Specific Features of an Open Resonator of Orotron with a Two-Row Periodic Structure

Abstract

The electrodynamic characteristics (EDCs) of open resonators (ORs) formed by multifocal spherical and plane mirrors are studied. The following periodic structures are used on the plane mirrors: a single-row quarter-wavelength comb in which the lamella height is h_s = λ/4 and two-row periodic structures (TRPSs) with h_s = λ/2 (half-wavelength) or λ/4 < h_s < λ/2 (intermediate) (λ is the wavelength). The corresponding EDCs are compared. It is shown that the OR with an intermediate TRPS exhibits a previously unknown feature of the distribution of the high-frequency field of the fundamental mode along the OR symmetry axis. Application of the OR with the intermediate TRPS in the orotron at h_s/λ < 0.3–0.32 makes it possible to increase the efficiency of the electron-wave interaction and, hence, the efficiency and generated power. Supporting experimental and calculated results are presented.

APPENDIX

Two formulas for the inrush current with low and high space charge have been derived in [9]. First, we must choose the appropriate formula for the regime under study. For this purpose, we must calculate parameter φ_p = h_pL, where L is the interaction length, h_p = ω_p/V_e, ω_p is the plasma frequency, and V_e is the velocity of beam electrons [9]. For φ_p ⪡ 1, the space charge can be disregarded and we can calculate the inrush current using formula (50) from [9] for the low space charge. For φ_p ⪢ 1, we must use formula (51) from [9]. Thus, parameter φ_p must be calculated to choose the formula. Following the approach of [9], we obtain an expression for parameter φ_p for a low space charge using formulas (49) and [50] from [9]:

$${{\varphi }_{p}} = {{({{2\pi \Gamma {{\pi }^{2}}} \mathord{\left/ {\vphantom {{2\pi \Gamma {{\pi }^{2}}} {8\psi Q}}} \right. \kern-0em} {8\psi Q}})}^{{0.5}}},$$

(A1)

here, Г is the coefficient of a decrease in the plasma frequency, Q is the loaded Q factor of the OR, ψ = W₁/W is the efficiency of the electric field in the OR (W₁ is the energy of the first space harmonic in the ribbon beam and W is the energy of the HF field stored in the OR) given by formula (43) from [9]

$$\psi = {{{{W}_{1}}} \mathord{\left/ {\vphantom {{{{W}_{1}}} W}} \right. \kern-0em} W} = {{\theta {{B}^{2}}{{S}_{0}}L} \mathord{\left/ {\vphantom {{\theta {{B}^{2}}{{S}_{0}}L} {8\pi W}}} \right. \kern-0em} {8\pi W}},$$

(A2)

and θ is the efficiency of the electron beam, which is 1 for the system under study, since the nonuniformity of the HF field with respect to width and thickness of electron beam can be disregarded.

For the device under study, the thickness of the electron beam is 2H = 0.1 mm and the TRPS period is l = 0.29 mm, so that the ratio is l/2H ≈ 3. Using formula (32) from [9]

$$B = \left[ {1 + \exp \left( {{{ - 2H\pi } \mathord{\left/ {\vphantom {{ - 2H\pi } {\text{l}}}} \right. \kern-0em} {\text{l}}}} \right)} \right]{{{{A}_{1}}} \mathord{\left/ {\vphantom {{{{A}_{1}}} 2}} \right. \kern-0em} 2},$$

(A3)

we obtain

$$\begin{gathered} B = \left[ {1 + \exp \left( { - 2.093} \right)} \right]{{{{A}_{1}}} \mathord{\left/ {\vphantom {{{{A}_{1}}} 2}} \right. \kern-0em} 2} \\ \approx \left( {{{1.123} \mathord{\left/ {\vphantom {{1.123} 2}} \right. \kern-0em} 2}} \right){{A}_{1}} = 0.5615{{A}_{1}}, \\ \end{gathered} $$

so that

$${{B}^{2}} = {{\left\{ {\left[ {1 + \exp \left( {{{ - 2H\pi } \mathord{\left/ {\vphantom {{ - 2H\pi } {\text{l}}}} \right. \kern-0em} {\text{l}}}} \right)} \right]{{{{A}_{1}}} \mathord{\left/ {\vphantom {{{{A}_{1}}} 2}} \right. \kern-0em} 2}} \right\}}^{2}} \approx 0.31528{{\left( {{{A}_{1}}} \right)}^{2}}.$$

(A4)

Substituting S₀ = 0.01 × 0.5 cm² and L = L_7cyl + 3r_c = 3.2 × 6 + 3 × 2.68 = 27.24 mm = 2.7245 cm in formula (A2) for parameter ψ, we obtain

$$\psi \approx 0.171 \times {{{{{10}}^{{ - 3}}}} \mathord{\left/ {\vphantom {{{{{10}}^{{ - 3}}}} W}} \right. \kern-0em} W}.$$

(A5)

To determine interaction length L, we add 3r_c (rather than 4r_c) to parameter L_7cyl (2r_c is not added on the plane mirror from the spherical surface on each side of the seven-focus spherocylindrical focusing mirror). Such an approach is based on the experimental results on the inrush current for ORs with different multifocal focusing mirrors. For an interaction length of L_7cyl + 4r_c, parameter W₁ = 0.171 × 10^–3 should be increased by a factor of 29.92/27.24 and the corresponding result is

$${{\psi }_{0}} = 1.075 \times 0.171 \times {{{{{10}}^{{ - 3}}}} \mathord{\left/ {\vphantom {{{{{10}}^{{ - 3}}}} W}} \right. \kern-0em} W} \approx 0.184 \times {{{{{10}}^{{ - 3}}}} \mathord{\left/ {\vphantom {{{{{10}}^{{ - 3}}}} W}} \right. \kern-0em} W}.$$

(A6)

Thus, parameter ψ₀ increases and, hence, parameter φ_p and inrush current I₀ decrease.

We must calculate quantity W using formula (44) from [9]

$$W = {{{{\nu }_{1}}{{{({{A}_{1}})}}^{2}}V} \mathord{\left/ {\vphantom {{{{\nu }_{1}}{{{({{A}_{1}})}}^{2}}V} {8\pi }}} \right. \kern-0em} {8\pi }},$$

(A7)

where V is the OR volume, A₁ is the amplitude of the first space harmonic, ν₁ = π²/8 for the optimal quarter-wavelength comb for which the slit-to-period ratio is d/l = 0.5 and the ridge height is h = b₀ = λ/4. Thus, the ratio is A₁/A₀ = 0.637 (A₀ is the amplitude of the HF field in the OR).

The first condition for the TRPS in the experimental model of the orotron is satisfied. However, the second condition is not satisfied. However, the orotron with such a TRPS works as if it should be satisfied. Hence, we assume that expression (A7) is valid with ν₁ = π²/8. Ratios b₀/λ = 0.3 and A₁/ A₀ = 0.5 for the device under study will be taken into account in the calculation of quantity W₁. We calculate

$$W = {{\pi {{{\left( {{{A}_{1}}} \right)}}^{2}}V} \mathord{\left/ {\vphantom {{\pi {{{\left( {{{A}_{1}}} \right)}}^{2}}V} {64}}} \right. \kern-0em} {64}} = {{[\pi {{{\left( {{{A}_{1}}} \right)}}^{2}}{{S}_{1}}]{{H}_{{{\text{OR}}}}}} \mathord{\left/ {\vphantom {{[\pi {{{\left( {{{A}_{1}}} \right)}}^{2}}{{S}_{1}}]{{H}_{{{\text{OR}}}}}} {64}}} \right. \kern-0em} {64}},$$

where S₁ is the surface area of the plane mirror and H_ОR is the OR height.

The area of the plane mirror is given by

$$\begin{gathered} {{S}_{1}} = ({{l}_{{{\text{cyl}}}}} + 4{{r}_{{\text{c}}}}){{L}_{{{\text{m}}{\text{.pl}}}}} = \left( {4\,\,{\text{mm}} + 4 \times 2.68} \right) \times 29.92 \\ = 14.72 \times 29.92\,\,{\text{mm}} = 440.422\,\,{\text{m}}{{{\text{m}}}^{2}} \approx 4.404\,\,{\text{c}}{{{\text{m}}}^{2}}, \\ \end{gathered} $$

$${{H}_{{{\text{OR}}}}} = 8.583\,\,{\text{mm}} = 0.8583\,\,{\text{cm}}.$$

Thus, we have

$$\begin{gathered} W = {{\pi {{{\left( {{{A}_{1}}} \right)}}^{2}}V} \mathord{\left/ {\vphantom {{\pi {{{\left( {{{A}_{1}}} \right)}}^{2}}V} {64}}} \right. \kern-0em} {64}} \\ = {{\pi {{{\left( {{{A}_{1}}} \right)}}^{2}}{{S}_{1}}{{H}_{{{\text{OR}}}}}} \mathord{\left/ {\vphantom {{\pi {{{\left( {{{A}_{1}}} \right)}}^{2}}{{S}_{1}}{{H}_{{{\text{OR}}}}}} {64}}} \right. \kern-0em} {64}} \approx 0.185{{\left( {{{A}_{1}}} \right)}^{2}}. \\ \end{gathered} $$

$$\begin{gathered} \psi = {{{{W}_{1}}} \mathord{\left/ {\vphantom {{{{W}_{1}}} W}} \right. \kern-0em} W} = 0.171 \times {{10}^{{ - 3}}}{{{{{({{A}_{1}})}}^{2}}} \mathord{\left/ {\vphantom {{{{{({{A}_{1}})}}^{2}}} W}} \right. \kern-0em} W} \\ = {{({{A}_{1}})}^{2}} \times 0.171 \times {{{{{10}}^{{ - 3}}}} \mathord{\left/ {\vphantom {{{{{10}}^{{ - 3}}}} {0.185}}} \right. \kern-0em} {0.185}}{{({{A}_{1}})}^{2}}. \\ \end{gathered} $$

(A8)

Consequently, we obtain

$$\begin{gathered} \psi \approx 0.924 \times {{10}^{{ - 3}}}, \\ {{\psi }_{0}} \approx 0.184{{{{{({{A}_{1}})}}^{2}}} \mathord{\left/ {\vphantom {{{{{({{A}_{1}})}}^{2}}} {0.185}}} \right. \kern-0em} {0.185}}{{({{A}_{1}})}^{2}} = 0.9935 \times {{10}^{{ - 3}}}. \\ \end{gathered} $$

(A9)

As was mentioned, the structure under study works as the quarter-wavelength comb with the parameters that differ from optimal ones. Indeed, for the optimal quarter-wavelength comb, we must have b/λ = 0.25 and, hence, A₁/A₀= 0.637. For the experimental system, we have b/λ = 0.3 and A₁/A₀ = 0.5. Energy storages in the electron beam and OR are W₁ ~ $A_{1}^{2}$ and W ~ $A_{0}^{2}$, respectively, so that coefficient ψ = W₁/W ~ (A₁/A₀)² must be decreased to ψ₁, and coefficient ψ₀ must be decreased to ψ₀₁:

$$\begin{gathered} {{\psi }_{1}} = {{{{W}_{1}}{{{\left( {{{0.5} \mathord{\left/ {\vphantom {{0.5} {0.637}}} \right. \kern-0em} {0.637}}} \right)}}^{2}}} \mathord{\left/ {\vphantom {{{{W}_{1}}{{{\left( {{{0.5} \mathord{\left/ {\vphantom {{0.5} {0.637}}} \right. \kern-0em} {0.637}}} \right)}}^{2}}} W}} \right. \kern-0em} W} \\ = {{W}_{1}} \times {{0.616} \mathord{\left/ {\vphantom {{0.616} W}} \right. \kern-0em} W} = 0.569 \times {{10}^{{ - 3}}}, \\ \end{gathered} $$

$$\begin{gathered} {{\psi }_{{01}}} = 0.9935 \times 0.616 \times {{10}^{{ - 3}}} \\ = 0.61199 \times {{10}^{{ - 3}}} \approx 0.612 \times {{10}^{{ - 3}}}. \\ \end{gathered} $$

Substituting ψ₁ = 0.569 × 10^–3 and Q = 2000 in formula (A1), we obtain ψ₁Q ≈1.14. Assuming that Г = 0.5 [6], we obtain the following result for a low space charge:

$$\begin{gathered} {{\varphi }_{p}} = {{({{2\pi \Gamma {{\pi }^{2}}} \mathord{\left/ {\vphantom {{2\pi \Gamma {{\pi }^{2}}} {8\psi Q}}} \right. \kern-0em} {8\psi Q}})}^{{0.5}}} = {{3.3946}^{{0.5}}} \approx 1.829, \\ {\text{or}}\,\,~1.966,\,\,{\text{i}}{\text{.e}}{\text{.}} \approx \,\, > 1. \\ \end{gathered} $$

However, expression q₂/q₁ = 64Г/π³ψQ is presented in [9], where q₂ and q₁ are calculated using formulas (51) and (50) from [9] for high and low space charges, respectively. When the ratio is less than 1, the space charge weakly affects the interaction of electron beam and HF field. For the system under study, we have q₂/q₁ = 64Г/π³ψQ ≈ 0.91 for ψ₁ or 0.978 for ψ₀₁, so that the inrush current can be calculated using formula (50) from [9] for the low space charge:

$${{I}_{0}} = 8 \times {{10}^{{ - 5}}}{{S}_{0}}{{U_{0}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}} \mathord{\left/ {\vphantom {{U_{0}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}} {{{L}^{2}}}}} \right. \kern-0em} {{{L}^{2}}}}\psi Q.$$

(A10)

Substituting parameters S₀ = 5 × 10^–3 cm², U₀ = 9 × 10³ V, L = 2.724 cm, ψ₁Q ≈ 1.14, we obtain

$$\begin{gathered} {{I}_{0}} \approx (8 \times {{10}^{{ - 5}}} \times 5 \times {{10}^{{ - 3}}} \\ \times \,\,{{{{{({{9}^{3}} \times {{{10}}^{9}})}}^{{0.5}}}} \mathord{\left/ {\vphantom {{{{{({{9}^{3}} \times {{{10}}^{9}})}}^{{0.5}}}} {{{{2.724}}^{2}}}}} \right. \kern-0em} {{{{2.724}}^{2}}}} \times 1.14){\text{ A}} \\ = 40.37 \times {{10}^{{ - 3}}}\,{\text{A}} \approx 40.4\,\,{\text{mA}}. \\ \end{gathered} $$

The calculated inrush current is close to the experimental result (40 mA). When an interaction length of 2.929 cm (instead of 2.724 cm) is used in the expression for quantity W₁, the calculated inrush current (32.5 mA) is less than the experimental current by a factor of (2.929/2.724)³ =1.243.