Semi-analytic Shape Function of High-harmonic Electron Cyclotron Emission in Tenuous Plasma

Semi-analytic expressions for the electron cyclotron emission (ECE) shape function are developed for arbitrary high harmonics. The integrand of the ${m}^{\mathrm{th}}$ harmonic ECE shape function is fitted with the readily-integrable test function which is parameterized by plasma temperature $T_e$, harmonic number $m$ and emission angle $\theta$. Semi-analytic formulae for high harmonic ECE emissivity are obtained by integrating the test integrand with the fitting parameters gained from the regression analysis as well as the inductive studies for arbitrary cases. The developed expression matches the numerically-integrated ECE shape function very well and the overall differences between original shape functions and semi-analytic shape functions are evaluated. The expression can be used for rapid analysis of high-harmonic ECE spectra.