1. Introduction

An accurate knowledge of the optical constants of materials is more and more pursued since optics and the interaction of light with matter is at the origin of incessant, new experiments and techniques. Pairs of optical constants are not fully independent, but linked through the Kramers-Kronig (KK) dispersion relations [1,2]. A set of optical constants is said to be self-consistent if it satisfies KK relations. Self-consistency is a property that is connected with accuracy in the sense that inconsistent data reveals inaccurate data. Self-consistency can be quantitatively measured through sum rules. Sum rules are integral tests of the optical constants, which must result in a value that can be predicted from basic assumptions [3,4,5]. Sum rules provide collective self-consistency evaluation through the full electromagnetic spectrum. As a negative consequence of this, some excess in an optical constant in one spectral range along with some defective value of such optical constant in another range could result in a sum-rule test that erroneously indicates self-consistency.

KK relations are useful tools in optical constant calculation. However, in order to apply them, knowledge of an optical constant has to be extended to the full spectrum, and this extension also applies to sum rules. Since the full spectral range cannot be measured with a single experiment, a given set of experimental optical constants requires data coming from other sources and/or some sort of extrapolation to reach the full spectrum. The main contribution to the sum-rule integral may come from outside the spectral range that was measured, which may result in a sum-rule test that is not informative of the self-consistency of the specific spectral range that was measured. The use of a window function (WF) can help adapt the main spectral range that contributes to sum rules, giving more weight to the desired spectral range with respect to more distant spectral ranges, in which one may have a more uncertain knowledge of the optical constants. The sum-rule with WF integrates the product of the WF times the complex optical constant function, either $\tilde{\varepsilon }(E )- 1 = {\varepsilon _1}(E )- 1 + i{\varepsilon _2}(E )$ or N(E)-1 = n(E)-1 + ik(E), where i is the imaginary unit. Several WFs have been proposed and used in the past, although WFs have still been used in few occasions so far. In a previous research article [6] we proposed the use of narrow WFs that are made to scan the spectrum of interest to check for self-consistency at each specific spectral range. They were seen to help avoid the aforementioned undesired spectral compensation of errors in different spectral ranges, and they do it by varying the weight of each spectral range in a continuous way.

At that time we proposed a simple WF consisting in one or two rectangle functions. The steep variation of the rectangle function at the edges can affect sum-rule integration and it can result in a larger-than-real apparent inconsistency. In this research we develop two new WFs that avoid this steep variation through the use of smoother envelope functions, which results in a more precise evaluation of self-consistency. Section 2 is devoted to develop these new WF’s. Section 3 presents examples of the benefit of the new WFs compared to the rectangle function both with calculated optical constants as well as with experimental data sets of Al and Au, whose self-consistency is evaluated.

2. Development of smooth window functions

Well-known KK dispersion relations are expressed by:

The optical-constant function must satisfy some requirements which arise from the fundamental assumptions of linearity and causality. Causality is at the heart of KK relations. The optical constant function being a complex function, Titchmarsh theorem demonstrates the reciprocal implications of causality, KK relations, and analyticity of such function. The latter means that the optical constant function must be analytic (holomorphic) in the upper complex half plane C₊ (z = x + iy∈C₊ if y > 0). It is noted that the optical constant function is usually required by the physicist only at the real axis, which means for a real photon energy, in what is a complex function (ɛ₁ and ɛ₂) of real variable (photon energy). The extension of the optical constant variable to complex energies in C₊ is a mathematical tool which should not confuse the reader since all final results will be limited to real energies. Other than the analyticity requirement for the optical-constant function, two further requirements for KK analysis, derived from causality and linearity restrictions, are parity and convergence. Parity for real photon energies involves the following conditions: ${\varepsilon _1}({ - E} )= {\varepsilon _1}(E )$ and ${\varepsilon _2}({ - E} )={-} {\varepsilon _2}(E )$. The optical constant function (in fact $\tilde{\varepsilon }$-1 or N-1) must decay to 0 fast enough at infinite photon energy both on the real axis and in any direction within C₊.

In order to develop a WF, say F, all above conditions must be satisfied by the product of the optical-constant function times F(E). In that case, KK relations can be extended to:

F(E) can be developed to weight the spectral range of interest with the desired profile. The WF will be developed through a linear superposition of Lorentz oscillators:

2.1 Weight function based on rectangles

A simple example was obtained in the previous research [6] by using wf = constant in [E₁-E₂] and 0 otherwise, which resulted in:

H₁, already developed in [6], decays at infinite energy. At zero energy, the imaginary part decays too but the real part tends to a nonzero constant. In order for the real part also to decay at small energies, a new function H₂ was developed in [6] from H₁ in two subranges, each with opposed sign weight factor:

 

Fig. 1. The weight functions based on two rectangles to obtain H₂ WF (a), based on three straight lines to obtain L₂ WF (b), and based on two 4-degree polynomials to obtain M₂ WF (c). In all cases, E₁=1 eV and E₂=3 eV. For L₂, the straight lines are seen slightly curved due to the abscissae log-axis.

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Fig. 2. Real (a) and (c) and imaginary part (b) and (d) of WFs H₂, L₂, and M₂ with E₁=1 eV, E₂=3 eV, γ₁=0.01 eV. (a) and (b): the WF. (c) and (d): absolute values in log-axis, along with the asymptotic decays.

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2.2 Weight function with straight lines

Both H₁ and H₂ were seen to be helpful to evaluate the self-consistency of various sets of experimental optical constants. However, the abrupt change of these functions at E₁ and E₂, and also at E_M for H₂, may derive in some instability when the WF is applied on experimental optical constant data. To reduce such instabilities, new functions with a smoother behavior are developed here. The new weight functions avoid discontinuities and are also derived with the superposition of Lorentz oscillators. The simplest weight function that avoids discontinuities is a set of straight lines that are valued zero at the edges of the desired spectral range. Figure 1 also plots the weight function with 3 straight lines. Three is the minimum number of straight lines to enable a WF to converge to 0 at infinity and also at E = 0. The E¹-term (at E = 0) of the imaginary part is also forced to cancel. Let [E₁-E₂], be the desired window of the weight function. Among various possible choices, a simple weight function is one that geometrically divides this range in three equal sub-ranges (one per straight line), so that E₂/E₁=f ³. The two intermediate energies where the small-energy line (l₁) switches to the middle-energy line (l₂-l₃) and the middle-energy line to the large-energy line (l₄) are E_1M=f·E₁ and E_M2=f ²·E₁=E₂/f, respectively. The middle-energy line, or central line, crosses the energy axis at E_M=(E_1M+E_M2)/2 = E₁f (1 + f)/2, that divides the line in two collinear segments l₂ and l₃. In Fig. 1, it was also selected E₁=1 and E₂=3, so that f = 3^1/3. To satisfy the asymptotic requirements, one needs to apply a condition on the widths of the Lorentz oscillator in the various subranges. For this to apply, the central line is divided into the two specified segments l₂ and l₃. With this definition, only two Lorentz-oscillator widths γ and γ’ are required. If γ in the [E₁-E_M] range is freely chosen, then γ’ must satisfy:

2.3 Weight function with 4-degree polynomials

The weight function for L₂ avoids the discontinuities of the weight function for H₂. However, the weight function for L₂ has a sudden slope change at each edge of a straight line, which might result in some source of instability. A WF that avoids such discontinuity in the first derivative has been also developed. Its purpose is a still smoother behavior. To develop this WF, the same above procedure is reapplied by using a weight function with, for instance, higher order polynomials instead of straight lines. If second-degree polynomials are used, then four polynomials are needed to cancel both the real part and the E¹ term of the imaginary part of the WF in the E = 0 limit. If one additionally requires that the slope at the connection energy between contiguous polynomials is not only continuous but zero in order to get an even smoother connection, it is necessary to use larger-degree polynomials. With 3-degree polynomials, four of them are required. With 4-degree polynomials, only two of them are necessary, and this option was adopted. By selecting the connection energy between the two polynomials at E_M, such weight function is immediately given by (E-E₁)²(E-E_M)² in the [E₁,E_M] range and -(E-E_M)²(E-E₂)² in the [E_M,E₂] range. Nevertheless, there is a necessary normalization factor between the two polynomials to force cancellation of the real part and of the E¹ term of the imaginary part of the WF at E = 0 in the same terms as referred above. The polynomials in the [E₁,E_M] and in the [E_M,E₂] range are named p₁ and p₂, respectively. Figure 1 also plots the weight function with two 4-degree polynomials.

As with the case of 3 straight lines, a simple weight function is adopted even though others may be possible. Let us define f so that E₂/E₁= f ² and then E_M=fE₁ and E_M=(E₁E₂)^1/2. In Fig. 1, E₁=1 eV and E₂=3 eV, so that f = 3^1/2. With the proper normalization between the two 4-degree polynomials to satisfy the aforementioned asymptotic requirements, the obtained WF is:

Let us now set the normalization factor for the Lorentz oscillator widths. The normalization factor applied to G₂ in the definition given in Eq. (9) already assures the cancellation of Re[M₂] at E = 0. To cancel the E¹-term of Im[M₂] at E = 0, one only needs to force that the ratio of the Lorentz oscillator widths corresponding to the two polynomials be:

Figure 2 also displays M₂ with E₁=1 eV, E₂=3 eV, and γ=0.01 eV. Table 1 summarizes the decay of H₂, L₂, and M₂ both at E = 0 and E=∞. In Fig. 2, the three WFs have been normalized for their imaginary parts to have equal maxima.

Table 1. Asymptotic decay of the real and imaginary part of H₂, L₂, and M₂.

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The new WFs have been named L₂ and M₂ because they share with H₂ their structure with two halves with inverted sign, which is necessary to cancel the real part of the WF and the E¹-term of its imaginary part at E = 0. In analogy to function H₁ in [6], functions L₁ and M₁ can be also defined starting with a single 4-degree polynomial (M₁) or with the pair of straight lines l₁ and l₂ (L₁). In spite of their lack of decay at E = 0, they have the advantage that its imaginary part is always positive and its real part changes sign only once, whereas H₂, L₂, and M₂ have one sign inversion in their imaginary part and two in their real part.

2.4 Sum rules with window functions

By virtue of the superconvergence theorem [3], the asymptotic decay of the optical constants at high energies provides the well-known sum rules:

WFs are used to generate new sum rules following the same procedure of [6], which is based in the procedure of Altarelli et al. [3], who used the superconvergence theorem.

First of all, the WFs developed in the previous subsections are complex analytic functions in the upper plane (for photon energies extended to the complex plane), i.e., all poles are located in the lower plane. The WFs have the correct parity, i.e., Re[F(-E)] = Re[F(E)] and Im[F(-E)]= -Im[F(E)] (for real-valued energies), F(E) being any of the above WFs. Finally, the WF multiplied by the refractive index N-1 decays at large photon energies. The decay at large energies is already satisfied by N-1, but the new WFs add a further decay, as given in Table 1. This enables to construct new sum rules based on Eqs. (13) and (14) in which the integrand is multiplied by powers of the photon energy as long as the decay of the WF times N-1 is fast enough. The large-energy asymptotic behavior of the optical constants given in [3,4] will be used:

By replacing n with E^mRe[F(N-1)] and k with E^mIm[F(N-1)] in Eqs. (13) and (14) for any m integers that provides the correct parity, assures convergence at E=∞ and avoid poles at E = 0 going with E^-1 or faster, by using the asymptotic behaviour of the optical constants [Eqs. (15) and either (16a) or (16b)], as well as of the WFs (Table 1), and by applying the superconvergence theorem [6], one gets to the following sum rules:

This research focuses on isotropic materials. The above sum rules can be also applied in the case of anisotropic crystals by just replacing $\tilde{\varepsilon } - 1$ with the dielectric function tensor ${\tilde{\varepsilon }_{ij}} - {\delta _{ij}}$ for i,j = 1 to 3. For conductors, this function has the same sort of divergence at E = 0 that isotropic materials [3], so that, again, the divergence for conductors is cancelled for sum-rules (18) to (22) when applied to the dielectric tensor.

The above sum rules can be also applied to other optical functions, such as optical conductivity and loss function. Optical conductivity is expressed by $\sigma ={-} {\textrm{i}\varepsilon_0}{\; }({\textrm{E}/{\hbar}} ){\; }[{\tilde{\varepsilon }(E )- 1} ]$ (SI units) [10]. By modifying the window function to F’(E)=(i/E)F(E), then $F^{\prime}\sigma = ({\varepsilon_0/{\hbar}} )\textrm{F}\; [{\tilde{\varepsilon } - 1} ]$, so that sum rules (17) to (22) can be immediately modified to express them in terms of σ. Regarding the optical loss, $({1/\tilde{\varepsilon }} )- 1$ has the same asymptotic behavior at large E that $\tilde{\varepsilon } - 1$ and has no divergence at E = 0 even for conductors. Therefore, sum rules (17) to (22) are all satisfied for $({1/\tilde{\varepsilon }} )- 1$ [with the aforementioned factor of 2 in Eq. (22)].

We need a criterion to evaluate the sum rules that result in a null integral when it is applied to an experimental data set. Altarelli and Smith [5] defined a verification parameter to evaluate how close to zero is the calculated integral for experimental n data which are known only at discrete photon energies and involve inaccuracies:

Shiles et al. [11] proposed that ζ be considered acceptable if within ±0.005. In a straightforward generalization of Shiles’s criterion, as performed in [6], we propose the same acceptable range for ζ given by Eq. (25) to evaluate the compliance of the above inertial-like sum rules given by Eqs. (17) to (21). Shiles’s validity range seems not to have been justified in the literature. To have a picture in mind, a simple case with ζ=+.005 can be figured out if one starts with 100% accurate optical constant n and scales n-1 up by 0.5% (factor of 1.005) for n > 1 and down by 0.5% (factor of .995) for n < 1. ζ= -.005 is obtained by inverting the above factors. Hence for this fabricated case, the [-0.005-+0.005] range of ζ corresponds to [-0.5%-+0.5%] range of inaccuracy of n-1.

Summarizing this sub-section, Eqs. (17) to (21) provide five inertial-like sum rules and Eqs. (23a) or (23b) provide one f-like sum rule that can be applied with the new WFs developed in subsections 2.2 and 2.3. The WFs are tunable in the desired energy range through the selection of E₁ and E₂, whereas the choice of γ drives the decaying speed outside the [E₁, E₂] band. The new sum rules are exemplified in the next section.

3. Application of the new window functions

3.1 A Lorentz oscillator

The benefit of sum rules with WFs was already demonstrated in [6]. It was seen that the contribution to the sum-rule integral could be tuned to the desired spectral range. A further benefit was seen by calculating the sum rules not for a single WF but for the latter scanning the spectrum to calculate optical-constant self-consistency as a function of photon energy. In this section several examples are presented with situations in which the new WFs are beneficial compared to the one in [6].

The weight function used to generate H₂ involves three discontinuities. Such discontinuities are not present in H₂, although the latter turns very steep at the three main energies of the window (the two edges and the center). Steepness increases for a decreasing γ parameter. This may result in instabilities in sum-rule calculation, which in the end limits how small γ can be made and therefore how fast the WF can decay outside the band. To highlight this problem, a self-consistent optical-constant function, a Lorentz oscillator, already presented in Eq. (3), was selected and the density of data was used as a variable. Such density of data in the photon energy axis will be referred to as the sampling rate. Exact values of the optical constants for the oscillator were calculated in a large energy range (from very small to very large energies) for sampling rates consisting in a constant ratio of each consecutive pair of energies E[i + 1]/E[i], i.e., the spacing between the photon energies selected is linear on a log of energy plot. Below it is displayed how sum-rule instability depends on such ratio.

Figure 3 plots ɛ₁ and ɛ₂ of a Lorentz oscillator with two different sampling rates (expressed as energy factor that stands for the above energy ratio). Oscillator parameters [as per Eq. (3)] were E₀=1 eV (oscillator center) and γ=0.05 eV (oscillator width), along with A = 1 eV.

 

Fig. 3. A Lorentz oscillator with parameters E₀=1, γ=0.05 eV. The indicated sampling rate factors were used.

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Figure 4 plots ζ for sum rule (19) (calculated with the dielectric function) applied to a Lorentz oscillator with sampling energy factor of 1.007 and 1.015 using H₂, L₂, and M₂ WFs. It is noted that the variable in the abscissae is not precisely the energy at which the optical constant is calculated, but the central energy of the WF ${E_w} = \sqrt {{E_1}{E_2}} $. ζ for each E_w is calculated by performing the integral of Eq. (19) for the full set of optical constants of the oscillator, but the WF centered at E_w results in a strong weight of the spectral range surrounding E_w. Then, E_w is let to scan the spectrum and ζ is calculated for each E_w and represented in the figure. In this calculation, the following parameters were used: E₂/E₁=3, γ₁=0.01E₁. γ₂ is calculated with the formula given in section 2. It is noted that according to the definition of E_M for the three WFs, E_w=E_M is satisfied for H₂ and for M₂, but not for L₂. For the latter there is a difference between E_M/E₁=(f + f²)/2 and E_w/E₁=f^3/2, where f=(E₂/E₁)^1/3. For L₂, E_M and E_w converge to each other only in the limit f→1. In the below examples with L₂, f = 3^1/3 and hence E_w/E_M=0.983.

 

Fig. 4. The self-consistency parameter ζ of inertial sum rule (19) versus the window central energy E_w for the Lorentz oscillator plotted in Fig. 3 with energy sampling factor of 1.007 (a) and 1.015 (b). Three WFs were used: H₂, L₂, and M₂. The WF is defined with E₂/E₁=3, E_w=(E₁E₂)^1/2, γ=0.01E₁. The ±0.005 acceptable limits are highlighted with orange lines. The width of the window is displayed.

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Since there is no uncertainty in the optical constants, the deviation of ζ from 0 in Fig. 4 displays the dependence of each WF on the sampling rate. The figures show large peaks for H₂ at energies where the oscillator takes largest values, in combination with the three energies of fast variation of H₂. For 1.015-sampling factor, ζ parameter exceeds the ±.005 limit at such peaks. For this sampling rate, these peaks are ∼20 times larger than for the three-line function (L₂) and more than 100 times larger than for the two 4-degree polynomial function (M₂), so that L₂ and M₂ are much less dependent on the sampling rate than H₂, even for a coarse sampling. Similar relative values are obtained for the 1.007-sampling factor, although in that case, ζ for H₂ also stays within the admitted limits. This result was obtained for a small value of γ=0.01E₁. If γ is increased, for instance to 0.1E₁, then the sampling problem is relaxed, but this results in a slower decay of the WF in the out-of-band.

The five different inertial-like sum rules were applied on the same Lorentz oscillator with sampling energy factor of 1.015 for L₂ and M₂ with E₂/E₁=3 and γ₁=0.01E₁. Figure 5 compares ζ for the inertial-like sum rules given by Eqs. (17) to (21) (calculated with the dielectric function), which vary with the power of energy in the integrand. In the energy range close to the Lorentz oscillator central energy, ζ is smaller for M₂ than for L₂ for all five sum rules. In general, larger deviations and fluctuations are observed at low or high energies, i.e., at ranges far from the oscillator energy peak. For L₂ there are no significant differences over the five sum rules, except that sum rule with E² starts some deviation at E>∼200 eV. For M₂, deviations are larger, mostly for the E²- and the E^-2-sum rules. A tighter sampling was attempted for M₂ and also a larger γ, but no significant differences were observed. The reason for the increase in the deviation and fluctuation level is attributed to the limited precision used in the calculations. Calculations were performed in Fortran F77 (Force 2.0). Complex variables were defined as complex*16. L₂ seems to be less dependent on precision, so it might be the choice when precision is the limiting element.

 

Fig. 5. The self-consistency parameter ζ of inertial sum rules (17, m=-2) to (21, m = 2) versus the window central energy E_w for the Lorentz oscillator plotted in Fig. 3 with energy sampling factor of 1.015. Two WFs were used: L₂ (a) and M₂ (b). The WF is defined with E₂/E₁=3, E_w=(E₁E₂)^1/2, γ=0.01E₁. The width of the window is displayed.

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Let us display an example of f-sum rule. To do this, it is useful to rewrite Eq. (22) by introducing the above value of the plasma frequency ω_p²= N_ee²/mɛ_o, and by setting N_e=N_atZ, where N_e and N_at are the electron and atom number density, respectively, Z is the atomic number (for a single species material; for a compound, the total number of electrons would be used), m is the effective mass of the electron, and ɛ_o is the permittivity of vacuum. Equation (22) is rewritten as:

Figure 6 plots n_eff calculated with Eq. (27) (using the dielectric function) for a sampling factor of 1.015. As seen for the inertial-like sum rule, a limited sampling rate results in oscillations of H₂ around n_eff=1 again at energies of fast variation of the oscillator, whereas L₂ and even more M₂ reduce such oscillations by ∼two orders of magnitude.

 

Fig. 6. The effective number of electrons n_eff calculated with Eq. (27) vs. the window central energy E_w for the Lorentz oscillator plotted in Fig. 3 with energy sampling factor 1.015. The WF is defined with E₂/E₁=3, E_w=(E₁E₂)^1/2, γ=0.01E₁. The width of the window is displayed.

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The above results correspond to optical constants with no errors. Let us add some specific error to see whether there are different effects over the WFs. With an inertial-like sum rule, let us assume an inconsistency such that ζ parameter is close to the limiting value of +.005. An illustration was given above to interpret Shiles’s limit in the standard inertial sum rule [Eq. (14)]. This interpretation is applied here to intentionally obtain ζ=+.005 with sum rule (19). This is done by multiplying the positive values of Re[F($\tilde{\varepsilon }$-1)] by 1.005 and the negative ones by 0.995.

Figure 7 presents ζ (calculated with the dielectric function) for the three WFs where Re[F($\tilde{\varepsilon }$-1)] data were modified as indicated. Again, there is a strong difference in the application of H₂ and either L₂ or M₂ similar to what was found for optical constants with no errors: L₂ and even more M₂ stick very well to the pre-established value of ζ=+.005 over the spectrum, whereas H₂ has important fluctuations at the typical energies, which are attributed again to the limited sampling.

 

Fig. 7. The self-consistency parameter ζ of inertial sum rule (19) versus the window central energy E_w for the Lorentz oscillator plotted in Fig. 3 with energy sampling factor 1.015. The oscillator ($\tilde{\varepsilon }$-1) times the WF was scaled for ζ to stay around + 0.005. The WF is defined with E₂/E₁=3, E_w=(E₁E₂)^1/2, γ=0.01E₁. The width of the window is displayed.

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Summarizing, the two new WFs result in evaluation parameters much less dependent on sampling rate than the old WF, particularly in situations where sampling is modest and window decay is fast.

3.2 Experimental optical constants

First, the new WFs are used to test the experimental optical constants of Al. The latter were taken from the comprehensive work by Shiles et al. [11], which reported self-consistent optical constants of Al in a broad spectral range. That compilation was obtained using a large collection of optical constant data in the literature. These data had been already selected to check Al optical-constant self-consistency with H₂ [6], and now Al self-consistency is tested with the new WFs. Inertial-like sum rule (19) was again selected, which was applied on Al’s n,k data. Figure 8 displays ζ vs. E_w for the three WFs. Two γ parameters were used: 0.1E₁ and 0.01E₁. Reference [6] reported the application of H₂ with γ=0.1E₁.

 

Fig. 8. The self-consistency parameter ζ of inertial sum rule (19) versus E_w for Al optical constants [11]. Three WFs were compared: H₂ (a), L₂ (b), and M₂ (c). The WF is defined with E₂/E₁=3, E_w=(E₁E₂)^1/2. WFs were used with γ=0.1E₁ and 0.01E₁. The ±0.005 admitted limits are highlighted with orange lines.

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For γ=0.1E₁, there is no essential difference over the three WFs. However, for γ=0.01E₁, instability in the application of H₂ is again much larger than for L₂ and M₂, which is again attributed to the sharp edges of H₂ at E₁, E_M, and E₂, M₂ being the least noisy. The figure displays some spectral ranges where ζ exceeds the top acceptable values of ±.005, which was already analyzed in [6]. Such excess looks larger for H₂ with γ=0.01E₁, but the extra fluctuations in Fig. 8(a) are attributed to the larger instability of H₂ compared with L₂ and M₂ that was investigated in the previous sub-section. The coincidence of the application of H₂ with γ=0.1E₁, and of L₂ and M₂ both with γ=0.1E₁ and 0.01E₁ (with some small fluctuations for L₂ at γ=0.01E₁) takes us to conclude that the large ζ values in some ranges correspond to intrinsic data inconsistency at the specific ranges. The strong dependence found in [6] for ζ with respect to the specific extrapolation selected for Al optical constants at small energies was seen to invalidate the application of the standard inertial sum rule for Al when no WF is used, probably due to the divergence of the optical constants of metals at small energies. Hence the use of a WF can overcome this problem, and the new WFs are compatible with situations like limited sampling and fast decaying WFs. Summarizing, it is remarkable that optical constants of Al in [11] had been obtained with KK analysis and they had passed standard sum rules (13) and (14) (although k data were corrected to help them pass f-sum rule), but this does not guarantee the lack of inconsistency, and the present window functions and sum rules are helpful to highlight such inconsistency.

The optical constants of Au have also been analyzed with the new and old WFs. Two sets of optical constants were taken from the literature. They correspond to the two sets of data evaluated by Bimonte [12], who performed a careful analysis to select the best possible literature data set in order to calculate the dielectric function ɛ₁ of Au films at imaginary energies towards calculating the thermal Casimir effect. The two sets were built starting with data mainly from Lynch and Hunter [13] or from Svetovoy et al. [14]. Bimonte performed a careful discussion on the sources included in [13], from which he felt more confident with the data of Thèye et al. [15] than with the data of Dold and Mecke [16]. Sum rule (19) was applied on Au’s n,k data.

Figure 9 displays ζ vs. E_w for the two sets of data of Au with WFs H₂ and M₂ for γ=0.01E₁. As with previous examples, H₂ results in a noisier spectrum compared to M₂, although the difference over the two WFs is smaller than in the above examples. This suggests intrinsic inconsistencies of the data, particularly above ∼100 eV. ζ stays outside the acceptable range of ±0.005 almost in the full spectrum larger than ∼0.05 eV (Svetovoy et al. data) or ∼0.4 eV (Lynch and Hunter data). The main contribution to the calculation of the dielectric function for imaginary energies in [12] is expected to be at ∼1 or 1.5 eV (as a consequence of the application of a WF used in Bimonte’s paper, which is mentioned below). In this calculation, ζ for Svetovoy data almost fits within the ±0.005 limits using M₂, whereas for Lynch and Hunter data, ζ largely exceeds the top acceptable values.

 

Fig. 9. The self-consistency parameter ζ of inertial sum rule (19) vs. E_w for Au optical constants built by Bimonte [12] mostly based on data from Svetovoy et al. [14] (a and c) or Lynch and Hunter [13] (b and d). The WF is defined with E₂/E₁=3, E_w=(E₁E₂)^1/2, γ=0.01E₁. (a) and (b): the full spectrum. (c) and (d): the 0.1-10 eV spectral range. The ±0.005 usual limits are highlighted with orange lines.

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The research article by Bimonte [12] uses the WF:

 

Fig. 10. The self-consistency parameter ζ of inertial sum rule (19) vs. E_w calculated with M₂ for the optical constants of Lynch and Hunter [13] or Svetovoy et al. [14], both as raw data and multiplied by Bimonte function with b = 1 eV. The WF is defined with E₂/E₁=3, E_w=(E₁E₂)^1/2, γ=0.01E₁. The ±0.005 usual limits are highlighted with orange lines.

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Self-consistency for Au optical-constant data times B(E) displays a large oscillation at E_w=1 eV, i.e., around b. It reaches a top value of ζ=.33 at E_w=1.3 eV for Svetovoy data and ζ= -.80 at E_w=1.23 eV for Lynch and Hunter data. Such ζ values are roughly two orders of magnitude outside the accepted limits of ±0.005. Hence it would be interesting to check if the displayed deviation in self-consistency might result in uncertainty in the calculation of thermal Casimir effect. It is noted that calculations of the dielectric function at imaginary energies for the thermal Casimir effect involve a somewhat different integral compared to the standard KK analysis, and the common sum rules do not apply for such calculation, so that the strong deviation displayed in Fig. 10 does not prove any uncertainty in the calculations of [12]. Nevertheless, it would not be wise to use B(E) function as a WF in standard KK analysis and sum-rule calculation.

Summarizing, regardless of the optical-constants data being errorless or not, and of their being mathematically generated or truly experimental data, the new WFs L₂ and M₂ appear very stable to evaluate the spectrally-resolved self-consistency of optical constant sets. Even though H₂ is also a valuable WF in situations where sampling is tight enough, a coarser sampling requires that the H₂ width parameter be increased, which limits the decay of WF. This undesired widening can be avoided with L₂ and M₂. Finally, M₂ has been found to be somewhat more stable under modest sampling than L₂.

4. Conclusions

New WFs have been developed to precisely measure self-consistency of optical constant sets. The new WFs are seen to excel a previously developed WF. The old WF involved a steep transition at the window edges and center, which results in instability when performing sum-rule integrals with both a fast decaying WF outside the window and with coarse data sampling. Two new WFs were developed that avoid such steep transition by starting with a weight function that cancels at the window edges. The two new WFs use weight functions with three straight lines or with two 4-degree polynomials.

The new WFs were tested both with exact optical constants using sampling rate as a variable and with experimental data sets. For exact optical constants, the new WFs were seen to be much less affected by coarse sampling than the old WF. This was observed both for an inertial-like sum rule and for an f-like sum rule. The same difference was observed for data with intentionally generated errors. The new WFs were also superior to deal with experimental data sets. The larger stability of the new WFs compared to the old one helps decide that the inconsistency calculated with the new WFs can be properly attributed to inconsistency of data sets. Finally, a WF that has been used in the literature to help in the calculations of the dielectric function at imaginary energies for the thermal Casimir effect was seen to induce strong fluctuations when applied to an inertial-like sum rule, although this does not necessarily translate into errors in the application of such WF to calculate the thermal Casimir effect.