J Infrared Milli Terahz Waves DOI 10.1007/s10762-017-0386-x

Cylindrical Cavity with Distributed Longitudinal Corrugations for Second-Harmonic Gyrotrons Vitalii I. Shcherbinin 1

& Viktor I. Tkachenko

1,2

Received: 10 January 2017 / Accepted: 10 April 2017 # Springer Science+Business Media New York 2017

Abstract Metallic cavity with distributed longitudinal corrugations is proposed and studied for the use in a subterahertz second-harmonic gyrotron. The corrugated conducting walls are treated as a homogeneous surface with effective (averaged) anisotropic impedance. The theoretical study incorporates both single-mode and coupled-mode approaches. It is shown that the distributed longitudinal corrugations provide several-fold increase in the Q-value of the operating mode with respect to that of the fundamental competing mode at a reasonable level of ohmic losses and mode conversion in the gyrotron cavity. Keywords Gyrotrons . Surface impedance . Corrugated surface . Cavity resonators . Cyclotron harmonics

1 Introduction Nowadays, terahertz (THz) radiation (occupying the frequency range between 0.3 and 10 THz) is a subject of considerable interest for a number of applications [1] including wireless communications and networking, detection of concealed weapon, explosives and radioactive materials, remote high-resolution imaging, chemical spectroscopy, non-destructive material evaluation, earth and space research, biological spectroscopy, and biomedical diagnostics. The growth of such applications is directly tied to the progress in the development of terahertz sources. Among them, gyrotron is one of the most promising [2]. On the one hand, terahertz gyrotrons can provide rather high averaged output power (kilowatts in short-pulse regime). For comparison, the power-handling capability of conventional vacuum tubes (backward-wave oscillators, traveling-wave tubes, klystrons, orotrons, klinotrons, etc.) and solid-

* Vitalii I. Shcherbinin [email protected]

1

National Science Center BKharkiv Institute of Physics and Technology^ of National Academy of Science of Ukraine, 1 Akademicheskaya St., Kharkiv 61008, Ukraine

2

V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv 61022, Ukraine

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state devices is at least 10,000 times lower in the terahertz band. On the other hand, their cost and dimensions are much lower than those of the free-electron lasers uniquely capable of producing THz radiation with comparable level of output power. The major advances in THz gyrotrons are now associated with tubes based on the pulsed solenoids having maximum field strength of about 50 T. At present, the record-breaking frequency and output power of such tubes are 1.3 THz [3] (at the power of 0.5 kW) and 210 kW [4] (at the frequency of 0.67 THz), respectively. However, many terahertz technologies require continuous-wave radiation. Compared to pulsed solenoids, permanent magnets for gyrotrons possess much lower maximum field strength and are more cumbersome, energyconsuming, and expensive. The gyrotron operation at high cyclotron harmonics has potential to alleviate this limitation. Such operation requires magnetic field, which is inversely proportional to the harmonic number at a given gyrotron frequency. However, realization of harmonic gyrotrons often faces the problem of severe mode competition [5–8] between the operating mode and the first (fundamental) cyclotron harmonic modes. This leads to the narrowed operating region of harmonic gyrotrons. Besides, even though the operating mode is excited first, it becomes completely suppressed by the competitors at increased beam current (and output power). Therefore, for harmonic gyrotrons, it is vital to discriminate against the fundamental competing modes. For instance, this can be done by increasing their losses in the gyrotron cavity. The mode discrimination can be realized by the electrodynamical methods. As a rule, such methods involve appropriate profiling [9–13] of the gyrotron cavity. In brief, the effect of the profiled walls is to trap the operating mode of harmonic gyrotron more effectively than the fundamental competing modes. This allows the diffractive losses of the operating mode to be reduced relative to those of the competitors, thus providing the inverse change in their starting currents. Despite this advantage, the profiled cavities are not always feasible for THz gyrotrons due to the problems of miniaturization and machining tolerances. In particular, such is most likely [14] the case for the gyrotron cavity loaded with iris. Alternatively, the mode selectivity of the gyrotron cavity may be affected by the change in the eigenvalue spectrum. The change in the mode eigenvalue acts effectively as additional profiling of the cavity wall and therefore has effect on the diffractive losses of the mode. More importantly, this effect can be mode-dependent. For instance, such is the case for the coaxial gyrotron cavity [15–17] with tapered inner conductor, which acts unequally on different eigenvalues of cavity modes. Further enhancement of mode selectivity can be achieved with longitudinally corrugated coaxial insert [18–20]. Such insert can be treated as a smooth conductor with mode-dependent anisotropic surface impedance. With properly sized corrugations, it can provide favorable condition of gyrotron operation at high cyclotron harmonics [19, 21]. Besides, the conducting insert induces additional ohmic losses in the gyrotron cavity. These losses are also mode-dependent and can ensure the predominant suppression of the competing modes [19, 21–23]. To be efficient, such mode discrimination, however, requires the coaxial inset to be sufficiently lossy. This seems to be not very attractive for present-day THz gyrotrons, which have already enough high level of ohmic losses [24–26] (up to 90% of generated power). The mode eigenvalues can also be selectively affected by the dielectric coating of the gyrotron cavity. The inner surface of such coating is somewhat similar to the cavity wall with mode-dependent anisotropic impedance [27–30]. Therefore, the dielectric coating, when properly distributed along the cavity, can trap mainly the operating mode of harmonic gyrotron and thus reduce the ratio between the starting currents of the operating and the competing

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modes [31, 32]. The disadvantage of such coating is intrinsic dielectric losses, which result in additional unwanted attenuation of cavity modes. Besides, the question of thermal properties of the dielectric-loaded cavity is still an open question. For these reasons, the all-metal cavities may be more advantageous for use in gyrotrons. In our study, we investigate the all-metal cylindrical gyrotron cavity with distributed anisotropic surface impedance associated with longitudinal periodic corrugations of the cavity wall. The study is based on both single-mode and coupled-mode treatments. The former treatment is the widely accepted Vlasov approach [33], which is used as a first approximation to evaluate the diffractive and the ohmic losses of the cavity modes. The latter approach is the extension of the mode-matching technique [34, 35] aimed to reveal the effect of mode conversion, arising from inhomogeneity of the cavity radius and the surface impedance. As an illustrative example, the cavity with distributed surface impedance for 0.4-THz secondharmonic gyrotron is considered.

2 Normal Modes of Uniform Circular Waveguide with Anisotropic Surface Impedance In electromagnetic analysis of a gyrotron cavity [34–39], it is common practice to use the normal modes of a uniform waveguide as basis. For this reason, before proceeding to the case of a gyrotron cavity, we consider a finite length section of the uniform impedance waveguide with the constant radius R. The aim is to find the waveguide field {E(r, t), H(r, t)} = {E(r, z), H(r, z)} exp(−iωt + ilφ) subject to the impedance boundary conditions at r = R Eφ =H z ¼ ηφ ; E z =H φ ¼ −ηz ;

ð1Þ

where ω is the wave frequency, l is the azimuth wavenumber, and ηφ and ηz are the azimuth φ and the axial z components of the surface impedance dyadic η, which goes to zero in the case of smooth-walled waveguide made of perfect electric conductor (PEC). A distinguishing feature of the circular impedance waveguide is that its field has the radial structure described by the single scalar function ψ(r) = Jl(k⊥r) [27–29, 40, 41] known as membrane function. Thus, one can write: H z ¼ f ðzÞhz ðrÞ ¼ f ðzÞk 2⊥ ψðrÞ; E z ¼ yðzÞez ðrÞ ¼ −iyðzÞk 2⊥ PψðrÞ; E⊥ ¼ f ðzÞe⊥ ðrÞ ¼ f ðzÞðik ½∇⊥ ψ  z þ ik z P∇⊥ ψÞ; H⊥ ¼ yðzÞh⊥ ðrÞ ¼ yðzÞðik z ∇⊥ ψ−kP½∇⊥ ψ  zÞ;

ð2Þ

where k 2⊥ ¼ k 2 −k 2z , k is the wave vector in free space, kz is the axial wavenumber, Jl(x) is the lth order Bessel function, ∇⊥ = ir∂/∂r + iφil/r, f(z) = A(z) + B(z), and y(z) = A(z) − B(z) are the field amplitudes to be determined in the next sections, A(z) = A0 exp(ikzz), B(z) = B0 exp(−ikzz), and P = iez(r)/hz(r) is the hybridization parameter. The value of P in (2) is as yet unknown and needs to be found for each hybrid wave of the waveguide. Evidently, for pure TE (ez = 0) and TM (hz = 0) waves, it must be zero and infinity, respectively. On this basis, the guided waves with P = iez/hz satisfying either |P| < 1 or |P| > 1 can be identified as TE-like (HE) or TM-like (EH) waves, respectively.

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Rewriting (1) in terms of (2),

eφ =hz
r¼R ¼ ηφ ; ez =hφ
r¼R ¼ −ηz we obtain the boundary condition on membrane function [41]:   dψ ¼ 0; þ gψ dr r¼R

ð3Þ

ð4Þ

where g = (a + bP) = (c + bP−1), a ¼ −ik 2⊥ ηφ =k, c ¼ −ik 2⊥ η−1 z =k, b = − lkz/(kR), P equals P1 or P2 [27, 41],  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ P1;2 ¼ α 1∓ 1 þ α−2 ;

α ¼ ðc−aÞ=ð2bÞ; P1 P2 ¼ −1: It is easy to show [41] that P1 and P2 in (5) satisfy the conditions |P1| ≤ 1 and |P2| ≥ 1 for arbitrary value of α. Thus, we have two independent problems on the different membrane functions ψ(r), one for TE-like waves (P = P1) and another for TM-like waves (P = P2). As α tends to infinity, the guided waves transform to pure TE and TM waves. Such transformation occurs in each of the following limiting cases: kz → 0, l → 0, ηz → 0, or ηφ → ∞. With the provision that |α2| ≫ 1, P1 and P2 can be written as: P1 ≈−

1 iη lk z R 1  ; P2 ¼ − ; ¼ z 2α χ2 1−η η P1 φ z

ð6Þ

where χ = k⊥R is the mode eigenvalue. Clearly, in this case, the guided waves are almost pure (quasi) TE (|P| = |P1| ≪ 1) and TM (|P| = |P2| ≫ 1) waves. In particular, such are the waves

satisfying inequality
k 2z =k 2
≪1, which is generally valid in gyrotrons. Substitution of ψ(r) = Jl(k⊥r) into (4) gives the dispersion equation for either TE-like or TMlike waves 0

Dðω; k z Þ ¼ χ J l ðχÞ þ gRJ l ðχÞ ¼ 0:

ð7Þ

Together, these equations yield the axial wavenumbers kzs(ω), where the index s = 1 , 2… numbers the radial (normal) modes of a circular hollow waveguide with arbitrary anisotropic surface impedance. Consider now the normal modes of the same or two jointed circular waveguides with a single axis of symmetry. It can be shown [32, 42] that for arbitrary two modes, there exists an integral R

T sn ¼ ∫0 rdrhe⊥s ; h⊥n i ¼

iR k 2zn −k 2zs



   k zs ezs hφn −ezn hφs −k zn eφs hzn −eφn hzs r¼R ;

ð8Þ

where 〈e⊥s, h⊥n〉 = (ershφn + eφshrn) will be called the inner product of e⊥s(r) and h⊥n(r), the mode fields es(r) and hn(r) are found from (2) with kz = kzs(ω) and kz = kzn(ω), respectively, and kzs(ω) and kzn(ω) are the axial wavenumbers of the modes.

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For the normal modes of two jointed PEC waveguides, integral Tsn represents the coefficient [34, 38] of mode coupling at the junction between the guide of radius R and the larger guide. As the radius of the larger guide tends to R, relation (8) takes the well-known form [43] of mode orthogonality relation. In this case, φ and z components of both es and en vanish at PEC surface r = R and integral Tsn goes to zero, provided that k 2zs ≠k 2zn . If both modes satisfy impedance boundary conditions (3), then Tsn can be understood as the coefficient of mode coupling at the junction between two equal-sized waveguides with different surface impedances. In this case, integral (8) takes the form [27, 44, 45]:  i −iRhzs hzn jr¼R h −1 −1 ; ð9Þ k η −η P þ k η −η P zs s n zn φs φn zs zn k 2zn −k 2zs     where ηs ¼ ηφs ; ηzs and ηn ¼ ηφn ; ηzn are the surface impedance dyadics for modes T sn ¼

with kz = kzs(ω), P = Ps and kz = kzn(ω), P = Pn, respectively. Relation (9) also clearly shows that the normal modes of one and the same impedance waveguide are not necessarily orthogonal. The mode orthogonality only holds true for the waveguides, having kz-independent (or modeindependent) surface impedance [27, 44–46], which means that condition ηn ¼ ηs is fulfilled for arbitrary kzs and kzn. Under such condition, integral (9) is reduced to the orthogonality relation T sn ¼ T Rs δsn , where δsn is the Kronecker delta and T Rs is the normalization factor of sth mode in the waveguide of radius R. However, the opposite situation is rather common. An example of interest here is a circular longitudinally corrugated waveguide [27, 47–49] with imperfectly conducting wall. When the number N of periodic corrugations is large enough (N > 2l) and the wall conductivity σ is rather high, the field inside each corrugation can be approximated by that of a single axially symmetric TE mode. Assuming that the corrugations are almost rectangular and |k⊥R| > > 1, one can average the ratios between φ and z components of this field over the corrugation period p = 2πR/N to obtain the components of the effective impedance for the surface r = R:

ηφ ðω; k z Þ ¼ −i

  w k sinðk ⊥ d Þ þ k ⊥ d ct cosðk ⊥ d Þ w þ 1− Z; ηz ðωÞ ¼ Z ; p k ⊥ cosðk ⊥ d Þ−k ⊥ d ct sinðk ⊥ d Þ p

ð10Þ

where w and d are the corrugation width and depth, respectively, Z = − ikdc, dc = δ(1 + i)/2, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ ¼ 2=ðωμ0 σÞ, dct = δt(1 + i)/2, and δt is the effective skin depth for the top surfaces of corrugations. The meaning of δt is clarified below. It is evident that the surface impedance (10) is mode-dependent such that the inequality ηφ(ω, kzs) ≠ ηφ(ω, kzn) is true. From (9) and (10), it follows that the normal modes of the longitudinally corrugated waveguide are non-orthogonal. Each individual mode, having some axial wavenumber kzs(ω), is orthogonal to modes of another circular waveguide with modified surface impedance ηφ(ω) = ηφ(ω, kzs(ω)) and ηz (ω). Setting ηφ(ω) = ηφ(ω, kzs(ω)), we imply in fact that inside the corrugations, there is a single mode with the same axial wavenumber kz = kzs(ω) as for selected mode of the original guide. Such single-mode approximation (known as surface impedance model [18–20]) is used above to evaluate (10). It is definitely valid for a corrugated waveguide of constant cross section and may serve as a first approximation for the weakly irregular structures, supporting the dominant propagation of selected mode. Among such structures is a gyrotron cavity with corrugated walls. Note that, for the operating gyrotron

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mode, kz in (10) is sometimes [50] assumed to be zero. Under certain conditions [39], the same assumption can be used to treat dielectric-lined waveguide as impedance waveguide. The surface impedance (10) takes the well-known form for PEC corrugated waveguide [18, 19, 49] when σ = ∞ and that for smooth-walled conducting waveguide [51] when d = 0 and δt = δ. Besides, it is consistent with the fact that the eigenvalues of TM waves [18, 27] and their ohmic losses [52] are almost insensitive to the longitudinal corrugations. Note that, within the surface impedance model, we are forced to neglect the finite conductivity for the side surfaces of corrugations. The reason is that there are no single-mode solutions to Maxwell’s equations for a conducting rectangular guide [27, 53] and thus for the conducting rectangular (or wedgeshaped) corrugations. To overcome this shortcoming, we introduce additional effective ohmic losses in the top surfaces of corrugations and thus increase their actual skin depth δ to the higher effective value δt. This value can be obtained by the perturbation technique (|Z| ≪ 1) [43]. Such technique yields the following transverse wavenumbers: ( k 2⊥ ¼ k 2⊥0 1−

) 2d c ðp−wÞcos2 ðk ⊥0 d Þ þ w þ d ½1 þ sinð2k ⊥0 d Þ=ð2k ⊥0 d Þ 2 2 ð11Þ pR 1−l =χ0 cos2 ðk ⊥0 d Þ þ w2 =p2 sin2 ðk ⊥0 d Þ þ w=pd=R½1 þ sinð2k ⊥0 d Þ=ð2k ⊥0 d Þ

for quasi-TE modes of the corrugated conducting waveguide in relation to their values k⊥0 = χ0/R in the limiting case of σ = ∞ (dc = dct = 0). Under condition w/p = 1, relation (11) reduces to the known form [48] for a circular conducting waveguide loaded periodically with infinitesimally thin strips. Note that the similar relation for a smooth (non-corrugated) conducting waveguide is in active use [54, 55] and can be obtained from (11) with d = 0. The first, the second, and the third terms in the numerator of (11) are associated with the finite conductivity σ for the bottom, top, and side surfaces of the corrugations, respectively. Therefore, neglect of the finite conductivity for the side surfaces can be compensated, for example, by the increased effective value δt ¼ δð1 þ d=w½1 þ sinð2k ⊥0 d Þ=ð2k ⊥0 d ÞÞ

ð12Þ

of the skin depth for the top surfaces of corrugations. This conclusion is confirmed by the comparison between (11) and solutions to the dispersion equation (7) with (10), (12) in place of ηφ and ηz. Figure 1 shows the typical

solutions of the dispersion equation for a quasi-TE mode near the cutoff frequency (
k 2z =k 2
≪1, k⊥ ≈ k). All of these data are in close agreement with those obtained within the perturbation theory. As an illustration, their comparison is shown in Fig. 1b for w/p = 0.5. Let us look more closely at the dispersion properties of the corrugated conducting waveguide. It can be seen form Fig. 1a that the corrugations cause a decrease in mode eigenvalue (and cutoff frequency) [48]. To be more specific, whatever the ratio w/p is, the eigenvalue of the TEl,s mode tends to (s − m)-th zero of the Bessel function Jl(x) or its derivative as k⊥d approaches π(m − 1/2) (|ηφ| ≫ 1) or πm (|ηφ| ≪ 1), respectively. Hence, it follows that, with the corrugation depth d ≈ πm/k⊥, it is possible to match the TEl,s mode of the corrugated guide to the TEl,s − m mode of the smooth circular waveguide, having the same radius R. In addition to that, the corrugations have an effect on mode attenuation. It can be seen from Fig. 1b that the attenuation has local maxima and minima near k⊥d = π(m − 1/2) and k⊥d = πm, respectively. With increase in ratio w/p, these extrema are flattened. From the above description, it follows that the effect of corrugations on quasi-TE modes near cutoff frequencies is mode-dependent. It is clear from Fig. 1a that the corrugations, as a rule, induce larger downshift of the eigenvalue χ for modes with higher frequencies. Besides,

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Fig. 1 Normalized eigenvalue (a) and attenuation (b) of the TE8,9 mode in the circular conducting waveguide versus the depth of the wall corrugations, where R = 0.45 cm, N = 20, σ = 2.9 × 107 S/m, Rek 2z ¼ 15 cm−2, PT 0 means the results of the perturbation theory, and μl , s and μl;s are the roots of the functions Jl(x) and dJl(x)/dx, respectively

the corrugations provide mode-dependent wall losses. For instance, the corrugation depth can be selected such that it equals d ≈ π/k and thus corresponds to the local minimum of the attenuation for some mode with the desired frequency ω. In this case, the mode attenuation is close to the first local maximum for another mode, having the frequency twice as low as ω. It is clear that this effect can potentially be used for selective suppression of the fundamental competing modes in second-harmonic gyrotrons. Although such mode selection is of practical importance [19, 21–23], it is beyond the scope of the present paper.

3 Gyrotron Cavity with Distributed Longitudinal Corrugations Consider now a typical gyrotron cavity (Fig. 2a) in the form of a hollow circular waveguide with the slowly varying radius R(z) (arctan(R′(z)) ≤ 3∘) and the finite length L. The cavity consists of input, central (main), and output sections. The input and the main sections incorporate the longitudinal wall corrugations of invariant shape and are characterized by the effective anisotropic surface impedance (10), (12). The cavity is assumed to be made of copper with reduced conductivity σ = σCu/2 = 2.9 × 107 S/m due to the wall roughness [56].

Fig. 2 Geometry (a) of the gyrotron cavity with distributed longitudinal corrugations and profile (b) of the effective radius of the equivalent smooth-walled cavity for w/p = 0.5 and d = 0.012 cm

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3.1 Single-Mode Analysis of Impedance Gyrotron Cavity The essential features of the corrugated cavity are easier to understand without considering mode coupling due to variations of R(z) and ηφ(z). In such case, the cavity field can be approximated by that (2) of a single quasi-TE mode, where {e(r), h(r)} is now the radial structure of the field subject to the impedance boundary conditions (3) at r = R(z) and the field amplitudes f(z) and y(z) are governed by the following equations [31]: 0

f ″ ðzÞ þ k 2z ðzÞ f ðzÞ ¼ 0; yðzÞ ¼ −i f ðzÞ=k z ðzÞ :

ð13Þ

Here, kz(z) satisfies the dispersion equation (7) for quasi-TE mode (P = P1) at given ω, z, R(z), ηφ(z), and ηz. Equations (13) are of the standard form [15–17, 20, 22, 26, 33, 36, 50, 54] and should be supplemented by the outgoing-wave boundary conditions at both ends (z = 0 and z = L) of the cavity. We investigate two possible [8, 24] competing modes of the gyrotron cavity. The first is the TE8,9 mode adopted as the operating second cyclotron-harmonic mode. The second is the competing first cyclotron-harmonic TE4,5 mode. Table 1 lists the frequencies fr = Reω/2π and the diffractive Qd and the ohmic Qohm quality factors of the TE8,9 and the TE4,5 modes in the case of smooth-walled (non-corrugated) cavity (d = 0). Figure 3a shows the effect of the distributed corrugations on the frequencies of the TE8,9 and the TE4,5 modes. Both frequencies decrease with increasing corrugation depth d. This is because the eigenvalues of TE modes in the main section of the cavity diminish (see Fig. 1a) under the action of corrugations. As discussed above, the higher is the mode frequency, the more profound is the change in eigenvalue. That is why the frequency downshift of Fig. 3a is larger for the second cyclotron-harmonic TE8,9 mode. Generally, the frequency change may enlarge or diminish the mismatch between the operating frequency and the doubled frequency of the fundamental competing mode. Hence, it may affect either beneficially or adversely the mode selectivity of the gyrotron cavity. The latter is the case for the modes under consideration. But, as will be apparent from the following discussion, this fact is not of crucial importance. From the above discussion, it appears that the TEl,s modes have the reduced eigenvalues χ(z) inside the corrugated part of the gyrotron cavity. Except for their ohmic losses, these modes are identical to those of equivalent smooth-walled cavity with the effective radius Req (z) = R (z) μ′l,s / Reχ (z), where μ′l,s is the s-th zero of the function dJl(x)/dx. It can be seen from Fig. 2b that the resulting equivalent cavity is iris-loaded. It is well-known [12, 13] that such iris induces a rise in the diffractive Q-values, especially for the higher frequency (or higher cyclotron-harmonic) modes. What is more, unlike conventional iris, the equivalent iris has the larger depth for the higher frequency modes (Fig. 2b) and thus induces even higher increments of Qd for these modes when compared to the fundamental competitors. This explains the results shown in Fig. 3b. Note that it is also possible to provide the equivalent Table 1 Mode characteristics for smooth-walled gyrotron cavity Characteristic

TE8,9 mode

TE4,5 mode

Frequency fr (GHz) Diffractive Q-value Qd Ohmic Q-value Qohm

399.37 1252 29,561

204.23 321 21,498

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Fig. 3 Normalized frequencies (a) and diffractive Q-values (b) of the TE modes versus d (w/p = 0.5, N = 20)

iris with the required width by removing the corrugations from the same-sized segment of the main section adjacent to the output one. Although such possibility expands the number of optimization parameters, we will not consider it for simplicity. As the ratio w/p increases, both the eigenvalue (Fig. 1b) and the diffractive Q-value (Fig. 4a) of the TEl,s mode change more rapidly with the corrugation depth d. As can be seen from Fig. 4a, the total quality factor Ql , s of this mode undergoes saturation in the process. The greater is the ratio w/p, the higher is the saturation value of Ql , s. The reason is that the ohmic Q-value of the TEl,s mode decreases with increasing d and tends to the first local minimum as d approaches 0.5πR/μl , s − 1 (Fig. 1b). Let us suppose that the condition Qd ≤ Qohm is one of the design specifications, which serve to maintain the required level of gyrotron efficiency. Note that this condition is not necessarily fulfilled in THz gyrotrons [24–26]. The maximal total quality factors Q8 , 9, which meet this specification, are shown in Fig. 4a by dots. It can be seen that for the intermediate ratio w/p = 0.5, such value of Q8 , 9 roughly equals 11,000 and is attained at d ≈ 0.009 cm. The resulting corrugations are close in depth to those studied recently [57] as a possible means of further enhancing mode selectivity of the coaxial cavities for fusion-relevant fundamental-harmonic gyrotrons. Although even shallower corrugations are investigated theoretically [52], such corrugations may face the problem of low tolerance during machining. For this reason, we next consider the gyrotron cavity with the increased values of d.

Fig. 4 Diffractive and total quality factors of the TE8,9 mode versus the corrugation depth for small (a) and larger (b) values of d (N = 20)

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As discussed above, the TEl,s mode of a smooth guide can be matched almost perfectly to 0 the TEl,s + 1 mode of the corrugated guide, having the depth d ¼ πR=μl;s and the same radius R. For the output TE8,9 mode of our cavity, such matching requires d to be roughly equal to 0.038 cm. When perfectly matched, this mode (TE8,10 mode in the corrugated sections) has essentially the same electromagnetic properties as for the smooth (d = 0) cavity, except for the decreased value of Qohm around 25,000. As the corrugation depth increases further, the diffractive and the total quality factors of the output TE8,9 mode start to grow (Fig. 4b) in the similar way as before (Fig. 4a). For the intermediate ratio w/p = 0.5, the condition Qd = Qohm now holds true at d ≈ 0.047 cm and corresponds to Q8 , 9 ≈ 9700. The corrugation depth of 0.047 cm is above the design values used in the currently available gyrotron cavities [58, 59] with corrugated coaxial inserts. This gives us hope that the corrugations under consideration are practicable.

3.2 Mode Conversion in the Gyrotron Cavity with Distributed Surface Impedance Next, we will take into account the mode coupling induced by the cavity non-uniformity. Since the cavity of interest (Fig. 2a) is characterized by the slow variation in radius, the mode coupling is mostly attributed to the non-uniform distribution of the surface impedance ηφ(z). In this case, the coupling coefficients Tsn are close to those given by (9). From (9) and (10), it follows that, under condition |Z| ≪ 1, the coupling between quasi-TE and quasi-TM modes is extremely weak and goes to zero as Z → 0 (σ → ∞). That is why we can exclude quasi-TM modes from our consideration. We use a stepwise approximation [34, 35] to model the cavity radius R(z). As a result, the gyrotron cavity is represented as a finite number (j = 1 , 2 … M) of the jointed uniform cylindrical sections with close radiuses and different anisotropic surface impedances (10). Since the wall conductivity plays almost no part in the mode coupling, the small ring-shaped surfaces Rj < r < Rj + 1 of the junctions z = zj are assumed to be PEC. For each section with the radius R = Rj, the field can be expanded in terms of the eigenfields (2) of the normal modes h i h i Eð⊥jÞ ¼ ∑ Aðs jÞ ðzÞ þ Bðs jÞ ðzÞ eð⊥sjÞ ðrÞ; Hð⊥jÞ ¼ ∑ Aðs jÞ ðzÞ−Bðs jÞ ðzÞ hð⊥sjÞ ðrÞ s

s

ð jÞ

ð jþ1Þ

ð jÞ

ð14Þ ð jþ1Þ

and H⊥ ¼ H⊥ and must satisfy both the boundary and the continuity E⊥ ¼ E⊥ conditions at the interface z = zj between neighboring sections. Taking the inner product of ð jþ1Þ

ð jÞ

the first condition with h⊥n and that of e⊥s with the second condition, and then integrating the resulting expressions over the junction area, we obtain 

R  Ak z j þ Bk z j T k jþ1 ¼ ∑ T ik Ai z j þ Bi z j ; i   R Ai z j −Bi z j T i ¼ ∑ T ik Ak z j −Bk z j ;

ð15Þ

k

where R = Rj ≤ Rj + 1, i = {s, j}, and k = {n, j + 1} are the pairs of the lower and upper indices used in (14) for designation of the normal modes. In deriving (15), we have replaced the axial wavenumber kz in (10) with that for one of the modes of interest, assuming that this mode dominates over its radial satellites (spurious modes). As discussed above, under this assumption, the normal modes of impedance waveguide can be considered orthogonal.

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If (j + 1)-th section is semi-infinite (Bk = 0), (15) can be used to determine the amplitudes of the reflected Bi and the transmitted Ak (k = 1 , 2…) modes in relation to the amplitude Ai of the mode incident from the region z < zj on the discontinuity between two guides [44, 45]. The effect of energy transformation from the incident mode to a number of transmitted modes with k ≠ i is known as mode conversion. Relations (15) must be considered for j = 1 , 2 … M and supplemented with outgoing-wave boundary conditions at the cavity ends (z = 0 and z = L). This yields a system of linear algebraic equations with unknown amplity tudes of forward and backward waves. Cavity eigenfrequencies and eigenfields are obtained by nullifying the determinant of the system matrix. To avoid problems associated with large-sized matrices, we have implemented the scattering matrix formalism [34, 35]. Figure 5a depicts the influence of mode conversion on the normalized ratio G(d) = Q8 , 9/Q4 , 5 between the total quality factors of the operating TE8,9 and the competing TE4,5 modes for the relatively small values of the corrugation depth d and the intermediate ratio w/p = 0.5. For such d, the mode conversion in the cavity leads to additional diffractive losses associated with the leakage of spurious modes. As a result, the diffractive and the total quality factors of both the TE8,9 and the TE4,5 modes decrease. Since the eigenvalue of the TE8,9 mode changes more rapidly within the selected region of d, this mode is strongly affected by the mode conversion. That is why the ratio G(d) = Q8 , 9/Q4 , 5 shown in Fig. 5a decreases. Figure 5b shows the same as Fig. 5a, but for increased values of the corrugation depth d. As mentioned above, at d ≈ 0.038 cm, there is a good matching between the output operating TE8,9 mode and the TE8,10 mode of the main corrugated section of the cavity. However, such is not the case for the TE4,5 mode, having a distinct step discontinuity of ηφ(z). This discontinuity causes both the reflection and the conversion of the fundamental competing mode. At d ≈ 0.038 cm, their combined effects result in increased values of the diffractive and the total quality factors of the competing TE4,5 mode. Because of this, the starting value of G(d)/G(0) shown in Fig. 5b is less than unity. As the corrugation depth increases further, the TE4,5 mode is subjected to the surface impedance with diminishing discontinuity step and its total quality factor decreases. The opposite situation occurs with operating TE8,9 mode. The combined change of Q8 , 9 and Q4 , 5 with d yields the dependence G(d)/G(0) shown in Fig. 5b.

Fig. 5 Normalized value of G(d) = Q8 , 9/Q4 , 5 versus the corrugation depth for small (a) and larger (b) values of d, where w/p = 0.5, N = 20, and SMA and CMA designate the results of the single-mode and the coupled-mode approaches, respectively

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It is well-known [60, 61] that the starting current I stl;s of the TEl,s mode is inversely proportional to the total quality factor Ql , s. Therefore, the rise in G = Q8 , 9/Q4 , 5 implies a decrease of the ratio I st8;9 =I st4;5 . Thus, from Fig. 5, it may be concluded that, with distributed wall corrugations in the gyrotron cavity, it is possible to achieve severalfold decrease in the starting current of the operating mode with respect to that of the competing mode at a reasonable level of ohmic losses (Qd ≤ Qohm) and mode conversion (∼70% of output mode purity). It should be emphasized again that further optimization is still possible and can be achieved by increasing the width of equivalent iris. The larger is this width, the smaller is the corrugation depth d required to ensure the desired selective increase in the diffractive quality factors of cavity modes [12]. Thus, on the one hand, it becomes possible to achieve the same beneficial effect with shallower corrugations and, on the other hand, to reduce the ill effects of ohmic losses and mode conversion due to decrease in d. Such optimization of the gyrotron cavity with distributed longitudinal corrugations is the subject of our future research.

4 Conclusion Distributed longitudinal wall corrugations have been proposed as a means to enhance mode selectivity of the conventional cylindrical cavity for a second-harmonic gyrotron. The corrugated cavity walls have been treated as a homogeneous surface with effective (averaged) anisotropic impedance. The mode-dependent surface impedance has been derived with regard to the finite wall conductivity. The single-mode analysis has been used to reveal the most essential features of the gyrotron cavity with distributed wall corrugations. The first feature is the increased ohmic losses inherent in corrugated walls. The second one is the effective cavity profiling arising from non-uniform longitudinal distribution of the averaged surface impedance. The surface impedance has been distributed in such a way as to form the equivalent smooth-walled (non-corrugated) gyrotron cavity loaded with iris. This iris is aimed to prevent the leakage of the operating mode in the most effective manner. Therefore, it leads to increased ratio between the diffractive Q-values of the operating and the fundamental competing modes. Moreover, compared to conventional iris, the equivalent iris induces the higher increment of this ratio. This is because the equivalent iris has the mode-dependent depth, which is larger for higherfrequency modes. Such discrimination against the fundamental competitors, however, is limited by the ohmic losses in the cavity walls. Another limiting factor is the mode conversion caused mainly by the longitudinal variation in the effective wall impedance. To study the mode conversion in the gyrotron cavity, the mode-matching technique has been extended to the case of corrugated cavity walls. It has been found that conversion of the operating mode into spurious modes results in additional diffractive losses associated with the leakage of spurious modes through the cavity ends. These unwanted losses, together with unavoidable wall losses, limit the total Q-value of the operating mode. However, this limitation does not negate the beneficial effect of the distributed corrugations of the cavity walls. It has been shown that such corrugations enable several-fold increase in the total Q-value of the operating mode with respect to that of the fundamental competing mode at a reasonable level of ohmic losses and mode conversion in the gyrotron cavity.

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