A QED framework for nonlinear and singular optics A thesis submitted by: Matt M. Coles as part of the requirements for the degree of PhD in the School of Chemistry University of East Anglia Norwich NR4 7TJ © This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that no quotation from the thesis, nor any information derived therefrom, may be published without the author's prior, written consent. The research in this thesis has not been submitted previously for a degree at this or any other university. Except where explicitly mentioned, the work is of my own. M. M. Coles January 2014 Abstract The theory of quantum electrodynamics is employed in the description of linear and nonlinear optical effects. We study the effects of using a two energy level approximation in simplifying expressions obtained from perturbation theory, equivalent to truncating the completeness relation. However, applying a two-level model with a lack of regard for its domain of validity may deliver misleading results. A new theorem on the expectation values of analytical operator functions imposes additional constraints on any atom or molecule modelled as a two-level system. We introduce measures designed to indicate occasions when the two-level approximation may be valid. Analysis of the optical angular momentum operator delivers a division into spin and orbital parts satisfying electric-magnetic democracy, and determine a new compartmentalisation of the optical angular momentum. An analysis is performed on the recently rediscovered optical chirality, and its corresponding flux, delivering results proportional to the helicity and spin angular momentum in monochromatic beams. A new polarisation basis is introduced to determine the maximum values that an infinite family of optical helicity- and spin- type measures may take, and disproves recent claims of ‘superchiral light’. A theoretical description of recent experiments relate helicity- and spin- type measures to the circular differential response of molecules, and show that nodal enhancements to circular dichroism relate only to photon number-phase uncertainty relation and do not signify ‘superchiral’ regions. The six-wave mixing of optical vortex input, in nonlinear media, demonstrates the quantum entanglement of pairs of optical vortex modes. The probability for each possible output pair displays a combinatorial weighting, associated with Pascal’s triangle. A quantum electrodynamic analysis of the effect of a second body on absorption can be extended by integrating over all possible positions of the mediator molecules, modelling a continuous medium. This provides links with both the molecular and bulk properties of materials. To my family and friends. Contents Abstract iii Publications ix Preface xiii Acknowledgments xvi Chapter 1: Introduction 1 1. Background 2 2. The Macroscopic Electromagnetic Field 4 3. The Microscopic Electromagnetic Field 6 4. Gauge Transformations 8 5. Gauge Fixing 10 6. Lagrangian Formulation 14 7. The Minimal Coupling Hamiltonian 17 8. The Multipolar Hamiltonian 21 9. Quantisation 28 10. Perturbation Theory 32 11. Appendix A 40 12. Bibliography 41 Chapter 2: The Two-Level Approximation 46 1. Background 47 2. Perturbation Theory and The Two-Level Approximation 49 3. The Two-Level Expectation Value Theorem 51 4. Extensions and Implications of the Theorem 54 5. The Optical Susceptibility Tensors 58 6. ‘Push-Pull’ Chromophores 62 7. Two-Level Model for Elastic Scattering 65 8. Two-Level Model for Second Harmonic Generation 67 9. Other Hyperpolarisability Components 73 10. Counting Terms in Optical Susceptibility Tensors 75 11. Conclusion 78 12. Bibliography 81 Chapter 3: Measures of Optical Angular Momentum 87 1. Background 88 2. Symmetry 90 3. Optical Angular Momentum 91 4. The Spin Part of Optical Angular Momentum 95 5. Poincaré Sphere Representation of Polarisation 100 6. Electromagnetic Helicity 102 7. Light with Orbital Angular Momentum 103 8. Optical Chirality/The Lipkin Zilch 113 9. Family of Helicity-type and Spin-type Measures 118 10. Conclusion 122 11. Appendix B 124 12. Bibliography 126 Chapter 4: The Interaction of Twisted Light with Matter 130 1. Background 131 2. Bilinear Measures 133 3. Connecting Molecular and Optical Chirality 137 4. Differential Absorption from a Single Beam 140 5. Mirrors and Standing Waves 146 6. Circular Dichroism in Counterpropagating Beams 147 7. Analysis of Recent Experiments 159 8. Six-Wave Mixing of Optical Vortices 163 9. Conclusion 168 10. Appendix C 170 11. Bibliography 171 Chapter 5: Medium Modified Absorption 178 1. Background 179 2. Medium Modified Absorption 180 3. Free-Space Absorption 182 4. Static Correction Term 183 5. Dynamic Correction Term 184 6. Conclusion 190 7. Bibliography 191 Chapter 6: Future Work 193 1. Introduction 194 2. The Two Level Approximation 194 3. Measures of Helicity 195 4. The Interaction of Twisted light with Matter 196 5. Optical Vortex Generation from Nanoantenna Arrays 196 6. Summary 197 7. Bibliography 201 202 Pages Publications “We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work.” – Richard P. Feynman† †Feynman, Richard P., Nobel Lecture (1965). The following are published papers resulting from research undertaken for this thesis; 1. David L. Andrews, Luciana C. Dávila Romero, Jamie M. Leeder and Matt M. Coles, Optomechanical control of molecular motors, Proceedings of SPIE 7762, Optical Trapping and Optical Micromanipulation VII, 776202 (2010). 2. David L. Andrews, David S. Bradshaw and Matt M. Coles, Perturbation theory and the two-level approximation: A corollary and critique, Chemical Physics Letters 503, pp. 153-156 (2011). 3. David L. Andrews, David S. Bradshaw and Matt M. Coles, Limitations and improvements upon the two-level approximation for molecular nonlinear optics, Proceedings of SPIE 7917, Nonlinear Frequency Generation and Conversion: Materials, Devices, and Applications X, 79171K (2011). 4. Matt M. Coles, Jamie N. Peck, Vasily S. Oganesyan and David L. Andrews, Assessing limitations to the two-level approximation in nonlinear optics for organic chromophores by ab initio methods, Proceedings of SPIE 8113, Linear and Nonlinear Optics of Organic Materials XI, 81130K (2011); 5. Scott N. A. Smith, Matt M. Coles and David L. Andrews, Optical binding with anisotropic particles: resolving the forces and torques, Proceedings of SPIE 8097, Optical Trapping and Optical Micromanipulation VIII, 80971E (2011). 6. David L. Andrews and Matt M. Coles, Optical superchirality and electromagnetic angular momentum, Proceedings of SPIE 8274, Complex Light and Optical Forces VI, 827405 (2012).; 7. Matt M. Coles, Jamie N. Peck, Vasily S. Oganesyan and David L. Andrews, Failure of the two-level and sum over states methods in nonlinear optics, demonstrated by ab initio methods, Proceedings of SPIE 8434, Nonlinear Optics and Applications VI, 84340K (2012). 8. David L. Andrews and Matt M. Coles, Measures of chirality and angular momentum in the electromagnetic field, Optics Letters 37, 3009-3011 (2012). 9. Matt M. Coles and David L. Andrews, Chirality and angular momentum in optical radiation, Physical Review A 85, 063810 (2012). 10. Matt M. Coles and David L. Andrews, Directions in optical angular momentum, Proceedings of SPIE 8637, Complex Light and Optical Forces VII, 863707 (2013). 11. Matt M. Coles and David L. Andrews, Photonic measures of helicity: optical vortices and circularly polarized reflection, Optics Letters 38, 869-871 (2013). 12. Matt M. Coles, Mathew D. Williams and David L. Andrews, Second harmonic generation in isotropic media: six-wave mixing of optical vortices, Optics Express 21, 12783 (2013). 13. Matt M. Coles and David L. Andrews, Expanded horizons for generating and exploring optical angular momentum in vortex structures, Proceedings of SPIE 8813, Spintronics VI, 881333 (2013). 14. Mathew D. Williams, Matt M. Coles, Kamel Saadi, David S. Bradshaw and David L. Andrews, Optical vortex generation from molecular chromophore arrays, Physical Review Letters 111, 153603 (2013) 15. Matt Coles, Mathew Williams, Kamel Saadi, David Bradshaw and David Andrews, Chiral nanoemitter array: a launchpad for optical vortices, Laser & Photonics Reviews 7, 1088-1092 (2013). Preface Quantum electrodynamics emerges from the application of quantum field theory to problems of electromagnetic origin, and describes, with unparalleled accuracy, phenomena involving light and matter. In this thesis, quantum electrodynamics is applied to molecular systems, using the formulation developed by D. P. Craig, E. A. Power and T. Thirunamachandran. Moreover, quantum optical techniques are employed in the description of various conserved electromagnetic quantities of interest, notably optical angular momentum. Chapter One is intended as an introduction to (and derivation of) the methods used in this thesis to describe light, matter and their interaction, and loosely follows the structure of the first chapters of Molecular Quantum Electrodynamics by Craig and Thirunamachandran. It develops a fully quantised framework by which we may analyse problems in quantum and nonlinear optics. Formally, the QED Lagrangian is shown to lead to the