Formulating the output power

We begin with equation (1) which is sometimes taken as granted for calculating the electron power P _e (detailed derivation of the equations of this section will be presented elsewhere):

(1)$$P_e = {{NV_r ^2} / {(2R_r )}}.$$

In this equation, V _r is the voltage at the aperture of the side resonators and R _r is the equivalent resistance of them (see Fig. 1(b)). V _r and R _r can be calculated using

(2)$$V_r = {{A_{ - 1} R_r {\rm X}\cos \Delta \varphi} / {(N\eta _r d}}),$$

(3)$$\eqalign{ R_r &= {{\eta _0^2 \eta _r^2 d^2 \sin ^2 (kl)} {\bigg/} {\left( {R_s Hl(1 + \displaystyle{{\sin (2kl)} \over {2kl}} + \displaystyle{d \over l}} \right)}}, \cr k &= 2\pi f_n \sqrt {\mu _z \varepsilon _\varphi}, \;\eta _r = \sqrt {{{\mu _z} / {\varepsilon _\varphi}}}, \;R_s = \sqrt {{{\pi f_n \mu _0} / \sigma}}}$$

where X is related to the interaction space current density and as a result the cathode current, Δ_φ is the phase difference between the interaction space current and the electric field, f _n is the operating frequency, σ is the copper conductivity, and A ₋₁ is the amplitude of the synchronous harmonic, i.e. the first backward one, and it can be calculated using ([Reference Nasr Esfahani and Schünemann2] and [Reference Nasr Esfahani and Schünemann3])

(4)$$A_{ - 1} = \eta _r {{N\theta \sin \left( {3N\theta /4} \right)} / {(3N\theta /4)}}.$$

Considering equations (1)–(4), it is concluded that the electron power can be increased with increasing A ₋₁, for example, through introducing an epsilon near zero (ENZ) layer which can realize |ε_φ| <1 in the side resonators. This approximate analysis fails to predict the behavior of the output power when ε _φ is approaching zero. In fact, according to equation (1) the electron power (and as a result output power) in this scenario will approach infinity. Obviously, a saturation effect which limits the electron power when A ₋₁ is increased is missing in equation (1). Careful examination of the magnetron operation shows that the missing saturation effect is embedded in the electron bunching process. This process is briefly explained in the following sub-section.

Bunching process and the resulted saturation effect

The bunching process in SHMs is governed by a delicate balance between the radial DC field (E _DC) and the first backward harmonic of the radially directed electric field. It can be shown that the amplitude of this component is proportional to

(5)$$E_\rho ^{ - 1} \propto {{A_{ - 1} V_r} / {\eta _r d}}.$$

The importance of the mentioned balance between the DC and the synchronous RF radially directed fields can be understood by formulating the azimuthal velocity of electrons (Ѵ _φ) in the interaction space

(6)$$\upsilon _\varphi = {{(E_{DC} + E_\rho ^{ - 1} )} / B}.$$

where B is the axially directed DC magnetic field. It can be shown that for a radial DC field in the interaction space which corresponds to an applied DC voltage V _DC, $E_\rho ^{ - 1} $ is limited to a variation range (Δ$E_\rho ^{ - 1} $) out of which the electron bunching cannot take place. If we rewrite $E_\rho ^{ - 1} $ in terms of a product of a geometrical term (T _g) and a term related to the interaction space current (T _J), i.e.

(7)$$E_\rho ^{ - 1} = T_g T_J, \quad T_g = {{A_{ - 1}^2 R_r} / {N\eta _r^2 d}},\quad T_J = X\cos \Delta \varphi $$

any variation in T _g (for example increasing A ₋₁) which can potentially result in an $E_\rho ^{ - 1} $ out of the acceptable range is compensated by a decrease in X (which is related to the cathode current) and an increase in Δ_φ. This compensation act has several practically important result: (1) SWSs which are able to realize larger T _g values (for example an ENZ-loaded SHM with |ε_φ| <1) can produce power with smaller cathode current densities comparing with their conventional counterparts where A ₋₁ is restricted to values smaller than 0.42 (see equation (5)). This is very important for designing THz sources where the required large cathode current density in conventional SHMs is a limiting factor and (2) a structure with large T _g has no advantage over the conventional SWSs when it is used with a cathode that its current cannot be limited to small values. In fact, in this case, large values of T _g and X are compensated by Δ_φ → 90°. In the most extreme case, it is even possible that SWSs with large T _g generate even smaller output power than the corresponding conventional one because Δ_φ approaches 90° (see equations (1) and (2)). In other words, employing appropriate cathode is of key importance in taking full advantage of a structure with large T _g.

An approach for realizing |ε_φ| <1 in the side resonators which as a result increases T _g and reduces the required cathode current density is presented in the next section.

CSOLR-loaded SHM

Figures 2(a) and 2(b) show a CSOLR and adaptation of the SWS to accommodate it. Figure 2(c) shows that the electric field in the side resonators is to a large extent perpendicular to the surface of the CSOLRs. Hence, it can be concluded that the CSOLR can be properly excited by the electric field in the side resonator and as a result, a Lorenz-type permittivity with regions of negative, zero, and positive relative permittivity is realized. In fact, the vane-type side resonators of a conventional SWS are short-circuited “quasi”-parallel plate waveguides. In other words, a CSOLR-loaded side resonator is a short-circuited unit-cell of a medium composed of CSOLRs. This unit-cell is shown in Fig. 2(c). The effective permittivity and permeability of this unit-cell (ε _φ and μ _z, if we consider a cylindrical coordinate) are obtained from its scattering parameters (S ₁₁ and S ₂₁) using equation (8) based on the electromagnetic parameter retrieval method described in [Reference Smith, Vier, Koschny and Soukoulis7].

(8)$$\eqalign{ &\mu _r {\rm =} \displaystyle{{B \,{\rm cos}^{{\rm -} {\rm 1}} {\rm (}A{\rm )}} \over {ks}}{\rm ;}\quad \varepsilon _r {\rm =} \displaystyle{{{\rm cos}^{{\rm -} {\rm 1}} {\rm (A)}} \over {B \, ks}}{\rm ;} \cr &B{\rm =} \sqrt {\displaystyle{{{\rm (1 +} S_{{\rm 11}} {\rm )}^{\rm 2} {\rm -} S_{{\rm 21}}^{\rm 2}} \over {{\rm (1}{\rm -} S_{{\rm 11}} {\rm )}^{\rm 2} {\rm -} S_{{\rm 21}}^{\rm 2}}}} {\rm ;}\quad A{\rm =} \displaystyle{{\rm 1} \over {{\rm 2}S_{{\rm 21}}}} {\rm (1}{\rm -} S_{{\rm 11}}^{\rm 2} {\rm +} S_{{\rm 21}}^{\rm 2} {\rm )}} $$

where k is the free-space wave number and s is the thickness of the layer.

Fig. 2. (a) A CSOLR; (b) CSOLR-loaded SWS; (c) CSOLR-loaded unit-cell in which the CSOLR is etched on the upper wall of a quasi-parallel plate waveguide (quasi-parallel walls are at φ = 0 and φ = h ₁/(2πr _a)), input and output ports (shown by dashed lines) of the unit-cell are placed at ρ = r _a and ρ = r _a + l ₁, z = 0, and z = l ₂ planes are assumed to be perfect magnetic walls and an extra space with the height of h ₂ has been considered above the plane of the CSOLR. The dimensions of the CSOLR and the unit-cell are: l ₁ = 1.6 mm, l ₂ = 1.65 mm, g = 0.2 mm, w = 0.25 mm, D = 1.4 mm, t = 0.49 mm, h ₁ = h ₂ = 0.39 mm; (d) simulated scattering parameters of the CSOLR unit-cell (obtained using ANSYS HFSS); (e) retrieved constitutive parameters, (f) simulated (using ANSYS HFSS) and calculated susceptance; (g) cutaway view of the electric field of the π/2-mode inside the 35 GHz CSRR-loaded SHM; and (h) cutaway view of the electric field of the π/2-mode inside a 35 GHz conventional SHM.

In the next sub-section an analytical approach for calculating the resonant frequencies of this complicated SWS will be presented.

Magnetron mode resonant frequencies of a CSOLR-loaded SHM

A straightforward analytical method for calculating the resonant frequencies of a CSOLR-loaded SHM is as follows: at each resonant frequency the negative of the susceptance of one of the side resonators (−B _res) is equal to the susceptance of the interaction space (B _int) seen from the aperture of that resonator, i.e.

(9)$$B_{res} = - B_{int}. $$

Susceptance of the interaction space can be calculated using [Reference Collins8]

(10)$$B_{int} = \sum\limits_{m = - \infty} ^\infty {\displaystyle{{N\theta} \over \pi} \left( {\displaystyle{{\sin \gamma _m \theta} \over {\gamma _m \theta}}} \right)^2 \displaystyle{{Z_{\gamma _m} (kr_a )} \over {Z^{\prime}_{\gamma _m} (kr_a )}}} $$

where $Z_{\gamma _m} (k\rho )$ and $Z'_{\gamma _m} (k\rho )$ are determined using

(11)$$\eqalign{ Z_{\gamma _m} (k\rho ) &= J_{\gamma _m} (k\rho ) - \displaystyle{{J^{\prime}_{\gamma _m} (kr_c )} \over {Y^{\prime}_{\gamma _m} (kr_c )}}Y^{\prime}_{\gamma _m} (k\rho ); \cr Z^{\prime}_{\gamma _m} (k\rho ) &= J^{\prime}_{\gamma _m} (k\rho ) - \displaystyle{{J^{\prime}_{\gamma _m} (kr_c )} \over {Y^{\prime}_{\gamma _m} (kr_c )}}Y^{\prime}_{\gamma _m} (k\rho ), \cr k &= \omega \sqrt {\mu _0 \varepsilon _0}, \gamma _m = mN + n.} $$

In equation (11), $J_{\gamma _m} $, $Y_{\gamma _m} $, $J'_{\gamma _m} $, and $Y'_{\gamma _m} $ are Bessel functions and their derivatives. N is the total number of side resonators, m denotes the number of the synchronous space harmonic and 2πn/N shows the phase difference between the electric field at the aperture of side resonators and determines the operation mode. It is worth mentioning that in SHMs, the synchronous harmonic is the first backward one (m = −1) and the operation mode is π/2 or a neighboring one (correspond to n = N/4 or n = N/4 ± 1) while in classical magnetrons the synchronous harmonic is the fundamental harmonic (m = 0) and the operation mode is π (corresponds to n = N/2).

Susceptance of a CSOLR-loaded side resonators (B _res) is equal to the susceptance of a conventional side resonator which is filled with ε _φ and μ _z. ε _φ and μ _z are calculated using the scattering parameters of the corresponding CSOLR-loaded unit-cell and equation (8). In the case of vane-type side resonators B _res is determined using (12).

(12)$$B_{res} = \left\{ {\matrix{ { - \sqrt {\displaystyle{{\varepsilon _\varphi } \over {\mu _z}}} \displaystyle{{J_0(k_\rho r_a) - AY_0(k_\rho r_a)} \over {J_1(k_\rho r_a) - AY_1(k_\rho r_a)}},} & {A = \displaystyle{{J_1(k_\rho (r_a + l))} \over {Y_1(k_\rho (r_a + l))}}} & {{\mkern 1mu} {\rm if}\;k_\rho ^2 = k_0^2 \varepsilon _\varphi \mu _z{\rm > }0} \cr {\displaystyle{{\varepsilon _\varphi } \over {\left \vert {\varepsilon _\varphi } \right \vert}}\sqrt {\left \vert {\displaystyle{{\varepsilon _\varphi } \over {\mu _z}}} \right \vert} \displaystyle{{I_0(\left \vert {k_\rho } \right \vertr_a) + BK_0(\left \vert {k_\rho } \right \vertr_a)} \over {I_1(\left \vert {k_\rho } \right \vertr_a) - BK_1(\left \vert {k_\rho } \right \vertr_a)}},} & {B = \displaystyle{{I_1(\left \vert {k_\rho } \right \vert(r_a + l))} \over {K_1(\left \vert {k_\rho } \right \vert(r_a + l))}}} & {{\rm if}\;k_\rho ^2 = k_0^2 \varepsilon _\varphi \mu _z{\rm < }0} \cr } } \right.$$

where r _a is the anode radius and l is the length of the side resonator, J ₀, J ₁, Y ₀, and Y ₁ are Bessel functions of zero and first order, and I ₀, I ₁, K ₀, and K ₁ are modified Bessel functions of zero and first order. Hence, the fundamental assumption in the above-mentioned straightforward approach is that the CSOLR-loaded side resonator which is, in fact, a short-circuited CSOLR unit-cell shows a susceptance which is equal to the susceptance of a conventional side resonator which is filled with ε _φ and μ _z obtained using equation (8). As is expected from the effective medium approach to metamaterial unit-cells, and as it will be shown in the following example this assumption is valid. As an example, a CSOLR-loaded vane-type SHM (see Fig. 2(b)) with N = 16, r _a = 2.25 mm, r _c = 1.3 mm, and d = 0.49 mm is considered. Dimensions of the interaction space and the resonator aperture are chosen to be equal to those of an SHM with conventional SWS which was subjected to comprehensive numerical simulations and measurements (see [Reference Nasr Esfahani and Schünemann2] and [Reference Schuenemann, Serebryannikov, Sosnytskiy and Vavriv9]). Suppose that for this SWS it is desired to have the resonant frequency of the n = 4 (N/4 or π/2)-mode at around 35 GHz. Then the CSOLR unit-cell should be designed in a way to provide small values of ε _φ at frequencies of around 35 GHz. Considering the unit-cell of Fig. 2(c) and its constitutive parameters (see Fig. 2(e), the permittivity and permeability are calculated using the scattering parameters of the unit-cell shown which is shown in Fig. 2(d) and equation (8)), it can be deduced that the unit-cell of Fig. 2(c) can provide small values of permittivity (ε _φ) at around 35 GHz. Figure 2(f) shows the simulated susceptance of the short-circuited unit-cell of Fig. 2(c) (using ANSYS HFSS) and also the susceptance which is calculated using equation (12) and the constitutive parameters are shown in Fig. 2(e). As can be deduced from this figure, the simulated and calculated susceptances are in good agreement. This validates the assumption of considering a CSOLR-loaded side resonator as a short-circuited unit-cell of Fig. 2(c) filled with the ε _φ and μ _z calculated using equation (8). The resonant frequency of the n = 4-mode which is calculated using equation (9) is equal to 34.89 GHz. Simulations using ANSYS HFSS show that the resonant frequency of this mode is equal to 34.99 GHz which is in good agreement with the analytical calculations. It is also important to note that the electric field in the interaction space of the 35 GHz CSOLR-loaded SHM and a conventional SHM with the same operating frequency are quite similar (see Fig. 2(g) and 2(h)) which confirms that the CSOLR has no negative effect on the field profile of the interaction space.

Modified CSOLR-loaded SHM

Although the CSOLR-loaded unit-cell of the previous section is able to provide small values of relative permittivity, the large slope of the effective permittivity curve at the frequency where the curve crosses zero makes the final SWS sensitive to slight frequency changes. An example of this large permittivity variation can be seen in Fig. 2(e). In order to solve this issue, a copper strip with a thickness of (D _c) is employed above the unit-cell of the CSOLR (see Fig. 3(a)). A copper strip provides a Drude-type effective permittivity and thus can decrease the slope of the Lorenz-type effective permittivity of the CSOLR unit-cell at the frequency where the curve crosses zero. Hereafter this unit-cell will be referred to as the modified CSOLR (MCSOLR) unit-cell. Figure 3(b) shows the relative permittivity of this unit-cell for different values of copper thickness. Figure 3(c) shows B _int (ω) + B _res (ω) for the n = 3-mode (according to equation (9), B _int (ω) + B _res (ω) = 0 determines the resonant frequencies). As is seen in this figure, for each value of D _c two resonant frequencies exist, one below the resonant frequency of the CSOLR (35 GHz) and one above this frequency. Only the latter is not suffering from rapid permittivity variations in the areas where permittivity is small and therefore is a suitable operation frequency for a magnetron. In this section, we will examine the operation of the MCSOLR-loaded SHM with D _C = 1 mm, around its second resonant frequency (f = 40.9 GHz, see Fig. 3(c)). This frequency has been selected because no mode competition difficulties occur at this frequency and the relative permittivity is also small (around −0.5, see Fig. 3(b)). Simulations (using ANSYS HFSS) show that the corresponding simulated resonant frequency is 42.38 GHz. This shows that the approach presented in the previous section for calculating the resonant frequency has a reasonable accuracy (the error in calculating the resonant frequency is 3.37% [Reference Esfahani, Schünemann, Avtomonov and Vavriv6]). It should be mentioned that since the operating frequency lies in the region where |ε_φ| <1 which is far from the resonance of the CSOLRs, the effect of losses from metamaterials is negligible. Therefore, the quality factor (Q) of an ENZ-loaded SHM is close to the Q of a conventional SHM (at 43 GHz, Q is around 1300).

Fig. 3. (a) MCSOLR-loaded SWS. Dimensions of the MCSOLR, unit-cell, and the interaction space are (see Fig. 2(c)) l ₁ = 1.6 mm, l ₂ = 1.724 mm, g = 0.2 mm, w = 0.25 mm, D = 1.4 mm, t = 0.39 mm, h ₁ = h ₂ = 0.49 mm, r _c = 1.3 mm, and r _a = 2.25 mm; (b) retrieved relative permittivity; and (c) calculated B _int (ω) + B _res (ω).