Differences
This shows you the differences between two versions of the page.
en:fitting [2012/02/01 19:32] deinega |
en:fitting [2012/10/15 18:37] (current) deinega |
||
---|---|---|---|
Line 1: | Line 1: | ||
+ | ======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>. */ | ||
+ | |||