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]
![](http://up.wiki.x.io/wikipedia/commons/thumb/0/0d/Brendel-Bormann_Oscillator_Model.png/220px-Brendel-Bormann_Oscillator_Model.png)
Mathematical formulation
editThe general form of an oscillator model is given by[2]
where
- is the relative permittivity,
- is the value of the relative permittivity at infinite frequency,
- is the angular frequency,
- is the contribution from the th absorption mechanism oscillator.
The Brendel-Bormann oscillator is related to the Lorentzian oscillator and Gaussian oscillator , given by
where
- is the Lorentzian strength of the th oscillator,
- is the Lorentzian resonant frequency of the th oscillator,
- is the Lorentzian broadening of the th oscillator,
- is the Gaussian broadening of the th oscillator.
The Brendel-Bormann oscillator is obtained from the convolution of the two aforementioned oscillators in the manner of
- ,
which yields
where
- is the Faddeeva function,
- .
The square root in the definition of must be taken such that its imaginary component is positive. This is achieved by:
References
edit- ^ 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.
- ^ a b c 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.
- ^ a b 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.
- ^ a b 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.
- ^ a b 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.
- ^ Efimov, Andrei M.; Makarova, E. G. (1983). "[Vitreous state and the dispersion theory]". Proc. Seventh All-Union Conf. on Vitreous State (in Russian). pp. 165–71.
- ^ Efimov, Andrei M.; Makarova, E. G. (1985). "[Dispersion equation for the complex equation constant of vitreous solids and dispersion analysis of their reflection spectra]". Fiz. Khim. Stekla [The Soviet Journal of Glass Physics and Chemistry] (in Russian). 11 (4): 385–401.
- ^ 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.
- ^ 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.
- ^ 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.