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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

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

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

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

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

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

${\displaystyle \chi _{j}^{BB}(\omega )={\frac {i{\sqrt {\pi }}s_{j}}{2{\sqrt {2}}\sigma _{j}a_{j}(\omega )}}\left}$

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:

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

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

References

1. Rakić, Aleksandar D.; Djurišić, Aleksandra B.; Elazar, Jovan M.; Majewski, Marian L. (1998). "Optical properties of metallic films for vertical-cavity optoelectronic devices". Applied Optics. 37 (22): 5271–5283. Bibcode:1998ApOpt..37.5271R. doi:10.1364/AO.37.005271. PMID 18286006. Retrieved 2021-10-13.
2. ^ Brendel, R.; Bormann, D. (1992). "An infrared dielectric function model for amorphous solids". Journal of Applied Physics. 71 (1): 1–6. Bibcode:1992JAP....71....1B. doi:10.1063/1.350737. Retrieved 2021-10-13.
3. ^ Naiman, M. L.; Kirk, C. T.; Aucoin, R. J.; Terry, F. L.; Wyatt, P. W.; Senturia, S. D. (1984). "Effect of Nitridation of Silicon Dioxide on Its Infrared Spectrum". Journal of the Electrochemical Society. 131 (3): 637–640. Bibcode:1984JElS..131..637N. doi:10.1149/1.2115648. Retrieved 2021-10-20.
4. ^ Kučírková, A.; Navrátil, K. (1994). "Interpretation of Infrared Transmittance Spectra of SiO2 Thin Films". Applied Spectroscopy. 48 (1): 113–120. Bibcode:1994ApSpe..48..113K. doi:10.1366/0003702944027534. S2CID 98613649. Retrieved 2021-10-20.
5. ^ Hobert, H.; Dunken, H. H. (1996). "Modelling of dielectric functions of glasses by convolution". Journal of Non-Crystalline Solids. 195 (1–2): 64–71. Bibcode:1996JNCS..195...64H. doi:10.1016/0022-3093(95)00517-X. Retrieved 2021-10-20.
6. Efimov, Andrei M.; Makarova, E. G. (1983). "". Proc. Seventh All-Union Conf. on Vitreous State (in Russian). pp. 165–71.
7. Efimov, Andrei M.; Makarova, E. G. (1985). "". Fiz. Khim. Stekla [The Soviet Journal of Glass Physics and Chemistry] (in Russian). 11 (4): 385–401.
8. Efimov, A. M. (1996). "Quantitative IR spectroscopy: Applications to studying glass structure and properties". Journal of Non-Crystalline Solids. 203: 1–11. Bibcode:1996JNCS..203....1E. doi:10.1016/0022-3093(96)00327-4. Retrieved 2021-10-13.
9. Orosco, J.; Coimbra, C. F. M. (2018). "On a causal dispersion model for the optical properties of metals". Applied Optics. 57 (19): 5333–5347. Bibcode:2018ApOpt..57.5333O. doi:10.1364/AO.57.005333. PMID 30117825. S2CID 51760671. Retrieved 2021-10-14.
10. Orosco, J.; Coimbra, C. F. M. (2018). "Optical response of thin amorphous films to infrared radiation". Physical Review B. 97 (9): 094301. Bibcode:2018PhRvB..97i4301O. doi:10.1103/PhysRevB.97.094301. Retrieved 2021-10-14.

See also

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

• Condensed matter physics
• Electric and magnetic fields in matter
• Optics