Optical Properties of Iron to 30 eV ()

Jerome K. Butler^{1}, Maria Martinez^{2*}, Reyhane Kilci^{1}, Gary A. Evans^{1}

^{1}Southern Methodist University, Dallas, USA.

^{2}Federal Center for Technological Education Celso Suckow da Fonseca, CEFET/RJ, Avenida Maracan?, Rio de Janeiro, Brazil.

**DOI: **10.4236/msa.2021.1212042
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A modified approach closely related to the Drude, Lorentz-Drude and Brendel-Bormann theories is developed to fit the experimental data of the optical properties of metals. This work, while simplifying and redefining the parameters of previous models, can be directly compared with the parameters of the Brendel-Bormann model. As a test of validity, our model is compared with the Brendel-Bormann model and experimental data for gold. Our model shows excellent agreement with the experimental data for gold (up to 5 eV) and iron (up to 30 eV).

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Butler, J. , Martinez, M. , Kilci, R. and Evans, G. (2021) Optical Properties of Iron to 30 eV. *Materials Sciences and Applications*, **12**, 622-638. doi: 10.4236/msa.2021.1212042.

1. Introduction

This work is motivated by the need for the optical characteristics of ferromagnetic materials, especially iron, for use in waveguide optical isolators. These metals may be used as one or more layers of an optical waveguide or they may be used as metal “dopant” atoms or clusters of metal atoms in a host material such as a semiconductor, glass or polymer to form one or more ferromagnetic layers in an optical waveguide [1] [2] [3] [4] [5]. Ferromagnetic materials are anisotropic and are characterized by a susceptibility tensor with non-zero off-diagonal elements whose values change with an applied magnetic field. Designing waveguide isolators requires accurate knowledge of such susceptibility tensors which are obtained from the electronic band structure of materials [6] - [11].

Ehrenheich *et al.* [6] analyzed experimental data for the dielectric constants for silver and copper from 1 to 25 eV with the help of three mechanisms which are free electron effects, interband transitions, and plasma oscillations. In order to distinguish plasma transitions from interband transitions, theoretical values of the real
$\Re \left\{\epsilon \right\}$ and imaginary parts
$\Im \left\{\epsilon \right\}$ of the dielectric constant, as well as the loss function
$\Im \left\{{\epsilon}^{-1}\right\}$ were plotted as a function of photon energy. They acquire average optical mass values for silver and copper for the free electron effect region by combining the theoretical and experimental values of the dielectric constant. In 1987, Adachi used a harmonic oscillator model with a critical point-parabolic band model that incorporated Lorentzian broadening and temperature dependence to find optical constants as a function of alloy composition for Zinc-Blende semiconductors [7]. The resulting model showed that contributions from indirect transitions can be significant [7].

The Drude model for the permittivity (based on free electrons) [12] [13] [14] was extensively used until the late 1980s to obtain the optical constants of metals. An extension of this model, referred to as the Lorentz-Drude (LD) model included bound electrons by assuming damped harmonic oscillators at critical wavelengths that correspond to interband transitions [15] [16] [17] [18].

Brendel and Bormann (BB) extended previous work to obtain optical constants of amorphous solids in the infrared by including a superposition of oscillators at critical wavelengths with linewidths that were a convolution of Gaussian and Lorentzian linewidths (Voigt profiles) [10], resulting in good agreement with experimental values at room temperatures [11]. Rakic *et al.* applied the BB approach to obtain optical constants for various metals in the infrared, visible and ultraviolet regions [17].

In this work, we build on the BB model and the work of Rakics’ to obtain a model for the optical constants of iron based on experimental data [19] [20]. We verify our model by comparing our theoretical calculations to the experimental data for gold and to Rakics’ theoretical model for gold. Our modified BB model used a reduced number of parameters yet provides excellent agreement with experimental data.

2. Optical Properties of Iron

To evaluate the susceptibility of iron under the influence of an external magnetic bias it is necessary to model the valence electrons that play a major role in the characteristics of metals. The Drude model assumed that free electrons determined the susceptibility while the LD model included valence and other bound electrons in the susceptibility calculation [21].

The optical properties of iron have been extensively studied experimentally with results that are somewhat divergent. However, several data indicate similar interband transitions of bound electrons that produce undulations in the susceptibility as a function of photon energy.

The susceptibility in Figure 1 was calculated from the refractive index and extinction coefficients [19] [20] [22], obtained from optical reflection of light from films. The divergence of the results is probably due to oxide formations on the surface [22] which tends to reduce the reflections and thus lower the value of the dielectric constants. Note that the spurious data for ${\chi}_{\text{r}}^{b}$ and ${\chi}_{\text{i}}^{b}$ at $h\nu \approx 5.5\text{\hspace{0.17em}}\text{eV}$

Figure 1. The real, ${\chi}_{\text{r}}$, and imaginary, ${\chi}_{\text{i}}$, parts of the susceptibility of iron as a function of photon energy. Data for ${\chi}_{\text{r}}^{a}$ and ${\chi}_{\text{i}}^{a}$ is computed from references [19] [20] while data for ${\chi}_{\text{r}}^{b}$ and ${\chi}_{\text{i}}^{b}$ is computed from reference [22].

does not appear in the data for ${\chi}_{\text{r}}^{a}$ and ${\chi}_{\text{i}}^{a}$ while the spurious data in ${\chi}_{\text{r}}^{a}$ and ${\chi}_{\text{i}}^{a}$ at $h\nu \approx 2.25\text{\hspace{0.17em}}\text{eV}$ does not appear in the data for ${\chi}_{\text{r}}^{b}$ and ${\chi}_{\text{i}}^{b}$.

2.1. Electron Displacement in an Electromagnetic Field

The dielectric constant of metals such as nanoparticles composed of iron, cobalt, or nickel can be developed assuming some of the electrons from each atomic site are free to move about within the metal while some are bound to the nucleus at each atomic site. Furthermore, it is generally assumed that only valence electrons participate in the behavior of the dielectric constant. Drude’s theory assumed that the free electrons in metals [23] [24] explained the behavior of the dielectric constant or conductivity at low frequencies (below microwave frequencies). For example, the DC conductivity of metals is determined from the slope of the imaginary part of the susceptibility ${\chi}_{\text{i}}$ at low energy photons. At higher photon energies ( $E=h\nu >0.1\text{\hspace{0.17em}}\text{eV}$ ) the electrons bound to atomic sites greatly influence the behavior of the susceptibility and produce the undulations in the real and imaginary parts of the susceptibility such as that illustrated in Figure 1. Note that both sets of data for iron show a type of resonance at about 2.5 eV.

The force produced by an electromagnetic field acting on a (free or bound) electron of charge *q* and velocity
$v$ is given by the Lorentz force

$F=q\left[e\left(t\right)+v\times \left(b\left(t\right)+{B}_{\text{s}}\right)\right]\approx q\left[e\left(t\right)+v\times {B}_{\text{s}}\right]\mathrm{,}$ (1)

where
$e$ and
$b$ are a wave’s time varying electric and magnetic fields while
${B}_{\text{s}}=\stackrel{^}{x}{B}_{1}+\stackrel{^}{y}{B}_{2}+\stackrel{^}{z}{B}_{3}$ is an externally-applied static magnetic field. In (1) the force due to a wave’s magnetic field,
$b\left(t\right)$, may be neglected compared to force produced by the wave’s electric field,
$e\left(t\right)$. For example, a plane wave has the ratio
$\left|b\right|/\left|e\right|=n/c$, where *n* is the refractive index. Since the velocity of electrons is much smaller than *c*, (1) is a good approximation of the force acting on a free electron.

The Drude model assumes an unbound electron has an equation of motion that can be written as

${m}^{\mathrm{*}}\left(\frac{{\partial}^{2}r}{\partial {t}^{2}}+{\gamma}_{\text{c}}\frac{\partial r}{\partial t}\right)=F$ (2)

where $r$ is the electron position relative to an atom, ${m}^{\mathrm{*}}$ is the electron effective mass, ${\gamma}_{\text{c}}$ is a “damping factor” related to electron collisions with atomic sites.

Lorentz modified Drude’s theory by assuming electrons bound to the nucleus have a harmonic-like restoring force so that (2) was modified to include the restoring force $kr$. There are numerous electrons bound to the nucleus and each exhibits different resonant and collision frequencies. Quantum mechanically, an electron may occupy different discrete energy levels that are separated according to the solution of the Schrodinger equation for the harmonic oscillator and the electron may move from one level to another and the electron lifetime at a certain energy level is inversely related to the “damping constant”, ${\gamma}_{i}$ [25].

The modified equation of motion of, say, the *i*^{th} electron is

${m}_{i}^{\mathrm{*}}\left(\frac{{\partial}^{2}{r}_{i}}{\partial {t}^{2}}+{\gamma}_{i}\frac{\partial r}{\partial t}+{\omega}_{i}^{2}{r}_{i}\right)=F$ (3)

where the resonant frequency of the harmonic oscillator is
${\omega}_{i}={\left({k}_{i}/{m}_{i}^{*}\right)}^{1/2}$. Equation (3) has been frequently used to model dispersion in dielectrics, conductors, and plasmas [21] [25] [26] [27]. (The equation of motion for an electron in the Drude model can be obtained from (3) by placing
${\omega}_{1}=0$, *i.e.*,
$i=1$ denotes free electrons).

For a harmonic time variation $\mathrm{exp}j\omega t$ of the electromagnetic field, the displacement vector ${r}_{i}={R}_{i}\mathrm{exp}j\omega t$, ${v}_{i}=j\omega {R}_{i}\mathrm{exp}j\omega t$, $e\left(t\right)=E\mathrm{exp}j\omega t$ and $b\left(t\right)=B\mathrm{exp}j\omega t$, so that (3) can be written as

$\left({\omega}_{i}^{2}+j\omega {\gamma}_{i}-{\omega}^{2}\right){R}_{i}=\frac{q}{{m}^{\mathrm{*}}}\left(E+j\omega {R}_{i}\times {B}_{\text{s}}\right)\mathrm{.}$ (4)

The vector product may be written as

${R}_{i}\times {B}_{\text{s}}={\stackrel{\leftrightarrow}{B}}_{s}\cdot \text{\hspace{0.05em}}{R}_{i}\mathrm{,}$

where ${\stackrel{\leftrightarrow}{B}}_{s}$ is the asymmetric dyad (in matrix form)

${\stackrel{\leftrightarrow}{B}}_{s}=\left(\begin{array}{ccc}0& {B}_{z}& -{B}_{y}\\ -{B}_{z}& 0& {B}_{x}\\ {B}_{y}& -{B}_{x}& 0\end{array}\right)\mathrm{.}$ (5)

For free and bound electrons (4) may be written as

$\left(\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\right)\cdot {R}_{i}=\frac{1}{j\omega {\stackrel{\u02dc}{\Lambda}}_{i}}E\mathrm{,}$ (6)

where ${\stackrel{\u02dc}{\Lambda}}_{i}=j\omega {m}^{\mathrm{*}}{\stackrel{\u02dc}{\Omega}}_{i}/q$, ${\stackrel{\u02dc}{\Omega}}_{i}=1-{\Omega}_{i}^{2}-j{\Gamma}_{i}$, ${\Omega}_{i}={\omega}_{i}/\omega $ is the normalized resonant frequency, ${\Gamma}_{i}={\gamma}_{i}/\omega $ is the normalized collision frequency, $\stackrel{\leftrightarrow}{I}$ is the unit dyad and the frequency-dependent dyad ${\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \right)$ is

${\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \right)=\left(\begin{array}{ccc}0& {b}_{\text{z}i}\left(\omega \right)& -{b}_{\text{y}i}\left(\omega \right)\\ -{b}_{\text{z}i}\left(\omega \right)& 0& {b}_{\text{x}i}\left(\omega \right)\\ {b}_{\text{y}i}\left(\omega \right)& -{b}_{\text{x}i}\left(\omega \right)& 0\end{array}\right)\mathrm{,}$ (7)

where ${b}_{\text{x}i}\left(\omega \right)={B}_{\text{x}}/{\stackrel{\u02dc}{\Lambda}}_{i}\left(\omega \right)$. The components ${b}_{\text{y}i}\left(\omega \right)$ and ${b}_{\text{z}i}\left(\omega \right)$ satisfy similar expressions (Note that ${\stackrel{\u02dc}{\Lambda}}_{i}$ has the dimension of the magnetic field, which is Tesla in SI units). In the absence of a static magnetic field, ${\stackrel{\leftrightarrow}{B}}_{i}=\stackrel{\leftrightarrow}{0}$, the zero dyad.

The solution for the electron displacement from the nucleus is

${R}_{i}=\frac{1}{j\omega {\stackrel{\u02dc}{\Lambda}}_{i}}{\left(\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \right)\right)}^{-1}\cdot E\mathrm{,}$ (8)

and the inverse is

${\left(\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \right)\right)}^{-1}=\frac{1}{1+{\left({B}_{\text{s}}/{\stackrel{\u02dc}{\Lambda}}_{i}\right)}^{2}}\left(\begin{array}{ccc}1+{b}_{\text{x}i}^{2}& {b}_{\text{x}i}{b}_{\text{y}i}+{b}_{\text{z}i}& {b}_{\text{x}i}{b}_{\text{z}i}-{b}_{\text{y}i}\\ {b}_{\text{x}i}{b}_{\text{y}i}-{b}_{\text{z}i}& 1+{b}_{\text{y}i}^{2}& {b}_{\text{y}i}{b}_{\text{z}i}+{b}_{\text{x}i}\\ {b}_{\text{x}i}{b}_{\text{z}i}+{b}_{\text{y}i}& {b}_{\text{y}i}{b}_{\text{z}i}-{b}_{\text{x}i}& 1+{b}_{\text{z}i}^{2}\end{array}\right)\mathrm{,}$ (9)

where
${B}_{\text{s}}^{2}={B}_{x}^{2}+{B}_{y}^{2}+{B}_{z}^{2}$. When the static field is directed along *x*, *y*, or *z*, there is only one off-diagonal component of
${\left(\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\right)}^{-1}$. For the case of a static field directed along the *y* axis,
${B}_{\text{s}}=\stackrel{^}{y}{B}_{\text{s}}$,

${\left(\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \right)\right)}^{-1}\approx \left(\begin{array}{ccc}1& 0& {B}_{s}/{\stackrel{\u02dc}{\Lambda}}_{i}\\ 0& 1& 0\\ {B}_{s}/{\stackrel{\u02dc}{\Lambda}}_{i}& 0& 1\end{array}\right)\mathrm{.}$ (10)

The resulting dipole moment is ${p}_{i}=q{R}_{i}$ has a single off-axis component.

2.2. Electric Susceptibility from the Lorentz-Drude Model

When the dipole moment for a single charge is ${p}_{1}=q{R}_{1}$, the polarization produced by the free charges becomes

${P}_{1}={N}_{1}{p}_{1}={\epsilon}_{\text{o}}{\stackrel{\leftrightarrow}{X}}_{1}\cdot \text{\hspace{0.05em}}E\mathrm{,}$ (11)

where ${N}_{1}$ is the number of electrons per unit volume that have the dipole moment ${p}_{1}$. The Drude model for most metals does not generally fit to the computed values of susceptibility determined from experimental measurements used to estimate the index of refraction and extinction coefficients in the infrared. However, it does give a reasonable representation at DC to microwave frequencies.

When the static magnetic field is directed along *y*, there are 2 off-diagonal components of the susceptibility so the susceptibility dyadic can be written as

$\stackrel{\leftrightarrow}{X}=\left(\begin{array}{ccc}{\chi}_{\text{v}}& 0& {\chi}_{\text{o}}\\ 0& {\chi}_{\text{c}}& 0\\ -{\chi}_{\text{o}}& 0& {\chi}_{\text{v}}\end{array}\right)\mathrm{.}$ (12)

Defining the plasma frequency as ${\omega}_{\text{p}}{}_{1}^{2}={q}^{2}{N}_{1}/{\epsilon}_{\text{o}}{m}_{1}^{\mathrm{*}}$, and a complex normalized frequency for “free electrons” as ${\stackrel{\u02dc}{\Omega}}_{1}=1-j{\Gamma}_{1}$, and a normalized collision frequency as ${\Gamma}_{1}={\gamma}_{1}/\omega $, so that the electric susceptibility due to free electrons becomes

${\stackrel{\leftrightarrow}{X}}_{1}=\frac{{N}_{1}q}{j{\epsilon}_{\text{o}}\omega {\stackrel{\u02dc}{\Lambda}}_{1}}{\left(\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{1}\right)}^{-1}=-\frac{{\Omega}_{\text{p}}{}_{1}^{2}}{{\stackrel{\u02dc}{\Omega}}_{1}}{\left(\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{1}\right)}^{-1}\mathrm{.}$ (13)

In the absence of a static magnetic field, the dyad ${\stackrel{\leftrightarrow}{B}}_{1}=\stackrel{\leftrightarrow}{0}$ so the susceptibility is a scalar times the unit dyad, where the scalar value is

${\chi}_{1}=-\frac{{\Omega}_{\text{p}}{}_{1}^{2}}{{\stackrel{\u02dc}{\Omega}}_{1}}=-\frac{{\omega}_{\text{p}}{}_{1}^{2}}{{\omega}^{2}+{\gamma}_{1}^{2}}-j\frac{{\omega}_{\text{p}}{}_{1}^{2}{\gamma}_{1}}{\omega \left({\omega}^{2}+{\gamma}_{1}^{2}\right)}$ (14)

and thus ${\chi}_{1}$ is the susceptibility due to the free electrons. Because the harmonic time variation is of the form $\mathrm{exp}\left(j\omega t\right)$, the real and imaginary parts of ${\chi}_{1}$ are negative and its imaginary part has a singularity at $\omega =0$, implying the low frequency conductance is $\sigma ={\epsilon}_{\text{o}}\text{\hspace{0.05em}}{\omega}_{\text{p}}{}_{1}^{2}/{\gamma}_{1}$. The diagonal component is ${\chi}_{\text{c}}={\chi}_{1}$, the susceptibility in the absence of a magnetic bias, while the “variable” part

${\chi}_{\text{v}}=\frac{{\chi}_{\text{c}}}{1+{\left({B}_{\text{s}}/{\stackrel{\u02dc}{\Lambda}}_{1}\right)}^{2}}\approx {\chi}_{\text{c}}\mathrm{.}$ (15)

The off-diagonal component satisfies

${\chi}_{\text{o}}=\frac{{\Omega}_{\text{p}}{}_{1}^{2}}{{\stackrel{\u02dc}{\Omega}}_{1}}\frac{{B}_{\text{s}}/{\stackrel{\u02dc}{\Lambda}}_{1}}{1+{\left({B}_{\text{s}}/{\stackrel{\u02dc}{\Lambda}}_{1}\right)}^{2}}\approx j{\Omega}_{\text{p}}{}_{1}^{2}{\mathcal{M}}_{1}^{\mathrm{*}}{\Omega}_{\text{cy}}\frac{1}{{\stackrel{\u02dc}{\Omega}}_{1}^{2}}\mathrm{,}$ (16)

where ${\Omega}_{\text{cy}}$ is the normalized cyclotron frequency, and using $q=-e$, ${\Omega}_{\text{cy}}=e\text{\hspace{0.05em}}{B}_{s}/{m}_{\text{e}}/\omega $, and ${\mathcal{M}}_{1}^{\mathrm{*}}={m}_{\text{e}}/{m}_{1}^{\mathrm{*}}$ is the ratio of the mass of the electron and effective mass of the free electron. With a static field of 1 Tesla, the cyclotron frequency ${\omega}_{\text{cy}}\approx 0.116\text{\hspace{0.17em}}\text{meV}$ is just a measure of the static field strength. The diagonal components of the susceptibility dyad contain the electron effective mass in the expression of the plasma frequency. However, the off-diagonal elements depend explicitly on the effective mass. The relative anisotropic dielectric constant is obtained from (12) and the unit dyad.

There are only two parameters in the Drude model of the electric susceptibility, ${\omega}_{\text{p}}{}_{1}$ and ${\gamma}_{1}$ and they may be estimated from experimental data [14]. Here it is estimated that for iron ${\omega}_{\text{p}}{}_{1}\approx 3.5\text{\hspace{0.17em}}\text{eV}$, while the low-frequency conductivity of iron, $\sigma \approx 1.044\times {10}^{7}$ Siemens per meter, is used to determine ${\gamma}_{1}={\epsilon}_{\text{o}}{\omega}_{\text{p}}^{2}/\sigma \approx 0.0158\text{\hspace{0.17em}}\text{eV}$ (The real part of the susceptibility is rather insensitive to the small values of ${\gamma}_{1}$, $\Re \left\{{\chi}_{1}\right\}\approx -{\left({\omega}_{\text{p}}{}_{1}/\omega \right)}^{2}$, so that it can describe experimental data in the infrared region by adjusting only the plasma frequency for free electrons). Figure 2 illustrates the result of “fitting” ${\omega}_{\text{p}}{}_{1}$ and ${\gamma}_{1}$ to the experimental data. In the infrared region, the real part of the susceptibility determined from the Drude model is satisfactory to about 2 eV, however, the imaginary part, $\Im \left\{{\chi}_{1}\right\}$ diverges from experimental values near 0.1 eV. When $\lambda =1.55\text{\hspace{0.17em}}\mu \text{m}$ (≈0.8 eV), the susceptibility of iron is $\chi \approx -19-j41$ which produces ${\left(1+{\chi}_{1}\right)}^{1/2}\approx 3.66-j5.6$ while the Drude model has ${\chi}_{1}\approx -19-j0.37$ which produces ${\left(1+{\chi}_{1}\right)}^{1/2}\approx 0.44-j4.24$.

Figure 2. The real, ${\chi}_{\text{r}}$, and imaginary, ${\chi}_{\text{i}}$, parts of the susceptibility of iron using the Drude model. The plasma frequency is ${\omega}_{\text{p}}=3.5\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{eV}$ and the collision frequency is determined from the DC conductivity, $\sigma $, ${\gamma}_{1}={\epsilon}_{\text{o}}{\omega}_{\text{p}}^{2}/\sigma $. Experimental values were computed from the data in [19] [20] as in Figure 1.

To develop a more comprehensive model of the anisotropic susceptibility of iron the modification of the susceptibility due to the bound electrons of the Lorentz theory are added to the model. Accordingly, the anisotropic susceptibility will be written as

$\stackrel{\leftrightarrow}{X}={\displaystyle \underset{i=1}{\overset{Z}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\stackrel{\leftrightarrow}{X}}_{i}\mathrm{,}$ (17)

where

${\stackrel{\leftrightarrow}{X}}_{i}=-\frac{{\Omega}_{\text{p}}{}_{i}^{2}}{{\stackrel{\u02dc}{\Omega}}_{i}}{\left(\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\right)}^{-1}\mathrm{,}$ (18)

where *Z* represents the number of electrons that can be bunched into groups having populations
${N}_{i}$ electrons per unit volume within each group. As before,
${N}_{1}$ represents free electrons per unit volume while
${N}_{2}\mathrm{,}{N}_{3}\mathrm{,}\cdots $ represent the number of bound electrons that can be grouped with identical collision and resonant frequencies. Iron is a transition metal and has about
${N}_{\text{Fe}}=8.46\times {10}^{28}$ atoms per cubic meter and 26 electrons per atom so that
${N}_{1}+{N}_{2}+\cdots +{N}_{Z}$ cannot exceed
$26{N}_{\text{Fe}}$.

The off-diagonal component of the susceptibility, ${\chi}_{13}={\chi}_{\text{o}}$, satisfies

${\chi}_{\text{o}}=-{\displaystyle \underset{i=1}{\overset{Z}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\chi}_{i}\frac{{B}_{s}/{\stackrel{\u02dc}{\Lambda}}_{i}}{1+{\left({B}_{s}/{\stackrel{\u02dc}{\Lambda}}_{i}\right)}^{2}}\approx j{\Omega}_{\text{cy}}\left({\displaystyle \underset{i=1}{\overset{Z}{\sum}}}\frac{{\mathcal{M}}_{i}^{\mathrm{*}}{\Omega}_{\text{p}}{}_{i}^{2}}{{\stackrel{\u02dc}{\Omega}}_{i}^{2}}\right)$ (19)

In the absence of a magnetic bias, the susceptibility tensor is diagonal and that produced by the individual groups, ${\chi}_{i}$, of the LD model is given by

$\chi ={\displaystyle \underset{i=1}{\overset{Z}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\chi}_{i},$ (20)

where

${\chi}_{i}=-\frac{{\Omega}_{\text{p}}{}_{i}^{2}}{{\stackrel{\u02dc}{\Omega}}_{i}}\mathrm{.}$ (21)

The value
${\chi}_{1}$ represents the susceptibility of the free electrons while
${\chi}_{i}$,
$i=2,\cdots ,Z$ represent the susceptibilities due to various groups of bound electrons, and *Z* is the number of “bunchable” groups that can be identified by experimental data. The susceptibility tensor may be written in terms of the applied static field and the unbiased group susceptibilities,
${\chi}_{i}$ as

$\stackrel{\leftrightarrow}{X}={\displaystyle \underset{i=1}{\overset{Z}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\chi}_{i}{\left[\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \right)\right]}^{-1}\mathrm{.}$ (22)

The electron magnetic dipole moments are determined from a combination of the orbital and spin moments. Because the contribution to the magnetic field from the orbital path of the electron is insignificant compared to that of the electron spin, magnetic moments in iron tend to be dominated by the electron spin [21].

2.3. Susceptibility: Brendel-Bormann Model

The Brendel-Bormann model [10], BB, is a slight extension of the LD theory and was used previously to explain frequency response of the dielectric constant of metals [17]. The resonant frequencies of bound electrons were assumed to be random variables ${\stackrel{\xaf}{\omega}}_{i}$ that have Gaussian distributions centered about ${\omega}_{i}$, $i=2,3,\cdots ,Z$. Thus, the susceptibility given by (22) becomes a random function of the set of random transition frequencies ${\left\{{\stackrel{\xaf}{\omega}}_{i}\right\}}_{i=2}^{Z}$ whose mean values are given by the set ${\left\{{\omega}_{i}\right\}}_{i=2}^{Z}$, so that

$\stackrel{\leftrightarrow}{X}\left(\omega \mathrm{;}{\stackrel{\xaf}{\omega}}_{2}\mathrm{,}{\stackrel{\xaf}{\omega}}_{3}\mathrm{,}\cdots \mathrm{,}{\stackrel{\xaf}{\omega}}_{Z}\right)={\chi}_{1}\left(\omega \right){\left[\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{1}\left(\omega \right)\right]}^{-1}{\displaystyle \underset{i=2}{\overset{Z}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\chi}_{i}\left(\omega \mathrm{;}{\stackrel{\xaf}{\omega}}_{i}\right){\left[\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \right)\right]}^{-1}\mathrm{.}$ (23)

Assuming the random variables are independent, the joint probability density function is assumed to have the Gaussian form

$\begin{array}{l}p\left({\stackrel{\xaf}{\omega}}_{2}\mathrm{,}{\stackrel{\xaf}{\omega}}_{3}\mathrm{,}\cdots \mathrm{,}{\stackrel{\xaf}{\omega}}_{Z}\right)\\ ={\left[{\left(2\pi \right)}^{\left(Z-1\right)/2}{\sigma}_{2}{\sigma}_{3}\cdots {\sigma}_{Z}\right]}^{-1/2}{\text{e}}^{-\left[{\left({\stackrel{\xaf}{\omega}}_{2}-{\omega}_{2}\right)}^{2}/2{\sigma}_{2}^{2}+{\left({\stackrel{\xaf}{\omega}}_{3}-{\omega}_{3}\right)}^{2}/2{\sigma}_{3}^{2}+\cdots +{\left({\stackrel{\xaf}{\omega}}_{Z}-{\omega}_{Z}\right)}^{2}/2{\sigma}_{Z}^{2}\right]}\mathrm{.}\end{array}$ (24)

The expected value of the susceptibility tensor becomes

$\langle \stackrel{\leftrightarrow}{X}\rangle ={\chi}_{1}\left(\omega \right){\left[\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{1}\left(\omega \right)\right]}^{-1}+{\displaystyle \underset{i=2}{\overset{Z}{\sum}}}\langle {\chi}_{i}\left(\omega \mathrm{,}{\stackrel{\xaf}{\omega}}_{i}\right){\left[\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \mathrm{,}{\stackrel{\xaf}{\omega}}_{i}\right)\right]}^{-1}\rangle \mathrm{,}$ (25)

where

$\begin{array}{l}\langle {\chi}_{i}\left(\omega \mathrm{,}{\stackrel{\xaf}{\omega}}_{i}\right){\left[\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \mathrm{,}{\stackrel{\xaf}{\omega}}_{i}\right)\right]}^{-1}\rangle \\ ={\left(2\pi {\sigma}_{i}^{2}\right)}^{-1/2}{\displaystyle {\int}_{-\infty}^{\infty}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\chi}_{i}\left(\omega \mathrm{,}{\stackrel{\xaf}{\omega}}_{i}\right){\left[\stackrel{\leftrightarrow}{I}-{\stackrel{\leftrightarrow}{B}}_{i}\left(\omega \mathrm{,}{\stackrel{\xaf}{\omega}}_{i}\right)\right]}^{-1}{\text{e}}^{-{\left({\stackrel{\xaf}{\omega}}_{i}-{\omega}_{i}\right)}^{2}/2{\sigma}_{i}^{2}}\text{d}{\stackrel{\xaf}{\omega}}_{i}\mathrm{.}\end{array}$ (26)

when the static bias is along the *y* direction,

$\langle \stackrel{\leftrightarrow}{X}\rangle =\left(\begin{array}{ccc}\langle {\chi}_{\text{v}}\rangle & 0& \langle {\chi}_{\text{o}}\rangle \\ 0& \langle {\chi}_{\text{c}}\rangle & 0\\ -\langle {\chi}_{\text{o}}\rangle & 0& \langle {\chi}_{\text{v}}\rangle \end{array}\right)\mathrm{,}$ (27)

where the expected value of ${\chi}_{\text{c}}$ is

$\langle {\chi}_{\text{c}}\rangle =-\frac{{\Omega}_{\text{p}}{}_{1}^{2}}{{\stackrel{\u02dc}{\Omega}}_{1}}-{\displaystyle \underset{i=2}{\overset{Z}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\Omega}_{\text{p}}{}_{i}^{2}\text{\hspace{0.05em}}\langle \frac{1}{{\stackrel{\u02dc}{\Omega}}_{i}}\rangle \mathrm{,}$ (28)

which is independent of bias and is thus the susceptibility in the absence of a bias. The expected value of the diagonal component that is dependent on the bias is

$\langle {\chi}_{\text{v}}\rangle =-\frac{{\Omega}_{\text{p}}{}_{1}^{2}}{{\stackrel{\u02dc}{\Omega}}_{1}}\frac{1}{1+{\left({B}_{\text{s}}/{\stackrel{\u02dc}{\Lambda}}_{1}\right)}^{2}}-{\displaystyle \underset{i=2}{\overset{Z}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\Omega}_{\text{p}}{}_{i}^{2}\text{\hspace{0.05em}}\langle \frac{{\stackrel{\u02dc}{\Omega}}_{i}}{{\stackrel{\u02dc}{\Omega}}_{i}^{2}-{\Omega}_{\text{cy}}^{2}}\rangle \approx \langle {\chi}_{\text{c}}\rangle $ (29)

The expected value of the off-diagonal susceptibility tensor is

$\langle {\chi}_{\text{o}}\rangle \approx j{\Omega}_{\text{c}y}\left(\frac{{\mathcal{M}}_{1}^{\mathrm{*}}{\Omega}_{\text{p}}{}_{1}^{2}}{{\stackrel{\u02dc}{\Omega}}_{1}^{2}}+{\displaystyle \underset{i=2}{\overset{Z}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\mathcal{M}}_{i}^{\mathrm{*}}{\Omega}_{\text{p}}{}_{i}^{2}\langle \frac{1}{{\stackrel{\u02dc}{\Omega}}_{i}^{2}}\rangle \right)\mathrm{,}$ (30)

so that the expected value of ${\chi}_{\text{o}}$ depends on $\langle 1/{\stackrel{\u02dc}{\Omega}}_{1}^{2}\rangle $ (see Appendix).

3. Formulation, Results and Discussion

The objective here is to match the theoretical susceptibility governed by the LD and the BB theories to the experimental data obtained from unbiased samples of iron. Previous models were concerned with matching the theoretical dielectric constant governed by the LD and the BB theories with experimental data [15] [17]. The approach here somewhat resembles that of Rakic *et al.* [17]. The experimental values in Figure 1, given in Weaver *et al.* [19] [20], are used because they are more extensive and extend over a larger range of energies than those given in Johnson and Christy [22]. Some spurious data points were dropped because we were unable to associate neighboring points with the dropped point to produce a so-called resonant transition condition.

In fitting experimental data to theory, the number of variables depends on the number of grouped electrons in the LD model. As discussed earlier, bound electrons have only two unknown parameters:
${\omega}_{\text{p}}{}_{i}^{2}$, the square of the plasma frequency, determined from
${N}_{i}$, and the collision frequency
${\gamma}_{i}$. All bound electrons are collected into different groups according to their plasma, collision and resonant frequencies. Unknown variables are contained in the vector
${X}^{\text{}\mathrm{*}}$ and are grouped into sets, namely, the Group 1 set is
$\left\{{\omega}_{\text{p}}{}_{1}^{2}\mathrm{,}{\gamma}_{1}\right\}=\left\{{X}_{1}\mathrm{,}{X}_{2}\right\}$, while the Group 2 set is
$\left\{{\omega}_{\text{p}}{}_{2}^{2}\mathrm{,}{\gamma}_{2}\mathrm{,}{\omega}_{2}\right\}=\left\{{X}_{3}\mathrm{,}{X}_{4}\mathrm{,}{X}_{5}\right\}$, etc. For free electrons,
${\tau}_{1}=1/2\pi {\gamma}_{1}$ represents the mean time between collisions, while for bound electrons
${\tau}_{i}=1/2\pi {\gamma}_{i}$ is the electron life time at an energy level. The frequency
${\omega}_{i}$ represents the transition frequency when an electron changes from one state to another. Thus, if there are *Z* groups of electrons, then there will be
${N}_{Z}=3Z-1$ unknown parameters. The unknowns
${X}_{1}\mathrm{,}{X}_{3}\mathrm{,}{X}_{6}\mathrm{,}\cdots $ are
${\omega}_{\text{p}}{}_{i}^{2}$,
$i=1,3,6,\cdots $.

The BB model of the dielectric constant is an extension of the LD model [10] [17] and assumes the resonant/transition frequencies exhibit homogeneous/inhomogeneous broadening that can be described by a Gaussian distribution centered at resonant frequencies with a width of
$\sigma $. The *i*th oscillator centered at
${\omega}_{i}$ has a Gaussian width of
${\sigma}_{i}$ and thus adds a new parameter to the
${X}^{\mathrm{*}}$ vector that must be determined from fitting the theory to the experimental data. Accordingly, Group *i* data has the parameters
$\left\{{\omega}_{\text{p}}{}_{i}^{2}\mathrm{,}{\gamma}_{i}\mathrm{,}{\omega}_{i}\mathrm{,}{\sigma}_{i}\right\}$ so that the total number of parameters is
${N}_{Z}=4Z-2$.

To fit the theoretical susceptibility to experimental data we use a mean-square relative error function given by

$E\left({X}^{\mathrm{*}}\right)={\displaystyle \underset{m}{\sum}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}W\left({\omega}_{m}\right)\left[{E}_{\text{r}}^{2}\left({X}^{\mathrm{*}}\mathrm{,}{\omega}_{m}\right)+{E}_{\text{i}}^{2}\left({X}^{\mathrm{*}}\mathrm{,}{\omega}_{m}\right)\right]\mathrm{,}$ (31)

where the real part is
${E}_{\text{r}}\left({X}^{\mathrm{*}}\mathrm{,}{\omega}_{m}\right)=\left[{\chi}_{\text{r}}\left({X}^{\mathrm{*}}\mathrm{,}{\omega}_{m}\right)-{\chi}_{\text{re}}\left({\omega}_{m}\right)\right]/{\chi}_{\text{re}}\left({\omega}_{m}\right)$. The real part of the theoretical susceptibility at
${\omega}_{m}$ is
${\chi}_{\text{r}}\left({X}^{\mathrm{*}}\mathrm{,}{\omega}_{m}\right)$, while
${\chi}_{\text{r}e}\left({\omega}_{m}\right)$ is the experimental data at
${\omega}_{m}$. The imaginary part,
${E}_{\text{i}}\left({X}^{\mathrm{*}}\mathrm{,}{\omega}_{m}\right)$ in (31) is obtained by replacing *r* with *i* in the above expressions. The weight function
$W\left({\omega}_{m}\right)=\left({\omega}_{m+1}-{\omega}_{m-1}\right)/2{\omega}_{m}$ is designed so as to represent sparse data equally with bunched data as well as to weigh low energy points equally with high-energy ones. This approach is similar to representing equally spaced data on a log abscissa axis.

The optimization process that minimizes (31) with respect to vector ${X}^{\mathrm{*}}$ uses the NAG Mark 23 optimization library routine E04LBF [28]. The definition of components of ${X}^{\mathrm{*}}$ and their relation to the parameters of the BB model of the dielectric constant or susceptibility, defined in (28) are illustrated in Table 1 and Table 2.

Our model is implemented for gold for the sake of comparison with Rakic’s work and the data was obtained from Handbook of Optical Constants of Solids [29]. The output of the vector ${X}^{\mathrm{*}}$ from E04LBF was used to compute the theoretical dielectric of gold. The theoretical calculation of the dielectric constant of gold obtained by Rakic [17] was obtained for comparison to the results obtained by our model. Figure 3 shows the real and imaginary parts of the dielectric constant as a function of photon energy for both methods.

Table 1 lists the output of appropriate variables from both methods. Furthermore, the implementation of our model for iron data [19] [20] can be seen in Section 4 Figure 4 shows real and imaginary parts of susceptibility for iron. The computed parameters for the modified BB model of iron are shown in Table 2.

Rakics’ method used for the analysis of gold employs 6 groups of electrons and incorporates 23 unknowns while our method uses 5 groups of electrons and a total of 18 unknowns. Also, our model uses 1 less unknown for a given group of electrons because the oscillator strength ${\omega}_{\text{p}}^{2}{}_{i}={\omega}_{p}^{2}\text{\hspace{0.05em}}{f}_{i}$ is treated as a single variable without the constraint $\sum}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{f}_{i}=1$.

Table 1. The computed components of the ${X}^{\mathrm{*}}$ vector that describes the dielectric constant of gold using the BB model results from Ref. [17] (first three columns) and that obtained with the model discussed above. All frequencies, ${\omega}_{i}$, damping constants, ${\gamma}_{i}$, and linewidths, ${\sigma}_{i}$, have units of eV. Relative oscillator strengths ${f}_{i}$ s have no units. The term ${\omega}_{\text{p}\text{\hspace{0.05em}}i}^{2}$ in this work is equal to ${\omega}_{p}^{2}\text{\hspace{0.05em}}{f}_{i}$, which are Rakic’s parameters.

3.1. Dielectric Constant of Gold

The comparative fit of our model with that of Rakic’s for the relative dielectric constant of gold is shown in Figure 3. Rakic *et al.* pointed out the superiority of the BB model over the LD model [17].

3.2. Susceptibility of Iron

The main aim of this work is to develop a theoretical model for the susceptibility of iron using experimental data. In the original BB paper, the superiority of the Gaussian-Lorentzian convolution over just a Lorentzian profile was explained for amorphous solids in the infrared region [10]. In a similar fashion, Rakic *et al.* showed that the model is applicable for various metals not only for the infrared region, but also for visible and ultraviolet regions. However, Rakic *et al.* did not work on optical properties of iron. In our work, we apply our model to iron and the computed fit parameters can be seen in Table 2. The main difference between our analysis and Rakics’ analysis is the elimination of
$\sum}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{f}_{i}=1$ constraint from the system. In Table 2 the first three columns show Rakic’s parameters and the last three columns show the parameters used for this work.

The fit for the susceptibility data of iron in terms of photon energy can be seen in Figure 4. The data is acquired from Weaver *et al.* [19] [20]. The numerical

Table 2. The computed components of the ${X}^{\mathrm{*}}$ vector describe the susceptibility of iron using modified BB model on experimental data for iron [19]. The first three columns represent the vector ${X}^{\mathrm{*}}$ as described by Ref. [17] while the last three columns represent the vector ${X}^{\mathrm{*}}$ obtained with our modified BB model. All frequencies, ${\omega}_{i}$, damping constants, ${\gamma}_{i}$, and linewidths, ${\sigma}_{i}$, have units of eV. Relative oscillator strengths ${f}_{i}$ s have no units.

Figure 3. The real, ${\kappa}_{\text{r}}$, and imaginary, ${\kappa}_{\text{i}}$, parts of the dielectric constant of gold using the BB model. Experimental values (small circles) were computed from the data in [29]. The solid curves were obtained by our model while the dashed curves were obtained from the gold data given in Ref. [17].

Figure 4. The real, ${\chi}_{\text{r}}$, and imaginary, ${\chi}_{\text{i}}$, parts of the susceptibility of iron using the BB model. Experimental values (small circles) were computed from the data in [19] [20]. The numbers above the abscissa represent the locations of transitions and the dashes above the numbers indicate the various values of ${\sigma}_{i}$, the degree of line broadening in the BB model.

values above the abscissa represent the transition energies and the dashes above the numbers indicate the broadening width. Figure 4 shows that our improved BB model provides an accurate fit even for narrower inter-band transitions and validates that our model is applicable to iron up to 30 eV.

4. Conclusions

We analyzed the optical properties of Au, and Fe by using an improved Brendel-Bormann (BB) model. As an initial step, the Drude free electron theory is used to model the susceptibility of iron as illustrated in Figure 2. However, the differences between the Drude predictions and that of the experimental data are large, particularly in the near infrared region. The Drude-Lorentz extension places damped harmonic oscillators at critical points, referred to as inter-band transition points. In addition to the idea of the placement of oscillators at critical points, a Voigt lineshape assists in more accurately predicting susceptibility. Rakic *et al.* [17] used the BB method for various metals to show its accuracy for modeling the optical constants not only for amorphous solids but also for metals. We modified Rakic’s model by reducing the number of unknowns, and relaxing a constraint from the system. Table 2 shows that the largest concentrations of oscillators occur at
${\omega}_{2}\approx 0.2\text{\hspace{0.17em}}\text{eV}$ and
${\omega}_{7}\approx 20\text{\hspace{0.17em}}\text{eV}$. The large concentration at
${\omega}_{2}$, about 32%, brings the imaginary part of the susceptibility,
$-{\chi}_{\text{i}}$ in line with the experimental data, while the concentration at
${\omega}_{7}$ tends to render the susceptibility to almost straight lines on the log-log plot in Figure 4. The fit of our model to iron data can be seen in Figure 4. The iron plot results up to 30 eV indicates that our model could perform fairly well for transition metals up to a broad range of the spectrum.

The studies that have been mentioned so far are related to diagonal elements of the relative permittivity tensor. However, one needs to take into account the off-diagonal elements if there is an external magnetic field in the anisotropic medium if the material is ferromagnetic. The off-diagonal elements were also incorporated by the contribution of Magneto-Optic Kerr Effect which includes the Kerr rotation angle and ellipticity [30]. The investigation of the off-diagonal elements was conducted by Krinchik *et al.* [31] [32] [33] by introducing the equatorial Kerr effect (T-MOKE) to the experimental setup.

Appendix

It is convenient to write the closed form integrals that appear in the BB model of the dielectric constant/susceptibility tensors as a function of error integrals. In particular the expected value of the susceptibility given in (30) requires calculation of the expected value of the random variable $1/{\stackrel{\u02dc}{\Omega}}_{i}$

$\begin{array}{c}I\left({a}_{i}\right)=\langle \frac{1}{{a}_{i}^{2}-{\stackrel{\xaf}{\Omega}}_{i}^{2}}\rangle =\frac{1}{\sqrt{2\pi}{\sigma}_{i}}{\displaystyle {\int}_{-\infty}^{\infty}}\frac{{\text{e}}^{-{\left({\stackrel{\xaf}{\omega}}_{i}-{\omega}_{i}\right)}^{2}/2{\sigma}_{i}^{2}}}{{a}_{i}^{2}-{\stackrel{\xaf}{\Omega}}_{i}^{2}}\text{d}{\stackrel{\xaf}{\omega}}_{i}\\ =\frac{1}{\sqrt{2\pi}{S}_{i}}{\displaystyle {\int}_{-\infty}^{\infty}}\frac{{\text{e}}^{-{\left({\stackrel{\xaf}{\Omega}}_{i}-{\Omega}_{i}\right)}^{2}/2{S}_{i}^{2}}}{{a}_{i}^{2}-{\stackrel{\xaf}{\Omega}}_{i}^{2}}\text{d}{\stackrel{\xaf}{\Omega}}_{i}\mathrm{,}\end{array}$ (32)

where ${S}_{i}={\sigma}_{i}/\omega $ is the variance of the random variable ${\stackrel{\xaf}{\Omega}}_{i}$, ${a}_{i}^{2}=1-j{\Gamma}_{i}$. As described in Ref. [17], $I\left({a}_{i}\right)$ can be written as a sum of 2 integrals by partial fraction expansion

$\frac{1}{{a}_{i}^{2}-{\stackrel{\xaf}{\Omega}}_{i}^{2}}=\frac{1}{2{a}_{i}}\left(\frac{1}{{a}_{i}-{\stackrel{\xaf}{\Omega}}_{i}}+\frac{1}{{a}_{i}+{\stackrel{\xaf}{\Omega}}_{i}}\right)\mathrm{.}$

As required by convergence of the integral in (32), $\Im \left\{{a}_{i}\right\}$ must be positive so that

${a}_{i}=\frac{1}{\sqrt{2}}\left(-\sqrt{\sqrt{1+{\Gamma}_{i}^{2}}+1}+j\sqrt{\sqrt{1+{\Gamma}_{i}^{2}}-1}\right),$

*i.e.*,
${a}_{i}$ must lie in the 2nd quadrant of the complex plane as opposed to the 4th quadrant (Equation (8) in Reference [17] has a different value for the sign of
${{a}^{\prime}}_{j}$ ). The resulting integral in (32) produces

$I\left({a}_{i}\right)=-j\sqrt{\frac{\pi}{2}}\frac{1}{2{S}_{i}{a}_{i}}\left[w\left({z}_{1}\right)+w\left({z}_{2}\right)\right]\mathrm{,}$ (33)

where $w\left(z\right)={\text{e}}^{-{z}^{2}}\text{erfc}\left(-jz\right)$ is computed from the NAG Library, using function S15DDF, ${z}_{1}=\left({a}_{i}-{\Omega}_{i}\right)/\sqrt{2}{S}_{i}$, and ${z}_{2}=\left({a}_{i}+{\Omega}_{i}\right)/\sqrt{2}{S}_{i}$. The expected value of the off-diagonal element ${\chi}_{\text{o}}$ as given by (30) requires calculation of the expected value of the random variable $1/{\stackrel{\u02dc}{\Omega}}_{i}^{2}$, written as

$J\left({a}_{i}\right)=\langle \frac{1}{{\stackrel{\u02dc}{\Omega}}_{i}^{2}}\rangle =-\frac{1}{2{a}_{i}}\frac{\text{d}I\left({a}_{i}\right)}{\text{d}{a}_{i}}=-j\frac{\sqrt{\pi}}{8{a}_{i}^{3}{S}_{i}^{2}}\left(\sqrt{2}{S}_{i}\left({w}_{1}+{w}_{2}\right)-{a}_{i}\left({{w}^{\prime}}_{1}+{{w}^{\prime}}_{2}\right)\right)$ (34)

The error functions and their derivatives are ${w}_{k}=w\left({z}_{k}\right)$ and ${{w}^{\prime}}_{k}=j\sqrt{2}/\pi -2{z}_{k}{w}_{k}$, while $k=1,2$, respectively [34]. The recursion relation ${w}_{k}^{\left(n+2\right)}+2{z}_{k}{w}_{k}^{\left(n+1\right)}+2\left(n+1\right){w}_{k}^{\left(n\right)}=0$ can be used for higher order derivatives.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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