Some Brief Mathematical Notes
Below are some brief notes I've written up for my students and for
use in my courses, especially 18.369 (Mathematical
Methods in Nanophotonics) and 18.336 (Numerical
Methods for PDEs).
For the most part, they are reviews and introductions to topics
that appear in the literature, but may not have a compact pedagogical
presentation that includes all of the information that I would like my
students to be aware of. The references in the notes are by no means
exhaustive (these are not formal review papers), but I try to give at
least a pointer or two for more information.
- Notes on adjoint methods — adjoint methods provide ways to evaluate gradients
of complicated functions quickly, and are very important for
optimization and sensitivity analysis. Uses Matlab code here.
- Notes on the algebraic
structure of wave equations — a few notes pointing out how all of
the common wave equations can be written in a particular
anti-Hermitian algebraic form, from which their properties can be
derived in a unified way. (In some sense, this is the
definition of a wave equation.)
- Notes on total internal reflection and waveguides in "slow" media, describing an elementary variational proof of the existence of guided modes that is closely connected to analytical methods for localization of bound states in potential wells of quantum mechanics.
- Notes on Perfectly Matched Layers
(PML) (arXiv:2108.05348) — discussion of various viewpoints on PML absorbing
layers, and their properties and limitation, focusing on the
fundamentals rather than on the detailed implementation in, for
example, FDTD. (See also
Notes on PML in
Meep for the specific application of these ideas to our Meep Maxwell's equation FDTD
- Notes on coordinate
transformation and invariance in electromagnetism — a
concise derivation of a beautiful result by Ward & Pendry, showing
that any coordinate transformation of Maxwell's equations can be
absorbed into a change of materials ε and μ. (This proof
is also published as an appendix in a 2008 paper
by Kottke, Farjadpour, and Johnson in Physical Review E.)
- Notes on solar-weighted integration and accompanying Julia code: construction of accurate Gaussian quadrature schemes for integrating functions weighted by the solar irradiance spectrum (e.g. for solar-cell calculations).
- Notes on the convergence of
trapezoidal-rule quadrature: the error rate of trapezoidal-rule
quadrature, and by extension Clenshaw-Curtis quadrature, can be linked
in an elementary way to the convergence rate of a Fourier series and
hence to the smoothness of a function (with an
accompanying handout on cosine series).
- When functions have no value(s):
Delta functions and distributions. A gentle introduction to what
delta functions really are, weak derivatives, and other
distribution concepts that replace the classical "function" notion,
from my undergraduate PDE course.
- A brief survey of
computational photonics, excerpted from a draft manuscript of the
upcoming Photonic Crystals: Molding the Flow of Light, second
edition (published in spring 2008 and available online in its
- Saddle-point integration of
C∞ "bump" functions (arXiv:1508.04376) — discussion of
asymptotic Fourier analysis of the C∞ (infinitely
differentiable) "bump" functions that often appear in analysis, using
saddle-point methods (other common asymptotic methods fail here
because of the essential singularity in the integrand).
- Notes on a discontinuous f(x) satisfying
f(x+y) = f(x)⋅f(y) — a relatively elementary
discussion of how one can construct (though not explicitly) a
non-exponential discontinuous function satisfying
f(x+y) = f(x)⋅f(y), a simple real-analysis question
that is surprisingly non-trivial to answer
- Notes on modified Cooley-Tukey algorithms based
on a generalized-DFT framework, describing an interesting (and
potentially useful) class of FFT algorithms, blending decimation in
time and frequency Cooley-Tukey, that use generalized DFTs as
subproblems rather than DFTs. (This algorithm is not published, as
far as I know, at least not at this level of generality.)
- Notes on FFT-based (spectral)
differentiation, which is often a source of confusion because of aliasing.
- Coupling of modes and adiabatic processes
in dynamic eigenproblems: An undergraduate-level introduction to
the adiabatic theorem and eigenvector expansions in linear ODEs of the
form ∂x/∂t=Ax where A(t) is matrix that varies
slowly in time.