 | Kramers-kronig relations in optical materials research / Lucarini V., Saarinen J.J., Peiponen K.E., Vartiainen E.M. - Berlin; Heidelberg: Springer, 2005. - ix, 160 p. - (Springer series in optical sciences; 110). - ISBN 3-540-23673-2; ISSN 03242-4111
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1. Introduction ................................................. 1
2. Electrodynamic Properties of a General Physical System ....... 5
2.1. The Maxwell Equations ...................................... 5
2.2. The System: Lagrangian and Hamiltonian Descriptions ........ 6
2.3. Polarization as a Statistical Property of a System ......... 9
3. General Properties of the Linear Optical Response ........... 11
3.1. Linear Optical Properties ................................. 11
3.1.1. Transmission and Reflection at the Boundary
Between Two Media .................................. 14
3.2. Microscopic Description of Linear Polarization ............ 16
3.3. Asymptotic Properties of Linear Susceptibility ............ 17
3.4. Local Field and Effective Medium Approximation in
Linear Optics ............................................. 19
3.4.1. Homogeneous Media .................................. 19
3.4.2. Two-Phase Media .................................... 21
4. Kramers-Kronig Relations and Sum Rules
in Linear Optics ............................................ 27
4.1. Introductory Remarks ...................................... 27
4.2. The Principle of Causality ................................ 27
4.3. Titchmarsch's Theorem and Kramers-Kronig Relations ........ 28
4.3.1. Kramers-Kronig Relations for Conductors ............ 29
4.3.2. Kramers-Kronig Relations
for the Effective Susceptibility of
Nanostructures ..................................... 30
4.4. Superconvergence Theorem and Sum Rules .................... 31
4.5. Sum Rules for Conductors .................................. 33
4.5.1. Sum Rules for the Linear Effective
Susceptibilities of Nanostructures ................. 33
4.6. Integral Properties of Optical Constants .................. 34
4.6.1. Integral Properties of the Index of Refraction ..... 35
4.6.2. Kramers-Kronig Relations
in Linear Reflectance Spectroscopy ................. 39
4.7. Generalization of Integral Properties
for More Effective Data Analysis .......................... 44
4.7.1. Generalized Kramers-Kronig Relations ............... 45
4.7.2. Subtractive K-K Relations .......................... 47
5. General Properties of the Nonlinear Optical Response ........ 49
5.1. Nonlinear Optics: A Brief Introduction .................... 49
5.2. Nonlinear Optical Properties .............................. 51
5.2.1. Pump-and-Probe Processes ........................... 54
5.3. Microscopic Description of Nonlinear Polarization ......... 56
5.4. Local Field and Effective Medium Approximation
in Nonlinear Optics ....................................... 58
5.4.1. Homogeneous Media .................................. 58
5.4.2. Two-Phase Media .................................... 60
5.4.3. Tailoring of the Optical Properties
of Nanostructures .................................. 63
6. Kramers-Kronig Relations and Sum Rules
in Nonlinear Optics ......................................... 71
6.1. Introductory Remarks ...................................... 71
6.2. Kramers-Kronig Relations in Nonlinear Optics:
Independent Variables ..................................... 72
6.3. Scandolo's Theorem and Kramers-Kronig Relations
in Nonlinear Optics ....................................... 73
6.4. Kramers-Kronig Analysis of the Pump-and-Probe System ...... 77
6.4.1 Generalization of Kramers-Kronig Relations
and Sum Rules ............................................. 79
7. Kramers-Kronig Relations and Sum Rules
for Harmonic-Generation Processes ........................... 83
7.1. Introductory Remarks ...................................... 83
7.2. Application of Scandolo's Theorem
to Harmonic-Generation Susceptibility ..................... 83
7.3. Asymptotic Behavior of Harmonic-Generation
Susceptibility ............................................ 84
7.4. General Kramers-Kronig Relations and Sum Rules
for Harmonic-Generation Susceptibility .................... 87
7.4.1. General Integral Properties of Nonlinear
Conductors ......................................... 89
7.5. Subtractive Kramers-Kronig Relations
for Harmonic-Generation Susceptibility .................... 90
8. Kramers-Kronig Relations and Sum Rules
for Data Analysis: Examples ................................. 93
8.1. Introductory Remarks ...................................... 93
8.2. Applications of Kramers-Kronig Relations
for Data Inversion ........................................ 93
8.2.1. Kramers-Kronig Inversion
of Harmonic-Generation Susceptibility .............. 94
8.2.2. Kramers-Kronig Inversion of the Second Power
of Harmonic-Generation Susceptibility .............. 96
8.3. Verification of Sum Rules
for Harmonic-Generation Susceptibility .................... 98
8.4. Application of Singly Subtractive
Kramers-Kronig Relations ................................. 101
8.5. Estimates of the Truncation Error
in Kramers-Kronig Relations .............................. 104
8.6. Sum Rules and Static Second-Order Nonlinear
Susceptibility ........................................... 106
9. Modified Kramers-Kronig Relations
in Nonlinear Optics ........................................ 109
9.1. Modified Kramers-Kronig Relations
for a Meromorphic Nonlinear Quantity ..................... 109
9.2. Sum Rules for a Meromorphic Nonlinear Quantity ........... 112
10.The Maximum Entropy Method: Theory and Applications ........ 115
10.1.The Theory of the Maximum Entropy Method ................. 115
10.2.The Maximum Entropy Method in Linear Optical
Spectroscopy ............................................. 117
10.2.1.Phase Retrieval from Linear Reflectance ........... 117
10.2.2.Study of Surface Plasmon Resonance ................ 120
10.2.3.Misplacement Phase Error Correction
in Terahertz Time-Domain Spectroscopy ............. 126
10.3.The Maximum Entropy Method in Nonlinear Optical
Spectroscopy ............................................. 128
11.Conclusions ................................................ 133
A. MATLAB Programs for Data Analysis .......................... 137
A.l. Program 1: Estimation of the Imaginary Part
via Kramers-Kronig Relations ............................. 137
A.2. Program 2: Estimation of the Real via
Kramers-Kronig Relations ................................. 139
A.3. Program 3: Self-Consistent Estimate of the Real
and Imaginary Parts of Susceptibility .................... 141
A.4. Program 4: Estimation of the Imaginary Part
via Singly Subtractive Kramers-Kronig Relations .......... 142
A.5. Program 5: Estimation of the Real Part
via Singly Subtractive Kramers-Kronig Relations .......... 143
References .................................................... 145
Index ......................................................... 159
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