Lionel Simonot

Philippe Colantoni

Mathieu Hébert

Optimization of monochromatic primaries in RGB system: on the specific question of purples

In the CIE1931 xy chromaticity diagram, the set of visible colors is delimited by the spectrum locus representing the monochromatic radiations and by the purple line, set of nonspectral colors obtained by binary mixing between monochromatic radiations at the two extremities of the visible range. In contrast with the spectrum locus, which represents, for most wavelengths, bright and maximally saturated colors, the purple line rather represents colors at the perception limit, corresponding to spectral powers with very low luminous efficacy. The notion of "practical" purple lines is discussed. It comes down to the definition of the blue and red monochromatic primaries of RGB light emitting systems. The choice of monochromatic primaries allowing the widest gamut while maintaining a good luminous efficacy over its whole perimeter is still an open question: by choosing red and blue primaries further in the extremes of the visible spectrum, the gamut size is increased but the luminous efficacy of the primaries, thereby of all reproducible colors, is decreased. As optimal trade-offs, we suggest two RGB systems favoring either the gamut size, or the luminous efficacy.

Introduction

Created in 1931 by the CIE, the XYZ color space is a 3D space representing all perceptible colors. Simultaneously was created the xy chromaticity diagram, a 2D representation resulting from a projective transformation of the XYZ color space discarding the luminance component. This latter is suitable for lighting systems for which lightness makes no sense. For example, by decreasing the power of a white source, one still obtains a source of white color not grey. Its well-known shape is delimited by the spectrum locus, line drawn by the chromaticity values attached to the visible monochromatic radiations, and by the purple line, straight line joining the chromaticity values attached to the two extremities of the visible spectrum. Owing to its ingenious mathematical construction, this diagram has the interesting property to represent the chromaticities of light stimuli obtained by additive mixing of two primary stimuli as a segment of straight line, whose extremities are the chromaticities of the two primary stimuli, a linearity inherited from the linearity of the XYZ color space in respect to additive color mixing. These mathematical properties probably explain the success, still well observable today, that this diagram has met for the color characterization of displays, video projectors, and any other RGB system based on additive color mixing of three primary red, green and blue lights. Even though their color gamut is actually a 3D volume, more precisely an hexahedron, in the XYZ color space, its projection (a triangle) in the xy chromaticity diagram is often preferred, probably because a 2D diagram is more convenient to display than a 3D volume, and also because the triangle in the chromaticity diagram exhibits well the color reproduction quality in terms of chroma for each hue.

The widest possible gamut is generally sought so that the maximum chroma can be achieved for each hue angle. For RGB systems, increasing the size of the triangular gamut can hardly be done otherwise than by choosing three monochromatic primaries. The choice of the spectral color green will strongly influence the system's ability to reproduce with a high chroma the colors of hue ranging from orange to cyan through greens, favouring either yellows or turquoises. The choice of the blue and red spectral colors will mainly affect the system's ability to reproduce blue, purple, red and other shades with high chroma. To increase the gamut coverage, one is tempted to take the red and blue primaries at the extremes of the visible spectrum, but as the luminous efficacy is very low at these extreme wavelengths, some colors, especially white, cannot be reproduced with sufficient luminous efficacy. This is an obvious technical and energy issue for displays or projectors. A trade-off has therefore to be found between gamut size and luminous efficacy. In this paper, we detail the solutions that have been proposed and try to better describe the criteria for achieving an optimal solution.

Enlarging the gamut is therefore equivalent to trying to achieve the highest chroma possible for each hue. It is worth considering the best way to quantify this color attribute. However, even with the component Y representing luminance added to the chromaticity diagram, CIE xyY color space is not a color appearance model, in contrast with CIELab or CIECAM02 spaces. The notions of hue, saturation and lightness are not directly accessible. To overcome this difficulty, the concepts of dominant wavelength (for hue) and excitation purity (for chroma) have been defined. The dominant wavelength is the wavelength indicated by the meeting point S between the spectrum locus and the line meeting the chromaticity points C and W representing respectively the considered color and a white reference color selected beforehand. The excitation purity is then the ratio of Euclidian distances CW/SW in the diagram. Spectral colors therefore have an excitation purity unity and a dominant wavelength equal to that of their radiation. These notions make less sense for colors close to the purple line, whose hue does not benefit from a spectral color to define them. By convention, a negative dominant wavelength is assigned corresponding to the complementary spectral color, i.e. on the same segment and opposite to the reference white. The excitation purity is calculated in the same way, taking the purple line as boundary with unit purity. This solution is not satisfactory since colors on the purple line are practically imperceptible because it results from the mixture of two red and blue lights at the limit of the locus spectrum, which are themselves almost imperceptible. This difference between purples and other hues runs counter to the main ideas that have accompanied the evolution of color representation systems since Newton, notably the idea of a continuous color circle, where all hues are treated similarly. We therefore provide a history of these ideas, starting in Section 2 with the place of non-spectral purple colors in color representation systems. We present in Section 3 the difficulty of determining the maximum chroma location for each hue, including purples. In Section 4, we discuss the choice of monochromatic primaries in existing RGB systems for which a practical purple line is defined as the line joining the blue and red monochromatic primaries. In the trade-off between gamut coverage and luminous efficacy, we highlight two new RGB systems with monochromatic primaries. In Section 5, the proposed RGB system favours a high luminous efficacy to obtain the reference white by ternary mixing between the three primaries. In Section 6, we present an alternative that allows for a wider color gamut and a constant and not zero luminous efficacy on the line connecting the blue and red primaries.

On purple in the History of color science

In 1665, the Great Plague epidemic in England forced the universities to close. Students had to confine themselves into their homes. Isaac Newton, then student at Cambridge, went back home to Woolsthorpe. History does not say whether Newton gained weight during his confinement, but opinions agree with the fact that he began to revolutionize physics at that time. He made his first studies on the decomposition of white light through the prism. He then demonstrated the composite nature of white light and the fact that the colors coming out of the prism are related with the elementary components of light. By desired analogy with the Dorian scale, he divided the spectrum into 7 intervals, 5 main colors (tones) to which he added orange and indigo (semitones). The names of the colors did not change between Newton's various writings except at one end of the spectrum where Newton first choosed purple before opting for violet. In a few pages in Opticks published in 1704 [START_REF] Newton | Opticks[END_REF], Newton sketched the beginnings of colorimetry. The idea was not the one of a physicist. It was more a matter of intuition that the colors of the spectrum form a continuum. In this way, he connected the two ends of the spectrum and created a color disc (Figure 1.a). The periphery was made of the spectral colors while white was at the centre. Newton proposed a geometric method for determining the perceived color of a mixture to which he associated what we would today call the dominant hue and saturation. On the diagram in Figure 1 is shown an example of mixture made of the seven colors according to the proportions indicated by the size of the small circles on the periphery of the disc. The centre of gravity is obtained in Z. The line OZ cuts the disc periphery in Y: the hue of the mixture is an orange color that tends slightly towards red. The OZ distance gives an indication of the saturation of the color: here, less than half compared to the corresponding spectral color (OY). Maximum saturation is logically achieved by the spectral colors at the periphery of the disc. It should be noted that the non-spectral colors, finally called purples, are not represented on the circumference of the circle. However, Newton did not ignore them completely and placed them on the OD segment as a result of mixing violet and red. In contrast to Newton's disc, current color classification systems such as Munsell or NCS grant purples a large angular sector of between 25 and 30 %.

In the 19th century, the English scientist Young suggests that our visual system should contain three types of particles, sensitive to the colors red, green or violet [START_REF] Young | A course of lectures on natural philosophy and the mechanical arts[END_REF]. This trichromacy principle allowed to represent all colors perceptible in an equilateral triangle at the vertices of which are located the three primaries. A few decades later, Maxwell developed a "light box" that allows to visually equalize chromatic area by mixing spectral colors [START_REF] Maxwell | On the theory of compound colours, and the relations of the colours of the Spectrum[END_REF]. He choosed three primaries labelled 24, 44 and 68, corresponding to wavelengths respectively around 631 nm, 529 nm and 457 nm (Figure 1.b). This shows that the spectral colors are slightly outside the equilateral triangle defined by the three primaries. Similar observation applies with non-spectral colors, even though Maxwell did not explicitly draw a "purple line" between the extreme wavelengths of his representation, labelled 20 and 80, corresponding to 664 nm and 435 nm. Naturally, Maxwell does not choose these ends of the spectrum as primaries because the perceived lights are too weak. The results obtained by visual equalization using these extreme wavelengths are not very satisfactory.

The CIERGB color space defined in 1931 from the work of Wright [START_REF] Wright | A Re-Determination of the Trichromatic Coefficients of the Spectral Colours[END_REF] and Guild [START_REF] Guild | The Colorimetric Properties of the Spectrum[END_REF] was finally a Maxwell triangle. After discussions, the CIE adopted the monochromatic primaries 700 nm, 546.1 nm and 435.8 nm. The two latter wavelengths are mercury emission lines. For red, 700 nm was chosen arbitrarily because there is no reference emission line in this range, but this was not a big issue since only L-cones are excited in this spectral domain and the perceived hue is stable. The CIE defined the color matching functions that translate the amounts of red, green and blue lights needed to match with a monochromatic stimulus at any wavelength. In practice, each monochromatic light mixed with one primary (negative contribution) was visually equalized by mixing the other two primaries (positive contributions). The negative values generate computation issues at a time when computation was mainly done manually. The CIE thus proposed a linear transformation converting the RGB color space into the XYZ color space in which all tristimulus values are always positive. Consequently, allchromaticity values are also positive and contained within the triangle drawned by the chromaticity points attached to the virtual primary color stimuli X, Y and Z. In the xy chromaticity diagram, the set of perceptible colors is represented by a surface delimited by the spectrum locus for the spectral colors and by the purple line joining the coordinates of the monochromatic radiations of the two ends of the visible spectrum, 380 nm and 780 nm (Figure 1.c). Sometimes these standardized limits of the visible domain are even extended to 360 nm and 830 nm. In the chromaticity diagram, the two ends of the purple line correspond to monochromatic radiations between 360 and 380 nm for the blue end and between 700 and 830 nm for the red end. As with Maxwell, the purple line slightly differs from the line connecting the blue (435.6 nm) and red (700 nm) primaries of the CIE1931 RGB color space. Color annotations have been added to the original documents to make them easier to read.

The CIE xy chromaticity diagram retains Newton's distinction in Opticks between violet as the end of the visible spectrum and purple as a non-spectral color. In common parlance and in other color spaces, violet and purple are two shades of the same hue. The famous Tyrian purple that dressed the Roman emperors is close to red. The word "violet", probably derived from name of the flower, is more recent and designates colors closer to blue. In the 17th Century, François d'Aguilon was the first scientist to propose a color representation system based on mixtures [START_REF]Opticorum libri sex[END_REF]. The main hues were the red, blue and yellow triad, which would leave its mark Western culture for more than three centuries and is still a standard in art education today. Purple (purpureus) is presented as the result of the mixture between blue (caeruleus) and red (rubeus). Goethe choose same colors but arranged them into a chromatic circle [START_REF] Von Goethe | Zur Farbenlehre[END_REF]. Once again, porpora is placed between blu and rosso. There is a strong linguistic bias between French and English languages. French uses the word violet to designate a color obtained by mixing blue and red. This is the case, for example, in Chevreul's chromatic circle [START_REF] Chevreul | Cercles chromatiques de M. E. Chevreul : reproduits au moyen de la chromocalcographie, gravure et impression en taille douce combinées[END_REF]. In Anglo-Saxon countries, the word purple is more common. It is, for example, one of the five main hues in Munsell's color space [START_REF] Munsell | The Atlas of the Munsell Color System[END_REF]. In the 19th century, advances in chemistry allowed the production of synthetic dyes based on aniline, which bore different trade names such as mauve, fuchsia or magenta. Marketed in 1859 in reference to the Battle of Magenta, which had taken place one year earlier, the color magenta was to impose itself as the primary of the ideal subtractive synthesis with cyan and yellow. For a symmetrical presentation, it is also the secondary colors in red, green, blue additive synthesis (and vice versa). Magenta has replaced violet and purple!

Maximum color chroma in the CIE1931 xy chromaticity diagram

As mentioned in the introduction, the xy chromaticity diagram is not suitable for quantifying precisely the chroma of a color.

Nevertheless, excitation purity is a notion that comes close to it. It can be expressed as:

     (2)
It should be noticed that x coordinates may substitute the y coordinates in equations ( 1) and ( 2) when yb and yw are very close.

Both excitation purity and colorimetric purity vary between 0 for the reference white and 1 for the limits of the perceptible colors. Thus, by convention, the non-spectral colors of the purple line have a purity 1. This choice does not really make sense because the colors of the purple line are almost imperceptible.

This issue is perfectly stated in a paper by Pridmore [START_REF] Pridmore | Chromatic luminance, colorimetric purity, and optimal aperture-color stimuli[END_REF]. To remedy this issue, he suggested calculating the luminous efficacy for obtaining the reference white by a binary mixture between a monochromatic radiation and its spectral complementary, which is located at the intersection between the spectrum locus and the segment joining the coordinates of the radiation and those of the selected white. The luminous efficacy maxima are obtained at the visible ends for 442  1 nm and 613  1 nm depending on whether illuminant D65 or illuminant A for white light is chosen. The line connecting these two wavelengths of the spectrum locus is substantially offset from the purple line. Pridmore proposed to keep the definitions of excitation and colorimetric purities given by the equations ( 1) and (2). He simply replaced the purple line with this new line (442 nm -613 nm) as the unit and maximum purity border.

Another approach is to consider non-fluorescent surface colors and determine the limits of the color solid. For this purpose, Schrödinger [START_REF] Schrödinger | Theorie der Pigmente von grösster Leuchtkraft[END_REF] defined optimal colors, whose spectra only take on values 0 and 1 with a maximum of two transitions between these two values. MacAdam [START_REF] Macadam | The theory of the maximum visual efficacy of colored materials[END_REF] [13] defined the boundary of the optimal color solid in the CIExyY color space. For non-spectral colors, this boundary meets the purple line only for very low Y luminance values. In 1980, Pointer collected the colors of a big set of non-fluorescent surfaces [START_REF] Pointer | The gamut of real surface colours[END_REF]. Whereas MacAdam's limits set theoretical boundaries for optimal surface colors, Pointer's gamut is an attempt to establish the empirical limits of real surface colors. Although approximated, Pointer's gamut gives an idea of the colors that a color reproduction system should be able to render. The most saturated purples in the Pointer gamut are noticeably away from the purple line in the xy chromaticity diagram. It is worth noting that both MacAdam limits and Pointer's gamut are relative to the chosen illuminant and the relative luminance Y.

RGB monochromatic primaries

In his analysis, Pridmore "forgot" that he achieved a third maximum luminous efficacy in the green part of the spectrum, at 531.5 nm [START_REF] Pridmore | Chromatic luminance, colorimetric purity, and optimal aperture-color stimuli[END_REF]. He actually defined a system of three monochromatic RGB primaries with the line connecting the blue and red primaries constituting a "practical" purple line. There is an infinite number of solutions to choose a triplet of wavelengths in the visible range. One condition is that the reference illuminant(s) must be reproducible by additive mixing. Therefore, the xy coordinates of this (these) illuminant(s) must be included into the triangle at the vertices of which the coordinates of the primaries are located. Other considerations come into play.

In his article [START_REF] Maxwell | On the theory of compound colours, and the relations of the colours of the Spectrum[END_REF], Maxwell chooses the three primaries (631 nm, 529 nm, 457 nm) "because they are well separated from each other on the scale, and because the colour of the spectrum at these points does not appear to the eye to vary very rapidly". For the CIE1931 RGB color space (Fig. 1.c), the choice of mercury lines for the blue (435.8 nm) and green (546.1 nm) primaries was made because they are standards in metrology. The practical realisation of monochromatic primaries is also to be taken into consideration. Therefore, laser sources seem to be the most appropriate solution even if the spectral purity of the LED emissions would make it possible to approach the chosen configurations.

More recently, RGB color spaces with monochromatic primaries have been developed to quantify wide gamut devices. To estimate the extension of a color gamut, it is usual to give the percentage of coverage that this gamut represents in the xy chromaticity diagram compared to all perceptible colors. Counterintuitively, the use of the perceptually non-uniform xy diagram allows a satisfying estimate of the size of color gamuts [START_REF] Masaoka | Display gamut metrology using chromaticity diagram[END_REF]. The Adobe Wide Gamut RGB color space (700 nm, 525 nm, 450 nm) developed by Adobe System has a very wide gamut covering 71.5 % of all chromaticities. It should be noted that the widest gamut (73.8 % coverage) is obtained with the triplet (700 nm, 518 nm, 431 nm). Here, we compare it with the RGB system (780 nm, 518 nm, 380 nm) which includes the purple line and has a coverage rate of 73.6%. REC2020 is the standard for Ultra High Definition Television (UHD-TV). The primaries (630 nm, 532 nm, 467 nm) have been selected in order to cover Pointer's gamut (99.9 % in the xy chromaticity diagram). The 100% optimum would be obtained with a green primary at 527 nm but more difficult to obtain in practice with LEDs or laser technologies. Although it is fortuitous, the primaries in the REC. 2020 standard are quite close to the ones proposed by Maxwell in the 19th Century.

Another approach is to consider human vision at the retinal level. The cones of types L, M and S have spectral sensitivity curves presenting a peak at 569 nm, 541 nm and 448 nm, respectively. These curves largely overlap each other, which explains why it is not possible to choose the wavelengths of the maxima as primaries. Thornton noted that whatever monochromatic RGB primaries are chosen, the three color matching functions preferentially peak at three wavelengths called Prime Colors [START_REF] Brill | Prime colors and color imaging[END_REF]: 603 nm, 538 nm and 446 nm, with little dispersion on the wavelengths displayed in the literature. Worthey shows in an original way the existence of these Prime Colors [START_REF] Worthey | Color matching with amplitude not left out[END_REF] by using the Cohen works on the "locus of unit monochromats" [START_REF] Cohen | Color mixture and fundamental metamers: Theory, algebra, geometry, application[END_REF], 3D representation of the "orthonormal opponent color functions". Worthey explicitly defined a line of practical purples joining the blue and red Prime Colors. Pridmore's primaries, although based on a different approach, are close to the ones of Prime Colors.

Table 1 shows the properties of the main monochromatic RGB systems. For some of them, their triangular representation in the xy chromaticity diagram is given in Figures 2.a and 2.b. Many practical purple lines can be obtained according to red and blue primaries selected by joining their representation in the diagram with a straight line.

In the following two Sections, we propose two monochromatic RGB systems using the REC2020 standard as a reference. In Section 5, we optimize the luminous efficacy for the reference white by maintaining the gamut coverage value of the RGB REC2020 system. In Section 6, we keep the blue primary of the REC2020 standard and we get a wider gamut by having a practical purple line with constant luminous efficacy.

High luminous efficacy RGB gamut

We take up the idea of Pridmore, which retains the primaries that enable the highest luminous efficacy for the reference white to be achieved. But in contrast with Pridmore who considers a binary mixture between each primary and its complementary, we consider the ternary mixture between the three primaries. We call cR, cG and cB the relative concentrations in R, G and B primaries yielding the reference white color. The luminous efficacy of a monochromatic radiation being   m K V  with Km = 683 lm/W, the luminous efficacy ηw of the white color is given by:

      w m R R G G B B K c V c V c V            (3)
where 1 For all RGB systems, the luminous efficacies ηD65 and ηA for white corresponding to illuminant D65 and illuminant A respectively, are given in Table 1. There is a trade-off to be found between high luminous efficacy ηw and wide gamut coverage. Many choices are possible. Here, we impose a minimum coverage, and it seemed relevant to us to choose the one of the REC2020 standard, i.e., 63.3%. We then consider all sets of three monochromatic primaries, each of them being an integral number in nm, and keep the set for which the luminous efficacy ηD65 is the highest while achieving this coverage. We obtain the primaries λR = 604 nm, λR = 519 nm, and λB = 407 nm, for which the luminous efficacy ηD65 = 227 lm/W is to be compared with the value ηD65 = 179 lm/W for the RGB REC2020 standard.

R G B c c c    .
    / R G V V       / B G V V  
Figures 2.c and 2.d show the luminous efficacy as a function of the gamut coverage where the blue and red primaries are fixed. The green primary wavelength varies from 492 nm (triangle dots) to 567 nm (square dots) and each circle corresponds to the green primary of the corresponding RGB system. The gamut coverage goes through a maximum for 518 1 G    nm for all RGB systems. It can be seen from Figure 2.d that the choice of a practical purple line between λB = 467 nm and λR = 604 nm optimises the luminous efficacy criterion.

Wide RGB gamut

Another way of representing an RGB system is to express the luminous efficacy of the primaries relative to that of the green primary, which is the highest. Table 1 lists for each RGB system the relative luminous efficacies in % for the blue    

/ B G V V   and red     / R G V V  
primaries. The triangle representation in Figures 2.e and 2.f shows the relative luminous efficacy obtained by binary mixing between two primaries, in particular between blue and red for the purples.

In the case of wide gamuts (Fig. 2.e), this representation confirms that the purple line (λB = 380 nm, λB = 380 nm) corresponds to aluminous efficacy zero. With a relative luminous efficacy of less than 4.8% on the line joining λB = 450 nm and λB = 700 nm, Adobe's wide gamut system is not more interesting. RGB systems with higher luminous efficacy (Fig. 2.f) correct this defect. The system we have proposed in Section 5 presents the highest luminous efficacies for the blue λB = 467 nm and red λR = 604 nm primaries corresponding to respectively 11.6% and 83.9% relative to that of the green primary. Nevertheless, for the three RGB systems shown in Fig. 2.f, there is an imbalance between the red primary and the much less efficient blue primary. We propose another RGB system by keeping the blue primary of the REC2020 standard, λB = 467 nm. It is indeed the blue primary with the longest wavelength and therefore the highest luminous efficacy among the presented RGB systems (Table 1). We thus choose a red primary λR with same luminous efficacy, satisfying

    467 nm R B V V    
, and we found λR = 655 nm. Finally, we determine the green primary λG = 519 nm that maximizes the gamut coverage (Fig. 2.c). This RGB system has a coverage of 72.4%, higher than the one of the REC2020 standard (63.3%), and even higher than the one of Adobe wide gamut system (71.5%) . Notice that with significantly different primaries, the luminous efficacy   65 120 lm/W D   is very close to the one of the Adobe wide gamut system (Table 1). The advantage of this RGB system is that the line connecting the blue and red primaries has a constant and significant luminous efficacy, corresponding approximately to 11.7% of the one of the green primary.

Conclusion

The purple line in the chromaticity diagram connects the two ends of the spectrum locus. This is a borderline in the human color gamut, but it also corresponds to colors that are almost imperceptible due to a very weak luminous efficacy. It is therefore interesting for color practitioners to define another, more practical purple line. This one is defined by linking the blue and red monochromatic primaries of an RGB system. Various criteria can be used, while a trade-off is generally to be found between a wide coverage rate and high luminous efficacy for the white generated by ternary mixing between the primaries. We thus propose two RGB systems:  λR = 604 nm, λG = 519 nm, λG = 467 nm. For similar coverage as the REC 2020 RGB system, the luminous efficacy ηD65 of the white color corresponding to the illuminant D65 is more than 25% higher.  λR = 655 nm, λG = 519 nm, λG = 467 nm. By changing only the red primary in the previous system, a significantly higher coverage is obtained. The practical purple line has a constant and not negligible luminous efficacy (more than 11% of the one of the green primary).

However, the practicality of an RGB system with monochromatic primaries should not obscure the fact that there are colors outside the RGB triangle, in particular faintly perceptible purples. These practical considerations are therefore not suitable for artists working with color lights and exploring these limits of perception such as James Turrell, Olafur Eliasson or Adrien Lucca.

The next step would be to visualize iso-chroma curves in the CIE xy chromaticity diagram. Pridmore suggested redefining excitation and colorimetric purities in relation to the new practical purple line. This approach is pragmatic. However, it does not allow quantifying this purity of excitation for all colors between the line connecting the blue and red primaries and the 380 nm -780 nm purple line. Above all, we must remind that the chromaticity diagram is not the appropriate space for chroma characterization. There are alternatives to the excitation purity, which are more in line with the perception of color [START_REF] Schiller | An evaluation of different measures of color saturation[END_REF]. Chroma could be evaluated in a selected color appearance model like CIECAM02 or Jzazbz [START_REF] Safdar | Perceptually uniform color space for image signals including high dynamic range and wide gamut[END_REF]. It would then be possible to preserve the representation in the widely used xy chromaticity diagram and to discriminate the location of the most saturated colors from that of the perceptible colors, especially near the purple line.

Figure 1 .

 1 Figure 1. (a) Newton's chromatic disc [1]. The purples are on the OD segment that connects the two ends of the spectrum. (b) Maxwell's color triangle [3] with monochromatic primaries noted 24, 44 and 68. An angular sector is dedicated to purple between the two extreme wavelengths noted 20 and 80. (c) CIE xy chromaticity diagram. The purple line connects the extreme wavelengths of the visible: 380 nm and 780 nm. The monochromatic primaries of the CIE1931 RGB color space correspond to the wavelengths 700 nm, 546.1 nm and 435.8 nm.

  yb and yw are the CIE 1931 chromaticity coordinates y for respectively the color, the monochromatic or boundary color, and the reference white. The notions of excitation purity and dominant wavelengths are indeed relative to a chosen white light. This white light is generally chosen from among the illuminants defined by the CIE as equal-energy illuminant E today, the CIE also defined colorimetric purity with a slightly different normalization:

Figure 2 .

 2 Figure 2. Comparison between our proposed RGB systems and two existing wide gamut systems (column on the left) and two existing high efficacy. gamuts (column on the right). a-b) Gamut representation in the CIE xy chromaticity diagram. c-d) Parametric curve representing the gamut coverage and the luminous efficacy ηD65 of a D65 white obtained by additive mixing of the three monochromatic primaries where R and B are fixed (indicated in the chromaticity diagram) and G is varied from 492 nm (triangles) to 567 nm (squares) through the G primary indicated in the chromaticity diagram (circles).e-f) Relative luminous efficacy of the monochromatic RGB primaries with respect to the one of the green primary.

Table 1 .

 1 Comparison of RGB systems a

	RGB system	RGB primaries (nm)	Gamut	Luminous efficacy of D65	Relative primary luminous
					coverage (%)	or A white (lm/W)	efficacy (%)
		λR	λG	λB		ηD65	ηA


O Haines

J Cambon

P Bantignies

T Fertey

M Stuerzer

J Poienar

Rouquette

G Hearne
email: grhearne@uj.ac.za

V Ranieri

P Hermet

J Haines

O Cambon

J L Bantignies

P Fertey

T Stuerzer

M Poienar

J Rouquette
email: jerome.rouquette@umontpellier.fr

Interplay between H-bonding proton dynamics and Fe valence fluctuations in Fe 3 (PO 4 ) 2 (OH) 2 at high pressure

Interplay between H-bonding proton dynamics and

Fe valence fluctuations in Fe 3 ( P O 4 ) 2 ( OH ) 2 at high pressure

Introduction. There is considerable interest in iron hydroxyl phosphates, which occur in various mineral forms and are readily synthesised by hydrothermal methods. They are used as catalysts in methyl methacrylate synthesis, of importance in polymer manufacturing [START_REF] Millet | FePO Catalysts for the Selective Oxidative Dehydrogenation of Isobutyric Acid into Methacrylic Acid[END_REF]. They are also candidates for positive-electrode materials in Li-ion rechargeable batteries, in analogy to the well-known phospho-olivine LiFePO4 [2]. These important applications potentially arise from the coexistence of two intrinsic physical properties in these compounds: iron mixed-valence state and the occurrence of hydrogen bonds in lattice segments, e.g., in barbosalite (Fe 2+ Fe 3+

2)(PO4)2(OH)2.

Mixed-valence barbosalite Fe 2+ Fe 3+ 2(PO4)2(OH)2 has a six-fold coordination of Fe involving FeO4(OH)2 octahedra, each of which have (OH) -hydroxyl groups in a trans configuration [3]. Such (OH) -intermediaries form part of Fe-O-H•••O-P structural sequences involving hydrogen bonds and a double-well proton potential energy profile along the line of oxygens [START_REF] Hydrogen | bonds have been classified as weak , intermediate and strong according to the bond length constituted by the distance R between the oxygens; R > 2.6 Å ('weak'), 2. 4 Å < R < 2.6 Å ('intermediate'), R < 2.4 Å ('strong')[END_REF]. Pressurizing such mixed-valence iron hydroxy phosphates affords a unique opportunity for investigating how changes in O-H•••O hydrogen bonding may influence Fe 2+ → (OH) -→Fe 3+ inter-valence electron transfer, and vice-versa. Such an interplay between proton and d-electron dynamics is also expected to impact on magnetic properties, as demonstrated in the topical quantum spin liquid candidate H3LiIr2O6 [START_REF] Li | Role of Hydrogen in the Spin-Orbital-Entangled Quantum Liquid Candidate H3LiIr2O6[END_REF]. involving proton localization in one of the minima, to have a low enough barrier across which proton delocalization occurs via quantum tunnelling [START_REF] Hydrogen | bonds have been classified as weak , intermediate and strong according to the bond length constituted by the distance R between the oxygens; R > 2.6 Å ('weak'), 2. 4 Å < R < 2.6 Å ('intermediate'), R < 2.4 Å ('strong')[END_REF][START_REF] Li | Role of Hydrogen in the Spin-Orbital-Entangled Quantum Liquid Candidate H3LiIr2O6[END_REF][6]. Further reductions in O•••O distances may lead to symmetrisation of the hydrogen bond where the proton locates at the centre between the oxygens, in a pronounced anharmonic single-minimum potential [7].

The relevance of this, for example, is that nominally anhydrous minerals are a major source of H2O in planetary interiors, in the form of hydroxyl groups (OH) -incorporated in the polyhedral coordination of cations in these structures. The thermoelastic moduli of these We have pressurised monoclinic barbosalite Fe 2+ Fe 3+ 2(PO4)2(OH)2 in diamond-anvil cells (DACs) up to ~30 GPa at room temperature (RT). We deploy single-crystal X-ray diffraction (SC-XRD) and infrared (IR) spectroscopy to monitor unit-cell evolution and proton state in situ, respectively, as O-H•••O bonds evolve under pressure [6c]. 57 Fe Mössbauer spectroscopy (MS) is used to discern the Fe local atomic environment, as a complementary and direct probe of cations proximate to pressure-tuned hydrogen bonds [START_REF] Hearne | Pressureinduced suppression of charge order and nanosecond valence dynamics in Fe2OBO3[END_REF]. Methodological details are in the Supplemental Material (SM) [START_REF]structural data and analysis from SC-XRD and DFT calculations, further IR data and DFT calculations[END_REF].

Our pressure study of mixed-valence barbosalite follows on from previous interesting findings of the interplay between single Fe-valence perturbations and proton dynamics of hydrogen bonds in some inorganic assemblages. For example, in layered brucite-type Fe(OH2) [11], pressurisation leads to increased interaction amongst protons in the hydrogen bonds. The resultant breakdown of O-H axial symmetry creates an effective dipole potential at the iron site. This, in addition to the relatively weak binding energy of the minority-spin valence electron, triggers "self-oxidation" Fe 2+ → Fe 3+ + e -, onset at some cation sites at ~8 GPa and progresses with ever increasing abundance up to ~40 GPa.

Structural considerations. Our SC-XRD studies indicate barbosalite crystallizes in the monoclinic space group P21/n at ambient conditions. Alternating Fe 2+ and Fe 3+ octahedra sharing opposite faces, form infinite chains in <110> directions, depicted in Fig. 1(a). A typical charge-ordered sequence in a chain is …-V-Fe 3+ -Fe 2+ -Fe 3+ -V-Fe 3+ -V-Fe 2+ -Fe 3+ -Fe 3+ … where V represents disordered vacant octahedral sites, due to some low occupancy of Fe in the vacant octahedral locations separating trimer sequences in Fig. 1(a). Further renditions and details of the structure are in Sub-sect. S1.1.2 of the SM [START_REF]structural data and analysis from SC-XRD and DFT calculations, further IR data and DFT calculations[END_REF], see Fig. [6c, 12], in Fig. 1(c). Note, even at ambient pressure these hydrogen bonds are near to moderate strength [START_REF] Hydrogen | bonds have been classified as weak , intermediate and strong according to the bond length constituted by the distance R between the oxygens; R > 2.6 Å ('weak'), 2. 4 Å < R < 2.6 Å ('intermediate'), R < 2.4 Å ('strong')[END_REF], deduced from IR stretch frequencies lower than the ~3500 cm -1 typical of weak hydrogen bonds. Fig. 1(d) depicts the pressure evolution of O-H symmetric and asymmetric stretch mode frequencies and their average frequency, up to where the two stretch modes are visibly resolved at [START_REF] Li | Role of Hydrogen in the Spin-Orbital-Entangled Quantum Liquid Candidate H3LiIr2O6[END_REF][6].

Thereafter, excessive mode broadening occurs and the frequency is taken as the extremum value of the broad maximum. There is a weak negative pressure dependence of the average O-H stretch-mode frequency up to ~4 GPa. Then a change to a much steeper negative pressure dependence occurs up to ~7 GPa, after which there is a change to a different negative pressure dependence and appreciable line broadening is evident. By 8-10 GPa O-H stretch modes manifest as a broadened humped profile. structural segments, at longer ranges beyond the oxygen endpoints, i.e., at the cations. To this end, we deployed 57 Fe MS to directly probe the proximate cations under pressure, viz., Fe charge state and local atomic environment, from behaviour of the hyperfine interaction (HI) parameters [START_REF] Hearne | Pressureinduced suppression of charge order and nanosecond valence dynamics in Fe2OBO3[END_REF][START_REF]The hyperfine interaction (HI) parameters as well as Fe site abundances are derived from fitting the spectral envelope with an appropriate static (Lorentzian sub-components) or dynamical theory. The isomer (chemical) shift IS is proportional to the s-electron density at the Fe nucleus and is influenced by 3d-electron shielding. The quadrupole doubletsplitting, QS, is proportional to the aspherical distribution of surrounding electronic and nearest neighbor atomic charge. The IS, QS, and Hhf (from magnetically split hyperfine structure) parameters have distinct "fingerprint[END_REF], see also Sec. S4 of the SM [START_REF]structural data and analysis from SC-XRD and DFT calculations, further IR data and DFT calculations[END_REF].

Evolution of 57 Fe MS spectral profiles of Fe 2+ Fe 3+ 2(PO4)2(OH)2 under pressure up to ~30 GPa are shown in Fig. 2(a). At the lowest pressure notice the conspicuous signature Fe 2+ wide quadrupole splitting (QS, doublet) and large isomer shift (IS, centroid) values, compared with Fe 3+ signatures in the near-central zero-velocity region. Fe 2+ signatures show noticeable changes at P ≥ 10 GPa, manifested as progressively increasing line broadening and asymmetry, and reductions in resonance intensity. Less conspicuous changes occur in the Fe 3+ component, because some of the more abundant Fe 3+ sites are likely not influenced by whatever is causing electronic changes at Fe 2+ sites.

Spectral evolution in Fig. 2(a) is typical of a low pressure charge-ordered state progressing to a Fe 2+ ⇔Fe 3+ valence fluctuation regime at higher pressures; also suggested by the dynamical theory fitting in Fig. 2 (b) and simulations in Fig. S11 [START_REF]structural data and analysis from SC-XRD and DFT calculations, further IR data and DFT calculations[END_REF]. HI parameters initially fall in wellknown characteristic ranges for static charge-ordered situations of Fe 2+ and Fe 3+ [START_REF] Gütlich | Mössbauer Spectroscopy and Transition Metal Chemistry: Fundamentals and Applications[END_REF]. Whereas Fig. S11 simulations indicate HI parameters IS and QS of the spectral profile appear to be weighted average values of the static situation when fast valence fluctuations occur at an Fe site [START_REF] Hearne | Pressureinduced suppression of charge order and nanosecond valence dynamics in Fe2OBO3[END_REF][START_REF] Herber | Electron hopping in FeOCl intercalation compounds: A Mössbauer relaxation study[END_REF], Fe 2+ ⇔Fe 3+ , for example from inter-valence electron transfer Fe 2+ → Fe 3+ across face-sharing octahedra. This is where the fast relaxation time τR between two valence states is much shorter than the 57 Fe nuclear quadrupole precession time, τQ ~40 ns [17], in the electric field gradient (EFG) from surrounding aspherical distributions of electronic and lattice charges. Motional narrowing of spectral profiles ensue in this case of τR « τQ. When τR ~τQ , spectral profiles develop smeared out (broadened) features where distinct valence signatures are not readily discerned [START_REF] Herber | Electron hopping in FeOCl intercalation compounds: A Mössbauer relaxation study[END_REF][START_REF] Ellis | Small polaron hopping in Li xFePO4 solid solutions: coupled lithium-ion and electron mobility[END_REF]. The dynamical theory fitting exemplifies this in Fig. 2(b) [START_REF] Herber | Electron hopping in FeOCl intercalation compounds: A Mössbauer relaxation study[END_REF][START_REF]For example MossWinn 4.0, Mössbauer spectral analysis software[END_REF], where slow relaxation τR » τQ and evolution at the highest pressures toward the intermediate relaxation time regime τR ~ τQ are depicted. In all cases the 2:1 Fe 3+ :Fe 2+ original stoichiometry for barborsalite is taken into consideration. Spectral line shapes when τR is within a 1-1000 ns time window yield quantitative information on the valence state (electron) dynamics, as exemplified in Fig. S11 simulations [START_REF]structural data and analysis from SC-XRD and DFT calculations, further IR data and DFT calculations[END_REF]. Therefore, pressure evolution of spectra in Fig. 2 57 Fe MS registers a fluctuating valence at each Fe site through which the polaron quasiparticle migrates along <110> octahedral chains of the high pressure Cc structure in Fig. S8, similar to Fig. 1(a). Fitting MS spectra in Fig. 2(a) with an appropriate electronhopping (dynamical) theory yields valence fluctuation rates ω = 1/τR [16, 19a], e.g., Figs. 2(b) and S11 [START_REF]structural data and analysis from SC-XRD and DFT calculations, further IR data and DFT calculations[END_REF]. Deduced electron hopping rates as a function of pressure are in Fig. 3(a). The thermally activated jump frequency for the polaron arising from the electron transfer process as a function of pressure and temperature is [20a, 21]:

0 1 ( , ) exp . A A R B E PV T P k T τ   + Γ = = Γ -     (1)
In Eq. 1, EA is activation energy and VA is activation volume, i.e., changes in energy and volume, respectively, produced by thermal fluctuations for a jump to occur. Γ = 1/τR is obtained from fitting spectra at RT, e.g., in Fig. 2(b). The Γ0 pre-factor involves electrontransfer distance between neighbouring cations and a characteristic phonon frequency.

Although these factors are pressure dependent, they normally only make small (~10 %) corrections to VA extracted from the slope of a ln(Γ) versus P plot, at a fixed temperature [START_REF] Tracy | Polaron-ion correlations in LixFePO4 studied by x-ray nuclear resonant forward scattering at elevated pressure and temperature[END_REF].

The linearity of the ln(Γ) versus P plot in Fig. 3(a) for P ≥ 10 GPa attests to polaron hopping governed by Eq. ( 1). Best and worst slopes in Fig. 3(a) in the high pressure regime yield VA in the range -0.25 to -0.21 cm 3 .mol -1 . This small negative VA value is typical of small polaron hopping in iron oxides [20a, 22].

Polaron hopping is associated with electron transfer across face-sharing octahedra of <110> chains, Fig. S8, similar to those in Fig. 1(a). Such an electron transfer process from cation to cation occurs through an anion intermediary in a two-step process involving highest occupied 3d and 2p orbitals, Protons are then less exclusively associated with ligand nearest-neighbours of Fe, in shared faces of neighbouring octahedra. Such delocalised protons are on average more in centralised regions between oxygens in Fe-O-H-O-P configurations [START_REF] Li | Role of Hydrogen in the Spin-Orbital-Entangled Quantum Liquid Candidate H3LiIr2O6[END_REF]. Thus, dipolar contributions to the Coulombic potential at the Fe sites from (OH) --group ligands is disrupted, upon the likely advent of proton delocalization onset at 4-7 GPa from signatures in Figs. 1(c) and (d). This furnishes the driving force for 3d-electron (t2g minority-spin) delocalization from Fe 2+ sites.

The result then is electron charge transfer and polaron migration to an Fe 3+ site in a neighbouring face-sharing octahedron along octahedral chains in the structure (see Figs. 1(a) and S8).

Is there any evidence in the Fe MS data that reinforcement of H-bonds and the onset of proton dynamics under pressure affects the electric potential V at proximate cation sites? For this, we turn to the pressure evolution of one of the HI parameters, viz, quadrupole interaction parameter. This manifests in the quadrupole splitting QS of doublet profile signatures for Fe 2+ and Fe 3+ valences, seen in spectra at low pressure of Fig. 2. The QS is related to the electricfield-gradient (EFG) at the Fe nuclear site, emanated from asphericity of electronic and lattice charges of its local environment [START_REF] Gütlich | Mössbauer Spectroscopy and Transition Metal Chemistry: Fundamentals and Applications[END_REF]:

( ) 1/2 2 1 QS 1 2 3 zz eQ V η   = +     . (2)
The quadrupole moment is designated by Q, Vzz = ∂E/∂z = ∂ 2 V/∂ 2 z is the maximum value of the EFG in the principal axes system of the Fe nucleus at the origin and η is the asymmetry parameter 0 ≤ η ≤ 1, which is zero for an EFG with axial symmetry. The total EFG , Vzz, has two contributions: from any anisotropic valence electron distribution around the Fe nucleus designated Vzz valence and from more distant ionic charges at neighbouring atoms surrounding the Fe atom in noncubic symmetry, designated Vzz lattice . The Vzz valence contribution is by far dominant, from proximity of the valence shell to the Fe nucleus. This is typical for Fe 2+ :t2g 4 eg 2 (↑↓↑↑)(↑↑) compared with Fe 3+ :t2g 3 eg 2 (↑↑↑)(↑↑), where in Fe 3+ with a spherically symmetric arrangement of valence electronic charge only the smaller Vzz lattice contributes to the EFG. More detailed expressions for Vzz and η, involving multiplicative prefactors for Vzz lattice and Vzz valence include effects of these respective aspherical charge distributions on polarising inner electron shells [START_REF] Gütlich | Mössbauer Spectroscopy and Transition Metal Chemistry: Fundamentals and Applications[END_REF][START_REF] Grodzicki | Electronic structure of Fe-bearing lazulites[END_REF].

QS as a function of pressure is shown in Fig. 3( b) for the two valences. QS for the Fe 2+ valence, dominated by Vzz valence , shows small discontinuous changes at ~10 GPa and ~26 GPa.

Whereas QS for Fe 3+ , dominated by Vzz lattice , is nearly constant up to ~10 GPa and then increases monotonically to appreciably higher values. This is a consequence of changing configurations of lattice charge surrounding the cation in FeO4(OH)2 octahedra, including progressive pressure-induced distortions from cubic symmetry of iron octahedra and delocalization of the proton at hydroxyl group (OH) -nearest neighbours to Fe. Changes in QS for Fe 3+ above ~10 GPa cannot be accounted for by changes in axial symmetry, η parameter in Eq. ( 2), as this would increase QS by only ~15% for η spanning its full range.

Thus, the behaviour in Fig. 3 The pressure response of these mixed-valence hydroxy phosphates exemplify the interplay between proton (THz) and electron (MHz) dynamics on two disparate time scales in the same condensed phase. This is of widespread relevance to charge dynamics in hydrogen bonded systems (e.g., biomolecular complexes and planetary interiors). 2)(PO4)2(OH)2, at RT as a function of pressure. At 1.4 GPa, static Fe 2+ and Fe 3+ valence signatures are in blue (wide doublet) and red (narrow doublet), respectively, with intensities scaled by 0.5 for clarity. Grey shading highlights Fe 2+ features becoming progressively asymmetrically broadened beyond ~8 GPa and quite diffuse at ~18 GPa and beyond. (b) Exemplary dynamical theory fitting to some of the Mössbauer spectra in (a). Solid yellow line through data symbols is the overall fit based on a theory of electron transfer (hopping) between neighbouring ferrous and ferric sites leading to Fe 2+ Fe 3+ valence fluctuations. Valence fluctuation rates ω in the range 0.7-7 MHz are derived as a fitting parameter and provide the time interval between electron transfers (relaxation time) τR = 1/ω indicated in the panels. See Sect. S4 of the SM for further details [START_REF]structural data and analysis from SC-XRD and DFT calculations, further IR data and DFT calculations[END_REF]. Typical spectral doublet parameters of the charge-ordered phase at ambient and low pressures [3a], are depicted by the stick spectra at long electron relaxation times at 8 GPa.

A B R PV k T τ -   Γ =     
. Activation volume VA is obtained from the slope at P > 10 GPa, assuming activated-state rate theory is applicable [20a, 21]. Cartoon below the plot depicts polaron hopping and associated bond length changes from delocalization of t2g minority-spin electron (↑↓↑↑) between neighbouring sites of

  Controlled reductions of barbosalite's unit-cell volume by applied pressure will reduce O•••O distances in O-H•••O hydrogen bonds. This tunes the original double-well potential

  mantle minerals and the impact on seismic behaviour are influenced by these O-H•••O configurations, notably degree of O-H•••O asymmetry or whether symmetrisation has occurred under extreme pressure-temperature conditions [8].

  S5. A hydrogen bond O-H•••O forms between the (OH) -common vertex of mainly three (and sometimes four) octahedra and one of the oxygens involving a PO4 group, i.e., as part of Fe-O-H•••O-P segments of the lattice structure. At ambient conditions hydrogen bonds are asymmetric involving double-well proton potential energy profiles along the line of oxygens [5, 6d]. Proton configurations in O-H•••O bonds throughout the structure are likely to be ordered in one of the potential energy minima of a slightly asymmetric double-well potential, due to repulsive effects of proton next-nearest neighbour Fe 2+ and Fe 3+ cations. Results and discussion. At P ≥ 5 GPa, SC-XRD results in Fig. 1(b) indicate a conspicuous pseudo-symmetrization of original disparate lattice parameters of the monoclinic unit-cell. Although, the volume does not show a discernible discontinuity in this regime. Additionally, the hydrogen bond in Fe-O-H•••O-P structural sequences up to ~16 GPa is monitored via the IR-active O-H bond stretching vibration

  Fig.1(d) [START_REF] Li | Role of Hydrogen in the Spin-Orbital-Entangled Quantum Liquid Candidate H3LiIr2O6[END_REF], and as proton tunnelling frequencies on an experimentally determinable timescale.Entry into the regime of incipient proton delocalization at 4-7 GPa also manifests in the pseudo-symmetrization of lattice parameters in Fig.1(b). This involves a P21/n  Cc displacive phase transition, wherein <110> octahedral chains similar to Fig 1(a) are maintained but with a and b axes doubling in the unit-cell, delineated in Sub-sect. S1.2 of the SM [10]. The onset of such proton dynamics at 4-7 GPa, as part of the surrounding charge environment of the cations, alters the crystal field potential at the Fe sites. This crystal field potential initially had partly a dipole field contribution from the polar O-H intramolecular bond and contributions from other oxygens in the FeO4(OH)2 octahedral local environment. We then examine what are the physical effects of hydrogen bond reinforcement in Fe-O-H•••O-P

  signify a change from a slow relaxation (static) case tending to a regime of intermediate relaxation times of Fe valence fluctuations. Fe 2+ ⇔Fe 3+ valence fluctuations at high pressure and associated 3d electron configuration changes t2g 4 eg 2 (↑↓↑↑)(↑↑) ⇔ t2g 3 eg 2 (↑↑↑)(↑↑) , suggested by Fig. 2, are plausibly a result of t2g minority-spin electron transfer between face-sharing Fe 2+ and Fe 3+ octahedra. Such an inter-valence charge transfer will have an associated propagation of local lattice distortions, i.e., polaron motion [20], see Fig 3(a) cartoon of this process. Polaron hopping can be verified from the spectral analysis of Fig. 2(a) as follows.

  a) cartoon. Thus, Fig.3(a) is compelling evidence of minority-spin electron transfer from the ferrous ion involving the following Fe 2+ →(OH) -→Fe 3+ and Fe 2+ →(O)→Fe 3+ superexchange pathways.We suggest a sequence of charge dynamics involving proton delocalization in hydrogen bonds (Fig.1(d)) influencing electron transfer at cation sites (Fig.3(a)) in Fe-O-H•••O-P configurations, rationalized as follows. The proton in the (OH) -group ligand contributes a dipolar field to the Coulombic potential at the Fe site [5, 11]. Proton localisation in one of the potential energy minima along the O-H•••O configuration at P < 10 GPa, contributes to the Coulombic potential at the Fe site to the extent that the minority-spin electron of Fe 2+ remains trapped. Here, hydroxyl (OH) -ligands are on average in a trans configuration of FeO4(OH)2 octahedra. Compression reinforces the hydrogen bond in Fe-O-H•••O-P configurations.

  (b) is suggestive of appreciable changes to the EFG at Fe sites occurring at ~10 GPa onwards from changes in surrounding ligand charge configurations. This includes disruption of covalent O-H bonds from the evolving proton delocalization, onset below 10 GPa, which eventually triggers detrapping of a relatively weakly bound minorityspin electron at Fe 2+ sites. This picture is in corroboration of polaron hopping and the associated electron-transfer process commencing at ~10 GPa, suggested by activated behaviour in Fig.3(a).Concluding summary. Pressurization of (Fe 2+ Fe 3+2)(PO4)2(OH)2 mixed-valence barbosalite results in proton delocalization onset in the range 4-7 GPa, involving hydrogen bonds of Fe-O-H⋅⋅⋅O-P structural segments. This is discerned by IR spectroscopy and pseudosymmetrization of the lattice parameters. The evolving hydrogen bond reinforcement and proton dynamics impact on the crystal field at proximate Fe cations. This eventually triggers dynamical minority-spin electron exchange along Fe 2+ →L→Fe 3+ pathways (ligand L = O or (OH) -of shared octahedral faces), evidenced by57 Fe Mössbauer spectroscopy at 10-30 GPa.

FIG. 1

 1 FIG. 1 Hearne et. al SC-XRD and IR

Fig. 2

 2 Fig. 2 Hearne et. al MS spectra

  Fig. 3 Hearne et. al MS param

Fe 2+ -

 2+ Fe 3+ face-sharing octahedra. This leads to a Fe 2+ ⇔Fe 3+ valence fluctuation rate ω = Γ = 1/τR at an Fe site. Valence fluctuation rates are from fitting Mössbauer spectral line shapes with the appropriate dynamical theory mentioned in the text (Fig.2(b)).(b) Pressure evolution of the quadrupole splitting QS. Note different scales on the vertical axes. QS parametrizes the electric field gradient at the Fe nucleus from surrounding aspheric electronic and lattice charge distributions. Error bars are discussed in Sect. S4 of the SM[START_REF]structural data and analysis from SC-XRD and DFT calculations, further IR data and DFT calculations[END_REF].

Acknowledgments G.H. acknowledges financial support from the National Research Foundation of South Africa (Grant Number 129291). Financial support for this work was also provided by the joint French-Romanian project ANR-12-IS08-0003, COFeIn. (text incl footnotes 3444 w, figures 549 w, figure captions 481w: total 4475 w)


Dorian Saint-Pierre

Lionel Simonot

Mathieu Hébert

Reflectance computation for a specular only V-cavity

Keywords: Surface reflection, Light interreflections, Reflectance model

The color of a surface structured at the mesoscopic scale differs from the one of a flat surface of the same material because of the light interreflections taking place in the concavities of the surface, as well as the shadowing effect. The color variation depends not only on the surface topology but also on the spectral reflectance of the material, its matte or glossy finishing, and the angular distribution of the incident light. For an accurate prediction of the radiance perceived from each point of the object by an observer or a camera, we must take into account comprehensively the multiple paths of light which can be reflected, scattered or absorbed by the material and its surface. In this paper, we focus on the light reflection component due to the material-air interface, in the special case of a surface structured with parallel, periodical, specular Vshaped ridges, illuminated either by collimated light from any direction of the hemisphere, or by diffuse light. Thanks to an analytical model, we compute the radiance reflected in every direction of the hemisphere by accounting for the different interreflections, according to the angular reflectance of the panels and the aperture angle of the cavity. We can then deduce the apparent reflectance of the cavity when viewed from a large distance.

Introduction

It is well known that the structure of surfaces and materials has a crucial influence on the way they reflect light, thereby on their appearance. A same material structured in different ways can yield very different appearance attributes, from bright to dark, glossy to matte or transparent to opaque.

The influence of the material structure on appearance is mainly related to concept of light scattering, a concept which covers a wide variety of optical principles according to the size of the material structures and their periodicity. Regular or periodical structures whose characteristic size is comparable to the wavelength of light generate interferences or diffraction, and consequently colorations which are often called structural colors [START_REF] Kinoshita | Physics of structural colors[END_REF]. These effects have been widely explored in optics for more than one century, even though pseudo-periodical structures are still an active subject of investigation [see for example Ref . 2]. In opposition, irregular structures can generate both coherent and incoherent light scattering which mainly results in a reorientation of light in space and depolarization. For these randomly microstructured materials, many models have also been proposed in the last century to predict their reflection and transmission properties according to the wavelength, polarization, orientation and position of light. Among the most famous theories for the light scattering by volumes, we can mention the Kubelka-Munk model initially introduced for paints [START_REF] Kubelka | New contributions to the optics of intensely light-scattering material," part I[END_REF], the Melamed model for pigments powders and slurries [START_REF] Melamed | Optical properties of powders: Part I. Optical absorption coefficients and the absolute value of the diffuse reflectance[END_REF], the radiative transfer theory by Chandrasekhar [START_REF] Chandrasekhar | Radiative Transfer[END_REF], the multi-flux theory, the Van de Hulst works for scattering by particles [START_REF] Van De Hulst | Light Scattering by Small Particles[END_REF], etc. We can also evocate famous models for the scattering of light by surfaces with a random roughness, from Beckman and Spizzichino [START_REF] Beckmann | The scattering of electromagnetic waves from rough surfaces[END_REF] who modelled diffraction by such surfaces, through Torrance and Sparrow [START_REF] Torrance | Theory for off-specular reflection from roughened surfaces[END_REF] who modelled incoherent reflection by randomly organized microfacets, to the most advanced models which also take into account the multiple scattering between different facets [START_REF] Heitz | Multiple-Scattering Microfacet BSDFs with the Smith Model[END_REF][START_REF] Lee | Practical multiple scattering for rough surfaces[END_REF][START_REF] Xie | Multiple scattering from distributions of specular v-grooves[END_REF]. All these models form a large prediction toolbox for many visual attributes (color, translucency and opacity, gloss and matt aspect), applicable to a wide range of materials according to their optical properties (refractive indices, scattering and absorption coefficients…) and structural properties (surface roughness, particle size and concentration, layer thickness…), provided the material can be considered as homogeneous at the macro-or mesoscopic scale.

However, for many kinds of surfaces or objects, the multiscale structure of the matter do have to be taken into account in order to obtain accurate optical models and appearance predictions. Describing scattering at multiple scales is generally done by combining different models. The classical literature in physics shows various examples. Mie scattering model is used to describe the light scattering by one particle, and a radiative transfer model is then used to describe the light transport through a piece of medium with particles. For stacks of diffusing layers, the Kubelka-Munk model describes the light scattering at the microscopic scale within each layer and predicts its reflectance and transmittance, then the Kubelka layering model [START_REF] Kubelka | New contributions to the optics of intensely light-scattering materials," part II: Non homogeneous layers[END_REF] or more advanced models describe the flux transfers at the mesoscopic scale between the different layers with their respective interfaces [START_REF] Simonot | Extension of the Williams-Clapper model to stacked nondiffusing colored coatings with different refractive indices[END_REF][START_REF] Hébert | Compositional reflectance and transmittance model for multilayer specimens[END_REF]. For halftone prints, the optical properties of the paper and the inks can be both modeled by the Kubelka-Munk theory [START_REF] Emmel | A Unified Model for Color Prediction of Halftoned Prints[END_REF], then the scattering properties of the set of ink dots on top of the paper can be predicted by a number of models describing the flux transfers between the different inked and non-inked areas [START_REF] Clapper | The Effect of Multiple Internal Reflections on the Densities of Halftone Prints on Paper[END_REF][START_REF] Yule | The penetration of light into paper and its effect on halftone reproductions[END_REF][START_REF] Rogers | Effect of light scatter on halftone color[END_REF][START_REF] Rogers | A Generalized Clapper-Yule Model of Halftone Reflectance[END_REF]. But models are still missing for a volume made of an alternation of mesoscopic bricks of materials and the 3D flux transfers taking place between them, as we can find in 3D inkjet printing, and for a surface whose shape has been given a mesoscopic, possibly periodical structure.

In the latter case on which the present paper is focused, the multiple reflections between the different areas of the non-flat surface, also called interreflections, give to the object specific reflection properties according to the illumination conditions that the models mentioned above cannot render properly. As shown in recent studies dedicated to ridged Lambertian materials (ridges with V-profile) [START_REF] Saint-Pierre | Light Interreflections and Shadowing Effects in a Lambertian V-Cavity under Diffuse Illumination[END_REF][START_REF] Deeb | Spectral reflectance estimation from one RGB image using self-interreflections in a concave object[END_REF], the presence of periodical ridges modifies the color of the material in comparison to the color of the flat surface, in different ways according to the ridge aperture and the illumination conditions: the color of the ridge surface is brighter and more saturated than the one of the flat surface under frontal collimated illumination, but it is darker and less saturated under diffuse illumination. Interreflection models taking explicitly into consideration the microscopic optical properties of the material and the mesoscopic structure of the surface are capable to predict these color variations, thus also allowing the prediction of the irradiance repartition at these two scales. The present paper follows this investigation on materials with periodical V-shaped ridges under different illumination conditions, by considering this time a nonscattering material and describing the multiple specular reflections undergone by each ray between faces of the structure, behaving like mirrors. As for the model dedicated to Lambertian materials, we adopt a radiometric approach, yielding analytical expressions for the angular and bihemispherical reflectance of the structured surface, as a function of the material refractive index and the ridge aperture α (see Figure 1). The paper is structured as follows: we first present the useful concepts for describing our model, to then introduce the formalization of multiple light reflections happening in a V-cavity, with specular surfaces as pannels. After this we move on to the fourth section where we present the results obtained by using the presented model, and we end with our conclusions.

Reflectance

The fraction of light reflected by the structured surface is characterized by the reflectance concept, which is defined for an area of the surface much larger than the width of the cavities. This concept relies on radiometric quantities related to the amount of incident and reflected light, recalled below, which can also be used to describe the multiple reflection process within each cavity.

The light power, or flux, denoted as F, can be regarded as a collection of light rays propagating from the source to the objects, then from areas of the objects to other areas, then from the objects to the observer. The distribution of the light flux over a given surface is described by the concept of irradiance (for incoming light) or exitance (for outgoing light), defined as the density of received or emitted flux dF per elementary area dA:

dF E dA  (1)
Radiance, denoted as L, is defined by the density of light power (or flux) d²F per elementary geometrical extent 2 d G :

2 2 d F L d G  ( 2
)
where the geometrical extent defines the flux transfer volume between the two elementary areas.

Reflectance denotes any ratio of reflected flux to incident flux relative to the same surface element, defined for a given illumination and observation geometry. In this paper, reflectance is generically denoted as R. In the special case of air-medium interfaces, the angular reflectance is denoted as R01(θ) for collimated light coming from medium 0 (in our case, it is air, of refractive index n0 = 1) at the interface with medium 1 (of refractive index n1, which can be either real or complex), with an angle of incidence θ. The term n denotes the relative optical index of the interface, i.e., the ratio of the refractive indices as follows:

1 0 n n n  (3)
When a light ray with radiance Li is reflected on a flat interface, the reflected radiance Lr is simply given by:

  01 r i L R L   ( 4
)
where θ is the angle between the incident radiance and the normal of the interface.

Multiple reflections of a light ray in a specular cavity

We can notice from Figure 1 that a light ray entering into one cavity is reflected, possibly multiple times, in this cavity only. Therefore, we can focus on the reflection of light by one cavity, and consider that all cavities reflect light in the same way. In this section, we propose to present the analytical model permitting to accurately predict the amount and directions of light reflected by the cavity. The model is based on geometrical optics, with an approach comparable to ray tracing. It describes the path of the light after the different reflections across the structure, and takes into account the precise number of bounces that the light undergoes on the panels.

Geometry of the cavity

Each cavity is formed by two specular panels of infinite length along the x axis of the 3D Cartesian space (Figure 2). The width of both panels is set to unity (it could be equivalently any other value: the width has no impact on the interreflection phenomenon as shown in [START_REF] Rogers | Effect of light scatter on halftone color[END_REF] and on the computation of the specular radiance that we want to perform here). The angle between the two panels, also called "aperture of the cavity", is denoted as α. Hence, each panel forms a dihedral angle α/2 with the (xOz)-plane, where the z axis corresponds to the normal of the average structured surface.

                           1 N N (5)
The incident light ray is characterized by unit radiance, and a unit vector e with spherical coordinates (θ, φ) represented in Figure 2. In this Cartesian coordinate system, the vector e is given by: sin sin sin cos cos

                e (6)

Multiple reflections in a cavity

Once a light ray enters into a cavity, it may undergo one or several successive reflections on the panels. After each reflection, the direction of the ray is modified according to Snell's laws. However, in geometrical optics, it is classical to represent the image of the ray reflected by a mirror which is aligned with the incident ray, as shown on the left of Figure 3 through the example of two rays. By using this representation for the cavity, we can draw a straight line aligned with the incident ray, crossing the successive images of the panels: after a reflection on panel 1, the ray reaches the im-age of panel 2 (which forms an angle α with panel 1), then the image of panel 1 (which also forms an angle α with the image of panel 2, and so on).

The number of reflections depends on both orientation and position of the ray. This is visible in Figure 3 where the two rays are parallel (thus characterized by the same vector e) and strike panel 2 in different positions: one ray (represented in red) undergoes 4 reflections, whereas the other ray undergoes 3 reflections. The ray light paths in broken straight lines are featured on the right of the figure, in a projection onto the (y0z) plane of the 3D scene represented on the left of the figure. In this plane, the projection of vector e, denoted as  e , is:,

sin cos             e with   arctan tan cos      (7)

Number of reflections

The number of reflections according to the orientation and position of the ray is computed according to the following geometrical considerations, in the (yOz) plane.

The orientation of the ray is denoted by the angle   given by Eq. ( 7). Its position is described by the point P where the ray meets the line (AB) which joins the extremities of the panels in the (yOz) plane, drawn in

 

, 0 q Q y  of the ray and the y-axis has a negative abscissa q y , panel 1 is met first, otherwise, panel 2 is met first. With some geometrical calculation, we find that abscissa q y is given by

  sin cos cos H G q G H y        ( 8
)
where the angles G  and H  are computed as follows.

G P G y                   (9)
After some calculation, Eq. ( 9) can be written

    2 sin cos cos sin G P y            ( 10
)
and by noticing that

/ 2 G       , we obtain   2 arcsin cos cos sin G P y                (11)
H P y                  (11)
Figure 4 illustrates the fact that H  is a reflex angle, i.e., higher than π, when the ray strikes first panel 1 ( 0 q y  ), and a salient angle, i.e., lower than π, when it strikes first panel 2 ( 0 q y  ). We may prefer using the angle H  , obtuse in any case, defined as: 2 when 0 when 0

H q H H q y y            (12)
Finally, the number of reflections occurring after the first reflection of the first panel met is the number of times angle / 2 H    contains α. Hence, the total number of reflections is given by

1 floor 1 2 H m            (13)
where symbol floor [.] gives the integral part of the number in argument.

3.4

Radiance attenuation for one ray Now that the number of light reflections has been determined, we can express the global attenuation undergone by the radiance, by multiplying the successive Fresnel reflectances

  12 i
R  corresponding to the different reflections. For each reflection, we need to compute the local incidence angle i  . This local incidence angle can be easily obtained through the dot product between vector e, which describes the direction of the ray, and the normal of the panel, or image of panel, on which the considered reflection occurs.

The panels have the normal vectors N1 and N2 given by Eq. ( 5). The local incident angle for the first reflection depends on whether the ray first meets panel 1 or panel 2, therefore on the sign of the parameter q y defined by Eq. ( 8):

    2 arccos if 0 arccos if 0 q i q y y            1 e N e N (14)
where the symbol "  " denotes the dot product

The following reflections, if any, occur on images of panels whose normal vector denoted as  

1 j N or   2 j
N if the first reflection occurs on panel 1, respectively on panel 2. These normal vectors, for 2 j  to the number of reflections m given by Eq. ( 13), are defined as

                    1 2 0 0 cos / 2 1 and cos / 2 1 sin / 2 1 sin / 2 1 j j j j j j                                        N N ( 15
)
and the local incident angle is given by

          1 2 arccos if 0 arccos if 0 j q j i j q y y            e N e N (16)
Finally, the global attenuation of the radiance according to its position P y between   sin / 2   and   sin / 2  and its orientation   ,   , is given by the reflectance:

    01 1 , , m j P i j R y R           (17)
where

  1 i

 denotes the local angle i

 for the first reflection given by Eq. ( 14). Notice that according to the Helmholtz reciprocity principle, a ray following the same path within the cavity but in opposite direction would undergo exactly the same attenuation. Hence,  

, , P R y   can denote the attenuation for the ray coming or exiting the cavity at the angle   ,   through the position P y .

Reflectance of the structured surface

From the reflectance attached to each incident ray within the cavity, we can derive the reflectance of the structured surface for a Lambertian illumination. It can be a hemispherical-directional reflectance, also called angular reflectance, being a function of the observation direction. Another interesting type is the bi-hemispherical reflectance. It is also possible to compute directional-hemispherical reflectance, equivalent to the hemispherical directional reflectance in this case, thanks to the reversibility of light principle.

Angular reflectance

Let us consider that the cavity is illuminated over a band of width Δx along the x axis, perpendicular to the cavity, i.e. illuminated along the y axis, by collimated light from a direction   ,   . The illumination is uniform, i.e., same radiance i L arrives in each point of the band, which receives a uniform irradiance . The different elementary fluxes are reflected in various directions according to the panel that each one meets first and the number of reflections. By collecting the whole reflected flux, in practice with a measurement device equipped with an integrating sphere, the captured flux FR is given by

cos i i i i E L    (
      sin 2 sin 2 , p r i P y F xE R y dy         (19)
The directional-hemispherical reflectance of the band, and by extension to the whole structured surface, associated with this orientation of the incident light, is therefore:

          sin 2 sin 2 1 , , , 2 sin / 2 p r P y i F R R y dy F            (20)
Again, according to the Helmholtz reciprocity principle, the angular function   , R   given by Eq. ( 20) also corresponds to the hemispherical-directional reflectance function of the structured surface when it is illuminated by Lambertian light over the hemisphere (same radiance Li comes from every direction) and observed in the direction   ,   . Notice that since the specular reflections on the panels do not modify the geometrical extent of the rays, the radiance r L perceived in one direction   , r r   is:

    , , r r r r r i L R L      (21)
It is possible to display the reflectance given by Eq. ( 20) according to the observation direction on a 2D map thanks to the Lambert azimuthal equal area projection. To every direction   ,   corresponds a point   , u v within a disk of radius 2 whose coordinates are given by:

    2 sin /2 cos 2 sin /2 sin u v            (21)
The advantage of this transformation is that it conserves the areas by mapping a portion of the hemisphere of a given area, into a portion of the disk with same area.

In Figure 5, we present the results given by Eq. ( 20) for two different materials, for aperture angle values of 45, 60, 90, 120, 150° and 180°. One material is dielectric, with a refraction index of 1.

 = 45°  =60°  =90°  =120°  =150°  =180°
Copper under a LED lighting (visible spectrum from 400 to 700 nm)

 = 45°  = 60°  = 90°  = 120°  = 150°  = 180°
For the dielectric material, we can see that the reflectance is globally very weak, except at high incidence angles (periphery of the graphs) when the cavity aperture angle is large. This is coherent with the angular variations of the Fresnel formulae. The highest angular reflectances peaks are located near the zones where the azimuthal angle 2    , i.e. when the incident plane contains the x-axis. We can also see that the radiance gradients have some discontinuities, correspond to the directions at which the number of reflections within the cavity is incremented by one. For example, on the map attached to an aperture of 120°, a central area is lighter than the rest of the graph: it corresponds to rays undergoing one reflection, whereas in the rest of the graphs, rays undergo two reflections.

It is even more visible in the case of the copper. This material being more reflective, the reflected light appears more sliced into specific areas. It is also important to notice the saturation of the color increasing when the aperture angle decreases, as also shown in the case of diffusing surfaces in [START_REF] Saint-Pierre | Light Interreflections and Shadowing Effects in a Lambertian V-Cavity under Diffuse Illumination[END_REF].

Bi-hemispherical reflectance

Now, we want to investigate the bi-hemispherical reflectance of the V-cavity and the influence of the surface structure (aperture angle α).

The bi-hemispherical reflectance corresponds to a uniform illumination over the hemisphere (Lambertian illumination, characterized by a constant radiance Li from every direction), and a capture of the whole reflected light over the hemisphere. It is obtained by integrating over the hemisphere the angular reflectance studied previously, as follows.

The irradiance on the structured surface is related to the radiance Li by: 2 2 0 0 cos sin

i i i E L d d L              (22)
and the incident flux on a band of the   2sin / 2 x   area cavity is:

  2 sin / 2 i i F xE    (23)
The exitance is the sum of the reflected radiances expressed by Eq. ( 21):

  2 2 0 0 , cos sin r r r r r r r r r M L d d                (24)
where

  , r r r i L R L   
is the radiance reflected by the cavity according to the reflectance defined by Eq. [START_REF] Saint-Pierre | Light Interreflections and Shadowing Effects in a Lambertian V-Cavity under Diffuse Illumination[END_REF].

Finally, the bi-hemispherical reflectance is given by:  

2 2 0 0 1 , cos sin r r r r r r r r r i i M R L d d E L                  (25)
which yields, according to Eqs. ( 20) and ( 21),

        sin 2 2 2 0 0 sin 2 1 , , cos sin 2 sin 2 r r p r r P r r r r y R R y dy d d                      
Using Eq. ( 26), we computed the bi-hemispherical reflectances for various aperture angles of the specular V-cavity, for the dielectric material previously studied, and for a cavity of silver at 550 nm (n = 0.1249 + i3.3391). The values are presented in Table 1. The bi-hemispherical values in Table 1 confirm the tendencies drawn by the angular reflectance maps. As the aperture angle gets smaller, the reflectance is lower and the structured surface has a darker appearance, which is due to the increase of the number of light reflections in the cavities, each reflection introducing a radiance attenuation. It is illustrated by Table 2, where we computed, in the case of a cavity of silver at 550 nm (n = 0.1249 + i3.3391) with an aperture of 45°, the bi-hemispherical reflectance by taking into account only 1 light reflections, then adding the paths of the light where a second reflection happens, then a third one, up to the maximum number of 4 light reflections possible in this structure. We observe from these results that if we take into account only one or two light reflections, as it is often done in light scattering models by rough metallic surfaces, we underestimate the reflectance. The error is sensible in the case of media with high refractive index, like metals. It is also visible through Figure 6, where the angular reflectance for silver at 550 nm with an aperture of 45° is represented in the cases where we only consider one reflection of the light, or all the possible reflections. In order to obtain a more precise prediction of the appearance, especially in the case of a small aperture angle in the concavities of the surface topography, it is necessary to compute it with a sufficient number of light reflections, as also shown recently in the domain of computer graphics [START_REF] Lee | Practical multiple scattering for rough surfaces[END_REF][START_REF] Xie | Multiple scattering from distributions of specular v-grooves[END_REF].

Conclusions

In this paper, we analyzed the interreflections happening in a structured surface made of parallel, specular V-cavities under a Lambertian illumination. We proposed a model taking into account the exact number of light reflections occurring in the structures, in order to accurately predict the reflectance according to the observation angle. We saw that the material and the angle of the cavity have a strong impact on the interreflections and the reflectance of the concave surfaces, in particular because of the number of light reflections. We also showed that it is crucial to model correctly the number of light reflections happening in surfaces presenting concavities with small aperture angle, as it has a great influence on the final visual appearance. This constitutes an extension for the modelization of the light being reflected by complex surfaces, to better predict the visual appearance of given surfaces. It could be combined in the future with a model predicting the interreflections in similar cavities made of a Lambertian material, in order to predict the appearance of a diffusing material presenting a structured interface with air.

Figure 1 -

 1 Figure 1 -Structured surface with parallel and periodical V-shaped ridges of aperture angle α.

Figure 2 -

 2 Figure 2 -3D geometry of one cavity, and vector e representing the direction of illumination.The normal of panels 1 and 2 are respectively:

Figure 3 -

 3 Figure 3 -Left: 3D representation of two parallel light rays oriented according to a same vector e striking the cavity in different positions on panel 2. Right: 2D representation of the two same light rays projected onto the (yOz) vertical plane. The light path can be represented by a straight line meeting the successive images of the panels from each other. The projection of real light paths in broken straight lines is also represented on the figure (left).

Figure 4 .

 4 This point P has the coordinates two examples for the same position P y but two different orientations of the ray. On the left of the figure, the ray strikes first panel 1, on the right of the figure, it strikes first panel 2. The panel first met is determined by the following condition: if the meeting point

Figure 4 -

 4 Figure 4 -Geometry for the calculation of the number of reflections, for a same position yP of the ray, and two different orientations. Since



  denotes the small solid angle of illumination. Since the illuminated area is   2sin / 2 x   , the incident flux on the band is  each elementary area within the band, centered around the position and of size

5 .

 5 Its spectral reflectances are converted first in CIE 1931 XYZ tristimulus values then into L*a*b* color values, for a better visualization. The other one is made of copper, with tabulated values for the refractive index in the visi-ble spectrum of light (400 -700 nm), the spectral reflectances being converted in CIE 1931 XYZ tristimulus values, and then into sRGB color values.

Figure 5 -

 5 Figure 5 -Maps of hemispherical-directional reflectance (in %) for cavities of dielectric material, and color maps for cavities of copper, obtained with different aperture angles of cavities, represented with the Lambert azimuthal equal area projection.

Figure 6 -

 6 Figure 6 -Maps of angular reflectance for a 45° V-cavity made of silver, at 550 nm (n = 0.1249 + i3.3391), represented with the Lambert azimuthal equal area projection, where only one reflection (left) and all the possible reflections (right) of rays in the cavity are rendered.

Table 1 -

 1 Bi-hemispherical reflectances for various cavity aperture angles.

	Aperture angle	45°	60°	90°	120°	150°	180°
	Silver	0.87	0.91	0.94	0.95	0.96	0.97
	Dielectric (n = 1.5)	0.01	0.02	0.04	0.05	0.08	0.09

Table 2 -

 2 Bi-hemispherical reflectance of a 45° V-cavity made of silver at 550 nm

	Maximal number of light reflections	Bi-hemispherical reflectance
	1	0.14
	2	0.40
	3	0.70
	4	0.97