

APPENDIX F: FLOQUET THEORY

APPENDIX G: HAMILTONIAN SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS

1. Symmetry of the spectra.

2. Special Hamiltonian systems.

APPENDIX H: FOLDED WAVEGUIDE TWT DISPERSION RELATIONS

APPENDIX I: CAPACITANCE

REFERENCES

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