Meep Introduction
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Meep implements the 'finite-difference time-domain (FDTD) method for computational electromagnetism. This is a widely used technique in which space is divided into a discrete grid and then the fields are evolved in time using discrete time steps—as the grid and the time steps are made finer and finer, this becomes a closer and closer approximation for the true continuous equations, and one can simulate many practical problems essentially exactly.
In this section, we introduce the equations and the electromagnetic units employed by Meep, the FDTD method, and Meep's approach to FDTD. Also, FDTD is only one of several useful computational methods in electromagnetism, each of which has their own special uses—we mention a few of the other methods, and try to give some hints as to which applications FDTD is best suited for and when you should consider a different method.
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Maxwell's equations
Meep simulates Maxwell's equations, which describe the interactions of electric (E) and magnetic (H) fields with one another and with matter and sources. In particular, the equations for the evolution of the fields are:
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Where D is the displacement field, ε is the dielectric constant, J is the current density, and JB is the magnetic-charge current density. (Magnetic currents are a convenient computational fiction in some situations.) B is the magnetic flux density (often called the magnetic field), but currently μ (the magnetic permeability) is always 1 in Meep so we don't need to distinguish between B and H.
You may have noticed the lack of annoying constants like ε0, μ0, c, and 4π — that's because Meep uses "dimensionless" units where all these constants are unity (you can tell it was written by theorists). As a practical matter, almost everything you might want to compute is expressed as a ratio anyway, so the units end up cancelling; see Units in Meep, below.
Material dispersion and nonlinearity
The material structure is determined by the dielectric function ε(x), but ε is not only a function of position — in general, it also depends on frequency (material dispersion) and on the electric field E itself (nonlinearity). Both effects can be simulated in Meep, with certain restrictions.
Dispersion
Physically, material dispersion arises because the polarization of the material does not respond instantaneously to an applied field E, and this is essentially the way that it is implemented in FDTD. In particular, is expanded to:
where 
is the instantaneous dielectric function (the infinite-frequency response) and P is the polarization density in the material. P, in turn, has its own time-evolution equation, and the exact form of this equation determines the frequency-dependence ε(ω). In particular, Meep supports any material dispersion of the form of a sum of harmonic resonances:
where ωn and γn are user-specified constants and 
is a user-specified function of position. This corresponds to evolving P via the equations:
That is, we must store and evolve a set of auxiliary fields 
along with the electric field in order to keep track of the polarization P. Essentially any ε(ω) could be modeled by including enough of the these polarization fields — Meep allows you to specify any number of these, limited only by computer memory and time (which must increase with the number of polarization terms you require).
Units in Meep
Finite-difference time-domain methods
The illusion of continuity in Meep
Although FDTD inherently uses discretized space and time, as much as possible Meep attempts to maintain the illusion that you are using continuous system.
