**Nonlinear optics, Optics 568**

**Mondays and Wednesdays, 14:30 – 15:45**

First class: room 1160 PAIS

## Instructor

Jean-Claude Diels

Physics & Astronomy room PAIS 2236, phone 277-4026

CHTM, room 114A, phone 272-7830

Email: jcdiels@unm

Oﬃce hours by appointment

## Teaching Assistant

**Yankang Liu**, yankangliu@unm.edu

## Reference material

Reference material will be posted on the Website:

## Introduction

While the book of Robert Boyd may still serve as a useful background/reference, I

will post on my Web site the complete text of this course, paralleled with powerpoint

presentations. Glancing at the table of content of any nonlinear optics book, there seem

to be as many unrelated theories of nonlinear optics as chapters. First a classical chap-

ter where the polarization is expanded in a power series. Then a quantum mechanical

perturbation treatment. Then another chapter suggests that it is the index of refraction

that should be expanded in a power series of the intensity. Then it is suggested that the

power expansion is sometimes invalid. Finally, you may be led to believe that continuous

radiation and short pulses are two different worlds, as there is generally a chapter on short

pulse nonlinear optics.

It is the purpose of this course to show that all these apparently unrelated aspects can

be put under a single umbrella. The main actor in all these interactions is the electron.

Successive approximations from the more general response of the electron in time varying

high electric eld, to linear optics, is a journey that will bring us through all the aspects

of nonlinear optics. The classical situation treated in the rst chapter(s) of most nonli-

near optics books results from a stationary weak eld approximation of a more general

interaction, with all real atomic level off-resonance with the radiation frequency.

While different aspects of nonlinear optics may be taught in a different order, *the ma-terial covered will be the same as that of previous nonlinear optics classes.*

My approach is analogous to one used in analytical geometry. Some of you may have

been exposed in high-school to a similar method in your course of analytical geometry.

The old school taught only cartesian coordinates, in which circles, ellipses hyperbolae

are totally unrelated objects. Going from cartesian to projective coordinates one realizes

that circles, ellipses and hyperbolae are just one object. Teaching analytical geometry in

particularizing from the general projective coordinates towards the more narrow minded

cartesian gives one a much richer and elegant understanding of geometry.

The last chapter of the class will deal with quantum aspects of nonlinear optics, with

a study of solitons, noise in measurements, and squeezing.

Reference material: book

In the semiclassical approach, we use classical Maxwell's equations to describe light propagation,

Powerpoint version of Lecture 1, January 17, 2024, including syllabus

```
From real field to complex representation - You can forget the comlplex conjugate in linear optics. You cannot in nonlinear optics. See homework 1.
Polarization instantaneous in linear optics. But nothing is truly instantaneous.
Maxwell's propagation equation is second order - we work with a first order equation.
```

Lecture 2, January 22, electron response, ATI, tunneling, Drude model

Lecture 3, January 24,Multiphoon ionization versus tunnel ionization; electron trajectories after ionization, linear versus circular polarization, recollision.

Lecture 4: January 29. More on attosecond generation. There are two recollisions/light period, leading to a frequency comb with 2 photon energy as mode spacing.

Polarization gating: method to extract a single attosecond pulse. (Homework 2)

```
Lecture 5: January 31. Kramers-Kroenig relations applied to Lorentz lines and Gaussian beams. Discussion about absorption, dispersion and group dispersion.
Warning about numerical packages.
```

**SEMI-CLASSICAL APPROACH TO NONLINEAR OPTICS. Book Chapter 3**

Derivations of Bloch's equations for a two level system. This is the approach followed by

R. P. Feynman and F. L. Vernon and R. W. Hellwarth, "Geometrical representation of the Schroedinger equation for solving maser problems",J.~Appl.~Phys., 28: 49-52 (1957).

The following 7 through 12 deal with coherent effects

Lecture 7: More on two-level systems, - self-induced transparency, 2 pi pulses, zero zrea pulses, photon echoes

Lecture 8: Free-induction decay; the absorbed energy is the overlap of the pulse spectrum and the inhomogeneous line for weak pulses; frequency pushing.

Lecture 9: Multilevel systems, starting with cascade multiphoton transitions, appropriate for molecules. Next two-photon coheent excitation; application to the sodium guidestar.

Lecture 10: Multiphoton reduced to two levels, adiabatic approximation.

Lecture 11:Coherent harmonic generation.

Lecture 12: More 2-photon coherent effects. Nonlinear index is the Kramers'Kroenig correspondant of two photon absorption. Experiment to determine the phase relaxation of a liquid dye identiies a 2-photon rather than single photon coherence (DFWM).

Raman as 2-photon coherent interaction. Vibrational Raman. Impulsive Raman. Rotational-Vibrational excitation.

Reading: Weiner 1990: Impulsive Raman scattering by pulse sequences.

**STEADY-STATE LIMIT OF THE PREVIOUS LECTURES: TRADITIONAL NONLINEAR OPTICS Book Chapter 4**

Lecture 13: Rotational spectra of diatomics. Pump-probe experiments on the nitrogen cation. Brillouin sattering. Application to high power pulse compression.

Lecture 14: Third order susceptibility (instantaneous_), nonlinear index, self-phase modulation, shock, self focusing, self trapping, self-filamentation.

Lecture 15: Filaments in air - various physical models.

Lecture16: Degenerate four wave mixing, phase conjugation.

Lecture 17: Phase matching, parametric generation, phase/group maching, Second harmonic generaion and sum frequency.

Lecture 18: Second harmonic generaion and sum frequency; crystal optics.

Lecture 19: Optical parametric oscillaators; pump and OPO cavities synchronization

## Quantum

lect21 Wigner to Harmonic_oscillator

**LAB TOUR: CHTM 163A - 161B**

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HOMEWORK

- Homework 1
- Homework 2 and solution2
- Homework 3 and solution3
- Homework 4 and solution4
- Homework 5

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PRESENTATION

Stone, Brandon Dispersive solitons April 22

Jalouli, Alireza SHG of 2D semiconductors April 22

Hooper, Charles Thomas two-color and THz generation via laser filamentation.April 24

Shock, Stephen Raman Scattering within hollow core fibers. April 24

Islam, Md Sakibul Attosecond polarization control April 29

Garcia, Alejandro Attosecond polarization control April 29

Zamani, Elaheh Ultrastable mode-locked lasers? May 1st

Parker, Meagan Atomic traps? May 1st

Gratings in air - Cannot upload zipped file - I can send them by e-mail if someone is interested.