The Nature of X-Rays and Their Interactions with Matter

Preface
	References
Contents
About the Author
1 Introduction and Overview
	1.1 About the Present Book
		1.1.1 Motivations
		1.1.2 Objectives
	1.2 The Nature of Light: From Light Rays to QED
		1.2.1 Early Concepts of the Nature of Light
		1.2.2 The Quantum Era and Wave-Particle Duality
		1.2.3 New Insight in the Period 1930–1950
		1.2.4 Beyond Quantum Mechanics: Quantum Electrodynamics
		1.2.5 Development of the Laser and Quantum Optics Around 1960
	1.3 The X-Ray Revolution
		1.3.1 The Discovery and Early Utilization of X-Rays
		1.3.2 Development of Synchrotron Radiation Sources
		1.3.3 The Advent of X-Ray Free Electron Lasers
	1.4 The Scientific Power of X-Rays
		1.4.1 From Optical to X-Ray Response of Matter
		1.4.2 The Importance of X-Ray Resonances
		1.4.3 X-Ray Spectro-Microscopy
		1.4.4 Summary of Key Capabilities of Synchrotron Radiation
	1.5 Science with XFELs
		1.5.1 Snapshots of the Atomic Structure of Matter: Probe Before Destruction
		1.5.2 Creation and Characterization of Transient States of Matter
		1.5.3 Creation and Probing High Energy Density Matter
		1.5.4 Non-linear X-Ray Interactions with Matter
	References
Part I Production of X-Rays and Their Description
2 Production of X-Rays: From Virtual to Real Photons
	2.1 Introduction
	2.2 Relativistic Concepts and Electron Bunch Compression
	2.3 The Fields of a Moving Charge: Liénard–Wiechert Equations
	2.4 Fields of a Charge in Uniform Motion: Velocity Fields
		2.4.1 Spatial Dependence of Velocity Fields of a Single Electron
		2.4.2 Temporal Dependence of Velocity Fields of a Single Electron
		2.4.3 The Fields of a Relativistic Gaussian Electron Bunch
		2.4.4 Generation of Huge Field Pulses: THz Fields and Radiation
		2.4.5 Frequency Spectrum of Gaussian Electron Bunches
	2.5 Weizsäcker-Williams Method: Virtual Photon Spectrum
		2.5.1 Virtual Photon Spectrum of a Gaussian Electron Bunch
		2.5.2 Coherent Virtual Spectrum: THz Photons
		2.5.3 Incoherent Virtual Spectrum: X-Ray Photons
	2.6 Acceleration Fields: Creation of EM Radiation
		2.6.1 Distortion of Field Lines: Radiation
		2.6.2 The Angular Spectrum of a Single Accelerated Charge
		2.6.3 Frequency Spectrum of a Single-Electron Bending Magnet Source
		2.6.4 Frequency Spectrum of a Single-Electron Undulator Source
		2.6.5 The Angular Distribution of Undulator Radiation
	2.7 The Synchrotron Radiation Spectrum of Electron Bunches
		2.7.1 The Spectrum of an Electron Bunch and Its Temporal Coherence
		2.7.2 The Emitted Radiation Cone and Lateral Coherence
		2.7.3 Toward Ultimate Storage Rings
		2.7.4 Polarization of Synchrotron Radiation
	2.8 X-Ray Free Electron Lasers
		2.8.1 The Radiation Emitted by SASE XFELs
		2.8.2 Realistic Pulse Structure in SASE XFELs
		2.8.3 From SASE to Transform Limited Pulses
	References
3 From Electromagnetic Waves to Photons
	3.1 Introduction and Overview
	3.2 Classical Description of Light
		3.2.1 Maxwell's Equations and Their Symmetries
		3.2.2 The Electromagnetic Wave Equation
		3.2.3 Energy, Momentum, Intensity, Flux, and Fluence
		3.2.4 Field Strength of X-Ray Beams
		3.2.5 Field Normalization Volume
		3.2.6 The Description of Polarized EM Waves
		3.2.7 The Degree of X-Ray Polarization
	3.3 Quantum Theoretical Description of X-Rays
		3.3.1 The Quantized Electromagnetic Field: Birth of the Photon
		3.3.2 Zero-Point Energy and Virtual Photons
		3.3.3 The Renormalized Hamiltonian of the Radiation Field
		3.3.4 The Basic Quantum States of Light
		3.3.5 Single Mode Number States
	3.4 The Properties of Single Mode Quantum States
		3.4.1 Definition of Key Properties
		3.4.2 Properties of Single Mode Number States
		3.4.3 Properties of Single Mode Coherent States
		3.4.4 Properties of Single Mode Chaotic States
	3.5 Photon Modes, Density of States, and Coherence Volume
		3.5.1 Photon Modes
		3.5.2 Number of Modes per Unit Energy
		3.5.3 Number of Modes per Unit Volume
		3.5.4 Coherence Volume per Mode
		3.5.5 Summary
	3.6 Link of Classical and Quantum Properties of Radiation
		3.6.1 Multi-mode Number States
		3.6.2 Expectation Value of the Squared Electric Field
	References
4 Brightness and Coherence
	4.1 Introduction
	4.2 Brightness and Coherence
		4.2.1 Introduction to the Concept of Brightness
		4.2.2 Formal Definition of Brightness
		4.2.3 Average Brightness
		4.2.4 Peak Brightness
		4.2.5 Brightness Reduction Through Partial Coherence
		4.2.6 Link of Brightness and Coherence
		4.2.7 Photon Degeneracy Parameter
		4.2.8 Brightness of Storage Rings and XFELs
		4.2.9 Summary
	4.3 Historical Descriptions of Coherence
		4.3.1 Geometrical Coherence Concepts in Time and Space
	4.4 Fourier Optics Description of Coherent Sources
		4.4.1 Fourier Transformation of Temporal Coherence
		4.4.2 Time-Bandwidth Product Definitions
		4.4.3 Fourier Transformation of Spatial Coherence
	4.5 Statistical and Quantum Description of Partially Coherent Sources
		4.5.1 Degree of First Order Coherence
		4.5.2 Statistical Forms of Light Behavior
		4.5.3 The Concept of Partial Lateral Coherence
	4.6 The van Cittert–Zernike Theorem: Propagation  of Field Correlations
		4.6.1 van Cittert–Zernike Theorem for a Schell Model Source
		4.6.2 Coherent Limit of a Circular Flat-Top Source
		4.6.3 Chaotic Limit of a Circular Flat-Top Source
		4.6.4 Coherent and Chaotic Limits of a Circular Gaussian Source
		4.6.5 Summary
	4.7 Quantum Derivation of the van Cittert–Zernike Theorem
		4.7.1 The Case of Two Source Points
		4.7.2 Finite Area Sources
	4.8 Comparison of Classical and Quantum Diffraction
		4.8.1 Link of Waves and One-Photon Probability Amplitudes
		4.8.2 Link of Probability Amplitude and Operator Formalisms
	4.9 X-Ray Measurement of First Order Lateral Coherence
	References
5 The Complete Description of Light: Higher Order Coherence
	5.1 Introduction
	5.2 The Existence of Photons and Their Interference
		5.2.1 The Grangier–Roger–Aspect Experiment
		5.2.2 The Hanbury Brown–Twiss Experiment
		5.2.3 The Hong–Ou–Mandel Experiment
	5.3 Light and Detection Timescales
		5.3.1 ``Light'' Timescales
		5.3.2 Detector Timescales
	5.4 The Description of Second Order Coherence
		5.4.1 Introduction
		5.4.2 Second Order Spatial Coherence Functions
		5.4.3 Link of First and Second Order Coherence
	5.5 The Propagation of Second Order Coherence
		5.5.1 Propagation of Two Photons from Two Source to Two Detection Points
		5.5.2 Fundamental Two-Photon Patterns of Two-Point Sources
		5.5.3 The New Paradigm of Quantum Diffraction
		5.5.4 Finite Source Areas
		5.5.5 Second Order Pattern of a Circular Flat-Top Source
		5.5.6 Power Conservation in Two-Photon Diffraction
		5.5.7 X-Ray Measurements of Second Order Lateral Coherence
	5.6 Detection of First and Second Order Coherence Patterns
		5.6.1 The Coherent Quantum States of Light
		5.6.2 Implication for the First Order Pattern
		5.6.3 Implications for the Second Order Pattern
	5.7 Higher Order Coherence Propagation
		5.7.1 Higher Order Patterns of a Coherent Source
		5.7.2 Higher Order Patterns of a Chaotic Flat-Top Source
	5.8 Higher Order Brightness
	5.9 From Rays to Waves to Photons to Rays—Going  Full Circle
	References
Part II Semi-classical Theory of X-Ray Interactions with Matter
6 Semi-classical Response of Atoms to Electromagnetic Fields
	6.1 Introduction
	6.2 X-Ray Thomson Scattering by Electrons and Spins
		6.2.1 Response of an Electron and Its Spin to the Incident Field
		6.2.2 The Scattered Dipole Field
		6.2.3 Thomson Scattering Length and Cross Sections
		6.2.4 Thomson Scattering By An Atom
	6.3 X-Ray Resonant Scattering and Absorption
		6.3.1 Dispersion Corrections to the Atomic Scattering Factor
		6.3.2 Resonant Scattering by Atomic Core Electrons
		6.3.3 Distinction of Thomson, Rayleigh, and Resonant Scattering
	6.4 The Classical Link of Resonant Scattering and Absorption
		6.4.1 Relative Size of Thomson, Resonant Scattering, and Absorption
	6.5 The First Born Approximation
		6.5.1 The Breit–Wigner Atomic Cross Section
	6.6 The Kramers-Kronig Relations
	6.7 The Henke-Gullikson Formalism
	6.8 Atomic Shell Sum Rules
	References
7 Semi-classical Response of Solids  to Electromagnetic Fields
	7.1 Introduction
	7.2 Static Magnetic and Electric Fields Inside Materials
	7.3 Frequency Response of Materials: Microwaves  to X-Rays
		7.3.1 Permittivity: Electric Field Response
		7.3.2 Permeability: Magnetic Field Response
		7.3.3 From Permittivity and Permeability to Optical Parameters
		7.3.4 Penetration Depth of EM Waves: From Microwaves to X-Rays
		7.3.5 Optical Parameters: From the THz to the X-Ray Regime
	7.4 Dielectric Response Formulation of X-Ray Absorption
		7.4.1 Optical Parameter Formulation
		7.4.2 Absorption Coefficient Formulation
		7.4.3 Beer-Lambert Formulation
	7.5 From Dielectric to Atom-Based X-Ray Response
		7.5.1 Brief Review of X-Ray Scattering Factors
		7.5.2 From Single Atoms to Atomic Sheets
		7.5.3 Atomic Scattering Factors: Born Approximation Versus Huygens–Fresnel Principle
		7.5.4 Scattering Phase Shifts
		7.5.5 Link of X-Ray Scattering Factors and Optical Parameters
	7.6 The Optical Theorem
	7.7 Coherent Versus Incoherent X-Ray Scattering from Solids
		7.7.1 Incoherent Versus Coherent Field Superposition
		7.7.2 X-Ray Forward Scattering and Absorption
		7.7.3 The Total Transmitted Intensity
		7.7.4 Relative Size of the Absorbed and Scattered Intensities
	7.8 The Response of a Thick Sample: Dynamical Scattering Theory
		7.8.1 Snell's Law and Total X-Ray Reflection
		7.8.2 Darwin-Prins Dynamical Scattering Theory
		7.8.3 The Transmitted Field and Intensity
		7.8.4 The Reflected Field in the Soft X-Ray Region
	7.9 Polarization Dependent Absorption: Dichroism
		7.9.1 History of Polarization Dependent Effects
		7.9.2 Chiral Versus Magnetic Orientation
		7.9.3 Fundamental Forms of X-Ray Dichroism
	7.10 Natural Dichroism and Orientational Order
		7.10.1 Orientation Factors and Saupe Matrix
		7.10.2 Determination of Orientation Factors
	7.11 Magnetic Dichroism and Faraday Rotation
		7.11.1 Polarization Dependent Scattering Length and Optical Parameters
		7.11.2 Phenomenological Model
		7.11.3 Transmission of Circularly Polarized X-Rays: XMCD
		7.11.4 Transmission of Linearly Polarized X-Rays: XMLD
		7.11.5 Faraday Rotation
	References
8 Classical Diffraction and Diffractive Imaging
	8.1 Introduction and Chapter Overview
	8.2 Real Space X-Ray Imaging
		8.2.1 X-Ray Microscopes
		8.2.2 Polarization Dependent Microscopy
	8.3 Diffractive Imaging
		8.3.1 Introduction to Diffractive Imaging
		8.3.2 Historical Development of Diffraction Theories
		8.3.3 The Huygens–Fresnel Principle
		8.3.4 The Rayleigh-Sommerfeld Diffraction Formula
	8.4 Approximate Solutions of the Rayleigh-Sommerfeld Formula
		8.4.1 Paraxial Approximation
		8.4.2 Fresnel Diffraction
		8.4.3 Fraunhofer Diffraction
		8.4.4 Fourier Theorem
	8.5 Diffraction and Wave-Particle Duality
		8.5.1 Feynman's Probability Amplitudes
		8.5.2 De Broglie–Bohm Pilot Wave Theory
	8.6 Consequences of Fraunhofer Diffraction
		8.6.1 The Diffraction Limit
		8.6.2 The Arago-Fresnel-Poisson Bright Spot
		8.6.3 Babinet's Principle
	8.7 Formulation of Polarization Dependent Diffractive Imaging
		8.7.1 Charge Domains with Orientational Order
		8.7.2 Charge Domains of Different Chemical Composition
		8.7.3 Diffraction by Ferromagnetic Domains
		8.7.4 Separation of Charge and Magnetic Contrast
	8.8 Illustration of the Phase Problem in X-Ray Diffractive Imaging
	8.9 Fourier Transform Holography: FTH
		8.9.1 Illustration of FTH
		8.9.2 Improved Reference Beams
		8.9.3 Application of FTH: Magnetic Domains
	8.10 Non-holographic Solutions to the Phase Problem
		8.10.1 Brief History of X-Ray Crystallography
		8.10.2 MIR, SAD, and MAD Image Reconstruction
		8.10.3 Sampling of the Diffraction Pattern
		8.10.4 Ptychography
	8.11 Multiple-Wavelength Anomalous Diffraction—MAD
		8.11.1 MAD of Macromolecular Crystals
		8.11.2 Formulation of MAD in Protein Cystallography
		8.11.3 MAD Imaging of Non-crystalline Samples
		8.11.4 Implementation of MAD for Non-periodic Samples
		8.11.5 Phase Contrast Imaging: Combining FTH and MAD
	References
Part III Quantum Theory of Weak Interactions
9 Quantum Formulation of X-Ray Interactions with Matter
	9.1 Introduction and Overview
	9.2 The Photon-Matter Interaction Hamiltonian
		9.2.1 The Pauli Equation Including the EM Field
		9.2.2 Evaluation of the Spin Dependent Part of the Pauli Equation
		9.2.3 The Complete Interaction Hamiltonian
		9.2.4 Relative Size of the Interactions
	9.3 Perturbation Treatment of X-Ray Scattering and Absorption
		9.3.1 On the Use of Time-Dependent Perturbation Theory
	9.4 Kramers-Heisenberg-Dirac Perturbation Theory
		9.4.1 The Kramers-Heisenberg-Dirac Formula
	9.5 Overview of First Order Processes
		9.5.1 X-Ray Absorption
		9.5.2 Resonant X-Ray Absorption
		9.5.3 X-Ray Emission
		9.5.4 X-Ray Thomson Scattering
	9.6 Overview of Second Order Processes
		9.6.1 Spontaneous X-Ray Resonant Scattering
		9.6.2 Stimulated X-Ray Resonant Scattering
		9.6.3 Two-Photon Absorption and Photoemission
	References
10 Quantum Theory of X-Ray Absorption Spectroscopy
	10.1 Overview
	10.2 Quantum Formulation of X-Ray Absorption Spectroscopy (XAS)
		10.2.1 Photon Flux, Intensity, and Absorption Cross Section
	10.3 Non-resonant Absorption: Excitation into Continuum States
		10.3.1 Wavefunctions
		10.3.2 Continuum Cross Section
		10.3.3 Simple Model Calculation
		10.3.4 The Core Level Photoemission Spectrum and Its Linewidth
	10.4 Resonant X-Ray Absorption
		10.4.1 Natural Linewidth of XAS Resonances
		10.4.2 The Natural Shape of XAS Resonances
		10.4.3 Natural Linewidth of Optical Versus X-Ray Transitions
		10.4.4 The Dipole Matrix Element
		10.4.5 One-Electron/Hole Model
		10.4.6 Polarization Dependence of the Angular Transition Matrix Element
		10.4.7 Sum Rules for the Angular Transition Matrix Element
	10.5 Resonant XAS in Experiment and Theory
		10.5.1 K-Shell Resonance in the Low-Z Atom Ne
		10.5.2 K-Shell Resonances in the N2 and O2 Molecules
		10.5.3 L-Shell Resonances in 3d Transition Metal Atoms
		10.5.4 L-Shell XAS Intensities and Valence Shell Occupation
		10.5.5 Resonant Lineshapes in Atoms and Solids
		10.5.6 Dipole Matrix Element, Oscillator Strength, and Sum Rules
	10.6 Multi-electron Formalism: Multiplet Structure
		10.6.1 Evolution of One-Electron to Multiplet Theory
	References
11 Quantum Theory of X-Ray Dichroism
	11.1 Overview
	11.2 Introduction to the Quantum Theory of Dichroism
	11.3 X-Ray Natural Linear Dichroism—XNLD
		11.3.1 The Search Light Effect
		11.3.2 XNLD and the Quadrupolar Valence Charge Density
		11.3.3 Application of XNLD
	11.4 X-Ray Natural Circular Dichroism—XNCD
		11.4.1 The Two Types of XNCD
	11.5 X-Ray Magnetic Circular Dichroism—XMCD
		11.5.1 Key Concepts of Magnetism and Magnetic Alignment
		11.5.2 XMCD Sum Rule for the Orbital Moment
		11.5.3 Experimental Studies of Orbital Magnetism
		11.5.4 XMCD Sum Rule for the Spin Moment
	11.6 Test of the Sum Rules: Cu-Phthalocyanine
		11.6.1 Electronic Structure of Cu-Pc
		11.6.2 Treatment of the Spin-Orbit Interaction
		11.6.3 Comparison of Orbital Momenta in Theory and Experiment
		11.6.4 Spin Momenta in Theory and Experiment
	11.7 Application of XMCD to the Study of Transient Spin Effects
		11.7.1 Spin Accumulation in Cu upon Injection from Co
		11.7.2 Spin-Orbit Induced Spin Currents in Pt, Injected into Co
	11.8 X-Ray Magnetic Linear Dichroism—XMLD
		11.8.1 Introduction
		11.8.2 Theoretical Formulation in One-Electron Theory
		11.8.3 XNLD Versus XMLD in Cu-Phthalocyanine
		11.8.4 XMLD in Ferromagnetic Transition Metals
		11.8.5 Enhanced XMLD Through Multiplet Effects
	References
12 Quantum Theory of X-Ray Emission and Thomson Scattering
	12.1 Introduction and Overview
	12.2 Quantum Formulation of X-Ray Emission Spectroscopy (XES)
		12.2.1 XES History and Terminology
		12.2.2 The Photon Part of the Transition Matrix Element
		12.2.3 XES Decay Time and Linewidth
		12.2.4 Decays to Excited Final States
		12.2.5 Auger Contribution to the XES Linewidth
		12.2.6 Putting It All Together: The X-Ray Emission Rate and Width
		12.2.7 Atomic Decay Time: Core Hole-Clock
	12.3 Fundamental X-Ray Emission Experiments
		12.3.1 K-shell Emission in Ne
		12.3.2 K-Shell Emission in N2
		12.3.3 L-Shell XES in 3d Metals
		12.3.4 L3-Shell XES in Cu Metal
	12.4 X-Ray Fluorescence Yield, Linewidths, and Strengths
		12.4.1 Radial Dipole Matrix Element
	12.5 Quantum Theory of Thomson Scattering
		12.5.1 Quantum Theoretical Formulation of Thomson Scattering
		12.5.2 Elastic Thomson Scattering: Atomic Form Factor
		12.5.3 Inelastic Thomson Scattering: Dynamical Structure Factor
		12.5.4 Core Shell Excitations: X-Ray Raman Scattering (XRS)
		12.5.5 Example: Typical O K-shell Cross Sections
	References
13 Quantum Theory of X-Ray Resonant Scattering
	13.1 Introduction and Overview
	13.2 Formulation of Resonant Scattering: REXS and RIXS
		13.2.1 Evaluation of the Double Matrix Element
		13.2.2 One-Electron Versus Configuration Picture
		13.2.3 Coherent Second Order Versus Consecutive First Order Processes
		13.2.4 REXS/RIXS Terminologies
	13.3 Quantum Formulation of REXS
		13.3.1 The Fundamental REXS Cross Section
		13.3.2 REXS with Finite Instrumental Resolution
	13.4 Spontaneous and Stimulated REXS Versus XAS
		13.4.1 Spontaneous REXS Versus XAS
		13.4.2 Stimulated REXS Versus XAS
		13.4.3 Link to Semi-classical Results
	13.5 Intermediate State Interference Effects in REXS
		13.5.1 REXS Interference Contour Map
		13.5.2 REXS Scattering Time
		13.5.3 REXS Interference in Molecular Spectra: N2 and O2
	13.6 Polarization and Spin Dependent Spontaneous REXS
		13.6.1 The Polarization Dependent Scattering Length
	13.7 Spontaneous Versus Stimulated REXS by an Atomic Sheet
		13.7.1 Forward Scattering by an Atomic Sheet
	13.8 Resonant Inelastic X-Ray Scattering: RIXS
		13.8.1 Two-Step RIXS
	13.9 RIXS with Finite Instrumental Resolution
		13.9.1 The Case of Small Final State Width
		13.9.2 Reduction of RIXS to XES
	13.10 Examples of RIXS Capabilities
		13.10.1 K-Shell RIXS of N2 and O2
		13.10.2 L-edge RIXS of Transition Metal Oxides
		13.10.3 L-Edge RIXS of Transition Metals
		13.10.4 RIXS of Chemisorbed Molecules: Polarization Dependence
		13.10.5 Utilization of the Scattered Polarization
	13.11 RIXS and Reduced Linewidth XAS (HERFD)
		13.11.1 HERFD XAS at the Pt L3-Edge
	References
Part IV Multi-photon Interaction Processes
14 Resonant Non-linear X-Ray Processes in Atoms
	14.1 Introduction
	14.2 X-Ray Induced Atomic Core to Valence Transitions
		14.2.1 Interaction Energy and Hamiltonian: The Rabi Frequency
		14.2.2 The Rabi Frequency in the X-Ray Regime
	14.3 The Optical Bloch Equations
		14.3.1 Time-Dependent Transitions in a Two-Level System: Density Matrix Formulation
		14.3.2 Damping Constants: Longitudinal Versus Transverse Relaxation
	14.4 Definition of Transition Rates in the BR Theory
		14.4.1 X-Ray Interaction Parameters for Model Calculations
		14.4.2 Practical Units and Beam Parameter Conversions
	14.5 Analytical Solutions of the Bloch Equations
		14.5.1 Arbitrary Bandwidth: Low Incident Intensity
		14.5.2 Exact Resonance: Arbitrary Incident Intensity
		14.5.3 Excitations by Transform-Limited and SASE Pulses
		14.5.4 Solution for the Steady-State or Long Time Limit
		14.5.5 Power Broadening of the BR Linewidth
	14.6 Link of KHD and Low Intensity BR Rates
		14.6.1 The KHD Transition Rates
		14.6.2 Link of Bandwidth in KHD and Time in BR Rates
		14.6.3 Mode-Based Versus Atom-Based Coherence Volumes
		14.6.4 Zero-Point Field in the Bloch-Rabi Formalism
	14.7 Link of BR Rates and KHD Rates in the Steady-State
		14.7.1 Steady-State Rate Expressions
		14.7.2 Illustration of the BR Rates and Their Saturation
		14.7.3 Time Dependence of Rates at Resonance
	14.8 Optical Theorem: Sum Rule for Absorption and Scattering
		14.8.1 XAS and REXS Cross-Section Sum Rule
		14.8.2 Atom Transmission Sum Rule
		14.8.3 BR Versus KHD Stimulated Enhancement: Saturation
	14.9 BR, KHD, and Einstein Treatment of a Two-Level System
		14.9.1 Einstein's Model
		14.9.2 Reduction of the BR to the Einstein Theory
	14.10 Resonance Fluorescence
		14.10.1 Introduction
		14.10.2 Second Quantization of the p p p pcdotA A A A Interaction Hamiltonian
		14.10.3 The Degree of First Order Temporal Coherence
		14.10.4 g(1)(t) From Numerical Solutions of Bloch Equations
	14.11 The Resonant Fluorescence Spectrum
		14.11.1 Form of the Spectrum
		14.11.2 Fourier Transform of g(1) (t): The Fluorescence Spectrum
		14.11.3 The Low Intensity Spectrum: ΓR < Γ/4
		14.11.4 High Intensity Spectrum: ΓR > Γ/4
		14.11.5 Calculated Fluorescence Spectra
		14.11.6 Coherent and Incoherent Parts of the Equilibrium Spectrum
	14.12 Second Order Coherence of the Fluorescent Photons
		14.12.1 Weak Incident Beam
		14.12.2 Large Dephasing: Einstein Result
		14.12.3 Resonant Case of Arbitrary Intensity
		14.12.4 Photon Antibunching in Resonance Fluorescence
	References
15 Non-linear Absorption and Scattering Processes in Solids
	15.1 Introduction and Chapter Overview
		15.1.1 Brief Introduction
		15.1.2 Chapter Overview
	15.2 The Fundamental Damage Issue of XFEL Radiation
		15.2.1 X-Ray Beam Parameters
		15.2.2 Temporal Evolution of Matter after X-Ray Excitation
		15.2.3 Energy Transfer to the Electronic System
		15.2.4 From Electronic to Lattice Temperature
		15.2.5 Ablation Threshold
		15.2.6 Summary
	15.3 Fluence-Dependent Changes of XAS Spectra
		15.3.1 Redistribution of Valence Electrons
		15.3.2 X-Ray Transparency: An Introduction
	15.4 BR Theory of the Stimulated Response of a Thin Sheet
		15.4.1 Non-linear Response of an Atomic Sheet
		15.4.2 Effective Excited State Population and Enhancement Factor calGcoh
	15.5 Non-linear Transmission Through a Film of Finite Thickness
		15.5.1 From Thin Sheet to Finite Thickness Film
		15.5.2 Summary: From Single Atom to Film Transmission
		15.5.3 Sum Rule for Non-linear Film Transmission
	15.6  Polarization and Time-Dependent NL Transmission
		15.6.1 The Polarization Dependent Generalized Beer-Lambert Law
		15.6.2 Non-linear Polarization Dependent Transmission by the Magnetic 3d Metals
		15.6.3 Dependence on X-Ray Pulse Coherence Time
		15.6.4 From Collective to Independent Atomic Response
	15.7 X-Ray Transparency
		15.7.1 Resonant Case: Co Metal
		15.7.2 Resonant Versus Non-resonant X-Ray Transparency
		15.7.3 Non-resonant Transparency Above the Al L-Edge
		15.7.4 Non-linear Transparency Above the Fe K-Edge
	15.8 Polarization Dependent NL Transmission  at Resonance
		15.8.1 The Maximum NL Transmission Effect
		15.8.2 Polarization Dependent Transmission
	15.9 Competition Between NL Transmission  and Diffraction
		15.9.1 The NL Airy Pattern of a Film in a Circular Aperture
		15.9.2 Change of the Spontaneous to the Stimulated Pattern
		15.9.3 X-Ray Soliton Model: Mode-Dependent Stimulation
	15.10 Polarization Dependent NL Diffraction
		15.10.1 NL Diffraction by Magnetic Domains
		15.10.2 Narrow Bandwidth Resonant Case
		15.10.3 Broad Bandwidth Case
	15.11 Stimulated Resonant Inelastic X-Ray Scattering
		15.11.1 Stimulated L3 REXS and RIXS for Co Metal
		15.11.2 Observation of Stimulated RIXS in a Solid
		15.11.3 The Stimulated REXS/RIXS Model
	References
16 Quantum Diffraction: Emergence of the Quantum Substructure of Light
	16.1 Introduction
	16.2 Generation of Different States of Light
	16.3 The Formulation of Quantum Diffraction
		16.3.1 First Order Diffraction Formulation
		16.3.2 Second Order Diffraction Formulation
		16.3.3 Order-Dependent Degree of Coherence
	16.4 The Quantum States of Light
		16.4.1 Two-Mode Collective Quantum States
		16.4.2 The Collective Coherent State and Its Substates
		16.4.3 The Collective Phase-Diffused Coherent State and Its Substates
		16.4.4 The Collective Chaotic State and Its Substates
		16.4.5 Plots of the Substate Distributions
		16.4.6 Other Fundamental Quantum States
		16.4.7 Summary of Key Multi-photon Quantum States
	16.5 First Order Double-Slit Diffraction Patterns
		16.5.1 Calculation of the First Order Patterns
		16.5.2 Coherent State
		16.5.3 Plots of the First Order Patterns
		16.5.4 Degree of First Order Coherence
		16.5.5 Reduction of First-Order Quantum to Wave Formalism
	16.6 Second Order Double-Slit Diffraction Patterns
		16.6.1 Coherent State
		16.6.2 Plots of the Second Order Patterns
		16.6.3 Degree of Second Order Coherence
		16.6.4 The Evolution from First to Second Order
	16.7 Summary
	References
Appendix
A.1 The International System of Units (SI)
A.2  Resonance Lineshapes
A.2.1 Lorentzian Lineshape and Integral
A.2.2 Gaussian Lineshape and Integral
A.2.3 Voigt Lineshape
A.3  Dirac δ-Function
A.4  Fourier Transforms and Parseval's Theorem
A.4.1 1D Fourier Transform
A.4.2 1D Transformation Under Preservation of Dimension
A.4.3 Effective 1D Distribution Widths and Transform Limit
A.4.4 2D Fourier Transform
A.4.5 2D Transformation Under Preservation of Dimension
A.4.6 Effective 2D Distribution Areas and Diffraction Limit
A.5  Spherical Harmonics and Tensors
A.5.1 Relations between First and Second Order Tensors C(1)m and C(2)m
A.6  s, p, and d Orbitals
A.7  Spin-Orbit Basis Functions and Matrix Elements
A.8  Matrix Elements of Spherical Tensors
A.8.1 Polarization Dependent p rightarrowd Transition Probabilies
A.8.2 Sum Rules For Matrix Elements of C(1)m and C(2)m
A.9  Quantum States and Diffraction Patterns
A.9.1 Coherent State
A.9.2 N-Photon Substate of Coherent State
A.9.3 Phase-Diffused Coherent State
A.9.4 N-Photon Substate of Phase-Diffused Coherent State
A.9.5 Chaotic State
A.9.6 N-Photon Substate of Chaotic State
A.9.7 N-Photon Entangled (NOON) State
A.9.8 N-Photon Number State
A.10  Matrix Element of Second Order Coherence Operators
A.10.1 Coherent State
A.10.2 N-Photon Substate of Coherent State
A.10.3 Phase-Diffused Coherent State
A.10.4 N-Photon Substate of Phase-Diffused Coherent State
A.10.5 Chaotic State
A.10.6 N-Photon Substate of Chaotic State
A.10.7 N-Photon Entangled (NOON) State
A.10.8 N-Photon Number State
A.11  Evaluation of the Degree of Second Order Coherence
	References
Index