Photographs of the Lycurgus Cup with a source light inside and outside exhibit purple and green colors, respectively (dichroism). A model relying on the scattering of light to colors in the photographs is proposed and used within an inverse problem algorithm, to deduce radius and composition of metallic particles, and the refractive index of the surrounding glass medium. The inverse problem algorithm is based on a hybridization of particle swarm optimization and of the simulated annealing methods. The results are compared to experimental measurements on a small sample of glass. The linear laws that are deduced from sets of possible parameters producing the same color in the photographs help simplify the understanding of phenomena. The proportion of silver to gold in nanoparticles is found to be in agreement, but a large proportion of copper is also found. The retrieved refractive index of the surrounding glass is close to 2.
© 2015 Optical Society of America
For centuries, the color of glass has been known to depend on small proportions of embedded metals. The stained windows in churches, as well as the Lycurgus Cup, are historical evidences of advanced expertise for the manufacture of colored glass. One of the most exciting properties of such an artifact is probably dichroism: a color change because of a change in the illumination direction. The Lycurgus Cup can be seen in the British Museum , and the two photographs show that “the opaque green cup turns to a glowing translucent red when light is shone through it. The glass contains tiny amounts of colloidal gold and silver, which give it these unusual optical properties.” The composition of the Lycurgus Cup was analyzed deeply from two small samples: sputter-thinned sections and dispersion of finely crushed glass particles, with analytical transmission electron microscopy (TEM) and energy dispersive x-ray (EDX) spectrometry . Former results from chemical analysis of glass were summarized in that paper. These results will be used in the discussion on the results of the inverse problem.
More recent studies discuss the origin of color in ruby , stating that the metallic nanoparticles of both copper and gold are the coloring agents in ruby glasses. The Cu-rich nanoparticles were proved to contribute to the purple color in glass with complex chemical composition . Moreover, the variation of concentration of metals as a function of the deepness of the analyzed zone in glass was proved [4,5]. Therefore, the dichroism can reveal bulk and near surface properties of glass.
The role of silver nanoparticles was studied in Ref. . The inclusion of silver nanoparticles leads to yellow color. The role of the particle size was described as follows: a redshift and an increase of the bandwidth of the spectrum are observed for bigger particles. This phenomenon was explained by the surface plasmon resonance of Au, Ag, and Cu nanoparticles, and a unique resonance wavelength, by assuming that the phenomenon can be explained by a single dipole in the particle . Therefore, the position of the maximum in the visible spectrum curve is considered commonly as the source of color [5,7,8]. However, the whole visible spectrum is involved in the color formation, especially when the brightness/darkness of the color must be taken into account, and when the bandwidth of the resonance is wide. Moreover, the dipole approximation fails to describe accurately the plasmon resonance if the particle size exceeds about 10 nm . Therefore, the full Mie theory is necessary to compute the visible spectrum of nanoparticles.
The purpose of this paper is to use photographs of the Lycurgus Cup to solve the inverse problem, leading to an inexpensive method of characterization . The proposed method relies on the Mie theory, the sRGB coding of color, and the particle swarm optimization (PSO) method to retrieve the size, the proportion of metals (Cu, Ag, and Cu), and the optical index of embedding glass. The entire visible spectrum of the efficiency is used to generate the sRGB color. Section 2 describes the physical model, the sRGB encoding, and the inverse problem strategy. Results and discussions are given in Section 3. Results are compared to those found in the literature, which are obtained from analytical TEM .
2. FROM EFFICIENCY SPECTRUM TO COLOR IN PHOTOGRAPHS
Colors and vision have been sources of theories throughout centuries (Newton, Goethe, Maxwell, Kandinski, and Klee). Colorimetry has been a domain of research since the beginning of the 20th century [12,13]. A first challenge was to link the wavelength to the color vision, but with the emergence of digital technology, an additional difficulty was to relate the wavelength to the red–green–blue (RGB) encoding of color in the photographs. Both the physical model of interaction between light and the Lycurgus Cup and the sRGB encoding are described in the following. The goal is to use this model to retrieve the physical properties of embedded metallic nanoparticles from the color in the photographs by using an inverse problem method.
A. Physical Model
The chemical analysis of a small piece of glass of the Lycurgus Cup revealed a complex composition, with inclusions of various materials in (73.5%): O (13%–15%), CaO (6.5%), (2.5%), (1.5%), CuO (0.04%), and PbO (2.2%) and traces of other materials, including K, Mg, Mn, P, Sb, Sn, Ti and gold (0.004%), and silver (0.03%) . The refractive index of glass was not measured but can be supposed to be larger than that of fused silica (1.45).
In the investigated samples of the Lycurgus Cup, the dichroism was related to nanoparticles made of gold and silver alloy [1,2]. According to the small concentration of these materials and to the difficulty to find such nanoparticles with TEM and SEM, the interaction of nanoparticles can be neglected . A first analysis of a small piece of glass reveals that the separation between particles is large (about 10 μm). The tiny density of metal particles in glass allows one to consider their interaction as negligible. Therefore, the following model considers isolated nanoparticles.
Processing the EDX spectra gave the atomic proportions of silver, gold, and copper: , , and , respectively . Therefore, the proportions of silver and gold are physical parameters of the model. The proportion of copper is .
The color of an object is related to the whole spectrum of recorded light in the visible domain of wavelengths. The white source illuminates the cup, and the interaction between light and matter produces the color. Consequently, a physical model is required to describe this interaction. In this study, the illumination is assumed to be a perfectly white light source over the whole domain of wavelengths. The white light impinges from air onto the glass, interacts with nanoparticles, and emerges from glass to be recorded by the camera. Therefore, the model takes into account both the dual transmission from air into glass (and from glass into air) and the interaction of light with spherical particles. The physical quantities of interest are the transmittance and the efficiency spectrum.
The transmittance describes the transmission of light from the external medium (air) into the glass and conversely. It is deduced from the Fresnel coefficient of flat surfaces. The main contribution of light in the photographs is supposed to come from the light impinging under normal incidence, and, therefore, can be written as a function of the refractive index of glass :15–17]. The spectrum of transmission of the light by glass is calculated from the scattering efficiency of a single particle of radius [17–19]:
The spectrum of the scattering efficiency is the physical source of the inverse problem for the translucent ruby color . The extinction efficiency  was used, but did not account for giving the purple color. The green tone  is supposed to be revealed by the above scattering efficiency or by the back scattering :16] for : 17]: 20]. A shape-preserving interpolant for refractive indices and extinction coefficients is used to get 81 values of the efficiencies for regularly spaced wavelengths in [380;780] nm and, thus, for copper, silver, and gold.
Finally, the sRGB color of the cup is deduced directly from the efficiency spectrum and the transmittance coefficient with the method described in Section 2.B.
B. sRGB Encoding of Photographs
The relation between spectrum and color was investigated by using CIELAB coordinates [21,22]. The CIE 1931 coordinates were used recently to describe the colorimetric sensing of photonic crystal . The present method pursues the same objective: to deduce the unknown physical parameters from color in the photographs. In this study, the goal is to find the parameters of the above described model (the radius of nanoparticle, the refractive index of embedding glass, and the proportions of silver and gold) that produce the same color as in the reference photographs.
The conversion of a spectrum in the visible domain of wavelengths to sRGB color requires the following steps according to the recommendations of the CIE (IEC 61966-2-1) :
- • Shape-preserving interpolants of data from the CIE1931 data (CIE Standard Primaries) on the 81 investigated wavelengths . The initial data from the International Commission on Illumination (CIE 1931 standard colorimetric observer) that can be downloaded from the official website of the commission are fitted to describe the considered spectrum: the wavelength varies from 380 to 780 nm by step .
True colors , , and must be greater than 0 and lower than 1. If the above described algorithm leads to values out of this interval, the color is virtual and does not correspond to a true color in the photographs. Some attempts to transform virtual colors in true colors were proposed (e.g., adding white, thresholding), with drawbacks such as the Abney effect [24,25]. Therefore, no additional processing of virtual colors is included in the model. The complete model of computation of sRGB colors from a physical model is included in an inverse problem algorithm described in Section 3.
3. INVERSE PROBLEM FOR THE LYCURGUS CUP
A similar algorithm of dichromatism measurement was proposed to explain the changes of color as a function of dilution of various materials [21,26]. The present approach uses sRGB data computed from the whole spectrum, obtained from the Mie model, including the transmission of light by the surrounding glass. The challenge is to obtain the same color as that in the photographs. In this paper, the reference data are the sRGB colors extracted from both photos of the Lycurgus Cup that were obtained courtesy of the British Museum [Figs. 1(a) and 1(b)].
A. Fitness Function
Twenty samples of color are picked from these pictures in various zones, and their mean value is calculated [Figs. 1(a) and 1(b)]. Therefore, the pixel resolution is not available and the noise is averaged . The mean values and standard deviations of the picked sRGB colors are indicated in Table 1.
The resolution of the inverse problem consists of finding the physical parameters of the model that minimize a fitness function [Eq. (12)]
The fitness function is the Euclidean distance of the computed color relative to the reference color [Eq. (10)].
B. Decision Parameters
Retrieving experimental parameters was pursued in  with a fascinating evolutionary procedure using recombination, mutation, and selection. In Ref. , the challenge was to retrieve the unknown experimental parameters from the measurement of the reflectance of a thin aluminum layer deposited on a glass prism . In that paper, the particle swarm optimization method was used. The measurement of thicknesses and the refractive indices of a thin gold layer with those of the chromium adhesion layer on a glass substrate was achieved with the same method . This method also is efficient to directly optimize complex structures . A similar approach was used to relate different codings of colors .
In the present case, the inverse problem resolution consists in finding the physical parameters of the model that minimize the fitness function [Eq. (12)]. The physical parameters are the decision parameters, and the targets are (, , and ) given in Table 1. A hybrid PSO-simulated annealing method is used to solve the inverse problem.
C. Inverse Problem Method: The Hybrid PSO—Simulated Annealing Method
This method was proposed by Kennedy and Eberhart . The method is described fully in Ref . The decision parameter mimics the behavior of a swarm of bees in search of pollen, in a domain of search. The position of the pollen corresponds to the minimum of a fitness function. Another point of view is closely related to physics. Any input of the model (or possible parameters or variables of decision) forms a vector whose size depends on the degree of freedom of the model. At each iteration step , this vector is updated and corresponds to a particle that moves randomly in the domain of search (hypercube) to fall in a potential well:14) is a finite difference expression of the velocity. Therefore, can be called “time” and is related directly to the execution time of the algorithm.
The particles communicate good positions to each other, and each adjusts its own position and velocity based on these best positions following15) from its previous velocity and from the distances between its current position and its own best position and the swarm global best . This new velocity is used to move the particle and to find its new position at step following Eq. (14). The swarm randomly investigates the space of search, but keeps memory of the “best” decision parameters (corresponding to the minimum of the fitness function over all and of the best decision parameters for each particle for all previous steps. Therefore, PSO often is classified as a collaborative method, the best particles and contributing equally to the next swarm if [Eq. (15)]. This specific aspect of the method distinguishes it from blind Monte Carlo methods  and from those based on the selection of the best decision parameters .
Exogenous parameters of the PSO method are fixed: , , and a linear decrease of the inertial weight from 0.9 to 0.4 from step 1 to . As mentioned in a previous study , the results are hardly dependent on the three parameters (, , and ). The initialization of the PSO algorithm and the stop criteria are specified in the following. A stop criterion must be defined at least to end the loop.
D. Initialization and Stop Criteria for PSO
Uniform law (pseudo)random initialization of the first vector of decision parameters and velocity [Eqs. (14) and (15)] is known to increase the success rate of each realization of the algorithm . Therefore, no a priori hypothesis is made on the distribution of size, proportions of metals, and refractive index of the surrounding medium. Moreover, the statistical distribution of shapes of radius of nanoparticles is supposed to be unknown for the Lycurgus Cup. Therefore, the whole space of search is investigated with the decision parameters, at each realization of the algorithm, through uniform law of probability.
The first stop criterion is [Eq. (12)]. Indeed, in the processed 24-bit color images, each of red, green, and blue colors is coded on 256 levels. For example, two red colors and are indistinguishable if , with and greater than 0 and lower than 1. Therefore, the minimum of the fitness function is 0.006766. Thus, three decimal places are kept to define reference colors, and the target of the inverse problem is chosen . This value is in agreement with the standard deviation of reference colors quoted in Table 1.
The second stop criterion is the maximum number of allowed iterations of the PSO . If when reaches , the loop ends and the best decision parameters are stored.
Each realization of the algorithm gives a single set of best decision parameters, verifying the stop criteria for the fitness function defined by Eq. (12). Then the best decision parameters obtained from PSO are used as starting points in a simulated annealing algorithm . Simulated annealing is an unbounded method, and, therefore, its solution can get out of the domain of physical parameters (for instance, the proportion of metal found can be negative) if the PSO solution is not close enough to any acceptable optimum of the problem. Consequently, the method is successful if the decision parameters fall within the domain of parameters with physical meaning, and if reference and computed colors are indistinguishable. From the model, virtual colors can be obtained, but only the true colors are supposed to correspond to physical parameters. The success rate is an indicator not only of the appropriateness of the model, but also of the effectiveness of the method of resolution of the inverse problem.
4. RESULTS AND DISCUSSION
The relationship between sRGB color in the photographs and the efficiency spectrum can be considered to be a constrained problem. The constraints are that the decision parameters must have physical sense: , , and are the constraints. Let us note that these constraints are not included in the inverse problem method. Therefore, the algorithm is free to give nonphysical parameters if the physical model is not relevant. Therefore, the convergence of the algorithm is a first proof of validity of the physical model. Even if Eqs. (8)–(10) appear to be continuous functions of the considered efficiency , the limited number of levels for encoding each color in the photographs (256) induces a thresholding of the result. Therefore, the inverse problem can be considered reasonably as multivalued, and a set of acceptable parameters is awaited. Since the uniqueness of the solution is not guaranteed, the inverse problem gives sets of possible decision parameters; therefore, a statistical analysis of the sources of dichroism is possible.
Three approaches are proposed in the following with a gradual increase in degree of freedom. First, the sRGB colors in transmission (purple color) and in reflection (green color) can be computed from experimental measurements given in Ref. , with the model described in Section 4.A. Since the results are not fully satisfactory, the above described inverse problem method is applied in a wider range of decision parameters and discussed.
A. Resolution of the Inverse Problem Using Literature Data
In a first part, the knowledge on the composition of the Lycurgus Cup is summarized in Ref.  and the references therein. Some information on particle size, refractive index of glass, and composition of nanoparticles is given successively. These physical parameters are used for first calculations.
The particle size was in the range 7–60 nm . In [2, Fig. 5], a TEM image of a typical metallic nanoparticle (radius about 25 nm) and the EDX spectrum are shown. The investigated domain of search for the radius of particles is 0–100 nm. Therefore, the maximum number of terms used in the Mie series to calculate the efficiencies is [Eq. (4)].
The optical property of the embedding glass was not measured. However, the chemical analysis summarized in Ref.  revealed the glass composition. The following gives proportions and refractive indices [38–41]:
- • (73.5%, 1.55),
- • (13-15%, 1.75),
- • CaO (6.5%, 2.0),
- • (2.5%, 1.77),
- • (1.5%, 3.1),
- • (0.9%, 1.46),
- • MgO (0.5–0.6%, 1.74),
- • MnO (0.45%, 2.16),
- • CuO (0.04%, 2.63).
Particles of NaCl (refractive index 1.544) are embedded in glass. The refractive index of glass was not measured, but can be evaluated to 1.63 using Eq. (7) and the above data. Typically, a glass containing 64% of , 12.6% of , 9.5% of CaO, and 3.0% of has a refractive index of about 1.55. The refractive index of flint can reach 1.96 . The optical index of glass is larger than that of : 1.46. The inclusion of metal oxide of high refractive index can suggest larger values. For example, a larger proportion of Pb deep in glass could increase the refractive index to around 2 as in crystals.
The nanoparticles were analyzed with EDX. Strong peaks corresponding to copper arise from the support grid. The Ag/Au ratio of this single nanoparticle is larger than 2. Further studies show that copper was also present in the nanoparticles. The discussion [2, p. 42] states that the purple color is mainly because of gold and that a high proportion of silver shifts the color toward the blue. The role of copper is supposed to be negligible, even if the chemical analysis of glass reveals a quantity of CuO that is larger than that of silver. The x-ray spectra of the sample on the Ni substrate showed peaks corresponding to Ag, Au, and Cu. Processing the spectra gave the atomic proportions of silver, gold, and copper as, respectively,
The dispersion of particles did not allow reduction of the uncertainties on these values. In , the gold-to-silver ratio of the alloy particles was mentioned as and differed from that in the glass as a whole (). Consequently, a substantial proportion of silver remains dissolved in the silicate matrix after precipitation of the alloy particles. The chemical proportions of Au, Ag, and CuO were, respectively, 0.004, 0.03, and 0.04. Depending on the mode of measurement, the ratio of silver to gold varies from 2 to 7.5. The variability of is much higher. The role of copper in the purple color of glass was underlined in Ref. .
According to these results, a first approach consists of calculating the best radius and the best refractive index of the embedding glass by assuming the above mentioned proportions of metals in nanoparticles. The best parameters and are given in Table 2. The large values of may be disappointing. Even if the results are close together in the green case, the size of nanoparticles that should produce purple color is far from that of the green color.
The corresponding spectra and colors are shown in Fig. 2. The sRGB color of white light spectrum is also shown as reference. The agreement between reference colors in the photographs and the best computed colors obtained from spectra as a function of both the radius of nanoparticles and the refractive index of the surrounding glass is not fully satisfactory. The maximum of green spectra is not in the green zone, and the green color is too light. The whole spectrum contributes to the final sRGB color. The purple color obtained from these parameters is pushed to a yellow color.
Therefore, additional degrees of freedom must be considered in the inverse problem. The inverse problem method described in Section 3.C is used considering , , and . The refractive index of glass is searched between 1.44 and 2.5. Table 3 gives the results of the inverse problem. The small standard deviation shows the stability of the method, but the mean radii found are still different in the three cases. The uncertainty intervals that could be defined from the mean value and the standard deviation have an empty intersection in the purple and green cases. However, except for the radius , the scattering and the back scattering models give the same proportion of metals and the same refractive index of glass . Nevertheless, the ratio of the mean values of the proportions of silver to those of gold is between 5 and 36, far away from the measurements . The proportion of metals is outside the domain of search. Moreover, the refractive index of glass is found close to 2.
Let us note that the reliability of solutions to the inverse problem further deteriorated, by considering that copper is missing in the nanoparticles. In that case (not shown here), the retrieved proportion of silver is always negative. Therefore, it seems that copper contributes to color, and Table 3 suggests that the particles that contribute to the green color do not have the same characteristics as those that contribute to the purple color. These results could suggest that the mode of fabrication of the glass was not the same for the bulk glass and for the glass near the surface. Nevertheless, the refractive indices of the bulk and of the surface are found to be very close together.
The above results are considered to be unsatisfactory as they are outside the domain of search. Thus, the domain of search should be enlarged to find eventual alternative solutions to the inverse problem. This enlargement is coherent with the alternative proposed by Barber and Freestone: “Another possibility is that the glassmakers actually attempted to equilibrate their batch in some way with metallic gold, when the more readily oxidizable impurities, such as lead, copper and silver, were preferentially incorporated into the glass” . Therefore, the part of oxidized silver may have been forgotten or lessened, as in [29,43], and the fusibility of copper in gold. Moreover, the analysis that is limited to very small pieces of glass could not be representative of the bulk glass (for the purple color) or of the whole near surface part of the cup (for the green color). As explained in , a multilayer structure can be observed in ancient glasses. Therefore, the reflected light could result from interaction with different particles than the transmitted light. Since a reasonable doubt exists as to the composition of glass as well as of the nanoparticle, in the following, the decision parameters still are the fraction of gold and silver, the radius of nanoparticles, and the refractive index of glass, but within a wider range of search. Indeed, the color is known to be highly sensitive to the fraction of metals [5, Fig. 5], and, therefore, in the following inverse problem, the whole domain (between 0 and 1) of proportions is investigated. A wider domain of search is also considered for the refractive index of glass.
B. Inverse Problem within a Wide Range of Parameters
The decision parameters are searched within a wide domain of physical parameters: , , nm, and . A hundred realizations of the same algorithm give the statistical results and the best values (Table 4). The inverse problem method gives colors that are always indistinguishable from the target, and within the interval of search. The maximum of the fitness function among all realizations is lower than . Therefore, the success of the method is 100% in the three cases, and the resolution of the inverse problem gives sets of possible solutions for the composition of nanoparticles. The mean radius of particles is about the same in the three cases. The dispersion of results is wider than in the previous case, but the scattering and the back scattering models give the same radius and the same material properties. The metal proportions and differ in the purple and green cases, and the ratio of proportions of silver and gold can be compatible with measurements. . Thus, a selection of the solutions that would approach measurements might be appropriate.
The statistics of the selected solutions that are compatible with the measurements in Ref.  () are also shown in Table 4. This case involves a supplementary constraint on the decision parameters. About 25%–30% of decision parameters correspond to a proportion of silver larger than that of gold. The ratio of silver to gold is roughly between 2 and 3 if the constraint is imposed. This ratio is close to that found in Ref.  (0.662/0.312). If a single manufacturing process was used, without glass coating of the surface of the cup, these results confirm that a high refractive index of the surrounding glass is probable: in the three cases, it is found to be near 2. The proportion of copper remains much larger than that measured in a small sample . For the green color, the material parameters are about the same assuming back scattering or scattering. The radii differ slightly. The back scattering and the scattering are almost equivalent. For the purple color, the ratio of silver to gold is about the same, but the proportion of copper and the refractive index is a little bit smaller, being always higher than that of pure . The three intervals of uncertainties are overlapping: a common value could be . Although this value may seem high, the exact determination of the refractive indices and layer thicknesses in a resonant surface plasmon experience is reassuring for the adequacy of current results . The embedding glass is probably enriched with lead and metal oxides. The proportion of Cu is typically around 50% or larger. The small standard deviations of the solutions and the agreement between metal proportions show the stability of the inverse problem method and that a set of particles can produce the same color. Even if the radii of particles are about the same for purple and green colors, the proportion of copper is larger to get the purple color, but the ratio of silver to gold remains about the same. The major influence of copper in transmission (purple color) is confirmed [3,4]. Before doing a more complete statistical analysis of the above results, some tests of the relevance of the model are proposed.
1. Relevance of the Model
The relevance of the model proposed in Section 2 is investigated by varying any of the underlying assumptions:
An alternative model of the effective refractive index of metallic alloy is proposed. The combination of refractive indices has also been considered:2]. The radii of the particles are about the same as those obtained from the linear combination of the relative permittivities [Eq. (7)]. Again, the retrieved refractive index of the surrounding glass is near 2. Therefore, even if the model of mixing of refractive indices is not appropriate, the analysis of results confirms the stability of the inverse problem method as well as the main results discussed above.
A second test consists of using a dipole approximation of the Mie model by using only the first term of the series (), whatever the radius is. The results are the same as those shown in Table 4. This fact confirms that the scattering by small metallic particles can be described by considering only the first term in the Mie series.
Further approximation can be done for small radii [17, Eqs. (5.8) and (5.9), p. 135]:9,44]. Using these approximations in the inverse problem method always gives negative values of the proportion of silver. Therefore, the dipole-small radius approximation is not relevant, even if the retrieved radius of metallic particles is about 7–8 nm. On the contrary, the dipole approximation using only the first Mie coefficient (without small radius approximation) enables us to find a solution to the inverse problem within the wide domain of search.
As a final test, the relevance of the transmittance of light is investigated. If the transmittance is removed from the model, the fitness function [Eq. (12)] and the success rate deteriorate somewhat. The mean radius is increased by 1 nm, but the metal proportion is about the same. The refractive index of the surrounding glass is still near 2. Therefore, the transmittance is kept in the model. Handling the transmittance in the model can improve the accuracy of the radii found.
With the existence of solutions established, the statistical data obtained from the resolution of the inverse problem (Table 4) are used to improve the knowledge of the possible composition of the Lycurgus Cup. The transmission case (purple color) and the reflection case (green color) are studied successively.
2. Purple Color
Figure 3 shows the scattered spectra and the comparison between the corresponding sRGB color and the reference color. The good agreement between colors and the small dispersion of spectra (error bars) open the way for statistical processing of the parameters retrieved by the inverse problem procedure. Figures 4–8 show histograms of radii, silver proportion , gold proportion , copper proportion , and refractive index of the surrounding glass. The smaller proportions of silver are the most probable, as well as the refractive index of glass. The minimum of is about 2. The most probable radii and proportions of gold and copper are the largest ones. The size of the particles contributing to the purple color is typically 15–16 nm. Figure 3 shows that the statistical dispersion of the scattered spectra is small. The peak in spectra is around 610 nm. The sets of solutions exhibit most probable values around (Fig. 4), (Fig. 5), (Fig. 6), (Fig. 7), and (Fig. 8). Other solutions give the same purple color with a larger radius gold and copper proportion, but with a smaller silver proportion. The possible intervals for each parameter are relatively tight for each solution under the constraint (light histogram). A high correlation between the parameters can be observed in Fig. 9. The proportion of silver appears to be the less correlated to the other parameters.
The strong correlation between decision parameters suggests linear dependence. The linear polynomials fit in a least-squares sense of sorted data provides behavioral trends within the domains of validity observed in Figs. 4–8. The following equations give these linear trends under the constraint , with the size of nanoparticles ( nm). The norm of the residuals is indicated between brackets:
For example, assuming known proportions of silver or gold, the size of particles and the refractive index of surrounding glass can be deduced:
The ratio of the proportion of silver to gold () that was important data for the discussion in Ref.  can be deduced:2], the purple color in the photographs is obtained for and .
3. Green Color
For the green color in the photographs, the light source is outside the cup. Therefore, the particles near the external surface of the cup are supposed to contribute to the color. The physical model that gives the spectrum is the scattering and the back scattering efficiencies in the following figures.
Figure 10 shows the scattered spectra and the comparison between the corresponding sRGB color and the reference color for the green color. The agreement between colors is satisfactory, but the dispersion of spectra (error bars) is larger than in the purple case. Figures 11–15 show histograms of radii without (black) and with (light) the constraint . The data are the same as those used in Table 4. The histograms show that, within the sets of solutions, (resp. ) is dominant for the scattering (resp. back scattering) model. The size range of particles is 12–19 nm, but radii are smaller than 7.5 and 6.6 nm under constraint. A wide range of silver proportions is available between 0 and 0.5. The scattering and back scattering models give the same ranges of proportions of metals and of refractive index of surrounding glass. The main gold and copper proportions and are about 0.28 and 0.20, respectively. The largest values of are retained under constraint, and the proportion of copper is around 0.49. The same conclusion can be drawn for the refractive index of the surrounding glass, and all solutions verify and under constraint. The correlations between each series of couples of parameters in the solutions of the inverse problem remain close to 1, and are minimum for (Fig. 16) as in the purple case. Therefore, the proportion of silver also has less influence on the green color, even if is between 0.25 and 0.45 (Fig. 12). The proportion of copper is lower than that of the purple case but close to 0.5. The ratio remains close to 2, as measured experimentally . The silver–gold materials appear to be more present in the region near the surface of the glass.
The correlation of decision parameters is larger than 92% (Fig. 16). Therefore, the same method of linear polynomials fitting as that used for the purple case can be used. The following equations, with (nm), are obtained from the sorted scattering results, the norm of the residuals (between brackets) being slightly smaller than for the back scattering:
These equations look similar to those obtained in the purple case. The proportion of copper is , which is larger than 55% and larger than that for the purple case.
For example, assuming known the proportions of silver or of gold, the size of particles and the refractive index of surrounding glass can be deduced:
The ratio of the proportion of silver to gold () that was important data for the discussion in Ref.  can be deduced:2] the green color in the photographs is obtained for and .
The same refractive index is obtained for the purple and green photographs. The ratio of silver to gold found experimentally by Barber and Freestone  corresponds to a solution of the problem for particles with radii of about 7 nm, and to the proportion of copper about 49% in the green case, and about 81% in the purple case.
From colors in the photographs of the Lycurgus Cup, the inverse problem has been solved, thus providing sets of solutions for the radii and the proportions of metals in nanoparticles, and for the refractive index of the surrounding glass. To reach this goal, both the Mie model of scattering in the visible spectrum and its conversion to sRGB colors have been used within a loop for solving the inverse problem with a hybrid particle swarm-simulated annealing method. A wide domain of search was selected, and some variants of the models have been tested. In particular, the dipole approximation for small radii is found to be irrelevant, even for nanoparticles with a radius of about 7 nm. The stability of the proposed algorithm was verified. The main results are as follows.
The correlation of decision parameters under experimental constraints considering a proportion of silver larger than that of gold led to simple linear laws. These laws help to identify behavior trends and explicit dependence of physical parameters, giving the same color in the photographs. The refractive index of the surrounding glass is close to 2. The proportion of silver to gold is mainly found to be about 2, as in experiments , and a large proportion of copper contributes to the color. The optical properties of the glass near the surface of the cup do not differ from those deep inside the glass.
It would be interesting to test systematically the proposed method on photographs of controlled samples of embedded nanoparticles with low concentration, in a wider range of colors. Moreover, some improvements could be made to the model by including, for example, electromagnetic interaction between particles and the data of measurement of the white source spectrum, but the stability of the method and the agreement with some experimental results provide reassurance about its efficiency to describe the problem of dichroism of the Lycurgus Cup.
European Research Council (ERC) (Health-F5-2009-241818).
The author thanks the European Union for computational facilities. The present study was made possible because of the high quality of photographs provided by the British Museum.
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