Undergraduates Who are Considering Taking Nonlinear Optics
Those undergraduates who are considering taking Nonlinear Optics should read the following.
I introduce everything that is needed along the way, then I build on the previous materials. The more you know, the easier the task of linear Nonlinear Optics; but, if an undergraduate student with solid preparation makes an effort, she or he will be able to get a lot out of the course. I think Miller does a very good job teaching QM, so if you did well in 450, your preparation in quantum will be solid. Below, I outline some of the topics that you may want to fortify before the start of classes:
1. Quantum Mechanics: Since the nonlinear response of a molecule can only be solved using perturbation theory, have at least a feeling for first and second-order time-dependent perturbation theory. When a molecule is inside of a material, it is buffeted around by thermal fluctuations. So, quantum statistical mechanics is used. Thus, in the latter half of the course, we will use density matrices. I will introduce these in a physically intuitive way. Once you understand the idea, applying density matrices is straightforward. As a student, I always had trouble with understanding how to use density matrices because they were always introduced in a very formal way. And finally, I like to use Dirac notation, which is usually introduced in a good undergraduate QM course.
2. E&M: I try to solve all of the wave prorogation stuff from scratch, so even if you have not done it before, it should not be too bad. However, it might make life easier if you go through an E&M textbook to see how you get plane electromagnetic waves from Maxwell's equations. If you are taking E&M II at the same time as NLO, the synergism between the two courses will be a plus for both. Finally, we make extensive use of tensors. I introduce tensors from scratch (though quickly) so you may want to read up on them. Also read about dipole radiation and the interaction energy between an electric field and a dipole.
I will throw in a little bit of quantum field theory and the diagrammatic approach to calculating the nonlinear response, so being familiar with energies of the electromagnetic field and harmonic ladder operators is useful.
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3. Mechanics: Be familiar with the driven harmonic oscillator. We will add a nonlinear term to the differential equation as we do for the static dipole on this web site.
4. Thermal physics: Understand how to use partition functions to calculate mean values of observables.
I don't mind doing some mini tutorials for undergraduates who have a deficiency in a particular area of physics that is important to NLO. Also, it would be beneficial if there were at least two of you in the class so that you could talk to each other and work together. Such interactions are a good way to flush out ideas.
I use Boyd's book on Nonlinear Optics (3rd Edition). Over the three editions, his book is converging to the way I normally teach the course. However, I do not follow it religiously, and, I tend to make up most of my own problems. There is lots of material in Boyd's textbook, so we will cover perhaps 60%. I try to avoid the mathematically complicated stuff (by complicated, I mean lots of algebra that that does not lead to any deeper understanding), and prefer to dwell on the more neat ramifications of the topics that I cover.
I do not think that the math we use is very advanced. We will not prove any esoteric theorems, and will mostly use approximation techniques when we need to solve an equation. Having a knowledge of some linear algebra (as used in quantum mechanics) and partial differential equations (and of course calculus and algebra) should be sufficient.
Nonlinear Optics is often taught from an engineering bent. I prefer to approach it from the perspective of understanding nonlinear-optical phenomena. However, we will certainly talk about applications, but understanding the underlying physics rather than the applications will be our motivation.
In conclusion, a highly motivated undergraduate student with the background I describe above should be able to get through the course. All students will be judged on the same criteria, and grades will be assigned accordingly. |
