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Fitting of dielectric function

Frequency dependent dielectric function tex:\varepsilon(\omega) cannot be specified in FDTD in tabular form.

However, it can be substituted into FDTD scheme using following approximation

tex:\varepsilon(\omega) = \varepsilon_{\infty} + \sum_{p=1}^P\varepsilon_p(\omega),

tex:\varepsilon_p(\omega) = \frac{a_{p,0}+ia_{p,1}(-i\omega)}{b_{p,0}+b_{p,1}(-i\omega)-b_{p,2}\omega^2}

Number of terms tex:P and coefficients tex:\varepsilon_{\infty}, tex:a_{p,j}, tex:b_{p,j} should be chosen in order to approximate given tex:\varepsilon(\omega) with sufficient accuracy and do not necessary have a physical meaning.

In physical literature following models are commonly used:

  • Debye term tex:\frac{\Delta \varepsilon}{1-2i\omega\gamma_p}
  • Drude term tex:\frac{\Delta \varepsilon \omega_p^2}{-2i\omega\gamma_p-\omega^2}
  • Lorentz term tex:\frac{\Delta \varepsilon \omega_p^2}{\omega_p^2-2i\omega\gamma_p-\omega^2}

Case tex:a_{p,1} \ne 0 does not correspond to any of physical model, but allows to obtain more accurate fittings. For example, two (tex:P=2) terms of this case are sufficient to fit silicon dielectric function over the wavelength range from 300 to 1000 nm, whereas even a large number of Debye, Drude or Lorentz terms (tex:a_{p,1}=0) is inadequate there 1). Previous fitting for silicon by 3 Lorentz terms (see paper on textured antireflective coatings 2)) is accurate only for the visible range and no fitting with Lorentz terms was found for both visible and near ultraviolet ranges.

The way to substitute Debye, Drude and Lorentz terms into FDTD scheme is described in 3)). FDTD scheme for tex:a_{p,1} \ne 0 terms using ADE (auxliliary differential equation) technique can be found in our work 4).

You can fit arbitrary dielectric function with simple MatLab program that can be download from here. This program is well commented and easy to understand. You should specify all necessary parameters (number of terms tex:P, file with tabular dielectric function, etc.) in file 'fitting.m' . Initial settings in 'fitting.m' were used to fit experimental data for silicon dielectric permittivity.

1) , 4) A. Deinega and S. John, “Effective optical response of silicon to sunlight in the finite-difference time-domain method,” Opt. Lett. 37, 112-114 (2012) http PDF
2) A. Deinega, I. Valuev, B. Potapkin and Yu. Lozovik, “Minimizing light reflection from dielectric textured surfaces,” JOSA A 28, 770-777 (2011) http PDF
3) A. Taflove and S. H. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method, Artech House, Boston (2005
 
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