Brendel–Bormann oscillator model

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 and amorphous insulators, 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, despite first being applied to optical constants in the literature by Andrei M. Efimov and E. G. Makarova in 1983. Around that time, several other researchers also independently discovered the model. 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.

Mathematical formulation
The general form of an oscillator model is given by


 * $$\varepsilon(\omega) = \varepsilon_{\infty} + \sum_{j} \chi_{j}$$

where
 * $$\varepsilon$$ is the relative permittivity,
 * $$\varepsilon_{\infty}$$ is the value of the relative permittivity at infinite frequency,
 * $$\omega$$ is the angular frequency,
 * $$\chi_{j}$$ is the contribution from the $$j$$th absorption mechanism oscillator.

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


 * $$\chi_{j}^{L}(\omega; \omega_{0,j}) = \frac{s_{j}}{\omega_{0,j}^{2} - \omega^{2} - i \Gamma_{j} \omega} $$
 * $$\chi_{j}^{G}(\omega) = \frac{1}{\sqrt{2 \pi} \sigma_{j}} \exp{\left[ -\left( \frac{\omega}{\sqrt{2} \sigma_{j}} \right)^{2} \right]}$$

where
 * $$s_{j}$$ is the Lorentzian strength of the $$j$$th oscillator,
 * $$\omega_{0,j}$$ is the Lorentzian resonant frequency of the $$j$$th oscillator,
 * $$\Gamma_{j}$$ is the Lorentzian broadening of the $$j$$th oscillator,
 * $$\sigma_{j}$$ is the Gaussian broadening of the $$j$$th oscillator.

The Brendel-Bormann oscillator $$\left(\chi^{BB}\right)$$ is obtained from the convolution of the two aforementioned oscillators in the manner of
 * $$\chi_{j}^{BB}(\omega) = \int_{-\infty}^{\infty} \chi_{j}^{G}(x-\omega_{0,j}) \chi_{j}^{L}(\omega; x) dx$$,

which yields
 * $$\chi_{j}^{BB}(\omega) = \frac{i \sqrt{\pi} s_{j}}{2 \sqrt{2} \sigma_{j} a_{j}(\omega)} \left[ w\left( \frac{a_{j}(\omega) - \omega_{0,j}}{\sqrt{2}\sigma_{j}} \right) + w\left( \frac{a_{j}(\omega) + \omega_{0,j}}{\sqrt{2}\sigma_{j}} \right) \right]$$

where
 * $$w(z)$$ is the Faddeeva function,
 * $$a_{j} = \sqrt{\omega^{2}+i \Gamma_{j} \omega}$$.

The square root in the definition of $$a_{j}$$ must be taken such that its imaginary component is positive. This is achieved by:
 * $$\Re\left( a_{j} \right) = \omega \sqrt{\frac{\sqrt{1+\left( \Gamma_{j}/\omega \right)^{2}}+1}{2}}$$
 * $$\Im\left( a_{j} \right) = \omega \sqrt{\frac{\sqrt{1+\left( \Gamma_{j}/\omega \right)^{2}}-1}{2}}$$