Ray transfer matrix analysis

Summary

Ray transfer matrix analysis (also known as ABCD matrix analysis) is a mathematical form for performing ray tracing calculations in sufficiently simple problems which can be solved considering only paraxial rays. Each optical element (surface, interface, mirror, or beam travel) is described by a 2×2 ray transfer matrix which operates on a vector describing an incoming light ray to calculate the outgoing ray. Multiplication of the successive matrices thus yields a concise ray transfer matrix describing the entire optical system. The same mathematics is also used in accelerator physics to track particles through the magnet installations of a particle accelerator, see electron optics. This technique, as described below, is derived using the paraxial approximation, which requires that all ray directions (directions normal to the wavefronts) are at small angles θ relative to the optical axis of the system, such that the approximation \sin \theta \approx \theta remains valid.

Official source

https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis

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In this thesis, we present development of the first magnetic model of the main Proton Synchrotron magnets. The operation of the machine and conducting research on increasing its performance require a significant amount of the magnetic field data for the optics modelling. For magnets of the Proton Synchrotron, such amount of data cannot be provided by the magnetic measurements, therefore, we have to rely on magnetic models. We have developed the model based on the numerical analysis of the magnetic field in the magnet and performed the validation with the use of the available magnetic measurements data. The results of the 2D quasi-static analysis were used to decompose the multipolar contributions of different magnet circuits on different levels fo the iron core magnetization within the whole range of the PS operation energies. Established formulas of every circuit contributions take into account the global magnetization of the yoke and its saturation as well as a local magnetization changes caused by contributions of other circuits. The 3D numerical analysis was setup to study the influence of the magnet stray field and gaps between the magnet blocks on the integrated multipolar field. Its results were used to derive appropriate model corrections of the integrated multipolar field with respect to the field in the reference cross-section and to approximate the effective magnetic lengths of the auxiliary circuits of the PS magnet. In order to perform a non-linear chromaticity analysis and to take the full advantage of the developed model, we have modified the existing optics model of the PS lattice. We have integrated magnetic model into the optics calculation with two simulation adjustment methods, which optimize the main coil current in similar way as it is done for the real magnets and adjust four free parameters to match the measured parameters of the initial working point. We have successfully validated the combined magnetic and optics models for various beam momentum levels and magnet powering conditions. With the use of the new magnetic model, for the first time in the history of the PS we are able to predict a higher-order chromaticity function for any energy. Base on the optics modelling results, we have derived working point transfer matrices for the non-linear chromaticity, which allow to predict a set of powering current offsets required to apply foreseen working point adjustment. We were also able to model an offset of the measured transfer matrices due to a change of the radial position chosen for the beam production.

EPFL2013

Discrete Holomorphicity and Ising Model Operator Formalism

Clément Hongler, Ali Zahabi

We explore the connection between the transfer matrix formalism and discrete complex analysis approach to the two dimensional Ising model. We construct a discrete analytic continuation matrix, analyze its spectrum and establish a direct connection with the critical Ising transfer matrix. We show that the lattice fermion operators of the transfer matrix formalism satisfy, as operators, discrete holomorphicity, and we show that their correlation functions are Ising parafermionic observables. We extend these correspondences also to outside the critical point. We show that critical Ising correlations can be computed with operators on discrete Cauchy data spaces, which encode the geometry and operator insertions in a manner analogous to the quantum states in the transfer matrix formalism.

2014

Experimental method for the evaluation of the dynamic transfer matrix using pressure transducers

François Avellan, Andres Müller, Keita Yamamoto

This paper introduces an experimental method for the evaluation of dynamic transfer matrices using only pressure transducers. The discharge fluctuations are evaluated from the fluctuation of the pressure difference at different streamwise locations. The transfer matrices of the resistance, the inertance and the compliance elements are determined by using simple flow configurations. This method is then validated by comparing the transfer matrix components to theoretical values. The results show that the direct measurement of the transfer matrices produces good results below the first structural eigenfrequency of the system. Furthermore, a deviation from the mass continuity in the amplitude ratio of the fluctuating upstream and downstream discharges is investigated. This behaviour can be explained with a simple model taking into account the compliance in the system.

Taylor & Francis Ltd2015

EPFL2013