Nonlinear Optical Polarization Analysis in Chemistry and Biology

Description

This rigorous yet accessible guide presents a molecular-based description of nonlinear optical polarization analysis of chemical and biological assemblies. It includes discussion of the most common nonlinear optical microscopy and interfacial measurements used for quantitative analysis, specifically second harmonic generation (SHG), two-photon excited fluorescence (2PEF), vibrational sum frequency generation (SFG), and coherent anti-Stokes Raman spectroscopy/stimulated Raman spectroscopy (CARS/SRS). A linear algebra mathematical framework is developed, allowing step-wise systematic connections to be made between the observable measurements and the molecular response. Effects considered include local field corrections, the molecular orientation distribution, rotations between the molecular frame, the local frame and the laboratory frame, and simplifications from molecular and macromolecular symmetry. Specific examples are provided throughout the book, working from the common and relatively simple case studies through to the most general scenarios. Cover Half-title Series information Title page Copyright information Table of contents List of Contributors List of Terms 1 Introduction 1.1 Goals of the Book 1.2 Jones Vectors and Jones Matrices 1.2.1 Jones Vector Representations for the Polarization of Light 1.2.2 Jones Matrices and Reference Frame Rotation 1.2.2.1 Alternative Rotation Formulation 1.2.3 Jones Tensor and Reference Frame Rotation 1.3 Classical 1D Linear Polarizability 1.3.1 The Damped, Driven Harmonic Oscillator 1.4 Classical 1D Nonlinear Polarizability 1.4.1 The Damped, Driven Anharmonic Oscillator 1.4.2 Power Series Expansion of the Induced Polarization 1.4.3 Symmetry and the Polarizability Tensor 1.5 Further Reading 2 The Molecular Nonlinear Polarizability 2.1 Contracted Expressions for the Molecular Tensor 2.1.1 Simplified Expressions for Resonance-Enhanced Second-Order Nonlinear Optics 2.1.2 Expressions for Third-Order Nonlinear Optics Based on Inspection 2.2 Sum-over-States Expressions for the Molecular Tensor 2.2.1 General Sum-over-States Expressions from Time-Dependent Perturbation Theory 2.2.2 Contracted Sum-over-States Expressions 2.2.3 Two-Wave Mixing 2.2.4 Three-Wave Mixing 2.2.5 Four-Wave Mixing 2.3 Simplification Based on Molecular Symmetry 2.3.1 Simplification of Parametric Polarizability Tensors by Inspection 2.3.2 Simplification of Parametric Polarizability Tensors by Matrix Multiplication 2.3.3 Parametric vs. Nonparametric Processes 2.4 Further Reading 3 Visualization of the Molecular Tensor 3.1 Sagittary Representations of the Resonance-Enhanced α(1) 3.2 Antisymmetric Pseudovector Representation of α(1) 3.3 Extension of Visualization Techniques to Second-Order NLO Phenomena 3.4 Vector-Sphere Representations 3.5 Recovery of the Tensors from the Sagittary Representations 3.6 Further Reading 4 NLO Properties of Coupled Oscillators and Crystals 4.1 Negligible Coupling Limit 4.2 Exciton States of Identical Coupled Oscillators: Symmetry-Additive Model 4.3 SHG of Chromophore Aggregates 4.3.1 Dimerization to Aggregation 4.3.2 Aggregation in Assemblies of ‘‘Rodlike’’ Chromophores 4.3.3 Aggregation in Assemblies of ‘‘Λ-Like’’ Chromophores 4.3.4 Aggregation in the General Case of Planar Chromophores: C2-Symmetry Dimer 4.3.5 Aggregation in the General Case of Planar Chromophores: C1-Symmetry Dimer 4.3.6 Aggregation in Monolayer Films of Chromophores 4.4 Coupling between Dissimilar Oscillators 4.5 Predicting/Interpreting the SHG Activity of Molecular Crystals 4.5.1 Symmetry-Additive Model for the Molecular Frame to Crystal Frame Transformation in SHG and SFG 4.5.2 Resonance-Enhanced SHG and SFG of P212121 Crystals 4.5.2.1 Case 1: Rodlike βzzz-Dominated NLO Chromophores 4.5.2.2 Case 2: Λ-like βxxz = βxzx-Dominated Chromophores 4.5.2.3 Extension to P21 Crystal Classes 4.5.2.4 Effects from Linear Optical Properties 4.6 Further Reading 5 Second-Order Nonlinear Optical Properties of Proteins 5.1 Amide Building Blocks 5.2 Symmetry-Additive Model for Weakly Coupled Amides 5.3 Symmetry-Additive Model for Coupled Amides 5.4 The Symmetry-Additive Model Applied to the α-Helix 5.4.1 The Role of Amide Coupling 5.5 The Symmetry-Additive Model Applied to Collagen 5.6 The Symmetry-Additive Model Applied to β-Sheets 5.7 Side Chain Effects 5.8 Complete Proteins 5.9 Further Reading 6 Surface SHG and SFG 6.1 The Goals of the Measurement 6.1.1 Measured Observables 6.2 The Jones-to-Surface Transformation 6.3 Fresnel Factors 6.3.1 Fresnel Factors for Measurements in Transmission 6.3.2 Fresnel Factors for Measurements in Total Internal Reflection: Hemisphere 6.3.3 Fresnel Factors for Measurements in Total Internal Reflection: Prism 6.3.4 Estimation of the Interfacial Optical Constants in Evaluating the Fresnel Factors 6.4 The Cartesian Tensor and the Molecular Tensor 6.5 Simplification from Symmetry in the Molecular Tensor 6.6 Orientation Parameters and the SHG/SFG Magic Angle 6.7 Case Studies 6.7.1 Case Study 1: Vibrational SFG from a Uniaxial Assembly of Chromophores with C2v Symmetry 6.7.1.1 Molecular Orientational Averages 6.7.1.2 Experimental Recovery of Cartesian Tensors 6.7.1.3 Explicit Evaluation from Measured SFG Intensity 6.7.1.4 Interpretation of the Orientation Distribution 6.7.1.5 Raman Depolarization Ratio 6.7.2 Case Study 2: Electronic SHG from a Uniaxial Assembly of Chromophores with C2v Symmetry. 6.7.2.1 Molecular Orientational Averages. 6.7.2.2 Experimental Recovery of Cartesian Tensors 6.7.2.3 Explicit Evaluation from the Measured SHG Intensity. 6.7.2.4 Interpretation of the Orientation Distribution 6.7.2.5 Two-Photon Absorption Tensor 7 Chirality in Nonlinear Optics 7.1 Isotropic Chirality in Sum Frequency Spectroscopy 7.2 Intrinsic Chirality in Interfacial SHG and SFG 7.3 Orientational Chirality in Interfacial SHG and SFG 7.4 Relative Magnitudes of the Chiral Contributions in SHG and SFG 7.5 Isotropic SFG Chirality of Coupled Oscillators 7.6 Chiral-Specific Second-Order NLO Activities Of Proteins 7.6.1 Isotropic Chirality of Protein Secondary Structures 7.6.1.1 Electronic SFG of Protein Secondary Structures 7.6.2 Intrinsic and Orientational Chirality of Protein Secondary Structures 7.7 Chiral-Specific Fluorescence and Four-Wave Mixing at Interfaces 7.8 Further Reading 8 Nonlinear Optical Ellipsometry 8.1 Definition of Nonlinear Optical Ellipsometry and Goals of the Measurement 8.2 NOE from Measurements of ρsum or ρ2ω 8.2.1 Jones Tensors for SHG and SFG 8.2.2 SHG from Uniaxial Achiral Interfaces 8.2.3 SHG from a Uniaxial Chiral Interfaces 8.2.4 Vibrational SFG from Uniaxial Achiral Interfaces 8.2.5 Vibrational SFG from Uniaxial Chiral Interfaces 8.2.6 Combining Pooled SFG Measurements of ρ from Uniaxial Chiral and Achiral Interfaces 8.2.7 Combining Pooled SHG Measurements of ρ from Uniaxial Chiral and Achiral Interfaces 8.2.8 Combining Pooled Measurements of ρ in SHG Microscopy 8.3 Polarimetry Measurement Approaches 8.3.1 Relationship between the Jones vector and ρ 8.3.2 Null Polarimetry (Figure 8.2a and 8.2b) 8.3.3 Rotating Analyzer Polarimetry (Figure 8.2c) 8.3.4 Rotating Quarter Wave Plate Polarimetry (Figure 8.2d and 8.2e) 8.4 NOE Based on Tensor Product Measurements 8.4.1 Phase Modulation for SHG Microscopy: Real-Valued Tensor Elements 8.4.2 Phase Modulation for SHG Microscopy with Complex-Valued Tensor Elements 8.4.2.1 Additional Postsample Polarization-Dependent Elements 8.4.3 Phase Modulation of the Incident Fundamental Beam for SHG of Achiral Uniaxial Interfaces 8.4.4 Phase Modulation of the Incident Fundamental Beam for Surface SHG of Chiral Interfaces 8.4.4.1 Additional Postsample Polarization-Dependent Elements 8.4.5 Rotation of a Linearly Polarized Incident Fundamental Beam for SHG from Chiral Interfaces 8.5 Further Reading 9 Bridging the Local to Laboratory Frames in SHG and SFG Microscopy 9.1 Mathematical Formalism for the General Case of SFG 9.2 Connection between the Measured Observables and χJ(2) 9.2.1 SHG with Linearly Polarized Incident Light 9.2.2 SHG with Phase-Modulated Incident Light 9.2.3 Generalization of F and P for Additional Elements in the Detection Path 9.2.3.1 Detection Using Phase Modulation 9.2.3.2 Detection Using Polarization Rotation 9.3 Recovery of the Local-Frame Tensor and/or Orientation from the Jones Tensor 9.3.1 The Influence of Local Symmetry and Parameter Reduction 9.3.2 Incorporation of Local Field Correction Factors Corrections for the Local Dielectric Medium 9.3.3 Rotation into the Jones Reference Frame, J 9.3.4 Use of a Focused Beam within the Paraxial Approximation: Thin Sample Limit 9.3.5 Use of a Focused Beam within the Paraxial Approximation: Thick Sample Limit 9.3.6 Influence of Tight Focusing on the Local Polarization: Beyond the Paraxial Approximation 9.3.7 The Influence of Birefringence 9.4 Specific Common Cases 9.4.1 Case 1: In-Plane Orientation with Azimuthal Rotation Only 9.4.1.1 Recovery of the Local Tensor by Nonlinear Fitting: Case 1 with Azimuthal Rotation Only 9.4.1.2 Recovery of the Local Tensor by Linear Fitting: Case 1 with Azimuthal Rotation Only 9.4.1.3 Assessment of Orientation 9.4.2 Case 2: Combined Azimuthal and Polar Rotation for Local Uniaxial Symmetry 9.4.3 Case 3: Combined Azimuthal, Polar, and Twist Rotation for a System of Arbitrary Local-Frame Symmetry 9.5 Concluding Remarks 9.6 Further Reading 10 Polarization-Dependent CARS/SRS/Raman Microscopy 10.1 Overview of Common Measurement Strategies in Uniaxial Systems 10.1.1 Case 1 - Determination of the Local-Frame Tensor 10.1.2 Case 2 - Determination of Local-Frame Orientation within the Field of View 10.1.3 Case 3 - Determination of the Molecular Orientation Distribution Relative to the Local-Frame Coordinates 10.1.4 Determination of the Raman and/or Two-Photon Absorption Tensor 10.2 Connecting the Measured Observables and χJ(3) 10.2.1 CARS/SRS with Linearly Polarized Incident Light 10.3 Rotation from Local to Jones Reference Frames 10.4 Incorporation of Local-Field Correction Factors 10.5 Local Symmetry and Parameter Reduction 10.5.1 Formalism for Contraction of the Local Frame χl(3) Tensor through Symmetry Operations 10.5.2 Achiral and Uniaxial Local-Frame Assemblies 10.5.3 Chiral Uniaxial Assemblies 10.6 Common Case Studies Revisited 10.6.1 Local Uniaxial Symmetry with In-Plane (Azimuthal) Orientation 10.6.1.1 CARS/CATS/SRS 10.6.1.2 Third Harmonic Generation 10.6.2 Arbitrary Symmetry with Arbitrary Orientation 10.6.3 Local Uniaxial Symmetry with Both Polar and Azimuthal Orientation 10.7 The Molecular γ(3) Tensors of Individual Chromophores in CARS/CATS/SRL/SRG 10.7.1 Molecular Tensor from Time-Dependent Perturbation Theory 10.7.2 The Symmetry Matrix S for Parameter Reduction 10.7.2.1 The Interchange Matrix Mint 10.7.2.2 The Raman Symmetry Matrix Sα 10.7.3 Using Character Tables to Simplify the Molecular Tensor 10.8 Local-Frame Molecular Orientation Distribution 10.8.1 Analytical Expressions for the Orientational Averages in a Uniaxial Assembly 10.8.2 Simplification through Molecular Symmetry 10.8.3 Bridging the Orientation Distribution and the Experimental Observables 10.8.4 Brute Force Computational Approach 10.9 Further Reading 11 Hyper-Rayleigh Scattering 11.1 Overview and Goals 11.2 The Molecular/Macromolecular Tensor 11.3 Experimental Design 11.4 Microscopic to Laboratory Frame: Simplest Case for a One-Dimensional ‘‘Rodlike’’ Molecule 11.5 The General Case of Arbitrary Molecular/Macromolecular Symmetry 11.6 Resonance-Enhanced HRS from Isotropic Assemblies of Planar Chromophores 11.7 HRS of Ordered Dimers, Aggregates, and Nanocrystals 11.8 Further Reading 12 Polarization-Dependent Single- and Multi-Photon Excited Fluorescence of Isotropic Assemblies 12.1 Motivation and Goals of the Measurement 12.2 The Molecular Tensor for 1PEF and 2PEF 12.2.1 Visualization of the Molecular Tensor 12.2.2 Relationship among Absorbance, the Transition Moment, and the Molecular Polarizability 12.3 Polarization-Dependent 1PEF from Isotropic Assemblies 12.3.1 1PEF in Isotropic Assemblies of ‘‘Rodlike’’ Fluorophores 12.3.2 1PEF in Isotropic Assemblies of Arbitrary Symmetry 12.4 Polarization-Dependent 2PEF from Isotropic Assemblies 12.4.1 2PEF Measurements by Polarization Rotation 12.4.2 2PEF from Isotropic Assembles of One-Dimensional ‘‘Rodlike’’ Fluorophores 12.4.3 2PEF from Isotropic Assemblies of Planar Fluorophores 12.4.4 Isotropic 2PEF Measurements of Chromophores with Arbitrary Symmetry 12.5 Isotropic 1PEF and 2PEF in Dimers and Aggregates; Symmetry-Additive Model 12.5.1 Sum and Difference States in the Symmetry-Additive Model 12.5.2 Spectral Shifts from Relative Positions in a Dimer: H and J Aggregates 12.5.2.1 Coparallel Transition Moments 12.5.2.2 Noncoparallel but Coplanar Transition Moments 12.5.3 1PEF from an Isotropic Assembly of Dimers 12.5.4 2PEF from an Isotropic Assembly of Dimers; Rodlike Monomers 12.5.4.1 2PEF Emission from the Sum State 12.5.4.2 2PEF Emission from the Difference State 12.5.5 2PEF from an Isotropic Assembly of Dimers; Monomers of Arbitrary Symmetry 12.6 Comparisons/Contrasts between SHG and 2PEF in the General Case of Arbitrary Local and Arbitrary Molecular Symmetry 12.7 Further Reading 13 1PEF and 2PEF from Uniaxial Interfaces 13.1 Goals of the Measurement and Experimental Design 13.2 1PEF from Stationary Rodlike Uncoupled Molecular Interfacial Assemblies 13.3 2PEF from Stationary Rodlike Uncoupled Molecular Interfacial Assemblies 13.4 The General Framework: 1PEF 13.4.1 1PEF from Achiral Uniaxial Assemblies 13.4.2 Chiral-Specific 1PEF from Chiral Uniaxial Assemblies 13.5 The General Framework: 2PEF 13.5.1 Laboratory-Frame Observables for 2PEF from Uniaxial Assemblies 13.5.2 2PEF from Achiral Uniaxial Assemblies 13.5.3 Chiral-Specific 2PEF from Chiral Uniaxial Assemblies 13.6 Fresnel Local-Field Correction Factors in Reflection, Transmission, and TIR 13.7 Recovering Moments of the Orientation Distribution 13.8 Further Reading 14 1PEF and 2PEF Microscopy 14.1 Goals of the Measurement and Experimental Design 14.1.1 General Architecture for Polarization-Dependent Fluorescence Microscopy 14.1.2 Illustrative Limiting Example for 1PEF 14.2 The General Framework: 1PEF 14.2.1 Molecular-Frame to Local-Frame Orientational Averaging for 1PEF 14.2.2 Fresnel Local Field Corrections for 1PEF 14.2.3 Local-Frame to Laboratory-Frame Rotation for 1PEF: Gentle Focusing Limit 14.2.4 Local-Frame to Laboratory-Frame Rotation with Tight Focusing for 1PEF 14.3 The General Framework: 2PEF 14.3.1 The Molecular Tensor for 2PEF 14.3.2 Molecular-Frame to Local-Frame Orientational Averaging for 2PEF 14.3.3 Fresnel Local Field Corrections for 2PEF 14.3.4 Local-Frame to Laboratory-Frame Rotation for 2PEF 14.4 Polarization-Dependent 1PEF and 2PEF Measurements 14.4.1 Measurements with Pure Polarizations 14.4.2 Polarization Rotation: 1PEF 14.4.3 Polarization Rotation: 2PEF 14.4.4 Phase Modulation: 1PEF 14.4.5 Phase Modulation: 2PEF 14.5 Specific Common Cases for 1PEF 14.5.1 1PEF from Rodlike Macromolecular Assemblies with In-Plane Orientation 14.5.2 1PEF from Uniaxial Assemblies of Rodlike Molecules with In-Plane Orientation 14.5.3 1PEF from Uniaxial Assemblies of Arbitrary Molecular Symmetry and In-Plane Orientation 14.5.4 1PEF from Uniaxial Assemblies of Arbitrary Symmetry and Arbitrary Orientation 14.6 Specific Common Cases for 2PEF 14.6.1 Case X: 2PEF from Rodlike Molecules/Aggregates with In-Plane Orientation 14.6.2 2PEF from Locally Uniaxial Highly Coupled Molecular Assemblies with In-Plane Orientation and Emission Polarized Parallel to the Unique Axis 14.6.2.1 A Symmetry 14.6.2.2 E Symmetry Absorption with Z-Polarized Fluorescence 14.6.2.3 Relationship to the Molecular Orientation Distribution 14.6.3 2PEF from Locally Uniaxial Highly Coupled Molecular Assemblies with In-Plane Orientation and Emission Polarized Orthogonal to the Unique Axis 14.6.4 2PEF from Locally Uniaxial Uncoupled Assemblies of Arbitrary Molecular Symmetry and Orientation 14.6.4.1 Direct Single-Step Approach 14.6.4.2 Symmetry-Based Intermediate Approach Using Q 14.6.5 2PEF from Locally Uniaxial Uncoupled Molecular Assemblies of ‘‘Rodlike’’ Molecules with In-Plane Orientation 14.7 Further Reading 15 Mueller Tensors 15.1 Mueller Matrices and Stokes Vectors in Linear Optics 15.1.1 Stokes and Jones Vectors 15.1.2 Mueller and Jones Matrices 15.1.3 Vectorization of Mueller Matrices 15.1.4 Generation of the Elementary Transformation Matrix E 15.1.5 Modeling Partial Depolarization 15.2 Jones and Mueller Tensors for Coherent Nonlinear Optics 15.2.1 Mueller and Jones Tensors for SHG and SFG 15.2.2 Mueller Tensors for n-Wave Mixing 15.2.3 Vectorization of the Mueller Tensor 15.2.4 Generation of the Elementary Transformation Matrix E 15.2.5 The Mueller Tensor for HRS and 2PEF in Isotropic Assemblies 15.2.5.1 General Framework for HRS Detected along the Optical Axis 15.2.5.2 Simplification from Kleinman Symmetry 15.2.5.3 Simplification for ‘‘Rodlike’’ Molecular Symmetry 15.2.5.4 Extension to 2PEF 15.2.6 Isotropic Model for Partial Depolarization in SHG Microscopy 15.2.6.1 General Model for the Mueller Tensor 15.2.6.2 Model 1 - Completely Depolarized Component 15.2.6.3 Model 2 - Purely Polarized Incident Light, Partially Depolarized Detected Light 15.2.6.4 Model 3 - Spatial Heterogeneity in the SHG-Active Source 15.3 ‘‘Super-Mueller’’ Matrix Approach 15.4 Uniaxial Symmetry in the Local-Frame Mueller Matrix 15.4.1 Coordinate Transformation of the Mueller Tensor 15.4.2 Dimension Reduction Based on Local-Frame Symmetry 15.4.3 Combining Azimuthal Orientation and Local-Frame Symmetry 15.4.4 Further Simplification by Connection to Jones Tensors 15.5 Bridging Mueller Tensors and Common Observables 15.5.1 Sets of Discrete Polarization States 15.5.2 Polarization Rotation with Partial Depolarization 15.5.2.1 Model 1 Purely Depolarized and Purely Polarized Contributions 15.5.2.2 Model 2 Purely Polarized Driving Field, Partially Polarized Detected Field 15.5.2.3 Model 3 Partial Depolarization from Out-of-Plane SHG Contributions 15.5.3 Phase Modulation with Partial Depolarization 15.6 Laboratory to Molecular Frame Transformations in SHG 15.7 Laboratory to Molecular Frame Transformations in 2PEF 15.8 Concluding Remarks 15.9 Further Reading Appendices Appendix A.1: Rotation Matrix Conventions for Intrinsic vs. Extrinsic Rotation Appendix A.2: SHG and SFG Character Tables for Molecules and Crystals Appendix A.3: Populating Sparse Matrices in SFG, SHG, and 2PEF Polarization Analysis A.3.1 Sparse Matrices A.3.2 Measured Intensity in the General Case of SFG, SHG, and 2PEF A.3.3 Binary Encoding/Decoding of Sparse Matrices A.3.4 Case Study: Polarization Rotation SHG and 2PEF Microscopy with No Postsample Optics A.3.4.1 Incorporation of Symmetry A.3.5 Case Study: Polarization Rotation SHG and 2PEF Microscopy with Additional Postsample Optics A.3.6 Case Study: Polarization Modulation SHG and 2PEF Microscopy with No Postsample Optics A.3.7 Case Study: Polarization Modulation SHG Microscopy with Additional Postsample Optics Appendix A.4: Populating Sparse Matrices in FWM Polarization Analysis A.4.1 Case 1 A 0-Polarized FWM Signal and a 0-Polarized Stokes Beam A.4.2 Case 2 A 1-Polarized FWM Signal and a 0-Polarized Stokes Beam A.4.3 Case 3 A 0-Polarized FWM Signal and a 1-Polarized Stokes Beam A.4.4 Case 4 A 1-Polarized FWM Signal and a 1-Polarized Stokes Beam Appendix A.5: Orientational Averages for Four-Wave Mixing of Uniaxial Assemblies A.5.1 Nonzero Tensor Elements in a Uniaxial Assembly in the Most General Case of Nondegenerate 4WM A.5.2 CARS, SRS, and CATS A.5.3 Third Harmonic Generation A.5.4 Single-Photon Excited Fluorescence A.5.5 Orientational Averages Connecting the Molecular and Uniaxial Reference Frames in the General Case of 4WM Appendix A.6: Identity Relations for Vectorization of Matrices and Tensors A.6.1 Vectorization of Matrices in Linear Optics A.6.2 Vectorization of Tensors and Tensor Products A.6.3 Identity Relationships with Kronecker Products of Tensors A.6.4 Explicit Evaluation of the Elementary Transformation Matrix for Kronecker Products of Matrices. A.6.5 Further Reading Index