Determination of Mueller Matrix Elements in the Presence of Imperfections in Optical Components

University of New Orleans ScholarWorks@UNO University of New Orleans Theses and Dissertations Dissertations and Theses 5-15-2009 Determination of Mueller matrix elements in the presence of imperfections in optical components Shibalik Chakraborty University of New Orleans Follow this and additional works at: https://scholarworks.uno.edu/td Recommended Citation Chakraborty, Shibalik, "Determination of Mueller matrix elements in the presence of imperfections in optical components" (2009). University of New Orleans Theses and Dissertations. 969. https://scholarworks.uno.edu/td/969 This Thesis is protected by copyright and/or related rights. It has been brought to you by ScholarWorks@UNO with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. 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Determination of Mueller matrix elements in the presence of imperfections in optical components A Thesis Submitted to the Graduate Faculty of the University of New Orleans In partial fulfillment of the requirements for the degree Master of Science In Electrical Engineering by Shibalik Chakraborty B.Tech, West Bengal University of Technology, Calcutta, India, 2007 May, 2009 ACKNOWLEDGEMENT First of all I would like to thank my graduate supervisor, Professor Rasheed Azzam, who helped and encouraged me to start my graduate studies at the University of New Orleans in the Electrical Engineering Department. I also thank him for his guidance, patient support, and for proposing to me a very interesting research subject. I would like to thank Dr. Vesselin Jilkov and Dr. Huimin Chen for serving as members of my thesis committee. Finally, I wish to dedicate this work to my family and friends who encouraged and supported me during all this time. ii TABLE OF CONTENTS List of Tables……………………………………………………………………………………………………………………………...v List of Figures…………………………………………………………………………………………………………………………….vi Abstract…………………………………………………………………………………………………………………………...........vii Chapter 1. Introduction……………………………………………………………………………………………………………..1 Chapter 2. Optical System and Modification………………………………………………………………………………9 2.1. Introduction………………………………………………………………………………………………………….9 2.2. Fourier analysis with ideal conditions………………………………………………………………….13 2.3. Mueller matrix ……….…………………………………………………………………………………………..13 Chapter 3. Offsets and Imperfections……………………….……………………………………………………………..14 3.1. Introduction………………………………………………………………………………………………………..14 3.2. Fourier analysis and determination of offsets and imperfection parameters from Fourier coefficients…………………………………………………………………………………….15 Chapter 4. Determination of Mueller matrix elements…………………………………………………………….26 4.1. Introduction…………………………………………………………………………………………………………26 4.2. Fourier coefficients in terms of imperfection parameters……………………………………27 iii 4.3. Determination of Mueller-matrix elements……….…………………………………………………29 5. Summary……………………………………………………………………………………………………………………………..32 6. Future Work……….………………………………………………………………………………………………………………..33 References………………….……………………………………………………………………………………………………………34 Vita………………………………….………………………………………………………………………………………………………35 iv LIST OF TABLES Table 2.1: Fourier coefficients of the detected signal for the modified optical arrangement in ideal conditions Table 3.1: Fourier coefficients for the straight-through configuration with an offset azimuth for the fixed analyzer Table 3.2: Fourier coefficients for the straight-through configuration when the phase angles (α and β) of the rotating components are introduced Table 3.3: Fourier coefficients for the configuration without rotating analyzer Table 3.4: Fourier coefficients in terms of the azimuth of fixed analyzer (A’) and its imperfection parameters (ρ and δ) Table 4.1: Interrelationship between the Mueller matrix elements mij (i,j=0,..,3) and the Fourier coefficients with imperfect optical components v LIST OF FIGURES Fig. 1.1: Four parameters that define the ellipse of polarization in its plane Fig. 1.2: Polarizer orientation and azimuth Fig. 1.3: System of n optical devices Fig. 2.1: The original Polarizer-Sample-Analyzer (PSA) arrangement Fig. 2.2: The modified arrangement by adding the fixed polarizer and analyzer at the ends Fig. 3.1: Straight-through configuration with the polarizer and analyzer in line Fig. 3.2: Straight-through configuration without the rotating analyzer Fig. 3.3: Straight-through configuration with all the parameters Fig. 4.1: Measurement configuration with the sample reintroduced between the rotating components for determination of Mueller matrix elements vi ABSTRACT The Polarizer-Sample-Analyzer (PSA) arrangement with the optical components P and A rotating with a fixed speed ratio (3:1) was originally introduced to determine nine Mueller matrix elements from Fourier analysis of the output signal of a photodetector. The arrangement is modified to the P’PSAA’ arrangement where P’ and A’ represent fixed polarizers that are added at both ends with the speed ratio of the rotating components (P and A) remaining the same as before. After determination of the partial Mueller matrix in the ideal case, azimuthal offsets and imperfection parameters are introduced in the straight-through configuration and the imperfection parameters are determined from the Fourier coefficients. Finally, the sample is reintroduced and the full Mueller matrix elements are calculated to show the deviation from the ideal case and their dependency on the offsets and imperfection parameters. Keywords: Polarization, Ellipsometry, Jones calculus, Mueller calculus, polarizer, Fourier series vii CHAPTER 1 INTRODUCTION Polarization is a property that is common to all vector waves. It is defined by the behavior with time of one of the field vectors appropriate to that wave, observed at a fixed point in space. Light waves are electromagnetic in nature and are described by four field vectors: electric field strength E , electric displacement density D , magnetic field strength H , and magnetic flux density B . The electric field strength E is used to define the state of polarization. This is because the force exerted on electrons by the electric field is much greater than that exerted by the magnetic field of the wave. The polarization of the three field vectors can be determined using the Maxwell’s field equations, once the polarization of E has been determined [1]. Ellipsometry is an optical technique for the characterization of interfaces and films and is based on the polarization transformation that occurs as a beam of polarized light is reflected from or transmitted through the interfaces or films [1]. We now present the four parameters that define an ellipse of polarization and we will describe the optical elements in the arrangement. 8 Fig 1.1. Four parameters defining the ellipse of polarization in its plane (1) the azimuth θ of the major axis from a fixed direction X, (2) the ellipticity e=±b/a=±tan ε, (3) the amplitude A=(a2 + b2)1/2 and (4) the absolute phase δ. (1) The azimuth θ is the angle between the major axis the ellipse and the positive direction of the X axis and defines the orientation of the ellipse. π π - ≤ θ < (1.1) 2 2 (2) The ellipticity e is the ratio of the semi-minor axis of the ellipse b to the semi-major axis a, b e= (1.2) a (3) The amplitude of the elliptic vibration can be defined in terms of the lengths a and b as 9 A= a2 + b2 1/2 (1.3) (4) The absolute phase vector δ determines the angle between the initial position of the electric vector at t=0 and the major axis of the ellipse. All possible values of δ are limited by -π ≤ δ < π (1.4) The electric vectors of light linearly polarized in the X and Y directions are given by, E x= Axcos(ωt+φx)i (1.5) E y= Aycos(ωt+φy)j (1.6) For elliptically polarized light the electric vector is then given by E = E x+E y= Axcos(ωt+φx)i + Aycos(ωt+φy)j (1.7) Or in matrix (column) form as, E A ei(ωt+φx ) E = x = x = ejωt퐉 (1.8) j(ωt+φy ) E y Aye Taking out the time dependence from Eq. (1.8), we have the Jones vector A eiφx 퐉 x = iφy (1.9) Ay e When light interacts with an optical system, the Jones vector transforms according to 10 E′ t t E x = 11 12 x (1.10) E′y t21 t22 Ey or 퐉’=퐓퐉 (1.11) Here 퐓 is defined as the Jones matrix of the optical system, the matrix elements t ij are determined from the known properties of the optical element encountered by the light and the equation is simply the transformation matrix that converts 퐉 into 퐉’. Similar to the Jones calculus, the Mueller calculus also represents the light by a vector; premultiplication of this vector by a matrix that represents an optical element yields a resultant vector that describes the beam after its interaction with the optical device. The vector representing the light is called a Stokes vector and is a 4 X 1 real vector instead of 2 X 1 complex Jones vector. The 4 elements of the Stokes vector represent intensities of the light. Therefore these elements are real. A common representation of the Stokes vector is given by 2 2 Ax + Ay S0 2 2 S1 Ax − Ay 퐒= = (1.12) S2 2A A cosδ x y S3 2Ax Aysinδ where δ =δy- δx and –π ≤ δ ≤ π. A deterministic complex Jones matrix

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