Bohdan Bieg

Stokes-vector evolution in a weakly anisotropic inhomogeneous medium

The equation for evolution of the four-component Stokes vector in weakly anisotropic and smoothly inhomogeneous media is derived on the basis of a quasi-isotropic approximation of the geometrical optics method, which provides the consequent asymptotic solution of Maxwells equations. Our equation generalizes previous results obtained for the normal propagation of electromagnetic waves in stratified media. It is valid for curvilinear rays with torsion and is capable of describing normal mode conversion in inhomogeneous media. Remarkably, evolution of the four-component Stokes vector is described by the Bargmann-Michel-Telegdi equation for relativistic spin precession, whereas the equation for the three-component Stokes vector resembles the Landau-Lifshitz equation describing spin precession in ferromagnetic systems. The general theory is applied for analysis of polarization evolution in a magnetized plasma. We also emphasize fundamental features of the non-Abelian polarization evolution in anisotropic inhomogeneous media and illustrate them by simple examples.

Evolution of complex amplitudes ratio in weakly anisotropic plasma

The equation for evolution of the complex amplitudes ratio (CAR) ζ = E y / E x in weakly anisotropic inhomogeneous media is derived on the basis of quasi-isotropic approximation (QIA) of the geometrical optics method. This equation is convenient for the description of electromagnetic wave polarization in magnetized plasma of thermonuclear reactors like the ITER. The equation for the CAR is in agreement with other approaches, analyzing polarization evolution in weakly anisotropic media, in particular, with the equation for complex polarization angle and, via QIA equations, with the Segre equation for Stokes vector evolution. Simple analytical solutions for the CAR, which relates to normal mode propagation in homogeneous and weakly inhomogeneous plasma, are obtained. Besides, the equation for the CAR is solved numerically to describe the phenomenon of normal wave conversion in magnetized plasma in the vicinity of the orthogonality point between the ray and the static magnetic field. In distinction to the line-averaged measurements in traditional plasma polarimetry, the phenomenon of normal wave conversion opens the way for measuring the local plasma parameters near the orthogonality point.

Invited Article: A novel calibration method for the JET real-time far infrared polarimeter and integration of polarimetry-based line-integrated density measurements for machine protection of a fusion plant

In this paper, we present the work in the implementation of a new calibration for the JET real-time polarimeter based on the complex amplitude ratio technique and a new self-validation mechanism of data. This allowed easy integration of the polarimetry measurements into the JET plasma density control (gas feedback control) and as well as machine protection systems (neutral beam injection heating safety interlocks). The new addition was used successfully during 2014 JET Campaign and is envisaged that will operate routinely from 2015 campaign onwards in any plasma condition (including ITER relevant scenarios). This mode of operation elevated the importance of the polarimetry as a diagnostic tool in the view of future fusion experiments.

New technique in plasma polarimetry: Evolution equations for angular parameters ‘amplitude ratio–phase difference’ of polarization ellipse

New technique is suggested in plasma polarimetry: Differential equations for angular parameters of polarization ellipse, characterizing the amplitude ratio and the phase difference between orthogonal components of the wave field. Equations for angular variables ‘amplitude ratio–phase difference’ are derived, which allow direct calculation of the parameters of polarization ellipse, omitting solutions for the Stokes vector. The simplest analytical solutions are presented for the pure Faraday and the pure Cotton–Mouton effects. Behavior of angular parameters in the homogeneous and inhomogeneous plasmas is illustrated by numerical modeling in conditions when the Faraday and Cotton–Mouton effects are large enough and comparable in strength.

Double passage of electromagnetic waves through magnetized plasma: approximation of independent normal waves

Polarization properties of electromagnetic waves, double-passed through magnetized plasma, are studied. Analyses are performed in the case of non-interacting normal modes, propagating in homogeneous and weakly inhomogeneous plasmas, and for three kinds of reflectors: metallic plane, 2D corner retro-reflector (2D-CR), and cubic corner retro-reflector (CCR). It is shown that an electromagnetic wave, reflected from a metallic plane and from a CCR, contains only “velocity-preserving” channels, whose phases are doubled in comparison with those of a single-passage propagation. At the same time, an electromagnetic wave reflected from a 2D-CR is shown to contain both “velocity-preserving” and “velocity-converting” channels, the latter converting the fast wave into the slow one and vice-versa. One characteristic feature of “velocity-converting” channels is that they reproduce the initial polarization state near the source, which might be of practical interest for plasma interferometry. In the case of circularly polarized modes, “velocity-preserving” channels completely disappear, and only “velocity-converting” channels are to be found.

Evolution of the polarization of electromagnetic waves in weakly anisotropic inhomogeneous media — a comparison of quasi-isotropic approximations of the geometrical optics method and the Stokes vector formalism

The main methods describing polarization of electromagnetic waves in weakly anisotropic inhomogeneous media are reviewed: the quasi-isotropic approximation (QIA) of geometrical optics method that deals with coupled equations for electromagnetic field components, and the Stokes vector formalism (SVF), dealing with Stokes vector components, which are quadratic in electromagnetic field intensity. The equation for the Stokes vector evolution is shown to be derived directly from QIA, whereas the inverse cannot be true. Derivation of SVF from QIA establishes a deep unity of these two approaches, which happen to be equivalent up to total phase. It is pointed out that in contrast to QIA, the Stokes vector cannot be applied for a polarization analysis of the superposition of coherent electromagnetic beams. Additionally, the ability of QIA to describe a normal modes conversion in inhomogeneous media is emphasized.

Localized plasma polarimetry based on circular modes conversion: theoretical prerequisites and practical limitations

The method of localized polarimetric measurements based on the phenomenon of modes conversion near the orthogonality point between the electromagnetic beam and one of the helical magnetic lines in a tokamak plasma is discussed. A simple model of toroidal magnetized plasma is suggested, which allows one to analyze the applicability of localized polarimetry for the ITER project. It is shown that an acceptable localization of polarimetric measurements of about 0.3?m can be achieved for very high densities of order Ne = 1017?cm?3, which are significantly higher than the plasma densities, Ne = 1014?1015?cm?3, envisaged for the ITER project.

Application of polarimetry to test the models of thermonuclear plasma and determination the safety factor profile

The paper presents the way that colour can serve solving the problem of calibration points indexing in a camera geometrical calibration process. We propose a technique in which indexes of calibration points in a black-and-white chessboard are represented as sets of colour regions in the neighbourhood of calibration points. We provide some general rules for designing a colour calibration chessboard and provide a method of calibration image analysis. We show that this approach leads to obtaining better results than in the case of widely used methods employing information about already indexed points to compute indexes. We also report constraints concerning the technique. Nowadays we are witnessing an increasing need for camera geometrical calibration systems. They are vital for such applications as 3D modelling, 3D reconstruction, assembly control systems, etc. Wherever possible, calibration objects placed in the scene are used in a camera geometrical calibration process. This approach significantly increases accuracy of calibration results and makes the calibration data extraction process easier and universal. There are many geometrical camera calibration techniques for a known calibration scene [1]. A great number of them use as an input calibration points which are localised and indexed in the scene. In this paper we propose the technique of calibration points indexing which uses a colour chessboard. The presented technique was developed by solving problems we encountered during experiments with our earlier methods of camera calibration scene analysis [2]-[3]. In particular, the proposed technique increases the number of indexed points points in case of local lack of calibration points detection. At the beginning of the paper we present a way of designing a chessboard pattern. Then we describe a calibration point indexing method, and finally we show experimental results. A black-and-white chessboard is widely used in order to obtain sub-pixel accuracy of calibration points localisation [1]. Calibration points are defined as corners of chessboard squares. Assuming the availability of rough localisation of these points, the points can be indexed. Noting that differences in distances between neighbouring points in calibration scene images differ slightly, one of the local searching methods can be employed (e.g. [2]). Methods of this type search for a calibration point to be indexed, using a window of a certain size. The position of the window is determined by a vector representing the distance between two previously indexed points in the same row or column. However, experiments show that this approach has its disadvantages, as described below. * E-mail: A.Nowakowski@ire.pw.edu.pl Firstly, there is a danger of omitting some points during indexing in case of local lack of calibration points detection in a neighbourhood (e.g. caused by the presence of non-homogeneous light in the calibration scene). A particularly unfavourable situation is when the local lack of detection effects in the appearance of separated regions of detected calibration points. It is worth saying that such situations are likely to happen for calibration points situated near image borders. Such points are very important for the analysis of optical nonlinearities, and a lack of them can significantly influence the accuracy of distortion modelling. Secondly, such methods may give wrong results in the case of optical distortion with strong nonlinearities when getting information about the neighbouring index is not an easy task. Beside this, the methods are very sensitive to a single false localisation of a calibration point. Such a single false localisation can even result in false indexing of a big set of calibration points. To avoid the above-mentioned problems, we propose using a black-and-white chessboard which contains the coded index of a calibration point in the form of colour squares situated in the nearest neighbourhood of each point. The index of a certain calibration point is determined by colours of four nearest neighbouring squares (Fig.1). An order of squares in such foursome is important. Because the size of a colour square is determined only by the possibility of correct colour detection, the size of a colour square can be smaller than the size of a black or white square. The larger size of a black or white square is determined by the requirements of the exact localisation step which follows the indexing of calibration points [3]. In this step, edge information is extracted from a blackand-white chessboard. This edge information needs larger Artur Nowakowski, Wladyslaw Skarbek Institute of Radioelectronics, Warsaw University of Technology, Nowowiejska 15/19, 00-665 Warszawa, A.Nowakowski@ire.pw.edu.pl Received February 10, 2009; accepted March 27, 2009; published March 31, 2009 http://www.photonics.pl/PLP

Polarization transformations in the vicinity of orthogonality point in the inhomogeneous magnetized plasma

New method in plasma polarimetry is considered, which is based on the phenomenon of normal wave transformation in the vicinity of orthogonality point, where the probing beam is perpendicular to the static magnetic. Phenomenon of normal wave transformation near orthogonality point is analyzed in the frame of quasi-isotropic approximation of geometrical optics, both theoretically and numerically. It is shown that measuring polarization state of the beam, passing through the orthogonality point allows estimating local electron density in the vicinity of that point. This property seems to be great advantage of the method under consideration as compared with traditional plasma polarimetry, which deals with line averaged measurements.

QUASI-ISOTROPIC APPROXIMATION OF GEOMETRICAL OPTICS METHOD WITH APPLICATIONS TO DENSE PLASMA POLARIMETRY

Basic equations of quasi-isotropic approximation (QIA) of geometrical optics method are presented, which describe electro- magnetic waves propagation in weakly inhomogeneous and weakly anisotropic media. It is shown that in submillimiter range of elec- tromagnetic spectrum plasma in all modern thermonuclear reactors, both acting and under construction, manifest properties of weakly in- homogeneous and weakly anisotropic medium, even for extreme elec- tron density Ne » 10 14 cm i3 and magnetic fleld B0 » 5T accepted for project ITER. In these conditions QIA serves as natural theoret- ical basis for plasma polarimetry in tokamaks and stallarators. It is pointed out that Stokes vector formalism (SVF), widely used in po- larimetry, can be derived from QIA in a generalized form, admitting the rays to be curvilinear and torsiened. Other important result of QIA is development of angular variables technique (AVT), which deals directly with angular parameters of polarization ellipse and operates with the system of two difierential equations against three equations in form of SVF.