Applications and Metrology at Nanometer-Scale 2

Pierre Richard Dahoo is Professor and Holder of the Chair of Materials Simulation and Engineering at the University of Versailles Saint-Quentin in France. He is Director of Institut des Sciences et Techniques des Yvelines and a specialist in modeling and spectroscopy at the LATMOS laboratory of the CNRS. Philippe Pougnet is a former expert in reliability and the technologyproduct-process of embedded mechatronic systems. He graduated from Université Grenoble Alpes and Grenoble INP in France. Abdelkhalak El Hami is Professor at the Institut National des Sciences Appliquées (INSA-Rouen Normandie) in France and is in charge of the Normandy Conservatoire National des Arts et Metiers (CNAM) Chair of Mechanics, as well as several European pedagogical projects.

Preface ix

Introduction xiii

Chapter 1. Measurement Systems Using Polarized Light 1

1.1. Introduction 1

1.2. Matrix optics 2

1.3. Photon emission and detection 12

1.4. Application exercises on interferometry 16

1.4.1. Propagation of electromagnetic waves in a Fabry–Pérot cavity 18

1.4.2. Propagation of electromagnetic waves in a material 19

1.4.3. Interferometry and optical lambda meter 21

1.4.4. The homodyne interferometer and refractometer 34

1.4.5. The heterodyne interferometer 40

1.4.6. Application exercises on ellipsometry 51

1.5. Appendices 56

1.5.1. Conventions used for Jones vectors and Jones ABCD matrices 56

1.5.2. 2×2 transfer dies 59

1.5.3. 2×2 matrix multiplication 59

1.5.4. Trigonometric forms 60

1.5.5. Solution by MATLAB (exercises 1.4.3, 1.4.4 and 1.4.5) 61

1.6. Conclusion 66

Chapter 2. Quantum-scale Interaction 67

2.1. Introduction 67

2.2. The spin through the Dirac equation 69

2.2.1. Theoretical background 69

2.2.2. Application: the Dirac equation and Pauli matrices 74

2.3. The density matrix for a two-level laser system 105

2.3.1. Definition of the density matrix 106

2.3.2. Density matrix properties 110

2.3.3. Equation of motion of the density matrix 113

2.3.4. Application to a two-level system 116

2.4. Ising’s phenomenological model for cooperative effects 123

2.4.1. The Ising 1D model 124

Chapter 3. Quantum Optics and Quantum Computers 135

3.1. Introduction 135

3.2. Polarized light in quantum mechanics 136

3.3. Introduction to quantum computers 140

3.4. Preparing a qubit 158

3.4.1. Application of the Bloch sphere 158

3.5. Application: interaction of a qubit with a classical field 172

3.5.1. Answer to question 1 173

3.5.2. Answer to question 2 176

3.6. Applying Ramsey fringes to evaluate the duration of phase coherence 181

3.6.1. Answer to question 1 181

3.6.2. Answer to question 2 183

Chapter 4. Reliability-based Design Optimization of Structures 185

4.1. Introduction 185

4.2. Deterministic optimization 186

4.3. Reliability analysis 187

4.3.1. Optimal conditions 189

4.4. Reliability-based design optimization 191

4.4.1. The objective function 192

4.4.2. Taking into account the total cost 192

4.4.3. Design variables 193

4.4.4. Response of a system by RBDO 193

4.4.5. Limit states 194

4.4.6. Solving methods 194

4.5. Applications 194

4.5.1. Application on a bending beam 194

4.5.2. Application on a circular plate with different thicknesses 196

4.5.3. Application: hook A 201

4.5.4. Application: optimization of the materials of an electronic board 211

4.6. Reliability-based design optimization in nanotechnology 222

4.6.1. Thin-film SWCNT structures 222

4.6.2. Digital model of thin-film SWCNT structures 224

4.6.3. Numerical results 225

4.7. Conclusion 231

Appendix 233

References 237

Index 245