ELLIPSOMETRY AND POLARIZED LIGHT

Transcription

1 ELLIPSOMETRY AND POLARIZED LIGHT R.M.A.AZZAM Department of Electrical Engineering University of New Orleans Lakefront, New Orleans, Louisiana, USA and N.M. В ASH AR A Electrical Materials Laboratory, Engineering Research Center, College of Engineering, University of Nebraska, Lincoln, USA Nil jjffl m ELSEVIER AMSTERDAM LAUSANNE NEW YORK OXFORD SHANNON TOKYO

2 Contents PREFACE то THE PAPERBACK EDITION PREFACE ACKNOWLEDGEMENTS CONTENTS v vi viii xi CHAPTER l. THE POLARIZATION OF LIGHT WAVES The concept of polarization Polychromatic and monochromatic waves of light Polarization of monochromatic waves of arbitrary spatial structure Elliptical polarization Linear and circular polarizations The phasor (complex) representation of time-harmonic fields Uniform ТЕ plane waves of light The Jones vector of a uniform ТЕ plane wave Wave intensity Jones vectors of some states of polarization Orthogonal and orthonormal pairs of Jones vectors Transformation of the Jones vector under the effect of a coordinate rotation Basis Jones vectors and basis states of polarization Cartesian basis vectors Circular basis vectors Arbitrary basis vectors The Cartesian and circular Jones vectors of a given elliptical polarization state Representation of the polarization states of light by complex numbers The Cartesian complex-plane representation of polarized light Orthogonal states Equi-azimuth and equi-ellipticity contours 33

3 Xll CONTENTS The circular complex-plane representation of polarized light The generalized complex-plane representation of polarized light The Poincare-sphere representation of polarized light The polarization of quasi-monochromatic waves The Jones vector of a quasi-monochromatic wave The Stokes parameters and the Stokes vector The coherency-matrix representation 60 CHAPTER 2. PROPAGATION OF POLARIZED LIGHT THROUGH POLARIZING OPTICAL SYSTEMS Polarizing optical systems The Jones-matrix formulation The significance of the elements Гц of the Jones matrix Cascade of optical systems Jones matrices of basic optical devices Transmission-type devices Reflection-type devices Transformation of the Cartesian Jones matrix under the effect of a coordinate rotation Circular and generalized Jones matrices Separability of information on the ellipse of polarization in the Jonesmatrix formulation-the polarization transfer function Transformation of the ellipse of polarization by an optical system in the complex-plane representation Transformation of polarization by an optical system in the Poincaresphere representation Separability of information on amplitude and phase in the Jones-matrix formulation-the amplitude and phase transfer functions The eigenvalue problem-general classification of optical devices Elliptic retarders Elliptic partial polarizers Elliptic ideal polarizers Otherdevices Polarization-dependent intensity transmittance of optical systems Loci of polarization states of equal attenuation or amplification Simplified expressions for the intensity transmittance Generalization of the Law of Malus Special cases of optical systems Ill 2.8. Resolution of an arbitrary polarization state into two given orthogonal states in the complex-plane representation Nearness of two polarization states in the complex-plane representation 113

4 CONTENTS XÜi Resolution of an arbitrary polarization state into two given orthogonal states in the Poincare-sphere representation Polarization-dependent phase shift introduced by an optical system Propagation of polarized light in anisotropic media Evolution of the ellipse of polarization The case of homogeneous anisotropic media The case of inhomogeneous anisotropic media Evolution of the complex amplitude Propagation of partially polarized quasi-monochromatic light through non-depolarizing optical systems-the coherency-matrix formulation Intensity transmittance of a linear non-depolarizing optical system for partially polarized incident light Propagation of partially polarized quasi-monochromatic light through depolarizing optical systems-the Mueller-matrix formulation CHAPTER 3. THEORY AND ANALYSIS OF MEASUREMENTS IN ELLIPSOMETER SYSTEMS Introduction Ellipsometric measurement of the Jones matrix of an optical system-the basis of generalized ellipsometry Ellipsometry on optical systems with known eigenpolarizations Ellipsometric measurement of the ratio of eigenvalues of an optical system with orthogonal linear eigenpolarizations Null ellipsometry The fixed-compensator-azimuth nulling scheme with C = ±W, * SC = -\TT and Tc = Alternate ellipsometer arrangement with the compensator placed after the optical system The ratio of eigenvalues ps in reflection and transmission ellipsometry Unified treatment of the PCSA and PSCA ellipsometer arrangements-example of the application of the bilinear polarization transfer function Ellipsometric measurement of the ratio of eigenvalues of an optical system with orthogonal circular eigenpolarizations Ellipsometric measurement of the ratio of eigenvalues of an optical system with elliptic eigenpolarizations Effect of azimuth-angle errors and component imperfections on the ellipsometric measurement of ps The imperfection plate of a non-ideal optical component Transposition of two optical components The PCWSW'A ellipsometer arrangement 192

5 XIV CONTENTS Azimuth-angle errors Component imperfections Errors in the ellipsometric angles фand Л Two-zone averages Four-zone averages Choice of compensator azimuth and position EUipsometry with imperfect components including incoherent effects Application to the PCWSW'A ellipsometer arrangement Polarizer and analyzer imperfections Compensator imperfection Entrance-window, system, and exit-window imperfections Zone averaging Component depolarization Discussion Generalized ellipsometry The polarizer-compensator combination as a controlled polarizationfilter Nulling schemes and conditions of compensation The fixed-analyzer nulling scheme The fixed-compensator nulling scheme The fixed-polarizer nulling scheme Multiple-null measurements Alternate PSCA ellipsometer arrangement Other ellipsometer arrangements Null ellipsometers that do not employ compensators An ellipsometer based on the detection of the azimuth of the polarization ellipse only, with no measurement of ellipticity Self-compensated ellipsometers based on the angle-ofincidence tunability of the reflection phase difference Л Photometric ellipsometers Static photometric ellipsometers Dynamic photometric ellipsometers Interferometric Ellipsometers (IE) Modulated ellipsometry PSA arrangement PCSA arrangement 267 CHAPTER 4. REFLECTION AND TRANSMISSION OF POLARIZED LIGHT BY STRATIFIED PLANAR STRUCTURES Introduction 269

6 CONTENTS XV 4.2. Reflection and refraction at the planar interface between two isotropic media Reflection and transmission by an ambient-film-substrate system The equations of reflection and transmission ellipsometry for ambientfilm-substrate systems Reflection ellipsometry Constant-Angle-of-Incidence Contours (CAIC) of the ellipsometric function рц>,а) Constant-Thickness Contours (CTC) of the ellipsometric function р(ф, d) Cartesian curves Transmission ellipsometry Linear approximation of the equations of reflection and transmission ellipsometry for ambient-film-substrate systems Reflection Transmission ф and Л sensitivity factors Validity of the linear approximations Numerical inversion of the exact equation of reflection ellipsometry Cases that allow analytical inversion General formulation of the numerical-inversion problem and choice of error function Multiple-Angle-of-Incidence (MAI) ellipsometry Independence (or interdependence) of the MAI system of equations: The parameter correlation test The Hessian matrix and rate of convergence The air-si0 2 -Si system - An example Reflection and transmission by isotropic stratified planar structures Reflection and transmission by anisotropic stratified planar structures Reflection and transmission by a finite anisotropic layer between semi-infinite isotropic ambient and substrate media Reflection and transmission by a semi-infinite anisotropic substrate in an isotropic ambient Explicit expressions for the reflection matrix in simple special cases Isotropic ambient and uniaxially anisotropic substrate Isotropic ambient and biaxially anisotropic substrate Uniaxially anisotropic film on an isotropic substrate in an isotropic ambient Biaxially anisotropic film on an isotropic substrate in an isotropic ambient Uniaxial film on a uniaxial substrate in an isotropic ambient Ellipsometry on surfaces covered with discontinuous films and on surfaces with rough boundaries 359

7 XVJ CONTENTS CHAPTER 5. INSTRUMENTATION AND TECHNIQUES OF ELLIPSOMETRY Introduction The basic instrument-the null ellipsometer The optical devices Linear polarizers Retarders Other polarizing optical devices Optical components of an ellipsometer that perform no polarizing function Light sources and detectors Operation of the null ellipsometer Alignment of the telescopes Calibration of the ellipsometer divided circles Nulling schemes Sources of error and their correction Azimuth-angle errors, component and cell-window imperfections Measurement of ellipsometer imperfection parameters Beam deviation errors Effect of auto-collimated parasitic beams from multiple reflections Errors that arise from certain assumptions concerning the ellipsometer light beam Bandwidth Source polarization and component depolarization Collimation and non-uniform polarization Polarization-dependent photodetector sensitivity Residual mechanical imperfections Othererrors Model errors Precision of null ellipsometers Automation of null ellipsometers Motor-driven self-nulled ellipsometers Electro-optic self-nulled ellipsometers Automatic photometric ellipsometers The rotating-analyzer ellipsometer The polarization-modulated ellipsometer 415 CHAPTER 6. APPLICATIONS OF ELLIPSOMETRY Introduction Optical properties of materials and spectroscopic ellipsometry Optical properties of materials in bulk phase Optical properties of materials in thin-film phase 428

8 CONTENTS XVÜ 6.3. Physical and chemical adsorption Oxidation of semiconductor and metal surf aces Electrochemistry Applications of ellipsometry in biology and medicine Interaction of blood with foreign surfaces-blood coagulation Antigen-antibody immunological reactions in thin films The immunoelectroadsorption test Measurement of cell-surface (coat) materials Other applications of ellipsometry 473 APPENDIX 487 REFERENCES 493 LISTINGS OF THE CONTENTS OF THE INTERNATIONAL ELLIPSOMETRY CONFERENCES 501 AUTHOR INDEX 517 SUBJECT INDEX 522