Generalized Radiation Boundary Conditions in Gyrotron Oscillator Modeling

S. Alberti; T. M. Tran; S. Brunner; F. Braunmueller; J. Genoud; J.-Ph. Hogge; M. Q. Tran

doi:10.1007/s10762-015-0204-2

Journal of Infrared, Millimeter, and Terahertz Waves

November 2015, Volume 36, Issue 11, pp 1043–1058 | Cite as

Generalized Radiation Boundary Conditions in Gyrotron Oscillator Modeling

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Authors and affiliations

S. AlbertiEmail author
T. M. Tran
S. Brunner
F. Braunmueller
J. Genoud
J.-Ph. Hogge
M. Q. Tran

Article

First Online: 14 September 2015

Received: 14 July 2015

Accepted: 24 August 2015

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2 Citations

Abstract

A numerical procedure to implement a frequency-independent generalized non-reflecting radiation boundary conditions, GNRBC, based on the Laplace Transform, is described in details and tested successfully on a simple 2 frequency test problem. In the case of non-stationary regimes occurring in gyrotron oscillators, it is shown that the reflection at frequencies significantly separated from the carrier frequency can be effectively suppressed by this method. A detailed analysis shows that this numerical approach can be consistently used only for models in which there is no assumed separation of time scales between the RF field envelope time-evolution and the electron time of flight across the interaction region. The GNRBC has been implemented in a nonlinear time-dependent self-consistent monomode model, TWANGpic, in which there is no time scale separation since the RF field envelope is updated at each integration time step of the electron motion. The illustration of the effectiveness of the GNRBC is made with TWANGpic on a gyrotron for which extensive theoretical and experimental results have been performed.

Keywords

Gyrotron oscillator Radiation boundary conditions

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Notes

Acknowledgments

This work is partially supported by EFDA under Grant WP13-DAS-HCD-EC and by Fusion for Energy under Grants F4E-GRT-432 and -553 within the European Gyrotron Consortium (EGYC). The views and opinions expressed herein do not necessarily reflect those of the European Commission. EGYC is a collaboration of CRPP, Switzerland; KIT, Germany; HELLAS, Greece; IFP-CNR, Italy. The authors would like to thank Dr. K. Avramides from KIT (Karlsruhe Institute for Technology) for precious scientific discussions.

Appendix: Inverse Laplace Transform of 1/i κ _∥(Ω)

Using the dispersion relation (11), the inverse Laplace transform can be computed as follow

$$ g(\tau)= \frac{1}{2\pi i} {\int}_{-\infty+i\gamma}^{\infty+i\gamma} ~\frac{e^{-i\varOmega \tau}}{\sqrt{2\varOmega+\kappa_{\parallel 0}^{2}}}~d\varOmega = \frac{e^{i\frac{\kappa_{\parallel 0}^{2}}{2}\tau}}{4\pi i} {\int}_{-\infty+i\gamma}^{\infty+i\gamma} ~\frac{e^{-i\varOmega \tau/2}}{\sqrt{\varOmega}}~d\varOmega $$

(35)

Splitting the integral path between its Re(Ω) < 0 and Re(Ω) > 0 parts, and noting that

$$\sqrt{\varOmega} = \sqrt{-(-\varOmega)} = i\sqrt{-\varOmega}, \qquad \text{for Re} (\varOmega) < 0, $$

the integral becomes, in the limit γ → 0

$${\int}_{-i\gamma}^{\infty-i\gamma}\!\!d\varOmega~\frac{e^{i\tau\varOmega/2}}{i\sqrt{\varOmega}} +{\int}_{i\gamma}^{\infty+i\gamma}\!\!d\varOmega~\frac{e^{-i\tau\varOmega/2}}{\sqrt{\varOmega}} = \sqrt{2\pi}\left[-i\sqrt{\frac{1}{-i\tau}} + \sqrt{\frac{1}{i\tau}}\right], $$

where we have used the following result

$${\int}_{0}^{\infty} \!\! du\,e^{-zu^{2}} = \frac{1}{2}\sqrt{\frac{\pi}{z}}, \qquad\text{Re}(z) \geq 0. $$

For τ < 0, the sum inside the bracket can be calculated as

$$-i\sqrt{\frac{1}{i|\tau|}} + \sqrt{\frac{1}{-i|\tau|}} = \frac{1}{\sqrt{|\tau|}} \left(\frac{e^{-i\pi/2}}{e^{i\pi/4}} + \frac{1}{e^{-i\pi/4}} \right) = 0, $$

while for τ > 0

$$-i\sqrt{\frac{1}{-i|\tau|}} + \sqrt{\frac{1}{i|\tau|}} = \frac{1}{\sqrt{|\tau|}} \left(\frac{e^{-i\pi/2}}{e^{-i\pi/4}} + \frac{1}{e^{i\pi/4}} \right) = \frac{2}{\sqrt{i\tau}}, $$

where the principal branch has been considered for computing $\sqrt {i}$.

The final result is thus

$$ g(\tau) = \left\{\begin{array}{ll} 0 & \text{for }\tau<0\\ \frac{1}{i}\sqrt{\frac{1}{2\pi i\tau }}\, \exp\left({i\frac{\kappa_{\parallel 0}^{2}}{2}\tau}\right). & \text{otherwise} \end{array}\right. $$

(36)

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Authors and Affiliations

S. Alberti
- 1
Email author
T. M. Tran
- 1
S. Brunner
- 1
F. Braunmueller
- 1
J. Genoud
- 1
J.-Ph. Hogge
- 1
M. Q. Tran
- 1

1.Ecole Polytechnique Fédérale de Lausanne (EPFL), Centre de Recherches en Physique des PlasmasLausanneSwitzerland

Generalized Radiation Boundary Conditions in Gyrotron Oscillator Modeling

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