Mathematical Models and Numerical Investigation for the Eigenmodes of the Modern Gyrotron Resonators Oleksiy KONONENKO RF Structure Development Meeting, CERN

2/36 Outline Introduction Mathematical model for the eigen TM modes Mathematical model of the corrugated gyrotron resonator with dielectrics for the eigen TE modes Mathematical model of the corrugated gyrotron resonator with dielectrics for the eigen TM modes Discrete mathematical model of the hypersingular integral equation of the general kind Numerical analysis of the gyrotron eigen modes Conclusion

3/36 Coaxial gyrotrons as a part of thermonuclear facility Introduction

4/36 Transverse and longitudinal cross-sections of the considered resonator Eigen electromagnetic oscillations are considered Arbitrary corrugation parameters are studied Introduction

5/36 Initial problem on the corrugation period Propagation constant Cut-off wave number Frequency 2D Dirichlet problem: ТМ modes

6/36 2D Helmholtz equation Mode representation of the solution: 2D Helmholtz equation in polar coordinates: Eigenvalue of the m-th TM mode ТМ modes

7/36 Fourier-series expansion of the solution Expansions in the cross-cut domains: Basis cylindrical functions expressions: ТМ modes

8/36 Continuity condition on the domains boundary Electromagnetic field continuity means: W functions can be expressed in the terms of the Φ ones: ТМ modes

9/36 Hypersingular integral equation of the problem The following unknown function is introduced: Problem is reduced to the hypersingular integral equation (HSIE): ТМ modes

10/36 Initial problem on the corrugation period 2D Neumann problem: ТЕ modes/dielectrics Eigen frequency

11/36 2D Helmholtz equation Mode representation of the solution: 2D Helmholtz equation in polar coordinates: ТЕ modes/dielectrics Eigen frequency of the m-th TE mode

12/36 Fourier-series expansion of the solution Solution expansions in the cross-cut domains: Basis cylindrical functions expressions: ТЕ modes/dielectrics

13/36 Continuity condition on the domains boundary Electromagnetic field continuity means: W functions can be expressed in the terms of the Φ ones: ТЕ modes/dielectrics

14/36 Singular integral equation of the problem The following unknown function is introduced: The problem is reduced to the singular integral equation (SIE) with the additional condition: ТЕ modes/dielectrics

15/36 Discrete mathematical model of the SIE To fulfill an edge condition F m function is considered in such a form: Discretization of the SIE is performed using quadrature formulas of the interpolative type based on the Chebyshev polynomials of the 1-st kind: ТЕ modes/dielectrics

16/36 Initial problem on the corrugation period Eigen frequency 2D Dirichlet problem: ТM modes/dielectrics

17/36 2D Helmholtz equation Mode representation of the solution: 2D Helmholtz equation in polar coordinates: Eigen frequency of the m-th TM mode ТM modes/dielectrics

18/36 Continuity condition on the domains boundary Solution expansions in the cross-cut domains: Electromagnetic field continuity means: ТM modes/dielectrics

19/36 Hypersingular integral equation of the problem The following unknown function is introduced: Problem is reduced to the hypersingular integral equation (HSIE): ТM modes/dielectrics

20/36 Discrete mathematical model of HSIE HSIE of the general kind Inhomogeneous HSIE is considered: In the polynomial spaces the following scalar products are considered:

21/36 Regularization of the integral operators The following integral operators are defined: The following regularized equation is considered: Discrete mathematical model of HSIE

22/36 Convergence of the discrete model The following estimations of the convergence are derived : Convergence of the approximate solution to the rigorous one : Discrete mathematical model of HSIE

23/36 Discretization of the HSIE for the TM modes To fulfill an edge condition F m function is considered in such a form: Discretization of the HSIE is performed using quadrature formulas of the interpolative type based on the Chebyshev polynomials of the 2-nd kind for the regularized integral operators: Discrete mathematical model of HSIE

24/36 Numerical investigation Operating modeTE 34,19 Frequency, f [GHz]170 Number of the corrugations, N75 Outer radius, R o [mm]29.55 Inner radius, R i [mm] Depth of the corrugation, h [mm]0.44 Width of the corrugation, L [mm]0.35 Output power, P [MW]2.2 Parameters of the ТЕ 34,19 coaxial gyrotron

25/36 Numerical investigation Gyrotron simulation software

26/36 Eigenvalue calculations for TE modes Relative accuracy of the TE 34,19 mode eigenvalue calculations depending on the number of the discretization points Numerical investigation

27/36 Eigenvalue calculations for the dielectrics and TE modes Eigenvalue of the traveling TE34,19 mode  TE 34,19  -  +  - Dependence of the eigenvalue upon the dielectric permittivity Numerical investigation

28/36 Field magnitude in the cross-cut Real part of the H z field component for TE 34,19 mode Numerical investigation

29/36 Absolute value of the H z field component for TE 34,19 mode in the corrugation Numerical investigation Field magnitude in the corrugation

30/36 Eigenvalue calculations for TM modes Relative accuracy of the TM 34,19 mode eigenvalue calculations depending on the number of the discretization points Numerical investigation

31/36 Eigenvalues for a fixed azimuthal mode number Eigenvalues of the ТЕ and ТМ modes for the fixed azimuthal mode number m=34 Numerical investigation ModeSIEHFSS TM 34, TE 34, TM 34, TE 34, TM 34, TE 34,

32/36 Eigenvalues for the cross-cut sets Dependence of the eigenvalue upon the longitudinal z coordinate. Problem is solved in each cross-cut separately. Numerical investigation

33/36 Field magnitude in the cross-cut Absolute value of the E z field component for TM 34,19 mode Numerical investigation

34/36 Field magnitude in the corrugation Absolute value of the E z field component for TM 34,19 mode in the corrugation Numerical investigation

35/36 Ohmic losses calculation Estimation of the Ohmic losses denisity on the corrugation walls for the operating TE 34,19 mode ρ,kW/cm 2 SIEIM top bottom side period h=0.44 Numerical investigation

36/36 Conclusion Mathematical model of the coaxial gyrotron resonator is developed for the eigen TM modes for the first time Mathematical models to study gyrotron resonators with dielectrics are derived for TE and TM modes Models are developed for the arbitrary corrugation parameters, radial and azimuthal mode indexes. This allows to use them for the analysis of the wide range modern gyrotron resonators. New discrete mathematical model is built and substantiated for the hypersingular integral equation of the general kind. Numerical investigation of the TM waves was carried out on its basis. This model can also be used for other applied physics problems. Basing on the developed models numerical analysis of the gyrotron resonators is performed. Comparison with the known results and validation is provided. Results of the numerical estimation for the Ohmic losses density are presented and suggestions for the geometry optimization are proposed. Conclusion

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