Laser - NonlinearOptics.pdf - PORADY- INSTRUKCJE - romt

Lecture Notes on

Nonlinear Optics

Fredrik Jonsson

Lectures presented at the Royal Institute of Technology

Department of Laser Physics and Quantum Optics

January 8 { March 24, 2003

Lecture Notes on Nonlinear Optics

Nonlinear Optics (5A5513, 5p for advanced undergraduate and doctoral students)

Course given at the Royal Institute of Technology,

Department of Laser Physics and Quantum Optics

SE{106 91, Stockholm, Sweden

January 8 { March 24, 2003

Author and lecturer:

Fredrik Jonsson

Proximion Fiber Optics AB

SE-164 40, Kista

Sweden

The texts and gures in this lecture series was typeset by the author in 10/12/16 pt Computer

Modern typeface using plain T E X and METAPOST.

This document is electronically available at the homepage of the Library of the Royal Institute of

Technology, at http://www.lib.kth.se.

or transmitted, in any form, or by any means, electronic, mechanical, photo-copying, recording, or

otherwise, without the prior consent of the author.

ISBN 91-7283-517-6

TRITA-FYS 2003:26

ISSN 0280-316X

ISRN KTH/FYS/- - 03:26 - - SE

Printed on July 7, 2003

T E X is a trademark of the American Mathematical Society

Contents

i. Structure of the course

ii. Errata for Butcher and Cotter's The Elements of Nonlinear Optics

iii. Notes on the \Butcher and Cotter convention" in nonlinear optics

iv. Recommended reference literature in nonlinear optics

Lecture 1

1. The contents of the course

2. Examples of applications of nonlinear optics

3. A brief history of nonlinear optics

4. Outline for calculations of polarization densities

4.1 Metals and plasmas

4.2 Dielectrics

5. Introduction to nonlinear dynamical systems

6. The anharmonic oscillator

Lecture 2

1. Nonlinear polarization density

2. Symmetries in nonlinear optics

2.1 Intrinsic permutation symmetry

2.2 Overall permutation symmetry

2.3 Kleinman symmetry

2.4 Spatial symmetries

3. Conditions for observing nonlinear optical interactions

4. Phenomenological description of the susceptibility tensors

5. Linear polarization response function

6. Quadratic polarization response function

7. Higher order polarization response functions

Lecture 3

1. Susceptibility tensors in the frequency domain

2. First order susceptibility tensor

3. Second order susceptibility tensor

4. Higher order susceptibility tensors

5. Monochromatic elds

6. Convention for description of nonlinear optical polarization

7. Note on the complex representation of the optical eld

8. Example: Optical Kerr-eect

Lecture 4

1. The Truth of polarization densitites

2. Outline

3. Quantum mechanics

4. Perturbation analysis of the density operator

5. The interaction picture

6. The rst order polarization density

7. The second order polarization density

Lecture 5

1. The second order polarization density

2. Higher order polarization densities

3. Assembly of independent molecules

iii

Lecture 6

1. Assembly of independent molecules

2. First order electric susceptibility

3. Overall permutation symmetry of rst order susceptibility

4. Second order electric susceptibility

5. Overall permutation symmetry of second order susceptibility

Lecture 7

1. Motivation for analysis of susceptibilities in rotated coordinate systems

2. Optical properties in rotated coordinate frames

2.1 First order polarization density in rotated coordinate frames

2.2 Second order polarization density in rotated coordinate frames

2.3 Higher order polarization densities in rotated coordinate frames

3. Crystallographic point symmetry groups

4. Schonies notation for the non-cubic crystallographic point groups

5. Neumann's principle

6. Inversion properties

7. Euler angles

8. Example of the direct inspection technique applied to tetragonal media

8.1 Does the 422 point symmetry group possess inversion symmetry?

8.2 Step one - Point symmetry under twofold rotation around the x-axis

8.3 Step two - Point symmetry under fourfold rotation around the z-axis

Lecture 8

1. Wave propagation

1.1 Maxwell's equations

1.2 Constitutive relations

2. Two frequent assumptions in nonlinear optics

3. The wave equation

4. The wave equation in frequency domain (optional)

5. Quasimonochromatic light - Time dependent problems

6. Three practical approximations

7. Monochromatic light

7.1 Monochromatic optical eld

7.2 Polarization density induced by monochromatic optical eld

8. Monochromatic light - Time independent problems

9. Example I: Optical Kerr-eect - Time independent case

10. Example II: Optical Kerr-eect - Time dependent case

Lecture 9

1. General process for solving problems in nonlinear optics

2. Formulation of the exercises

2.1 Second harmonic generation in negative uniaxial media

2.2 Optical Kerr-eect { continuous wave case

3. Second harmonic generation

3.1 The optical interaction

3.2 Symmetries of the medium

3.3 Additional symmetries

3.4 The polarization density

3.5 The wave equation

3.6 Boundary conditions

3.7 Solving the wave equation

4. Optical Kerr-eect - Field corrected refractive index

4.1 The optical interaction

4.2 Symmetries of the medium

4.3 Additional symmetries

4.4 The polarization density

4.5 The wave equation { Time independent case

4.6 Boundary conditions { Time independent case

4.7 Solving the wave equation { Time independent case

Lecture 10

1. What are solitons?

2. Classes of solitons

2.1 Bright temporal envelope solitons

2.2 Dark temporal envelope solitons

2.3 Spatial solitons

3. The normalized nonlinear Schrodinger equation for temporal solitons

3.1 The eect of dispersion

3.2 The eect of a nonlinear refractive index

3.3 The basic idea behind temporal solitons

3.4 Normalization of the nonlinear Schrodinger equation

4. Spatial solitons

5. Mathematical equivalence between temporal and spatial solitons

6. Soliton solutions

7. General travelling wave solutions

8. Soliton interactions

9. Dependence on initial conditions

Lecture 11

1. Singularities of non-resonant susceptibilities

2. Modication of the Hamiltonian for resonant interaction

3. Phenomenological representation of relaxation processes

4. Perturbation analysis of weakly resonant interactions

5. Validity of perturbation analysis of the polarization density

6. The two-level system

6.1 Terms involving the thermal equilibrium Hamiltonian

6.2 Terms involving the interaction Hamiltonian

6.3 Terms involving relaxation processes

7. The rotating-wave approximation

8. The Bloch equations

9. The resulting electric polarization density of the medium

Lecture 12

1. Recapitulation of the Bloch equations for two-level systems

2. The resulting electric polarization density of the medium

3. The vector model of the Bloch equations

4. Transient build-up at exact resonance as the optical eld is switched on

4.1 The case T 1

T 2 { Longitudinal relaxation slower than transverse relaxation

T 2 { Longitudinal relaxation approximately equal to transverse relaxation

5. Transient build-up at o-resonance as the optical eld is switched on

6. Transient decay for a process tuned to exact resonance

6.1 The case T 1 T 2 { Longitudinal relaxation slower than transverse relaxation

6.2 The case T 1 T 2 { Longitudinal relaxation approximately equal to transverse relaxation

7. Transient decay for a slightly o-resonant process

7.1 The case T 1 T 2 { Longitudinal relaxation slower than transverse relaxation

7.2 The case T 1 T 2 { Longitudinal relaxation approximately equal to transverse relaxation

8. Transient decay for a far o-resonant process

4.2 The case T 1

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