Materials in Meep
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The material structure in Maxwell's equations 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). Material dispersion, in turn, is generally associated with absorption loss in the material, or possibly gain. All of these effects can be simulated in Meep, with certain restrictions.
In this section, we describe the form of the equations and material properties that Meep can simulate. The actual interface with which you specify these properties is described in the Meep reference.
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Material 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, γn, and are user-specified constants and is a user-specified function of position (usually 0 or 1). 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 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).
To implement a Drude model of , in which ωn is zero, we just set ωn to be a very small number (e.g. 1e-20
) and make large to compensate in the numerator.
Loss and gain
If γ above is nonzero, then the dielectric function ε(ω) becomes complex, where the imaginary part is associated with absorption loss in the material if it is positive, or gain if it is negative.
If you look at Maxwell's equations, then plays exactly the same role as a current . Just as is the rate of change of mechanical energy (the power expended by the electric field on moving the currents), therefore, the rate at which energy is lost to absorption is given by:
- absorption rate
Meep can keep track of this energy, which for gain gives the amount of energy expended in amplifying the field. Using this energy, Meep supports the idea of a saturable gain (e.g. a situation in which there is a depletable population inversion causing the gain). For more information, see saturable gain in Meep.
Nonlinearity
In general, ε can be changed anisotropically by the E field itself, with:
where the ij is the index of the change in the 3×3 ε tensor and the χ terms are the nonlinear susceptibilities. The χ(2) sum is the Pockels effect and the χ(3) sum is the Kerr effect. (If the above expansion is frequency-independent, then the nonlinearity is instantaneous; more generally, Δε would depend on some average of the fields at previous times.)
Currently, Meep supports instantaneous, isotropic Kerr nonlinearities, corresponding to a frequency-independent . Thus,
Normally, for nonlinear systems you will want to use real fields E. (This is normally the default. However, Meep uses complex fields if you have Bloch-periodic boundary conditions with a non-zero Bloch wavevector k, or in cylindrical coordinates with . In the C++ interface, real fields must be explicitly specified.) For complex fields in nonlinear systems, the physical interpretration of the above equations is more subtle because one cannot simply obtain the physical solution by taking the real part any more.