# Lectures on nonlinear optics

This is the full set of twelve lectures in *Nonlinear Optics*
which I in the spring of 2003 gave at the Royal Institute of
Technology, Department of Laser Physics and Quantum Optics, Stockholm, Sweden.
The course was aimed towards advanced undergraduate and doctoral students and
gave 5 points, corresponding to five weeks of full-time study.
The lecture notes available below are a product from the continuous preparation
for the lectures.

Throughout the course, the convention on nonlinear susceptibilities and
degeneracy factors follows that of Butcher and Cotter's *The Elements
of Nonlinear Optics* (Cambridge University Press, 1990), for which a
separate errata and
summary of
conventions can be found.

These lecture notes have also been published electronically by the Royal
Institute of Technology, as
*Lecture
Notes on Nonlinear Optics*, ISBN 91-7283-517-6, TRITA-FYS 2003:26
(Transactions of the Royal Institute of Technology, Stockholm, 2003).

URL:
http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-9154.

## The complete series of lectures

**nlo2003.pdf** [2.96 MB]
The entire set of lecture notes, supplementary notes and home assignments
in a single PDF document.
[ download ]

**nlo2003.djvu** [1567 kB]
The entire set of lecture notes, supplementary notes and home assignments
in a single DejaVu document.
[ download ]

**nlo2003.ps** [10561 kB]
The entire set of lecture notes, supplementary notes and home assignments
in a single PostScript document.
[ download ]

Plug-in viewers of the PDF and DejaVu formats can be downloaded for free from the sites of Adobe and LizardTech, respectively. In order to display PostScript files on screen, you need a viewer like GhostView, which can be downloaded for free from the web site of the University of Wisconsin.

## Additional resources

**errata.pdf
**
[107 kB]
Errata for Butcher and Cotter's *The Elements of Nonlinear Optics*
(Cambridge University Press, 1990).
Also available online.
[ download ]

**bcconven.pdf
**
[102 kB]
Summary of the "Butcher and Cotter convention" on nonlinear susceptibilities,
adopted from *The Elements of Nonlinear Optics* (Cambridge University
Press, 1990).
TeX source for the document is available here.
[ download ]

## Lecture 1 - *Introduction to nonlinear optics*

Read this lecture online »
**lect1.tex** [31 kB]
Plain TeX source for Lecture 1.
[ download |
view source ]

### Contents

- The contents of the course
- Examples of applications of nonlinear optics
- A brief history of nonlinear optics
- Outline for calculations of polarization densities
- Introduction to nonlinear dynamical systems
- The anharmonic oscillator

## Lecture 2 - *The nonlinear susceptibilities and their symmetries*

Read this lecture online »
**lect2.tex** [22 kB]
Plain TeX source for Lecture 2.
[ download |
view source ]

### Contents

- Nonlinear polarization density
- Symmetries in nonlinear optics
- Conditions for observing nonlinear optical interactions
- Phenomenological description of the susceptibility tensors
- Linear polarization response function
- Quadratic polarization response function
- Higher order polarization response functions

## Lecture 3 - *Quasi-monochromatic fields and the degeneracy factor in nonlinear optics*

Read this lecture online »
**lect3.tex** [21 kB]
Plain TeX source for Lecture 3.
[ download |
view source ]

### Contents

- Susceptibility tensors in the frequency domain
- First order susceptibility tensor
- Second order susceptibility tensor
- Higher order susceptibility tensors
- Monochromatic fields
- Convention for description of nonlinear optical polarization

## Lecture 4 - *Quantum mechanics I: Formulation of linear optical interactions*

Read this lecture online »
**lect4.tex** [23 kB]
Plain TeX source for Lecture 4.
[ download |
view source ]

### Contents

- The Truth of polarization densitites
- Outline
- Quantum mechanics
- Perturbation analysis of the density operator
- The interaction picture
- The first order polarization density

## Lecture 5 - *Quantum mechanics II: Formulation of nonlinear optical interactions*

Read this lecture online »
**lect5.tex** [12 kB]
Plain TeX source for Lecture 5.
[ download |
view source ]

### Contents

## Lecture 6 - *Quantum mechanics III: Linking the microcscopic to the macroscopic*

Read this lecture online »
**lect6.tex** [27 kB]
Plain TeX source for Lecture 6.
[ download |
view source ]

### Contents

- Assembly of independent molecules
- First order electric susceptibility
- Second order electric susceptibility
- Overall permutation symmetry of second order susceptibility

## Lecture 7 - *Spatial symmetries in nonlinear optics*

Read this lecture online »
**lect7.tex** [29 kB]
Plain TeX source for Lecture 7.
[ download |
view source ]

### Contents

- Motivation for analysis of susceptibilities in rotated coordinate systems
- Optical properties in rotated coordinate frames
- Crystallographic point symmetry groups
- Schönflies notation for the non-cubic crystallographic point groups
- Neumann's principle
- Inversion properties
- Euler angles
- Example of the direct inspection technique applied to tetragonal media

## Lecture 8 - *The nonlinear electromagnetic wave equation*

Read this lecture online »
**lect8.tex** [14 kB]
Plain TeX source for Lecture 8.
[ download |
view source ]

### Contents

- Wave propagation in nonlinear media
- Two frequent assumptions in nonlinear optics
- The wave equation
- The wave equation in frequency domain (optional)
- Quasimonochromatic light - Time dependent problems
- Three practical approximations
- Monochromatic light
- Monochromatic light - Time independent problems
- Example I: Optical Kerr-effect - Time independent case
- Example II: Optical Kerr-effect - Time dependent case

## Lecture 9

Read this lecture online »
**lect9.tex** [36 kB]
Plain TeX source for Lecture 9.
[ download |
view source ]

### Contents

- General process for solving problems in nonlinear optics
- Formulation of the two case studies in this lecture
- Second harmonic generation
- Optical Kerr-effect - Field corrected refractive index
- References

## Lecture 10

Read this lecture online »
**lect10.pdf** [507 kB]
Lecture 10 in Portable Document Format.
[ download ]

**lect10.tex** [30 kB]
Plain TeX source for Lecture 10.
[ download |
view source ]

### Contents

- What are solitons?
- Classes of solitons
- The normalized nonlinear Schrödinger equation for temporal solitons
- Spatial solitons
- Mathematical equivalence between temporal and spatial solitons
- Soliton solutions
- General travelling wave solutions
- Soliton interactions
- Dependence on initial conditions
- References

## Lecture 11

**lect11.pdf** [173 kB]
Lecture 11 in Portable Document Format.
[ download ]

**lect11.tex** [32 kB]
Plain TeX source for Lecture 11.
[ download |
view source ]

### Contents

- Singularities of non-resonant susceptibilities
- Modification of the Hamiltonian for resonant interaction
- Phenomenological representation of relaxation processes
- Perturbation analysis of weakly resonant interactions
- Validity of perturbation analysis of the polarization density
- The two-level system
- Terms involving the thermal equilibrium Hamiltonian
- Terms involving the interaction Hamiltonian
- Terms involving relaxation processes

- The rotating-wave approximation
- The Bloch equations
- The resulting electric polarization density of the medium

## Lecture 12

**lect12.pdf** [805 kB]
Lecture 12 in Portable Document Format.
[ download ]

**lect12.tex** [37 kB]
Plain TeX source for Lecture 12.
[ download |
view source ]

### Contents

- Recapitulation of the Bloch equations for two-level systems
- The resulting electric polarization density of the medium
- The vector model of the Bloch equations
- Transient build-up at exact resonance as the optical field is switched on
- The case T
_{1}>> T_{2}- Longitudinal relaxation slower than transverse relaxation - The case T
_{1}≈ T_{2}- Longitudinal relaxation approximately equal to transverse relaxation

- The case T
- Transient build-up at off-resonance as the optical field is switched on
- Transient decay for a process tuned to exact resonance
- The case T
_{1}>> T_{2}- Longitudinal relaxation slower than transverse relaxation - The case T
_{1}≈ T_{2}- Longitudinal relaxation approximately equal to transverse relaxation

- The case T
- Transient decay for a slightly off-resonant process
- The case T
_{1}>> T_{2}- Longitudinal relaxation slower than transverse relaxation - The case T
_{1}≈ T_{2}- Longitudinal relaxation approximately equal to transverse relaxation

- The case T
- Transient decay for a far off-resonant process
- The case T
_{1}>> T_{2}- Longitudinal relaxation slower than transverse relaxation - The case T
_{1}≈ T_{2}- Longitudinal relaxation approximately equal to transverse relaxation - The case T
_{1}<< T_{2}- Longitudinal relaxation faster than transverse relaxation

- The case T
- The connection between the Bloch equations and the susceptibility
- The intensity-dependent refractive index in the susceptibility formalism
- The intensity-dependent refractive index in the Bloch-vector formalism

- Summary of the Bloch and susceptibility polarization densities
- Appendix: Notes on the numerical solution to the Bloch equations