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^[1] and amorphous insulators,^[2][3][4][5] 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,^[2] despite first being applied to optical constants in the literature by Andrei M. Efimov and E. G. Makarova in 1983.^[6][7][8] Around that time, several other researchers also independently discovered the model.^[3][4][5] 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.^[9][10]

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 ${\displaystyle \omega _{0}}$ = 500 cm${\displaystyle ^{-1}}$, ${\displaystyle s}$ = 0.25 cm${\displaystyle ^{-2}}$, ${\displaystyle \Gamma }$ = 0.05 cm${\displaystyle ^{-1}}$, and ${\displaystyle \sigma }$ = 0.25 cm${\displaystyle ^{-1}}$.

Mathematical formulation

The general form of an oscillator model is given by^[2]

${\displaystyle \varepsilon (\omega )=\varepsilon _{\infty }+\sum _{j}\chi _{j}}$

where

• ${\displaystyle \varepsilon }$ is the relative permittivity,
• ${\displaystyle \varepsilon _{\infty }}$ is the value of the relative permittivity at infinite frequency,
• ${\displaystyle \omega }$ is the angular frequency,
• ${\displaystyle \chi _{j}}$ is the contribution from the ${\displaystyle j}$th absorption mechanism oscillator.

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

${\displaystyle \chi _{j}^{L}(\omega$ ;\omega _{0,j})={\frac {s_{j}}{\omega _{0,j}^{2}-\omega ^{2}-i\Gamma _{j}\omega }}}

${\displaystyle \chi _{j}^{G}(\omega )={\frac {1}{{\sqrt {2\pi }}\sigma _{j}}}\exp {\left[-\left({\frac {\omega }{{\sqrt {2}}\sigma _{j}}}\right)^{2}\right]}}$

where

• ${\displaystyle s_{j}}$ is the Lorentzian strength of the ${\displaystyle j}$th oscillator,
• ${\displaystyle \omega _{0,j}}$ is the Lorentzian resonant frequency of the ${\displaystyle j}$th oscillator,
• ${\displaystyle \Gamma _{j}}$ is the Lorentzian broadening of the ${\displaystyle j}$th oscillator,
• ${\displaystyle \sigma _{j}}$ is the Gaussian broadening of the ${\displaystyle j}$th oscillator.

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

${\displaystyle \chi _{j}^{BB}(\omega )=\int _{-\infty }^{\infty }\chi _{j}^{G}(x-\omega _{0,j})\chi _{j}^{L}(\omega ;x)dx}$,

which yields

${\displaystyle \chi _{j}^{BB}(\omega )={\frac {i{\sqrt {\pi }}s_{j}}{2{\sqrt {2}}\sigma _{j}a_{j}(\omega )}}\left[w\left({\frac {a_{j}(\omega )-\omega _{0,j}}{{\sqrt {2}}\sigma _{j}}}\right)+w\left({\frac {a_{j}(\omega )+\omega _{0,j}}{{\sqrt {2}}\sigma _{j}}}\right)\right]}$

where

• ${\displaystyle w(z)}$ is the Faddeeva function,
• ${\displaystyle a_{j}={\sqrt {\omega ^{2}+i\Gamma _{j}\omega }}}$.

The square root in the definition of ${\displaystyle a_{j}}$ must be taken such that its imaginary component is positive. This is achieved by:

${\displaystyle \Re \left(a_{j}\right)=\omega {\sqrt {\frac {{\sqrt {1+\left(\Gamma _{j}/\omega \right)^{2}}}+1}{2}}}}$

${\displaystyle \Im \left(a_{j}\right)=\omega {\sqrt {\frac {{\sqrt {1+\left(\Gamma _{j}/\omega \right)^{2}}}-1}{2}}}}$