What is a good book to self-teach optics? I'm interested in camera lenses and how things like barrel distortion and lens flare happens, but know that one cannot just start there.  I'm a math major and I've worked out of Strauss PDE book so I know about the wave equation and some stuff beyond as well.  Are there any good books out there for me to self-teach so I can at least begin to understand all this?
 A: As @CuriousOne says, both the effects you refer to are explicable with ray optics. But both are subtle insofar that you won't find easy expressions for them in any undergrad physics course: you actually have to do accurate numerical ray simulations to see that spherical lenses impose geometrical distortion (barrelling and so forth). Very roughly, geometric distortion arises simply because Snell's law is $n_j\,\sin\theta_j = const$ at interfaces instead of $n_j\,\theta_j = const$. The latter, approximate form would mean that a lens system would impose conformal, linear maps on the object plane. The higher order terms in the sine Taylor series mess up conformalhood and linearity more and more as the ray angles increase, i.e. the system deviates from the so called paraxial approximation more and more.
There are rules of thumb - not very accurate - that designers used to estimate geometric distortion in the old days before computer design. All of these rules of thumb are grounded on the Seidel distortion measure, which is the geometric distortion that arises from spherical surface lenses calculated by assuming that Snells law is $n_j (\theta_j - \frac{\theta_j^3}{3!}) = const$. The five Seidel aberrations, of which geometric distortion is one, are treated thoroughly in Chapter 5 "Geometrical Theory of Aberration" in:
Born, M and Wolf, E, "Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light", Seventh edition with errors corrected 2002, Cambridge University Press
Ludwig von Seidel was the originator of this treatment, although he is not well kenned outside the German speaking world. I don't think there are English translations of his works and Born and Wolf are his most thorough messengers.
Lens flare is almost always modelled numerically if it is modelled at all. There isn't really a systematic treatment as the phenomenon itself is a mess. It arises from scattering from unwanted centers in lens glasses, from multiple reflexions in poorly coated lenses and scattering off lens train hardware - e.g. edges of apertures. Simply put, some unwanted light path forms a bright spot, much brighter than the surrounding image, and the affected pixels saturate given the relatively high CCD sensitivity you need to glean the rest of your image, which is at much lower intensity than the flare. You need a really advanced optical design software, that can handle multiple beam paths and reflexions, to analyze this phenomenon. Most designers treat lens flare like electrical engineers treat electromagnetic interference: it's hard to model and best reduced by following "codes of practice"- standards that dictate quality of blacking of lens hardware (i.e. making them rough and black so that they don't scatter) and similar flare mitigation measures.
