Brendel–Bormann oscillator model

Brendel-Bormann oscillator model. The real (blue dashed line) and imaginary (orange solid line) components of relative permittivity are plotted for a single oscillator model with parameters <math>\omega_{0}</math> = 500 cm<math>^{-1}</math>, <math>s</math> = 0.25 cm<math>^{-2}</math>, <math>\Gamma</math> = 0.05 cm<math>^{-1}</math>, and <math>\sigma</math> = 0.25 cm<math>^{-1}</math>.

The Brendel–Bormann oscillator model is a mathematical formula for the frequency dependence of the complex-valued relative permittivity, sometimes referred to as the dielectric function. The model has been used to fit to the complex refractive index of materials with absorption lineshapes exhibiting non-Lorentzian broadening, such as metals<ref name = "Rakić1998"/> and amorphous insulators,<ref name = "Brendel1992"/><ref name = "Naiman1984"/><ref name = "Kučírková1994"/><ref name = "Hobert1996"/> across broad spectral ranges, typically near-ultraviolet, visible, and infrared frequencies. The dispersion relation bears the names of R. Brendel and D. Bormann, who derived the model in 1992,<ref name = "Brendel1992"/> despite first being applied to optical constants in the literature by Andrei M. Efimov and E. G. Makarova in 1983.<ref name = "Efimov1983"/><ref name = "Efimov1985"/><ref name = "Efimov1996"/> Around that time, several other researchers also independently discovered the model.<ref name = "Naiman1984"/><ref name = "Kučírková1994"/><ref name = "Hobert1996"/> The Brendel-Bormann oscillator model is aphysical because it does not satisfy the Kramers–Kronig relations. The model is non-causal, due to a singularity at zero frequency, and non-Hermitian. These drawbacks inspired J. Orosco and C. F. M. Coimbra to develop a similar, causal oscillator model.<ref name = "Orosco2018"/><ref name = "Orosco2018a"/>

Mathematical formulation

The general form of an oscillator model is given by<ref name = "Brendel1992"/>

<math>\varepsilon(\omega) = \varepsilon_{\infty} + \sum_{j} \chi_{j}</math>

where

• <math>\varepsilon</math> is the relative permittivity,
• <math>\varepsilon_{\infty}</math> is the value of the relative permittivity at infinite frequency,
• <math>\omega</math> is the angular frequency,
• <math>\chi_{j}</math> is the contribution from the <math>j</math>th absorption mechanism oscillator.

The Brendel-Bormann oscillator is related to the Lorentzian oscillator <math>\left(\chi^{L}\right)</math> and Gaussian oscillator <math>\left(\chi^{G}\right)</math>, given by

<math>\chi_{j}^{L}(\omega; \omega_{0,j}) = \frac{s_{j}}{\omega_{0,j}^{2} - \omega^{2} - i \Gamma_{j} \omega} </math>

<math>\chi_{j}^{G}(\omega) = \frac{1}{\sqrt{2 \pi} \sigma_{j}} \exp{\left[ -\left( \frac{\omega}{\sqrt{2} \sigma_{j}} \right)^{2} \right]}</math>

where

• <math>s_{j}</math> is the Lorentzian strength of the <math>j</math>th oscillator,
• <math>\omega_{0,j}</math> is the Lorentzian resonant frequency of the <math>j</math>th oscillator,
• <math>\Gamma_{j}</math> is the Lorentzian broadening of the <math>j</math>th oscillator,
• <math>\sigma_{j}</math> is the Gaussian broadening of the <math>j</math>th oscillator.

The Brendel-Bormann oscillator <math>\left(\chi^{BB}\right)</math> is obtained from the convolution of the two aforementioned oscillators in the manner of

<math>\chi_{j}^{BB}(\omega) = \int_{-\infty}^{\infty} \chi_{j}^{G}(x-\omega_{0,j}) \chi_{j}^{L}(\omega; x) dx</math>,

which yields

<math>\chi_{j}^{BB}(\omega) = \frac{i \sqrt{\pi} s_{j}}{2 \sqrt{2} \sigma_{j} a_{j}(\omega)} \left[ w\left( \frac{a_{j}(\omega) - \omega_{0,j}}{\sqrt{2}\sigma_{j}} \right) + w\left( \frac{a_{j}(\omega) + \omega_{0,j}}{\sqrt{2}\sigma_{j}} \right) \right]</math>

where

• <math>w(z)</math> is the Faddeeva function,
• <math>a_{j} = \sqrt{\omega^{2}+i \Gamma_{j} \omega}</math>.

The square root in the definition of <math>a_{j}</math> must be taken such that its imaginary component is positive. This is achieved by:

<math>\Re\left( a_{j} \right) = \omega \sqrt{\frac{\sqrt{1+\left( \Gamma_{j}/\omega \right)^{2}}+1}{2}}</math>

<math>\Im\left( a_{j} \right) = \omega \sqrt{\frac{\sqrt{1+\left( \Gamma_{j}/\omega \right)^{2}}-1}{2}}</math>

See also

• Cauchy equation
• Sellmeier equation
• Forouhi–Bloomer model
• Tauc–Lorentz model
• Lorentz oscillator model