Mathematical Models and Numerical Investigation for the Eigenmodes of Modern Gyrotron Resonators

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

2. 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. Introduction Coaxial gyrotrons as a part of thermonuclear facility

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

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

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

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

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

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

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

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

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

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

14. ТЕ modes/dielectrics 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:

15. ТЕ modes/dielectrics Discrete mathematical model of the SIE • To fulfill an edge condition Fm 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:

16. ТMmodes/dielectrics Eigen frequency Initial problem on the corrugation period • 2D Dirichlet problem:

17. ТMmodes/dielectrics Eigen frequency of the m-th TM mode 2D Helmholtz equation • Mode representation of the solution: • 2D Helmholtz equation in polar coordinates:

18. ТMmodes/dielectrics Continuity condition on the domains boundary • Solution expansions in the cross-cut domains: • Electromagnetic field continuity means:

19. ТMmodes/dielectrics Hypersingular integral equation of the problem • The following unknown function is introduced: • Problem is reduced to the hypersingular integral equation (HSIE):

20. 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. Discrete mathematical model of HSIE Regularization of the integral operators • The following integral operators are defined: • The following regularized equation is considered:

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

23. Discrete mathematical model of HSIE Discretization of the HSIE for the TM modes • To fulfill an edge condition Fm 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:

24. Numerical investigation Parameters of the ТЕ34,19coaxial gyrotron

25. Numerical investigation Gyrotron simulation software

26. Numerical investigation Eigenvalue calculations for TE modes • Relative accuracy of the TE34,19 mode eigenvalue calculations depending on the number of the discretization points

27. Numerical investigation Eigenvalue calculations for the dielectrics and TE modes • Dependence of the eigenvalue upon the dielectric permittivity c 190 TE 34,19 - c 180 170 160 150 Eigenvalue of the traveling TE34,19 mode 140 130 120 + c 110 - e 100 1.0 1.5 2.0 2.5 3.0

28. Numerical investigation Field magnitude in the cross-cut • Real part of the Hz field component for TE34,19mode

29. Numerical investigation Field magnitude in the corrugation • Absolute value of the Hz field component for TE34,19mode in the corrugation

30. Numerical investigation Eigenvalue calculations for TM modes • Relative accuracy of the TM34,19 mode eigenvalue calculations depending on the number of the discretization points

31. Numerical investigation Eigenvalues for a fixed azimuthal mode number • Eigenvalues of the ТЕ and ТМ modes for the fixed azimuthal mode number m=34

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

33. Numerical investigation Field magnitude in the cross-cut • Absolute value of the Ez field component for TM34,19mode

34. Numerical investigation Field magnitude in the corrugation • Absolute value of the Ez field component for TM34,19mode in the corrugation

35. Numerical investigation h=0.44 Ohmic losses calculation • Estimation of the Ohmic losses denisity on the corrugation walls for the operatingTE34,19 mode

36. Conclusion 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.

37. Thank you for your attention!