Fitting of dielectric function

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======Fitting of dielectric function======

Frequency dependent dielectric function <tex>\varepsilon(\omega)</tex> 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>

<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}
</tex>

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

You can fit arbitrary dielectric function with {{:fitting.zip|fitting program}} written on MatLab.
This program is well commented and easy to understand.
You should specify all necessary parameters (number of terms <tex>P</tex>, 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.

In physical literature following models are commonly used:
  *Debye term <tex>\frac{\Delta \varepsilon}{1-2i\omega\gamma_p}</tex>
  *Drude term <tex>\frac{\Delta \varepsilon \omega_p^2}{-2i\omega\gamma_p-\omega^2}</tex>
  *Lorentz term <tex>\frac{\Delta \varepsilon \omega_p^2}{\omega_p^2-2i\omega\gamma_p-\omega^2}</tex>
  *modified Lorentz term <tex>\frac{\Delta \varepsilon (\omega_p^2 - i\omega\gamma_p')}{\omega_p^2-2i\omega\gamma_p-\omega^2}</tex>

Using modified Lorentz term allows to obtain more accurate fittings.
For example, two (<tex>P=2</tex>) modified Lorentz terms 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</tex>) is inadequate there 
((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://www.opticsinfobase.org/ol/abstract.cfm?uri=ol-37-1-112|http]]
{{:deinega_-_effective_optical_response_of_silicon_to_sunlight_in_the_fdtd_method.pdf|PDF}})).
Previous fitting for silicon by 3 Lorentz terms (see paper on textured antireflective coatings 
((A. Deinega, I. Valuev, B. Potapkin and Yu. Lozovik, "Minimizing light reflection from dielectric textured surfaces," JOSA A 28, 770-777 (2011) 
[[http://www.opticsinfobase.org/abstract.cfm?URI=josaa-28-5-770|http]] 
{{:deinega_-_minimizing_light_reflection_from_dielectric_textured_surfaces.pdf|PDF}}))) 
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 
((A. Taflove and S. H. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method, Artech House, Boston (2005) )).
FDTD scheme for modified Lorentz terms using ADE (auxliliary differential equation) technique can be found in our work
((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://www.opticsinfobase.org/ol/abstract.cfm?uri=ol-37-1-112|http]]
{{:deinega_-_effective_optical_response_of_silicon_to_sunlight_in_the_fdtd_method.pdf|PDF}})).


/* ((M. A. Green and M. Keevers, Optical properties of intrinsic silicon at 300 K, Progress in Photovoltaics 3, 189 (1995)
[[http://onlinelibrary.wiley.com/doi/10.1002/pip.4670030303/abstract|http]]))
(see file 'si.dat' in archive) with two terms <tex>a_{p,1} \ne 0</tex>. */