Meep Tutorial/Material dispersion

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In this example, we will perform a simulation with a frequency-dependent dielectric ε(ω), corresponding to material dispersion. (See Dielectric materials in Meep for more information on how material dispersion is supported in Meep.) In particular, we will model a uniform medium of the dispersive material; see also the material-dispersion.ctl file included with Meep. From the dispersion relation ω(k), we will compute the numerical ε(ω) via the formula:

\varepsilon(\omega) = \left( \frac{k}{\omega} \right) ^2

We will then compare this with the analytical ε(ω) that we specified.

Since this is a uniform medium, our computational cell can actually be of zero size (i.e. one pixel), where we will use Bloch-periodic boundary conditions to specify the wavevector k.

(set! geometry-lattice (make lattice (size no-size no-size no-size)))

We will then fill all space with a dispersive material:

(set! default-material
      (make dielectric (epsilon 2.25)
            (polarizations 
             (make polarizability
               (omega 1.1) (gamma 1e-5) (delta-epsilon 0.5))
             (make polarizability
               (omega 0.5) (gamma 0.1) (delta-epsilon 2e-5))
             )))


corresponding to the dielectric function:

\varepsilon(\omega) = 2.25 + \frac{1.1^2 \cdot 0.5}{1.1^2 - \omega^2 -i\omega \cdot 10^{-5}} + \frac{0.5^2 \cdot 2\cdot 10^{-5}}{0.5^2 - \omega^2 -i\omega \cdot 0.1}

The real and imaginary parts of this dielectric function ε(ω) are plotted below:

Real and imaginary parts of specified analytical ε(ω).

Here, we can see that the ω=1.1 resonance causes a large change in both the real and imaginary parts of ε around that frequency. In fact, there is a range of frequencies from 1.1 to 1.2161 where ε is negative. In this range, no propagating modes exist—it is actually a kind of photonic band gap associated with polariton resonances in a material. (For more information on the physics of such materials, see e.g. chapter 10 of Introduction to Solid State Physics by C. Kittel.)

On the other hand, the ω=0.5 resonance, because the delta-epsilon numerator is so small, causes very little change in the real part of ε. On the other hand, it generates a clear peak in the imaginary part of ε, corresponding to a resonant absorption peak.

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