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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 Spectral_line_shape#Lorentzian, non-Lorentzian broadening, such as metals and amorphous insulators, across broad spectral ranges, typically near-ultraviolet, visible spectrum, 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 Herm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Oscillator Model
The Lorentz oscillator model describes the optical response of bound charges. The model is named after the Dutch physicist Hendrik Antoon Lorentz. It is a classical, phenomenological model for materials with characteristic resonance frequencies (or other characteristic energy scales) for optical absorption, e.g. ionic and molecular vibrations, interband transitions (semiconductors), phonons, and collective excitations. Derivation of electron motion The model is derived by modeling an electron orbiting a massive, stationary nucleus as a spring-mass-damper system. The electron is modeled to be connected to the nucleus via a hypothetical spring and its motion is damped by via a hypothetical damper. The damping force ensures that the oscillator's response is finite at its resonance frequency. For a time-harmonic driving force which originates from the electric field, Newton’s second law can be applied to the electron to obtain the motion of the electron and expressions for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condensed Matter Physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the subject deals with "condensed" phases of matter: systems of many constituents with strong interactions between them. More exotic condensed phases include the superconducting phase exhibited by certain materials at low temperature, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, and the Bose–Einstein condensate found in ultracold atomic systems. Condensed matter physicists seek to understand the behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics, electromagnetism, statistical mechanics, and other theories to develop mathematical models. The diversity of systems and phenomena available for study makes condensed matter phy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tauc–Lorentz Model
The Tauc–Lorentz 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 the complex refractive index of amorphous semiconductor materials at frequencies greater than their optical band gap. The dispersion relation bears the names of Jan Tauc and Hendrik Lorentz, whose previous works were combined by G. E. Jellison and F. A. Modine to create the model. The model was inspired, in part, by shortcomings of the Forouhi–Bloomer model, which is aphysical due to its incorrect asymptotic behavior and non-Hermitian character. Despite the inspiration, the Tauc–Lorentz model is itself aphysical due to being non-Hermitian and non-analytic in the upper half-plane. Further researchers have modified the model to address these shortcomings. Mathematical formulation The general form of the model is given by :\varepsilon(E) = \varepsilon_ + \chi^(E) where * \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forouhi–Bloomer Model
The Forouhi–Bloomer model is a mathematical formula for the frequency dependence of the complex-valued refractive index. The model can be used to fit the refractive index of amorphous and crystalline semiconductor and dielectric materials at energies near and greater than their optical band gap. The dispersion relation bears the names of Rahim Forouhi and Iris Bloomer, who created the model and interpreted the physical significance of its parameters. The model is aphysical due to its incorrect asymptotic behavior and non-Hermitian character. These shortcomings inspired modified versions of the model as well as development of the Tauc–Lorentz model. Mathematical Formulation The complex refractive index is given by : \tilde(E) = n(E) + i \kappa(E) where * n is the real component of the complex refractive index, commonly called the refractive index, * \kappa is the imaginary component of the complex refractive index, commonly called the extinction coefficient, * E is the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sellmeier Equation
The Sellmeier equation is an empirical relationship between refractive index and wavelength for a particular transparent medium. The equation is used to determine the dispersion of light in the medium. It was first proposed in 1872 by Wolfgang Sellmeier and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion. The equation In its original and the most general form, the Sellmeier equation is given as : n^2(\lambda) = 1 + \sum_i \frac , where ''n'' is the refractive index, ''λ'' is the wavelength, and ''B''i and ''C''i are experimentally determined ''Sellmeier coefficients''. These coefficients are usually quoted for λ in micrometres. Note that this λ is the vacuum wavelength, not that in the material itself, which is λ/n. A different form of the equation is sometimes used for certain types of materials, e.g. crystals. Each term of the sum representing an absorption resonance of strength ''B''i at a wavelength . For example, the coe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Equation
In optics, Cauchy's transmission equation is an empirical relationship between the refractive index and wavelength of light for a particular transparent material. It is named for the mathematician Augustin-Louis Cauchy, who defined it in 1837. The equation The most general form of Cauchy's equation is : n(\lambda) = A + \frac + \frac + \cdots, where ''n'' is the refractive index, λ is the wavelength, ''A'', ''B'', ''C'', etc., are coefficients that can be determined for a material by fitting the equation to measured refractive indices at known wavelengths. The coefficients are usually quoted for λ as the vacuum wavelength in micrometres. Usually, it is sufficient to use a two-term form of the equation: : n(\lambda) = A + \frac, where the coefficients ''A'' and ''B'' are determined specifically for this form of the equation. A table of coefficients for common optical materials is shown below: The theory of light-matter interaction on which Cauchy based this equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faddeeva Function
The Faddeeva function or Kramp function is a scaled complex complementary error function, :w(z):=e^\operatorname(-iz) = \operatorname(-iz) =e^\left(1+\frac\int_0^z e^\textt\right). It is related to the Fresnel integral, to Dawson's integral, and to the Voigt function. The function arises in various physical problems, typically relating to electromagnetic responses in complicated media. * problems involving small-amplitude waves propagating through Maxwellian plasmas, and in particular appears in the plasma's permittivity from which dispersion relations are derived, hence it is sometimes referred to as the plasma dispersion function (although this name is sometimes used instead for the rescaled function defined by ''Fried and Conte'', 1961). * the infrared permittivity functions of amorphous oxides have resonances (due to phonons) that are sometimes too complicated to fit using simple harmonic oscillators. The Brendel–Bormann oscillator model uses an infinite superposition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit time (for example, in rotation) or the rate of change of the phase of a sinusoidal waveform (for example, in oscillations and waves), or as the rate of change of the argument of the sine function. Angular frequency (or angular speed) is the magnitude of the pseudovector quantity angular velocity.(UP1) One turn is equal to 2''π'' radians, hence \omega = \frac = , where: *''ω'' is the angular frequency (unit: radians per second), *''T'' is the period (unit: seconds), *''f'' is the ordinary frequency (unit: hertz) (sometimes ''ν''). Units In SI units, angular frequency is normally presented in radians per second, even when it does not express a rotational value. The unit hertz (Hz) is dimensionally equivalent, but by convention it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Permittivity
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insulator measures the ability of the insulator to store electric energy in an electrical field. Permittivity is a material's property that affects the Coulomb force between two point charges in the material. Relative permittivity is the factor by which the electric field between the charges is decreased relative to vacuum. Likewise, relative permittivity is the ratio of the capacitance of a capacitor using that material as a dielectric, compared with a similar capacitor that has vacuum as its dielectric. Relative permittivity is also commonly known as the dielectric constant, a term still used but deprecated by standards organizations in engineering as well as in chemistry. Definition Relative permittivity is typically denoted as (sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Function
In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: :f^*(x) = f(-x) (where the ^* indicates the complex conjugate) for all x in the domain of f. In physics, this property is referred to as PT symmetry. This definition extends also to functions of two or more variables, e.g., in the case that f is a function of two variables it is Hermitian if :f^*(x_1, x_2) = f(-x_1, -x_2) for all pairs (x_1, x_2) in the domain of f. From this definition it follows immediately that: f is a Hermitian function if and only if * the real part of f is an even function, * the imaginary part of f is an odd function. Motivation Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform: * The function f is real-valued if and only if the Fourier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
