Fitting optical properties of metals by Drude-Lorentz and partial-fraction models in the [0.5;6] eV range

Tasnim Gharbi; Tasnim Gharbi; Dominique Barchiesi; Sameh Kessentini; Ramzi Maalej

doi:10.1364/OME.388060

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 $\omega$:

(1)$$\varepsilon_{DL}(\omega)=\varepsilon_{\infty}-\sum_{i=1}^{NDL} \frac{A_i^2}{\omega^{2}-\omega_{i}^{2}+\imath |\Gamma_{i}|\omega},$$

where $\imath$ is the pure imaginary complex number, $NDL$ is the number of Drude-Lorentz terms (or model order), $\varepsilon _{\infty }$ is the contribution of interband transitions spectrally located beyond the energy range being considered to the real part of the relative permittivity, $A_i$ is the amplitude, actually the product of the oscillator strength by the square of the plasma frequency. The numerical values of the plasma frequencies can be found in Ref. [28]. $\omega _{i}$ is the resonant frequency, $|\Gamma _{i}|$ is the damping coefficient which governs the broadening of the resonance peak. The damping coefficient is in absolute value to ensure physical solutions from the fitting process. The values of $\omega _i$, $A_i$ and $|\Gamma _i|$ are expressed in electronvolt (eV). The Drude model is obtained by setting $\omega _i=0$.

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].

(2)$$\epsilon_{PF}(\omega) = \epsilon_\infty + \sum_{i=1}^{NPF} \frac{c_i}{\imath \omega - p_i} + \frac{c_i^*}{\imath \omega - p_i^*}$$

where $NPF$ is the model order, $p_i$ are poles, $c_i$ are the residues, and $c_i^*$ is the complex conjugate of $c_i$ . This model is similar to the Critical Points model [30] that can used in FDTD method, if the parameters verify a convergence condition. The model is the sum of $\epsilon _\infty$ and $NPF$ terms of the angular frequency $\omega$, each of them involving parameters $c_i$ and $p_i$. The parameters $c_i$ and $p_i$ being complex numbers, the number of parameters to be adjusted is therefore $D=4 NPF +1 = 17$ for $NPF=4$. Han et al [13] demonstrated the relationship between the DL and the PF model. The values of $|p_i|$ can be compared to $\omega _{i}$ and $2\Re (p_i)$ to $\Gamma _i$. In that paper, it is mentioned that the real part of $p_i$ should be lower than $0$ for lossy dispersive materials [13]. Indeed, introducing a complex amplitude $A_i$ in DL can help to identify the two formula [31].

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 $\Phi (\omega _j)$:

(3)$$\Phi(\omega_j)= \varepsilon_{DL}(\omega_j)-\varepsilon_{Ref}(\omega_j),$$

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:

(4)$$F=C_0 \sqrt{\frac{\displaystyle \sum_{j=1}^{N_\omega} \left| C_1 \Re(\Phi(\omega_j)) \right|^2 + \left| C_2 \Im(\Phi(\omega_j)) \right|^2}{2 N_\omega-(D+1)}},$$

where $C_0$ is a coefficient that will be set in what follows. $C_1$ and $C_2$ are weighting coefficients, which values will be set at each step of the fitting process. The weighting coefficients $C_1$ and $C_2$ are introduced to offset the predominance of the real part of the relative permittivity at low energies (typically $\omega < 3$ eV). This may prevent the attraction of fitting solutions toward the pure Drude model when neglecting the physical energies transitions, which appear as discreet peaks in the reference experimental data. This observation led us to introduce a three-step fitting method in Sec. 3.1. The denominator $2 N_\omega -(D+1)$ is the degree of freedom, and is equal to the number of reference data ($N_\omega$ real values and $N_\omega$ imaginary values of the reference relative permittivity) minus the number of parameters to be estimated minus $1$, a constraint being used. Actually, the results of fitting must verify a condition (or constraint) to be used in FDTD calculations [32,33]:

(5)$$C= \left| \frac{\epsilon_\infty}{\epsilon_\infty+\chi_0} \right| < 1.$$

For a Drude-Lorentz term (Eq. (1)), $\chi _0$ is a function of $\alpha =\Gamma _i/2$, $\beta =\sqrt {\omega _i^2-\alpha ^2}$, $\eta =A_i^2/\beta$ and $NDL$ is the number of the Drude-Lorentz terms (model order):

(6)$$\begin{aligned} \chi_0 &= - \left(\frac{\omega_i}{\Gamma_i}\right)^2 (1-\exp(-\Gamma_i \Delta t))+ \frac{\omega_i^2}{\Gamma_D} \Delta t\\ & +\sum_{i=1}^{NDL} \Re\left(-\imath \frac{\eta}{\alpha - \imath \beta} (1-\exp((-\alpha + \imath \beta) \Delta t)) \right). \end{aligned}$$

In the case of PF model [14,29]:

(7)$$\chi_0 = \sum_{i=1}^{NPL} \Re\left({-}2 \frac{c_i}{p_i} (1-\exp((-\Re(p_i) + \imath \Im(p_i)) \Delta t)) \right).$$

In Eqs. (6–7), the time step $\Delta t$ is evaluated from the size of the spatial grid $\Delta x$ of the domain of computation in FDTD. The discretization in the time domain follows: $\Delta t = \Delta x /(2 c)$, with $c$ the speed of light in vacuum. This is a criterion of convergence of the finite difference algorithm [32]. In the following we consider $\Delta x = 1$ nm [21,33]. The constraint (5) is handled by degrading $F$ by a factor $C_0=10$ if $C\geq 1$, otherwise $C_0=1$ (Eq. (4)). Thus, the objective function is penalized when the criterion (5) is not verified for the considered parameters. All results in this paper, verify the constraint given in Eq. (5). Fits of better quality could be obtained without introducing this constraint.

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:

(8)$$\mathbf{x}_i(t+1)=\mathbf{x}_i(t)+\mathbf{v}_i(t+1),$$

where $\mathbf {v}_i(t+1)$ is the updated velocity of particle $i$ given by:

(9)$$\mathbf{v}_i(t+1)= \omega \mathbf{v}_i(t)+U_{1}c_{1}(\mathbf{p}_i(t)-\mathbf{x}_i(t))+U_{2}c_{2}(\mathbf{g}(t)-\mathbf{x}_i(t)),$$

where $\omega$ is the inertia weight, $U_{i=1,2}$ are vectors of random uniform variables in the interval $[0,1]$, $c_{i=1,2}$ are acceleration coefficients and $\mathbf {g}(t)$ is the global best solution (the best among $\mathbf {p}_i(t)$).

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:

(10)$$nreal = \mathcal{N}^{{-}1}(0.975)^2/0.06^2 \approx 1000,$$

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.

The minimum number of the two sets of peculiar energies is considered to fix the number of Drude-Lorentz terms and the boundaries of the search domains. The search domains are $\omega _i \in [0.9 E; E+1]$ eV, $E$ being each energy of the set; $A_i \in [A/2; 2 A]$ eV, $A$ being the amplitude of the detected peak and $\Gamma _i \in [0; 5]$ eV. These intervals are deduced from a preliminary numerical study of the Drude-Lorentz function (Eq. (1)). The boundaries of the search domains are chosen to enable to get visible peaks in the relative permittivity.

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.

For PF model, the domain of search are: $[1;10]$ for $\epsilon _\infty$, $[-100;100]$ for $\Re (c_i)$, $\Im (c_i)$, $\Re (p_i)$ and $\Im (p_i)$. The preliminary detection of peaks is not used in this case. The hybrid method is used directly, in a single step.

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:

(11)$$S(\omega_i) = \frac{m(\omega_i)}{\sigma(\omega_i)}$$

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.

Fig. 1. Real parts of the permittivity of gold $\epsilon _{Ref}$ [4] (o), [39] (+), [21] ($\square$) and best fits for $NDL=9$ (thick solid line), $NDL=5$ (thin solid line), and PF (dashed line) ($NPF=4$). The parameters can be found in Tables 1–2.

Au: $F_{P S O + N M} = 0.92$ , $ϵ_{\infty} = 3.95$ .
$i$	1	2	3	4	5	6	7	8	9
$A_{i}$ (eV)	8.21	3.14	0.155	3.17	5.61	6.72	0.263	2.07	0.608
$ω_{i}$ (eV)	0.010	0.40	0.80	2.98	4.20	6.15	7.50	7.99	13.8
$Γ_{i}$ (eV)	0.0401	0.169	3.89	0.875	1.68	3.04	9.86	10.0	9.72
$S (ω_{i})$	0.095	6.8	2.2	3.9	4.7	4.5	4.9	6.1	9.2
Au: $F_{P S O + G R} = 0.99$ , $ϵ_{\infty} = 1.87$ .
$A_{i}$ (eV)	8.31	3.28	3.31	4.8	10.4
$ω_{i}$ (eV)	0.265	3.00	3.90	4.65	7.71
$Γ_{i}$ (eV)	0.0655	0.904	1.22	1.89	4.66
$S (ω_{i})$	6.9	8.3	13	6.3	7.6

Au: $F_{P S O + G R} = 0.39$ , $ϵ_{\infty} = - 1.41$ .
$i$	1	2	3	4
$ℜ (c_{i})$ (eV)	2.2	-7.41	4.94	6.27
$ℑ (c_{i})$ (eV)	-111	1.44	32.4	20.1
$ℜ (p_{i})$ (eV)	0.0416	0.997	-4.73	-1.5
$ℑ (p_{i})$ (eV)	0.279	2.27	3.41	0.767
$S (ℑ (p_{i}))$	0.82	0.64	0.97	0.38

Ag: $F_{P S O + N M} = 0.52$ , $ϵ_{\infty} = 1.77$
$i$	1	2	3	4	5	6	7	8	9
$A_{i}$ (eV)	8.53	0.939	0.362	1.89	3	0.00781	4.13	0.796	5.3
$ω_{i}$ (eV)	0.144	0.651	4.23	4.33	4.83	5.76	5.77	6.01	7.16
$Γ_{i}$ (eV)	0.0654	0.182	3.76	0.409	0.855	2.83	1.59	2.66	2.98
$S (ω_{i})$	2.34	0.966	4.56	8.58	10.5	7.45	2.78	1.99	1.08
Ag: $F_{P S O + G R} = 0.46$ , $ϵ_{\infty} = 1.12$ .
$A_{i}$ (eV)	7.89	3.66	3.14	6.12	8.43
$ω_{i}$ (eV)	0	0.346	4.59	5.91	9.28
$Γ_{i}$ (eV)	0.0205	0.290	0.854	2.30	0.267
$S (ω_{i})$	0.17	2.2	6.2	9.9	7.9

Ag: $F_{P S O + G R} = 0.72$ , $ϵ_{\infty} = - 2.83$ .
$i$	1	2	3	4
$ℜ (c_{i})$ (eV)	-15.7	-1.78	22.4	14.7
$ℑ (c_{i})$ (eV)	51.3	0.701	-16.3	-21.1
$ℜ (p_{i})$ (eV)	-0.108	0.559	-2.32	0.234
$ℑ (p_{i})$ (eV)	-0.333	-4.18	11.7	0.468
$S (ℑ (p_{i}))$	0.26	0.063	0.42	0.12

Au: $F_{P S O + N M} = 2.9$ , $ϵ_{\infty} = 2.68$ .
$i$	1	2	3	4	5	6	7	8	9
$A_{i}$ (eV)	7.87	2.81	3.07	5.44	3.14	5.43	0.168	0.110	0.885
$ω_{i}$ (eV)	0.010	0.417	2.99	3.98	4.83	5.96	8.54	9.60	9.80
$Γ_{i}$ (eV)	0.0200	0.551	0.696	1.30	0.853	1.67	5.31	7.27	2.99
$S (ω_{i})$	0.40	1.3	1.5	2.8	3.7	4.2	4.4	1.5	0.44
Au: $F_{P S O + N M} = 3.31$ , $ϵ_{\infty} = 2.28$ .
$A_{i}$ (eV)	8.11	1.87	2.04	5.46	6.91
$ω_{i}$ (eV)	0.000175	0.015	2.91	3.8	5.35
$Γ_{i}$ (eV)	0.0722	4.69	0.593	1.43	2.39
$S (ω_{i})$	0.83	1.3	4.8	6.4	3.7

Au: $F_{P S O + G R} = 1.7$ , $ϵ_{\infty} = - 5.77$ .
$i$	1	2	3	4
$ℜ (c_{i})$ (eV)	-1.5	5.82	-10.8	21.9
$ℑ (c_{i})$ (eV)	769	34.4	-8.04	7
$ℜ (p_{i})$ (eV)	0.0341	-3.02	1.19	-21
$ℑ (p_{i})$ (eV)	-0.0377	1.45	-2.24	-1.7
$S (ℑ (p_{i}))$	-0.2	0.23	0.35	0.24

Cu: $F_{P S O + G R} = 3.6$ , $ϵ_{\infty} = - 64.7$
$i$	1	2	3	4	5	6	7	8	9
$A_{i}$ (eV)	6.02	3.82	1.4	2.01	3.03	8.28	0.181	126	13.8
$ω_{i}$ (eV)	0	0.01	2.38	2.74	3.43	5.16	5.87	16.4	32.9
$Γ_{i}$ (eV)	2.39e-05	0.507	0.274	0.525	1.1	2.77	3.7	0.011	10.1
$S (ω_{i})$	NaN	1.4	2.6	3.7	3.5	3.4	3.4	2.5	1.8
Cu: $F_{P S O + G R} = 4.1$ , $ϵ_{\infty} = - 0.97$ .
$A_{i}$ (eV)	6.58	3.15	2.48	9.73	2.55
$ω_{i}$ (eV)	0	0.373	2.66	5.00	5.95
$Γ_{i}$ (eV)	1.44e-06	0.611	0.706	4.11	5.48
$S (ω_{i})$	0.49	1	2	1.3	0.91

Cu: $F_{P S O + G R} = 2.2$ , $ϵ_{\infty} = - 0.0297$ .
$i$	1	2	3	4
$ℜ (c_{i})$ (eV)	-12.7	-0.218	-19.4	20.5
$ℑ (c_{i})$ (eV)	8.35	11.2	18.6	20
$ℜ (p_{i})$ (eV)	-0.0597	0.843	-1.09	0.261
$ℑ (p_{i})$ (eV)	-0.368	1.55	0.656	-0.112
$S (ℑ (p_{i}))$	-0.21	-0.0019	-0.14	0.29

Al: $F_{P S O + N M} = 0.39$ , $ϵ_{\infty} = 0.727$
$i$	1	2	3	4	5	6	7	8	9
$A_{i}$ (eV)	8.36	10.6	1.36	3.66	3.66	5.83	0.563	0.296	15.8
$ω_{i}$ (eV)	0	0.0167	1.45	1.58	1.85	2.17	2.17	5.53	26.3
$Γ_{i}$ (eV)	0	0.200	0.163	0.329	0.800	2.70	17.7	17.9	0.886
$S (ω_{i})$	NaN	0.91	1.4	3.1	5.8	2.1	0.98	1.1	1.3
Al: $F_{P S O + G R} = 0.44$ , $ϵ_{\infty} = 1.04$ .
$A_{i}$ (eV)	10.2	8.83	4.29	3.8	5.3
$ω_{i}$ (eV)	0	0.0738	1.56	1.87	2.48
$Γ_{i}$ (eV)	0.224	0	0.378	0.822	2.76
$S (ω_{i})$	NaN	0.50	7.1	16	1.5

Al: $F_{P S O + G R} = 1.4$ , $ϵ_{\infty} = - 2.02$ .
$i$	1	2	3	4
$ℜ (c_{i})$ (eV)	-2.23	-3.74	-1.65	75.5
$ℑ (c_{i})$ (eV)	2.45e+03	10.4	10.6	-35.4
$ℜ (p_{i})$ (eV)	0.0552	18.3	0.285	-341
$ℑ (p_{i})$ (eV)	-0.0354	-18.5	-1.56	105
$S (ℑ (p_{i}))$	-0.18	-0.17	0.33	0.097

Cr: $F_{P S O + N M} = 0.74$ , $ϵ_{\infty} = 0$ .
$i$	1	2	3	4	5	6	7	8	9
$A_{i}$ (eV)	1.24	5.35	0.000449	3.13	4.86	4.23	9.44	5.84	21.8
$ω_{i}$ (eV)	0	0.00751	0.447	1.12	1.57	2.42	3.56	6.41	14.4
$Γ_{i}$ (eV)	25.6	8.84e-05	17.0	0.638	1.14	0.971	3.52	2.12	5.31
$S (ω_{i})$	NaN	0.72	1.0	2.3	2.7	1.8	1.8	6.1	3.7
Cr: $F_{P S O + N M} = 1.43$ , $ϵ_{\infty} = - 1.76$ .
$A_{i}$ (eV)	18	4.19	2.38	6.71	6.52
$ω_{i}$ (eV)	0.672	1.27	2.49	6.95	8.59
$Γ_{i}$ (eV)	11.7	3.04	0.631	1.07	3.15
$S (ω_{i})$	0.65	2.7	1.8	2.1	2

Pt: $F_{P S O + G R} = 0.15$ , $ϵ_{\infty} = - 0.00042$
$i$	1	2	3	4	5	6	7	8	9
$A_{i}$ (eV)	1.4	5.15	0.424	0.74	3.49	9.75	6.33	3.79	53.3
$ω_{i}$ (eV)	0	0.119	0.371	0.723	0.821	1.13	3.68	3.78	25
$Γ_{i}$ (eV)	10.6	0	17.9	0.139	0.561	2.49	4.67	16.6	43.6
$S (ω_{i})$	NaN	0.99	2.2	16	5.4	1.4	1.9	2	2.1
Pt: $F_{P S O + G R} = 0.26$ , $ϵ_{\infty} = 2.44$ .
$A_{i}$ (eV)	4.95	70.6	2.97	4.28	7.69
$ω_{i}$ (eV)	0	0	0.762	1	1.83
$Γ_{i}$ (eV)	0	324	0.411	0.966	3.66
$S (ω_{i})$	NaN	2.8	75	7.1	1.1

Pt: $F_{P S O + N M} = 0.47$ , $ϵ_{\infty} = 0.119$ .
$i$	1	2	3	4
$ℜ (c_{i})$ (eV)	-15.2	-4.69	-1.14	-0.949
$ℑ (c_{i})$ (eV)	19.5	-11.8	16.4	-14.3
$ℜ (p_{i})$ (eV)	-0.121	0.299	3.51	10.7
$ℑ (p_{i})$ (eV)	-0.286	0.708	3.05	-6.55
$S (ℑ (p_{i}))$	0.32	-0.25	-0.21	-0.27

Ti: $F_{P S O + G R} = 0.10$ , $ϵ_{\infty} = - 0.0229$ .
$i$	1	2	3	4
$ℜ (c_{i})$ (eV)	-10.8	0.271	0.818	0.66
$ℑ (c_{i})$ (eV)	2.09	-13.4	-1.96	-0.975
$ℜ (p_{i})$ (eV)	-0.00362	-3.93	0.395	0.615
$ℑ (p_{i})$ (eV)	-0.237	-0.149	1.55	-3.97
$S (ℑ (p_{i}))$	0.58	-0.31	0.32	-0.78

Cr: $F_{P S O + G R} = 0.46$ , $ϵ_{\infty} = 4.6$ .
$i$	1	2	3	4
$ℜ (c_{i})$ (eV)	5.46	-1.01	20.2	-35.8
$ℑ (c_{i})$ (eV)	328	-0.943	54.6	4.34
$ℜ (p_{i})$ (eV)	1.89	0.313	-2.31	-4
$ℑ (p_{i})$ (eV)	-0.157	2.26	1.19	3.04
$S (ℑ (p_{i}))$	-0.24	0.42	0.88	0.091

Ti: $F_{P S O + G R} = 0.17$ , $ϵ_{\infty} = 0.0021$
$i$	1	2	3	4	5	6	7	8	9
$A_{i}$ (eV)	8.91	2.81	3.19	4.09	11.0	0.00501	0.0357	0.182	0.0432
$ω_{i}$ (eV)	0	1.55	3.12	5.61	8.86	9.77	17.8	25.3	35.8
$Γ_{i}$ (eV)	3.28	0.882	1.51	1.67	3.22e-08	12.9	2.61	8.12	20.4
$S (ω_{i})$	NaN	2.0	2.9	3.0	2.8	3.7	2.5	2.1	2.1
Ti: $F_{P S O + G R} = 0.13$ , $ϵ_{\infty} = 1.82$ .
$A_{i}$ (eV)	7.44	3.28	3.16	2.19	6.27
$ω_{i}$ (eV)	0	1.55	2.7	3.31	6.09
$Γ_{i}$ (eV)	2.17	0.966	1.63	0.929	2.83
$S (ω_{i})$	NaN	9.0	12	8.9	1.5

		DL $N D L = 9$	DL $N D L = 5$	PF $N P F = 4$
Au [4]	$ϵ_{\infty} \geq 1$	yes	yes	no
	sign	no	yes	yes
	$F$	0.92	0.99	0.39
	Method	PSO+NM	PSO+NM	PSO+NM
Au [11]	$ϵ_{\infty} \geq 1$	yes	no	no
	sign	no	yes	no
	$F$	2.9	3.3	1.7
	Method	PSO+NM	PSO+GR	PSO+GR
Ag [11]	$ϵ_{\infty} \geq 1$	yes	yes	no
	sign	yes	no	no
	$F$	0.52	0.46	0.72
	Method	PSO+NM	PSO+GR	PSO+GR
Cu [11]	$ϵ_{\infty} \geq 1$	no	no	no
	sign	no	yes	yes
	$F$	3.6	4.1	2.2
	Method	PSO+GR	PSO+GR	PSO+GR
Al [11]	$ϵ_{\infty} \geq 1$	no	yes	no
	sign	yes	yes	yes
	$F$	0.39	0.44	1.4
	Method	PSO+NM	PSO+GR	PSO+GR
Cr [11]	$ϵ_{\infty} \geq 1$	no	no	yes
	sign	yes	yes	yes
	$F$	0.74	1.43	0.46
	Method	PSO+NM	PSO+NM	PSO+GR
Pt [11]	$ϵ_{\infty} \geq 1$	no	yes	no
	sign	yes	yes	yes
	$F$	0.15	0.26	0.47
	Method	PSO+NM	PSO+NM	PSO+GR
Ti [11]	$ϵ_{\infty} \geq 1$	no	yes	no
	sign	no	yes	no
	$F$	0.17	0.13	0.10
	Method	PSO+GR	PSO+GR	PSO+GR

Abstract

1. Introduction

2. Physical functions for fitting

2.1 Drude-Lorentz model

2.2 Partial-fraction model

3. Method for the fitting

3.1 The minimization method

3.2 Preliminary detection of Drude-Lorentz shapes in the reference relative permittivity

3.3 Three-step method of fitting

3.4 Sensitivity

3.5 First test of the method with Johnson and Christy data

4. Results and discussion

4.1 Silver

4.2 Gold

4.3 Copper

4.4 Aluminum

4.5 Chromium

4.6 Platinum

4.7 Titanium

4.8 Complementary discussion for noble metals and summary of results

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (33)

Tables (17)

Equations (11)

Optical Materials Express