A primer on the theory of Thomson scattering for high-temperature fusion plasmas

Abstract

This article has been written to provide an introduction to the theory of Thomson scattering for students embarking on a research career in plasma diagnostics. The presentation is introductory in nature and it provides a basis for the further study of Thomson scattering theory at a more advanced level. It is assumed that the reader is familiar with electromagnetic theory undertaken in typical undergraduate courses in either physics or electrical engineering, while familiarity with calculus and vector analysis is a prerequisite. It is acknowledged that students involved in this area of research will be concerned with the experimental aspects of the diagnostic and with the interpretation of experimental results; nonetheless, it is essential that they should have a good understanding of the theory of Thomson scattering at the outset. The purpose of this article is to give an introductory account of the theory of Thomson scattering and to present the subject matter at a level that is suitable for students that have just completed undergraduate courses in either physics or electrical engineering. The objective is to provide sufficient detail to make the presentation suitable for self-study as many of the equations are derived from first principles, and it is relatively easy to follow the equations line-by-line. Some of the tangential mathematical derivations are included in appendices so as not to distract from the main thrust of the mathematical argument. The theory of the Thomson scattering process presented here is based on the classical electrodynamics approach and only incoherent Thomson scattering is considered. Another approach, based on quantum electrodynamics, gives the same results at the classical electrodynamics approach but it also includes the Compton effect which is negligible in the case of Thomson scattering at laser wavelengths. The material presented is primarily concerned with the derivation of the Thomson scattered spectrum for very high-temperature plasmas encountered in modern-day experimental fusion plasmas. The strength of the scattered signal provides information on the electron density while the width of the scattered spectrum provides information on the plasma temperature. The plasma temperature is such that the electrons are moving at a large fraction of the velocity of light and it is, therefore, necessary to have a relativistic treatment of the Thomson scattering process. The derivation of the scattered spectrum is quite lengthy since it is necessary at various stages in the calculations to use equations that may not look too familiar, notably, those found in texts dealing with electromagnetic field theory. It is important to bear in mind, however, that Maxwell’s electromagnetic equations are consistent with the special theory of relativity. The material presented here is divided into the following sections: an introductory section outlines the Thomson scattering process and gives an indication of the magnitudes of various physical quantities. Section 2 is concerned with the equation of motion of a typical electron in the field of a light wave (that is, a high-power laser beam passing through a cross-section of the plasma), and the solution of the resulting equation gives the acceleration of the electron; noting that an accelerated charge particle emits electromagnetic radiation. The power radiated by an accelerated charge is discussed in section 3, as are polar diagrams to indicate the distribution of the scattered radiation. The typical scattering geometry routinely employed in Thomson scattering systems and the nature of the scattered signal is presented in section 4. The fundamental equation for the scattered signal in relation to incoherent scattering is also presented in this section. The remaining sections are concerned with the form of the scattered spectrum: in section 5 the spectrum is derived by neglecting a so-called ‘depolarization term’ which is a relativistic effect and only becomes significant in very high-temperature plasmas. LIDAR scattering is discussed in section 6 and finally in section 7, the scattered spectrum for all scattering angles is outlined and an alternative derivation for the spectrum, not previously published, is presented here for the first time.