## Abstract

Fitting optical properties of metals is of great interest for numerical methods in electromagnetism, especially finite difference time domain (FDTD). However, this is a tedious task given that theoretical models used usually fail to interlink perfectly with the experimental data. However, in this paper, we propose a method for fitting the relative permittivity of metals by a sum of Drude-Lorentz or a sum of partial-fraction models. We use the particle swarm optimization (PSO) hybridized either with Nelder-Mead downhill simplex, or with gradient method. The main electronic transitions in metals help to guide the fitting process toward the solution. The method is automatic and applied blindly to silver, gold, copper, aluminum, chromium, platinum, and titanium.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The measured refractive index of some metals as a function of a sample of illumination wavelengths (or photon energies) can be found in many papers [1–8], handbooks [9] and websites [10]. These properties strongly depend on both the preparation mode of the sample and the method of measurement. In this paper, we use the data from [11] for all investigated metals.

Fitting these optical properties by models of dispersion is of great interest for numerical simulations in spectroscopy, plasmonic, optical engineering, and to boost advanced materials applications. Actually, the Fourier transform of these models is used in FDTD calculations [12]. Therefore, different dispersion models were introduced, e.g., Drude, Drude-Lorentz, Partial-Fraction, and Critical Points [13–15]. Combinations of models were also commonly applied to get better results, e. g., combining Critical Points with Drude or Drude-Lorentz models [14,16]. The function corresponding to each model of dispersion, depends on different parameters that must be fixed [17] or recovered by fitting the experimental data [18–21]. Nevertheless, the results depend not only on the applied models, but also on the numerical fitting method, which may fail to get a satisfactory solution if the number of unknown parameters is large.

The non-unique solution of the fit may increase the computational time of the fitting process. The relative permittivities are complex numbers depending on the wavelength (or energy), then the balance between the magnitude of the real and imaginary parts can attract the fitting process toward a non-optimal solution [22]. To limit these drawbacks for fitting by Drude-Lorentz models, we propose a three-step approach, using the preliminary detection of peaks in the reference data.

We adopt the particle swarm optimization (PSO), a metaheuristic algorithm [23], to fit the optical properties of metals. The solution given by the PSO is then used as a starting point of the gradient or the Nelder-Mead methods. The purpose of this hybridization (PSO-gradient or PSO-Nelder-Mead) is to enhance the quality of the solutions [24]. This hybrid method is more straightforward than other approaches [25] and guarantees good fitting results. However, the parameters found can be outside the search domain fixed for the PSO, these two local search methods being not bound constrained.

The remainder of this paper is organized as follows. Sections 2. describes the considered Drude-Lorentz and the Partial-Fraction models, and the objective function that has to be minimized is given in section 3. We give an overview of the PSO algorithm, details on the three-step method of fitting, and a definition of sensitivity which is an indicator that helps to analyze the results; in section 4., we report and discuss the results as well as the comparison between the proposed models of fitting of the relative permittivity of gold, silver, copper, aluminum, chromium, platinum and titanium in the the $[0.5;6]$ eV range. Finally, Section 5. concludes the paper.

## 2. Physical functions for fitting

#### 2.1 Drude-Lorentz model

The classical models used to fit experimental data are Drude [26] and Drude-Lorentz models [27]. The first one describes the interaction of the free-electrons with light (intraband [28]). The second one describes interband (bound electron) effects [28]. Drude model suits well the optical properties of various types of metals, such as noble metals and semiconductors. In this paper, we consider the sum of Drude-Lorentz terms to fit the relative permittivity as a function of the angular frequency $\omega$:

In addition to $\varepsilon _{\infty }$, each Drude-Lorentz term involves three unknown real parameters. The number of terms $NDL$ (in Eq. (1)) is firstly fixed to 9 and then reduced to 5. This optimum number has been determined to get a good quality of fit for all investigated metals, without specific tuning of the method. Therefore, the total number of parameters to be adjusted for each experimental data is $D=3 NDL+1$.

#### 2.2 Partial-fraction model

The Partial-Fraction (PF) model (Eq. (2), [13]) was also proposed to fit Johnson and Christy data for metals [29].

A comparative study is done for all investigated metals. The fitting function being defined, the objective function that should be minimized can be introduced.

## 3. Method for the fitting

The key indicator of the quality of the fit is the difference between the numerical result of the model $\varepsilon _{DL}$ and the experimental (or reference) data $\varepsilon _{Ref}$. The reference data are a set of $N_\omega$ pairs of values $(\omega _j,\varepsilon _{Ref}(\omega _j))$. Therefore, for each $\omega _j$ of the experimental data, we define the complex number $\Phi (\omega _j)$:

In the following, the error on fit is either $|\Re (\Phi (\omega _j))|$ for the real part of the relative permittivity or $|\Im (\Phi (\omega _j))|$ for the imaginary part.The fitting method consists of minimizing the objective function given by:

The following general method used for the objective function minimization is implemented in three-step described in Sec. 3.3.

#### 3.1 The minimization method

To minimize the objective function (Eq. (4)), we use two hybrid methods: a combination of the particle swarm optimization with Nelder-Mead downhill simplex method [34] or with gradient method to end the descent toward the minimum. Therefore, the retrieved parameters for the best fit may get outside the search space defined for PSO. In particular, $\epsilon _\infty <1$ can be found.

PSO [23] is a trajectory-based and population-based optimization algorithm. It can be described as follows [35]: “it is an iterative metaheuristic based on the collaboration between particles representing the variable of the optimization problem.” Each particle $i$ of the swarm ($1 \leq i \leq N$, $N$ being the number of particles) is a vector, which components are the $D$ unknown parameters ($D=3 NDL+1$ and $D=4 NPF+1$ in the case of DL and PF respectively). It represents a candidate solution or position $\mathbf {x}_i(t)$. Particles memorize their best previous positions, denoted by $\mathbf {p}_i(t)$, and update their position at step $t+1$ as follows:

where $\mathbf {v}_i(t+1)$ is the updated velocity of particle $i$ given by:If particles get outside the search domains after position update (Eq. (8)), they are either stuck on, randomly regenerated, or reflected on the boundaries of the search domains [36]. Moreover, if the particle’s fitness is not decreasing for ten successive iterations, then the particle is regenerated randomly in the search domains to avoid stagnation [37].

The exogenous parameters ($c_1, c_2, \omega$, $N$, $T$ the maximum number of iterations, and $nreal$ the number of realizations) should be adjusted to improve the convergence and speed of the PSO algorithm. In the current study, we set $c_1=c_2=2$, $N=30$, $T=1500$, $nreal=1000$ and $\omega = (0.9-0.5 (t-1)/(T-1))$. The number of realizations is deduced from the optimal sample size, which is necessary to handle a given acceptable margin error (for example 6%) and a confidence level of 95%. The optimal sample size is approximated by:

where $\mathcal {N}^{-1}$ is the inverse function of the standard normal distribution of probability. A small standard deviation of the optimization results is a criterion of stability of the method.#### 3.2 Preliminary detection of Drude-Lorentz shapes in the reference relative permittivity

The fitting process may override the shallow peaks observed mainly for the imaginary part of the experimental data. These peaks have physical significance; therefore, they should be conserved. They are automatically detected in the reference relative permittivity data by investigating the change of derivative sign:

- • The detection of a maximum followed by a minimum in the real part of the reference relative permittivity when the energy increases enable to calculate the average energy between these particular energies $E$ and the amplitude $A$ of the corresponding Drude-Lorentz function.
- • The detection of a maximum in the imaginary part of the reference relative permittivity when the energy increases enables to identify the corresponding energy $E$ and the amplitude $A$ of the corresponding Drude-Lorentz function.

For each investigated material, the number of preliminary detected peaks varies from 1 to 3. Therefore, some terms of models can have a physical meaning and the others improve the fitting in the whole domain of energies. Generally, the amplitude of these peaks is small compared to the global variation of the relative permittivity in a wide range of angular frequencies; consequently, a standard adjustment would fail to depict them. Then, we propose a three-step method to keep this physical information. To characterize the method of fit, we set the number of terms to 5 and 9 for DL and to 4 for PF: $D=$ 16, 28 and 17 respectively.

#### 3.3 Three-step method of fitting

The real part of the relative permittivity of metals exhibits considerable variations and magnitude, unlike its imaginary part. Consequently, the objective function $F$ (Eq. (4)) with $C_1=C_2=1$ is highly sensitive to the variations of the real part of the relative permittivities. We propose a three-step approach to lead the fitting process to physical solutions:

- • The global shape of the reference relative permittivity is fitted by a single Drude function ($\omega _1=0$ in Eq. (1)). The weighting coefficients get equal values $C_1=C_2=1$ in Eq. (4). The search domains for PSO are $[0;60]$ for $A_1$, $[0;30]$ for $\Gamma _1$, $[1;20]$ for $\epsilon _\infty$.
- • The objective function is weighted such as the real and imaginary parts have equal contribution to the objective function: $C_1=1/(\max (\Re (\epsilon _{Ref}))-\min (\Re (\epsilon _{Ref})))$ and $C_2=1/(\max (\Im (\epsilon _{Ref}))-\min (\Im (\epsilon _{Ref})))$ over the spectral range. The parameters $A_1$, $\Gamma _1$ of the previously detected Drude-Lorentz term are used to determine the search domains for the first Drude-Lorentz term (Sec.3.2): $[\epsilon _\infty /2;2 \epsilon _\infty ]$, $[A_1/2;2 A_1]$ and $[\Gamma _1/2;2 \Gamma _1]$. The preliminary detection of Drude-Lorentz shapes is used as mentioned above.. The domains of search for the parameters of the remaining Drude-Lorentz terms are less constrained: the search intervals $[0;20]$ for $A_i$, $[0;10]$ for $\omega _i$ and $[0;20]$ for $\Gamma _i$.
- • $C_1=C_2=1$. The parameters of the Drude-Lorentz terms associated to detected peaks in the relative permittivity (step 2) are kept unchanged. Only the parameters of the remaining Drude-Lorentz terms are searched in the following search intervals: $[0;20]$ for $A_i$, $[0;10]$ for $\omega _i$ and $[0;20]$ for $\Gamma _i$. These terms may have no physical meaning but hopefully, they will improve the fit.

Any efficient method of optimization could be used to minimize the objective function. We considered a hybrid method: the PSO and either the Nelder-Mead downhill simplex method or the gradient method (Sec. 3.1). Both are no bound constrained methods; therefore, the result of minimization may leave the search domains and the solutions may have a non physical sense.

As a metaheuristic, the PSO uses pseudo-random generation. Thus, two successive realizations of the same algorithm could lead to different solutions, especially in the case of non-uniqueness of the solution. Therefore, the results obtained from the repeated realizations of the same algorithm can be considered as successive numerical experiments [38]. Consequently, the analysis of the best results can reveal the sensitivity of the fit to each decision variable (the parameters of fitting functions). Indeed, the dispersion of the results of optimization is an indicator of the contribution of each term used for fitting. The sensitivity to each contribution can be defined to facilitate the discussion of results.

#### 3.4 Sensitivity

The model used for fitting reference data is a sum of $NDL$ Drude-Lorentz (DL) terms (Eq. (1), or $NPF$ Partial-Fraction (PF) terms. Nevertheless, the number of terms with physical meaning may vary depending on the investigated metal. The $nreal$ realizations of the minimization of the objective function produce $T$ vectors of the best parameters ($A_i$, $\omega _{i}$, $\Gamma _{i}$) for each DL law. The ratio of the mean value $m(.)$ of each parameter to its standard deviation $\sigma (.)$ is an indicator of the sensitivity $S$ of the results to the corresponding function, for example:

As results differ from a realization to another, only the decision parameters found for values of the objective function lower than $1.5$ times its best value among all realizations are kept to calculate the sensitivity. If the standard deviation is equal to zero, the sensitivity is infinite (not a number NaN). In this case, all realizations give the same result. If it increases, the sensitivity decreases.We apply the above described method to the gold data from Johnson and Christy [4] and compare our results to previous studies.

#### 3.5 First test of the method with Johnson and Christy data

To test the efficiency of the proposed method, we consider a first set of reference data from Ref. [4]. We compare the fit to those given in Refs. [39] and [21]. In those papers, a combination of Drude-Lorentz models was used for fitting relative permittivity of gold. Figures 1 and 3 show the real and imaginary parts of the reference permittivity [4], the best fits using $NDL=9$ ($D=28$), $NDL=5$ ($D=16$) and $NPF=4$ ($D=17$), and the results reported in papers [39] and [21]. The fitting errors are shown in Figs. 2 and 4. The comparison of real (Fig. 1) and imaginary (Fig. 3) parts of the relative permittivities of reference (o) and calculated from data in Refs. [21,39,40] ($+$) and ($\square$) respectively, illustrates that the quality of fits depends both on the domain of considered energies and on the number of Drude-Lorentz terms that are used. The values of the error at high energies confirms this discussion (Figs. 2–4). Around $\omega =2$ eV, the quality of our fitting is comparable to that in [39] and [21,40], especially for the PF model. However, the quality of our fitting is more than one order of magnitude better than those in [39] and [21], for $\omega >3$ eV. The PF model is competitive and using $NDL=5$ can be satisfactory. For a fair comparison, the quality of fits parameters found in Ref. [13] is calculated with Eq. (4) where $C_0=C_1=C_2=1$. We obtain $F =$ 7.3, 7.75, 13.1, 21.5, 23.1 using a Drude function and 3, 4 and 5 DL respectively. Values of the objective function for PF with 4 and 5 terms are 8.0 and 8.3. Therefore, our method gives good results, but with a possibly value of $\epsilon _\infty$ less than one. The criterion (5) is verified for the three fits: $C=$ 0.67, 0.99962 and 0.98 respectively. The sign of the result of fit is the same as the reference data for all $\omega$ for $NDL=5$ and $NPF=4$. For $NDL=9$, the real part is positive in $[3.5;6]$ eV. Wrong sign of the real part of the fits with parameters from [39] and [21] is obtained for $\omega \in [4;6]$ eV and $[4.2;6]$ eV respectively. This supports the fact that data should be refitted, if the considered energy domain varies. However, using 9 Drude-Lorentz terms better describes the variations of imaginary part in a wide domain of energies. This peculiar behavior is imperceptible while fitting the real part because its curve shape hides the local behavior at high energies. The PF model is the most competitive to fit the real part for energies around 3 eV and the imaginary part for energies around 2 eV.

In Table 1, $\epsilon _\infty >0$, whereas it is negative for PF (Table 2). This non physical value is the result of the conjugate-gradient method that is used in the hybrid optimization scheme: the numerical results can get out the domain of search defined for PSO.

The position of th peaks in the imaginary part can be analyzed with results from Ref. [41]. The first peak in the imaginary part of the reference relative permittivity at 3 eV is due to transition from the d to s states. Transition from the d to p band just above the Fermi level is damped in the fitting. The second peak at 4 eV is a composite structure involving transitions from lower lying d bands to band 6 and transitions from s, p states to s, p-like states.). In Table 1, the nearest values of $\omega _i$ corresponding to these preliminary detected peaks (3.00 and 3.87 eV) are located at 2.98 and 4.20 eV (terms #4 and #5 for $NDL=9$) and at 3.00 and 3.90 eV (terms #2 and #3 for $NDL=5$). The corresponding $|p_2|$ is 2.47 eV (Table 2). Parameters of term #1 for $NDL=9$ correspond to a small variation of a Drude function. The value of $\omega _1$ for $NDL=5$ is close to that of $|p_2|$. Therefore, the proposed method of fit appears to be efficient for the two models. Using $NDL=9$ terms improves the quality of fit and, in this case the PF model is more adapted to the data.

To complete the discussion of the method, we report the results and the sensitivity defined in Sec. 3.4 in Tables 1–2. The sensitivity for DL #1 in Tables 1 ($NDL=9$) is smaller than for other DL. Therefore, we can conclude that the quasi Drude function ($\omega _1\approx 0$) is not critical in the fitting process. On the contrary, if the degrees of freedom are reduced, The sensitivities maintain of the same order of magnitude.

As a result of the above discussion, we can separate the terms of models in two families. The first family results from the preliminary detection of peaks in reference data. The second group has no physical sense but improves the quality of fit. Even if we limit the domain of search for PSO to $\epsilon _\infty \geq 1$, the best parameters that are retrieved by the hybrid method may not verify this condition. The plot of the contributions of each Drude-Lorentz term is shown in Fig. 5. The energies obtained from the automatic detection of peaks are located by vertical dashed lines ($3$ and $3.87$ eV) in Fig. 5.

Therefore, the three-step application of the hybrid optimization method enables to keep the peaks that have physical meaning.

In Sec. 4. the proposed method of fit is applied to silver, other data for gold, copper, aluminum, chromium, platinum and titanium.

## 4. Results and discussion

We report the results of fit of the optical properties extracted from the SOPRA database [11]. The study covers transition, noble metals, and post-transition metals. The investigated transition metals are chromium (Cr), platinum (Pt), titanium (Ti). The investigated noble metals are gold (Au), silver (Ag), and copper (Cu), and we considered the post-transition metal aluminum (Al) [19].

Chromium, platinum, and titanium have many applications in the field of electronics, e.g., manufacturing of catalysts. In plasmonics, Cr, Pt, and Ti are used as an adhesion layer to deposit gold on glass substrates, e.g., for the surface plasmon resonance biosensors [42–44]. Chromium is an essential element involved in the regulation of body metabolism. Chromium is also frequently used in painting, famous for its yellow (PbCr$O_{4}$) or green ($Cr_{2}O_{3}$) color. Platinum is highly recommended in jewelry, photography, and weapon manufacturing to reduce the sound of missiles. Platinum is used to minimize emissions of carbon oxides as well as nitrogen, not burned hydrocarbons and solid particles. Titanium is frequently used in medicine for orthopedic prostheses, aerospace and car manufacturing.

Noble metals are used for biomedical sensors, electronics, and plasmonics. They are also essential in the generation of new materials and enhancement of mirrors reflectivity and optoelectronic devices [45] and nanostructures.

Aluminum (Al), which has 3s$^{2}$3p$^{1}$ electronic configuration, is frequently used in the industry and craft. Aluminum has a remarkable lightness, corrosion resistance, and durable color [46]. Furthermore, it suits for mirrors and optoelectronic devices since these latter are sensitive to ohmic contact [45]. It is also used in plasmonics.

#### 4.1 Silver

Figures 6 and 8 show the real and imaginary parts of the relative permittivity of silver and their fits with 5 and 9 DL and with 4 PF, as a function of the photon energy. The Error on fit is shown in Figs. 7 and 9 as a function of $\omega$. For the plots, we use the data in Tables 3–4. The results of fit including the sensitivity to $\omega _i$ and $\Im (p_i)$ are reported in Tables 3–4.

The position of the automatically detected peak in the imaginary part of the relative permittivity is 4.1 eV corresponding to peaks #3 and #4 (for $NDL=9$), #3 ($NDL=5$) and #3-#4 ($NPF=4$, $|p_3|=4.8$, $|p_4|=5$). All are shifted toward higher energy. The highest sensitivities are found for these peaks and for greater $\omega _i$ and $|\Im (p_i)|$. For silver, some characteristic energies greater than 6 eV are found. They are related to an interband transition (see Sec. 4.8). The value of $\epsilon \infty$ are greater than one for fits by DL. The criterion (5) $C<1$ is verified: $C=$ 0.9997, 0.9994 and 0.97 respectively. The sign of the fit is the same as the reference over the whole domain of energy for $NDL=9$. For $NDL=5$, the sign of the real part of the reference data and of the fit differ in $[3.9;5.5]$ eV and for $NPF=4$, in $[3.9;6.6]$ eV.

The minimum value of the objective function is obtained from the PSO+NM for 9 DL terms, while for $NDL=5$, the minimum value of the objective function is obtained from the PSO+GR hybrid method. This is also the case as for the Partial-Fraction model. Using $NDL=5$ appears to be more efficient to fit silver data. The PF term #2 exhibits the smallest sensitivity. The real part being $0.559$, we can conclude that the position of the pole is close the imaginary axis in the complex plane and therefore, the sensitivity to variations of its position is critical [47,48]. The sensitivity for $\omega _1$ ($NDL=5$) shows that the result of fit depends slightly on the best parameters of the Drude function.

#### 4.2 Gold

The curves in Figs. 10 and 12 differ from those in Figs. 1 and 3. Figures 10–13 show that the fits are well superposed with the reference data (see also the Error on fit as a function of photon energy: Figs. 11 and 13). Both real and imaginary parts of the relative permittivity are satisfactory fitted, especially by the PF model, near 2 eV. The errors are close together for DL models with $NDL=9$ and $NDL=5$, but the real part is better fitted by the PF model for almost all $\omega$. The value of objective function confirms this observation. The data used to plot the fits are given in Tables 5–4. The criterion (5) $C<1$ is verified: $C=$ 0.9999, 0.97 and 0.99 respectively. The sign of the fit is the same as the reference over the whole domain of energy for $NDL=5$. For $NDL=9$, the sign of the real part of the reference data and of the fit differ in $[2.7;6]$ eV and for $NPF=4$, in $[3.1;6]$ eV.

The detected peaks in the imaginary part of gold data [11] are 3.16 eV and 4.31 eV. The transition energy for gold is around 2.6 eV, which is attributed to interband transition. They differ from those calculated from [4]. In Table 5, energies $\omega _3=2.99$ eV and $\omega _4=3.98$ eV ($NDL=9$), $\omega _3=2.91$ eV and $\omega _4=3.8$ eV ($NDL=5$) and $|p_3|=2.07$ eV correspond to a slight modification of this parameter set. The quasi-Drude term (#1) sensitivity to variations of $\omega$ is less than that for others. Parameters of terms #8 and #9 are outside the initial domain of search and $\epsilon _\infty \geq 1$, on the contrary of the PF model. The second energy transition corresponds to another interband transition at 6 eV ($\omega _6$). For gold, the intraband contribution remains low. For the interband term, in the case of noble metals, ‘d’ and ‘s’ bands are respectively valence and conduction bands. When ‘d’ is full, and ‘s’ band is filled around 3/4, allowable re-emission only applies to wavelengths higher than 550 nm (energy lower than 2.2 eV). This explains the yellow color of gold ($\omega _1\approx 0$, $\omega _2=0.417$, $|p_1|=0.245$, $|p_2|=1.17$, $|p_3|=2.1$).

#### 4.3 Copper

The last studied noble metal is copper. The numerical fits are well superposed with the experimental data as shown in Figs. 14 and 16. The error on the imaginary part (Fig. 17) is smaller than that on the real part (Fig. 15). The small peak in the real part (near 1 eV) is not revealed by the fits.

The position of the preliminary detected peaks are at 2.45, 3.2, 3.45, 4.9 and 5.1 eV. The main peaks near 2.4 eV and 5 eV. The first one is probably due to the displacement of free electrons. The transition from d states (valence band) to ‘s-p’ conduction band is near 2 eV. The latter is associated with the interband transition. Table 7 show three main peaks at 2.38 (#3), 3.43 (#5) and 5.16 eV (#6) for $NDL=9$, and #3-#5 for $NDL=5$. They do not appear in Table 8: $|p_i| = 0.28,$, $0.37$, $1.27$, $1.77$ eV. Table 8 show that three poles are close to the imaginary axis, but they cannot be related to peaks with physical meaning. The Drude terms are about the same ($NDL=9$, $NDL=5$, #1). The criterion (5) $C<1$ is verified: $C=$ 0.998, 0.99999 and 0.98 respectively. The sign of the fit is the same as the reference over the whole domain of energy for DL with $NDL=5$ and PF. For $NDL=9$, the sign of the real part of the reference data and of the fit differ in $[5.9;6.5]$ eV.

#### 4.4 Aluminum

Figures 18 and 20 show a good agreement between the experimental data and the best fits with DL and PF models. This agreement can be analyzed as a function of energy $\omega$ in Figs. 19 and 21 (error plots). The best values of the objective functions, the sensitivity and the parameters for fits of aluminum data can be found in Tables 9–10. These parameters are used to plot Figs. 18–21.

The best value of objective function is obtained from the PSO+NM hybrid method for $NDL=9$, but using $NDL=5$ may be considered as sufficient. The position of the preliminary detected peak is at 1.55 eV. This is explained with the excitation of the free electrons in the conduction band of Al, which has $3s^{2}3p^{1}$ electronic configuration. The associated terms in fits are #3 and #4, and #3 for $NDL=9, 5$ and $NPF=4$, respectively ($\omega _3 =1.45$, $\omega _4 =1.58$, $\omega _3 =1.56$, $|p_3|=1.58$ eV). The highest sensitivity is observed for these terms and the next one, at nearby energy for the DL model. The highest sensibility is for $\Im (p_4)$ that is far from the imaginary axis and therefore contribute to a background. The first term for DL is actually a Drude function with small damping. They correspond to the term #1 in the PF model. $\epsilon _\infty \geq 1$ only for $NDL=5$. The criterion (5) $C<1$ is verified: $C=$ 0.33, 0.9992 and 0.80 respectively. The sign of the fit is the same as the reference over the whole domain of energy for the three models.

#### 4.5 Chromium

Figures 22 and 24 show the real part and the imaginary part of the chromium relative permittivity and their fits. The corresponding error of fit is plotted in Figs. 23 and 25. The corresponding parameters of functions are given in Tables 11–12. In the case of chromium, the quality of fit appears less than for the previously investigated materials. However, all models are unable to fit correctly the reference data. The PF model is the most efficient to fit the optical properties of chromium even if the DL model ($NDL=9$) is a good candidate for the fitting of the real part. $\epsilon _\infty \geq 1$ only in the fit with the PF model. The criterion (5) $C<1$ is verified: $C=$ 0, 0.90 and 0.988 respectively. The sign of the fit is the same as the reference over the whole domain of energy for the three models.

The preliminary detected energies of peaks in the imaginary part are 2.2 eV and 5.8 eV. Nearby values appears in Tables 11–12: $\omega _6=2.42$ ($NDL=9$), $\omega _3=2.49$ ($NDL=5$) and $|p_2|=2.28$. The highest preliminary detected energy is shifted from more than 1 eV (#8 and #4 respectively), and it disappears in the parameter set of the PF fit. Chromium has a CC crystalline structure and electronic structure [Ar] $3d^{5} 4s^{1}$ [49]. We can notice that Cr has a similar electronic structure as Cu. However, its ‘d’ band is half full when ‘d’ band of Cu is full. Orbitals 3d act as heart orbitals. In fact, in the case of transitions metals, the bands ‘d’ and ‘s’ serve as bands of valence and conduction, respectively. These metals are good conductors of electricity and electronic transitions are between the band d and the band ‘s’. Therefore, the maximum around 2.2 eV correspond to the gap energy of electronic transition. The method appears to keep the related electronic transitions.

#### 4.6 Platinum

Figs. 26 and 28 show a good agreement between the fits and the experimental data over a wide range of energies. This observation is confirmed in Figs. 27 and 29 where the error of fit is plotted as a function of $\omega$. $\epsilon _\infty \geq 1$ only for the DL fit ($NDL=5$). The DL fit ($NDL=9$) appears to be the best one.

The preliminary detected energy of the peak in the imaginary part of the reference relative permittivity is 0.85 eV. This main peak is due to the inter-band transition. Actually, electronic configuration of Pt is [Xe] $6s^{1} 4f^{14} 5d^{9}$ and presents the same electronic properties as Au. Therefore, this peak corresponds to the interband transition of the electrons of conduction. This latter is related to the photo-excitation of electrons which are not in the d band near the X-point of Brillouin zone to the ‘s-p’ conduction band, [50,51]. The corresponding values of energies are $\omega _5=0.82$ ($NDL=9$), $\omega _3=0.76$ ($NDL=5$) and $|p_2|=0.77$ eV ($NPF=4$). The two first terms (#1 and #2) are combinations of Drude or quasi-Drude functions (value of $\omega _1$ close to $0$). The maximum of sensitivity if observed for $\omega _3$ for the DL fit with $NDL=5$. Sensitivities are of the same order of magnitude for the PF fit. The criterion (5) $C<1$ is verified: $C=$ 0.00088, 0.995 and 0.52 respectively. The sign of the fit is the same as the reference over the whole domain of energy for the three models.

#### 4.7 Titanium

Figs. 30–33 show reference data, best fits, and error of fit, as functions of the energy $\omega$. The best parameters for the fitting function are given in Tables 15–16, as well as the best value of the objective function. The best fit is obtained with the PF but the DL ones remain competitive. Note that $\epsilon _\infty \geq 1$ only for DL ($NDL=5$).

The preliminary detected energies are 4.51 and 5.3 eV. Titanium is a transition metal which electronic configuration is $[Ar]\ 3d^{2}4s^{2}$. The interband transition near 1.5 eV is not detected. In Ref. [52], it is claimed that absorption takes place from the partially filled ‘d’ band to the empty parts of the ‘d’ states. Therefore, this band is hybridized with ‘s-p’ characteristic. This corresponds to energies around 3 eV. The closest energies in the parameters of fit are : $\omega _4=5.61$ ($NDL=9$) and $\omega _3=3.12$, $\omega _4=3.31$ ($NDL=5$), and $|p_2|=3.92$ eV. Therefore, we can conclude that the energies of transition for the transition metals found are shifted toward low energies. The criterion (5) $C<1$ is verified: $C=$ 0.73, 0.9998 and 0.998 respectively. The sign of the fit is the same as the reference over the whole domain of energy for $NDL=5$. For $NDL=9$, the sign of the real part of the reference data and of the fit differ in $[4.9;5.8]$ eV and for $NPF=4$, in $[4.9;6]$ eV.

#### 4.8 Complementary discussion for noble metals and summary of results

It is well known that noble metals have a cubic type face-centered (CFC) crystalline structure [49]. Copper, gold, and silver have a similar electronic structure, the band ‘d’ is full and 1 electron in ‘s’ band:

The CFC crystalline structure means the presence of five dispersed valences that correspond to the band ‘d’ and the half-full band ‘s-p’ known as the conduction band. Two main transitions, interband and intraband transitions, are exhibited. In general, the interband involves the absorption of a photon between two electronic bands. The intraband transition couples the light absorption with the electrons inside the conduction band. Some parameters recovered from the proposed method can be interpreted physically. Other fitting functions only improve the quality of fit.In our case, the fitted curves for Ag and Au reveal a peak near 4.2 eV, that of Cu near 3.5 eV. The interband transition can explain the presence of this peak. Indeed, a photon absorption corresponds to the transition between the d valence band and the ‘s-p’ conduction band for copper, silver, and gold. Guerrisi *et al.* studied the optical properties of noble metals [53] and found that for each metal the interband transition took place at different energy. The obtained results match with bibliography results, 2.3 eV, 4.1 eV and 2.6 eV for copper, silver, and gold, respectively. The relative permittivities show that the threshold of the interband transition is observed for higher energies around 4 eV for copper and gold. In the case of silver, an important threshold is observed at 4-5 eV with the sum of 3 peaks. This related to the transition energy phenomenon. Moreover, the wide peak present at 6 eV is also related to an interband transition. The optical properties of the transition metals were discussed in Ref. [50], where the interband transition are found from the $\omega \approx 7$ eV, transition to high lying states above 7 eV, core level-excitations and plasmon effects.

The summary of results follow in Table 17. This could help to choose the adequate results of fit according to the purpose of any calculation using them. The condition $\epsilon _\infty \geq 1$, the facts that sign of real parts of the fit and of the reference data are the same over all the range of energies, the value of the best objective function and the best method are indicated.

## 5. Conclusion

We used a hybrid algorithm to fit the experimental relative permittivity of 7 metals in the range $[0,5;6]$ eV with a sum of 5 and 9 Drude-Lorentz terms, and a sum of 4 Partial-Fractional terms. The method uses a preliminary detection of peaks that helps to guide the hybrid algorithm finding solutions with physical meaning. The hybrid gradient-PSO algorithm was applied in a three-step method. First, the overall shape of the relative permittivity is fitted with a Drude function. Then, the best parameters of all Drude-Lorentz terms are searched around those of the detected peaks in experimental data. Finally, the previous results are adjusted by using an objective function that gives the same weight to the real and imaginary parts. This strategy guides the optimization to find the best set of parameters, recovering the electronic transitions. The PSO algorithm appears to be a useful tool to find the starting point for Nelder-Mead or gradient method that improve the quality of the fits.

The results of fit are satisfactory for the 8 investigated metals, on the condition of releasing the physical constraint $\epsilon _\infty \geq 1$. Finally, results of fit verify the necessary constraint of convergence for FDTD. The control of the connection of the sign of the real part of the fit with that of the reference data is of interest for future use of the results. Numerical results show that increasing the number of terms in the model is not always effective to improve the fit. Moreover, the observation of results show that the Drude model is not fully adapted for all metals.

Thus, we expect that the method could be used for other materials [54] and other models [22,55,56]. Especially, it would be useful to determine the dispersion of nanometric size materials, after adjusting slightly the model parameters to fit the spectroscopic experimental data.

## Funding

European Regional Development Fund (CUMIN CA0021200).

## Disclosures

The authors declare no conflicts of interest.

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