The Inuence of LightPath Length on the Color of Flame-Fusion Synthetic Ruby

Corresponding Author Abstract Colorimetric studies of different light path lengths from a new perspective of UV - Vis spectroscopy. The corrected ultraviolet - visible light spectrum was used to calculate the color of flame - fusion synthetic ruby, and the influence of light path length and standard light source on the color of flame - fusion synthetic ruby was studied. The results show that the L * , C * , h° under the A light source are higher than those under the D65 light source. In the ultraviolet - visible light spectrum, the strong absorption band of Cr 3+ at 545nm is the main cause of the color of the ruby. As the length of the light path increases, the absorption peak area at 545nm also increases, the lightness L * decreases, and the hue angle h° increases. The chroma C * first increases as the length of the light path increases, and then begins to decrease under the influence of the continuous decrease in lightness. The color difference ΔE *ab reaches the maximum when the light path length is around 10mm, and the standard light source has the greatest influence on the color difference ΔE *ab . As the length of the light path continues to increase, the influence of the standard light source on the color difference ΔE


Introduction
Ruby is a kind of corundum with a beautiful bright red color. It is widely distributed all over the world, such as Myanmar, Thailand, Sri Lanka, Tanzania, etc. When Al in corundum is replaced by various elements such as Cr, Fe, Ti, and V, the corundum will show various colors. The color of ruby is related to Cr 3+ . And as the replacement of Al 3+ by Cr 3+ increases, the color will change from light pink to red 1,2 .flame-fusion synthetic ruby is the most common type of synthetic ruby. Compared with natural ruby, it has a pure texture and is more suitable for chromaticity research.
There are many ways to evaluate the color of gems.The most common is the Cape series diamond color grading. The grader compares the diamond to be graded with a standard colorimetric stone to determine the color grade of the diamond 3,4,5 . However, in addition to diamonds, it is difficult to determine the color grade of other gemstones in this way, and because it is artificially graded, there may be large errors.The CIE standard colorimetric system can effectively solve this problem, of which the CIE1931-XYZ system and the CIE1976L * a * b uniform color space are the most widely used. Stockton 6 first used the Gem ColorMaster instrument to quantify the color of peridot in the CIE1931 color space.But the CIE1931-XYZ system is a non-uniform color space and cannot describe the color of gems well.The CIE1976L * a * b uniform color space is modified based on the CIE1964 uniform color space, and it is widely used in gem color evaluation 7,8,9,10,11,12 .
The main instrument currently used to measure the color of gemstones is a spectrophotometer.The portable spectrophotometer is used to measure the color of perdiot 13,14,15 , amethyst 16 , turquoise 17 , rubellite 18 , and green chrysoprase 19,20 .Color i5 and GemDialogue color cards are used to quantitatively describe the color of jadeite 21 .In addition, computer vision systems can also be used to measure the color of jadeite 22 .Because the portable spectrophotometer is a closed system, it cannot effectively reflect the pleochroism of some gemstones.The use of ultraviolet-visible spectrophotometer can effectively solve this problem 23 .Compared with the portable spectrophotometer, the UV-Vis spectrophotometer measures color more objectively and accurately.The UV-Vis spectrophotometer calculates the colors of leaves 24 and flowers 25 , garnets 26,27 , synthetic alexandrite 28 , dyed purple opal 29 , etc. by using CIE1931RGB and CIEXYZ color matching functions.
The human retina has three kinds of color photoreceptor cells, namely cone cells, that are sensitive to red, green, and blue light.S cones detect short wavelength (blue), M cones detects medium wavelength (green), L cones detect long-wavelength (red).When exposed to radiation,the spectral stimulus energy is absorbed by photoreceptors of the three cones.The cone cells produce different degrees of neurophysiological reactions.The International Commission on Illumination (CIE) has established a series of color matching functions through visual experiments. As a proxy for the cone response function, the CIE color-matching functions are used to express the linear combination of the average visual response 30 .The matching function can be used to calculate the energy of light that enters the human eye and produces the color perception.Sun 28 discussed the influence of different path lengths on the color of synthetic Cr-bearing chrysoberyl by calculating the color matching function.In addition, pleochroism is also a factor that needs to be considered when studying the color of gems 31 .
Two fully polished synthetic ruby cuboids R and Ru which have been oriented by an X-ray crystal orientation instrument are selected.Size is 5.89mm×5.97mm× 9.29mm and 5.46mm× 5.41mm × 8.9mm.The sample is shown in Figure 1. The height is the c-axis, and the surface is perpendicular to the optical axis. The length and width of the gemstone are selected as the a-axis and b-axis.The a, b, and c axes of the two samples are numbered respectively Roa, Rob, Rec, Ruoa, Ruob, Ruec.This paper mainly uses the color matching function to quantitatively characterize the color, and studies the influence of the light path length and the light source on the color of the ruby.

UV-Vis spectral analysis
The UV-Vis spectrum of synthetic ruby is shown in Figure 2.It can be seen that there are obvious absorption peaks at 693.50nm, 669nm, and 659nm, as well as broad absorption bands centered at 545nm and 416nm.The absorption peak at 693.50nm is the fluorescence emission peak caused by Cr 3+ , which is caused by 2 E→ 4 A2 of Cr 3+ , which is the reason for the strong fluorescence of synthetic ruby.The two weak absorption peaks at 669nm and 659nm are the 4 A2→ 2 T1 transition caused by Cr 3+ .The absorption peak near 581nm is related to the charge transfer of Fe 2+ -Ti 4+ . The absorption peak at 528nm is related to the 2 D splitting of the Ti 3+ spectrum item 32,33 .The absorption peaks at 545nm and 416nm are the 4 A2→ 4 T2 transition caused by Cr 3+ , which absorb yellow-green light, allowing red light and a small amount of blue-violet light to pass through, forming the color of synthetic ruby.

Correcting the UV-Vis Spectra
When light passes through the sample, energy is lost in three ways.A is the total absorbance of the sample measured directly from the spectrophotometer, including Ac (absorbance contribution of the absorber), Arl (absorption caused by light reflection at the boundary), Aisl (absorption caused by scattering of internal inclusions) 26,28 .
There are many different methods to correct the baseline. For example, Sun 26 subtracted the absorption spectrum at 800 nm to obtain a corrected spectrum.In this study, the Sellmeier equation is used to eliminate Arl and to correct the baseline of the ultraviolet-visible spectrum.At the same time because the sample is synthetic ruby with pure texture and no inclusions, Aisl does not affect the UV-Vis spectrum.Thus, the corrected UV-Vis spectrum is obtained.
The absorption (Arl) of light reflected at the boundary is related to the refractive index (n) of the sample.The Sellmeier equation is an empirical formula that describes the refractive index and wavelength in a specific transparent medium and is used to determine the dispersion of light in the medium.Different materials have different Sellmeier coefficients.According to the research of Malitson 32 , we get the following Sellmeier formula: Where n is the refractive index, λis the wavelength, and B1,2,3 and C1,2,3 are different Sellmeier coefficients.For corundum, there are the following Sellmeier equations in the direction of the corundum o light and the direction of the e light respectively: n 2 (λ) = 1 + (4) It is assumed that in the ideal state when unpolarized light is incident perpendicularly to the surface of the gemstone, a certain length of optical path will not be generated inside.In this case, the light absorption inside the gem can be ignored. Therefore, the transmittance through the surface of the gemstone can be expressed as: According to Lambert Beer's law, transmittance can be converted to absorbance: Where A is the absorbance, T is the transmittance, k is the molar absorbance coefficient, c is the concentration of the absorbing substance, and b is the optical path of the light.
R is the reflectivity, n0=1 is the refractive index of light in the air, n1 is the refractive index of light in the corundum, T is the transmittance, and A is the absorbance produced by single boundary reflection. Arl is the absorbance produced by the reflection of the two boundaries. Accurate visible spectroscopic measurements rely on correct calibration of the spectral baseline. Figure 3 shows the spectrum of baseline correction performed by the Sellmeier equation.By calculating the corrected ultraviolet-visible light spectrum, the ultraviolet-visible light spectrums corresponding to the light path length of 1 to 25mm are obtained,the color matching function can be used to calculate the tristimulus value XYZ29,30,and color space conversion can be used to obtain the color parameters L * , a * and b * in the CIE 1976 L * a * b * uniform color space 35,36 .The detailed conversion steps are detailed in the method. The synthetic ruby color parameters of the light path length from 1mm to 10mm are shown in Table 1.    The CIE XYZ color space is based on the uniform distribution of color perception, and colors will not be scattered in the Cartesian coordinate space,The calculated Euclidean distances between various color coordinates cannot be reasonably compared with each other.In the CIE 1976 L * a * b * color space, a * and b * represent horizontal and vertical Cartesian coordinate axes，L * represents the displacement perpendicular to the circular a * -b * grid.Different combinations of a * and b * can reproduce different tones, and the position of the color coordinate along L * indicates the brightness of the color.In the CIE 1976 L * a * b * color space, the tristimulus value XYZ is non-linearly converted into color parameters. And calculate the chromaticity value C * and hue angle h° under D65 light source and A light source.Taking sample R as an example, the absorption spectrum of light passing through synthetic ruby along any crystal direction can be calculated by the following formula: Am= × R down a+ × R down b+ × R down c (11) 2 + 2 + 2 = 2 (12) Where Am is the absorption spectrum in either direction, R down a is Roa's UV-Vis absorption spectrum,R down b is Rob's UV-Vis absorption spectrum,R down c is Rec's UV-Vis absorption spectrum, x, y, z are the Cartesian coordinates of the ray path length, and the ray path length is constrained to lie on a sphere with a radius of r. Figure 5. Synthetic ruby chromaticity diagram between Roa, Rob, and Rec of different light path lengths. The left column is the D65 light source, and the right column is the A light source. (a) 1mm and 5mm chromaticity diagram (b) 10mm and 20mm chromaticity diagram As shown in Figure 5, the color of the synthetic ruby under D65 and A light source increases with the increase of the light path length, from the original light pink to purple-red and finally to deep red.When the light path length is 1mm and 20mm, the effect of different light sources on the color of ruby is not obvious.When the light path length is 5mm and 10mm, the color difference between the two light sources can be observed from the figure.Therefore, only when the length of the light path is in the appropriate range, that is, the length of the light path is not too large or too small, the light source will have a significant effect on the color of the ruby.
When the light path length is 5mm, the color difference between a-axis (o-light direction) and c-axis (e-light direction) of synthetic ruby is the most obvious.In addition, when the light path length is 1mm, the color of the e-light direction is light pink. The color in the o-light direction is pink, and the color lightness in the e-light direction is greater than in the o-light direction.But when the light path length is increased to 5mm, the color in the e-light direction is already darker than the o-light direction.It shows that when the length of the illumination path is from 1mm to 5mm, the change of the color lightness L* in the e-light direction with the change of the illumination path length is greater than that in the o-light direction.This has to do with the pleochroism of ruby.It can be seen from Figure 5 that the length of the light path and the light source will affect the color of the synthetic ruby, and the colors of the synthetic ruby o-light and e-light directions are also different, so let's quantitatively discuss their influence on the color parameters of the synthetic ruby.   Figure 6 shows whether it is D65 light source or A light source, when the light path length increases, the lightness will decrease, showing a negative correlation (R 2 =0.952).This is consistent with the perception of the human eye.The lightness under the A light source is higher than the lightness under the D65 light source.When the length of the light path increases from 1mm to 5mm, the lightness drops rapidly, indicating that when the length of the light path is very small, its change has a significant impact on the lightness.  Figure 7 shows that under D65 and A light sources, the chroma C * of synthetic rubies with different light path lengths increases first and then decreases. Under the D65 light source, when the light path length is less than 7mm, the chromaticity C * increases with the increase of the light path length, reaching the maximum value of 63.58; When the path length is greater than 7mm, the color saturation reaches saturation and begins to decrease. Under A light source, it is bounded by 10mm.

The Influence of Light Path Length and Light Source on the Color of Synthetic Ruby
In addition, the chroma values of Rec and Ruec under the two light sources are significantly lower than Roa and Ruoa, which is caused by the pleochroism of the ruby.But under the D65 light source, when the light path length is 10mm to 15mm, the chroma value is close.This is because Roa and Ruoa reached the highest chroma and began to decline at 7mm, while Rec and Ruec continued to increase with the increase of the light path length because of the small chroma, and did not start to decrease until 12mm.The color temperature of A light source is lower than that of D65 light source, so the chroma C * of Roa and Ruoa starts to decrease at 10mm, while Rec and Ruec start to decrease at 14mm.  Figure 8 shows that the hue angle h° increases with the increase of the light path length under the two light sources, and the increasing amplitude gradually decreases.The hue angle under A light source is generally higher than that under D65 light source.But under the D65 light source, the hue angle h° decreases when Ruec (e light direction) is 1mm to 3mm, and h° begins to increase after 3mm,this phenomenon does not appear under the A light source.This is because when the length of the light path increases, the color changes from pink to light pink, and the chromaticity coordinate a * increases significantly, resulting in a decrease in hue angle h° .The A light source has a yellow hue, making the hue angle h° greater than that under the D65 light source, so the hue angle will not decrease due to the increase in the length of the light path.

The Influence of Light Path Length and Pleochroism on the Color Difference ∆E * ab
The color of synthetic ruby is plotted in CIE 1976 L*a*b* color space.In the three-dimensional space, the line connecting the points under the two light sources represents the Euclidean distance, which is the color difference ΔE * ab.Project the three-dimensional L * a * b * color coordinates to the two-dimensional a * b * plane (color circle), and draw a connecting line between the D65 light source and the A light source.In addition, link the change in the length of the light path with the color difference, draw a connecting line for each path length, and draw them in the same plane.The left end of the connecting line represents the color coordinate under the D65 light source, and the right end represents the color coordinate under the A light source.As shown in Figure 9, take Roa and Rec as examples for drawing. Figure 9 Roa (o light direction) and Rec (e light direction) color difference ΔE * ab are distributed in two-dimensional a * b * plane, the left end of the connecting line is the color coordinate of D65 light source, and the right end is the color coordinate of light source A Roa and Rec are the o-light direction and e-light direction of synthetic ruby respectively.It can be seen from Figure 9 that as the length of the light path increases, the connecting lines are distributed in an upward trend in the two-dimensional a*b* plane.The color difference ∆E * ab in Roa and Rec both increases first and then decreases. When the light path length is 12mm, the color difference ΔE * ab reach the maximum, which are 14.85 and 15.06 respectively.In order to further explore the influence of the light path length on the color when the light source changes, the light path length is increased to 25m. As shown in Figure 10, increasing the light path length to 25mm, the color difference ∆E * ab continues to decrease, indicating that as the light path length increases, the influence of the light source on the color is weakened.By comparing Figure 10(a) and Figure 10(b), it is found that the color difference ∆E * ab changes of the two samples in the o-light and e-light directions are different, which is related to the color and pleochroism of the samples. Effect of UV-Vis Absorbance Peak Area on the Color of Synthetic Ruby Figure 11 The absorption peak area (X) at 545nm.Taking the sample Roa as an example, the point where the first derivative is equal to zero is selected as the starting point and the end point of the 545nm absorption peak range, and the absorption peak area is obtained by integrating from 473nm to 652nm.The top right is the ultraviolet-visible light absorption spectra of different light path lengths The strong absorption band at 545nm in the ultraviolet-visible spectrum is caused by Cr 3+ , which absorbs yellow-green light in visible light and transmits red light, which has an important influence on the color of ruby.By calculating the first derivative of the ultraviolet-visible spectrum, the points with zero derivative near 473nm and 652nm are determined as the starting and ending points, and the absorption peak area at 545nm is calculated (Figure 11). Select a part of the data of different 545nm absorption peak areas under D65 light source to plot 12.At the same time, it can be seen from Figure 11 that the 545nm absorption peak area is positively correlated with the light path length. Figure 12 shows that the hue angle h° is positively correlated with the peak area at 545nm. When the absorption peak area increases, the hue angle h° increases.The R 2 of the hue angle under the D65 light source is 0.996. h° changed from -11.86 (that is, 348.14) to 36.75.The lightness L * is negatively correlated with the peak area at 545nm. As the peak area increases, the ruby absorbs more visible light, and the lightness decreases.The chroma C * first increases and then decreases with the absorption peak area, which is related to the changes of L * and h°.When h° increases, L * decreases. When the length of the light path increases to a certain length, the decrease in brightness L * has a significant impact on chroma C * , making chroma C * begin to decrease.

Conclusion
In the ultraviolet-visible spectrum of synthetic ruby, Cr 3+ causes a strong absorption broadband at 545nm, which is the main cause of ruby color. The color parameters of synthetic rubies under different light sources are obtained through the ultraviolet-visible spectrum. The L * , C * , and h°of light source A are higher than those of light source D65, so the color appearance of synthetic rubies under light source A is more vivid.
When the light source is unchanged, the absorption peak area at 545nm increases with the increase of the light path length, the absorption of visible light increases, and the lightness L * decreases.When the light path length is 1mm to 5mm, its subtle changes will cause a significant decrease in lightness;The hue angle h° increases with the increase of the light path length, but under the D65 light source, when the light path length is 1 to 3mm, the Ruec color shifts red, causing the hue angle h° to decrease;As the length of the light path increases, the chroma C * increases, but when the length of the light path increases to a certain length, the decrease of the lightness L * causes the chroma to start to decrease. At the same time, because ruby has pleochroism, the chromaticity value C * of Roa and Ruoa is lower than that of Rec and Ruec.As the length of the light path increases, the chroma C * increases, but when the length of the light path increases to a certain length, the decrease of the lightness L * causes the chroma to start to decrease. At the same time, because ruby has pleochroism, the chromaticity value C * of Roa and Ruoa is lower than that of Rec and Ruec. The color difference ΔE * D65-A first increases and then decreases with the increase of the light path length. When the length of the light path reaches about 10mm, the color difference reaches its maximum value. At this time, the light source has the greatest influence on the color. When the length of the light path continues to increase, the influence of the light source on the color is weakened.
From the above conclusions, we can know that the UV-Vis spectrum can be used to calculate the color of synthetic rubies with different light path lengths. Therefore, we believe that this method can be widely used in other gems and transparent minerals, providing great help for quantitatively describing their colors.

UV-Vis Spectroscopy.
The UV-3600 UV-VIS spectrophotometer(Shimadzu,Tokyo,Japan) was used to carrry out the UV-Vis spectra.The test conditions were described as follows:the range of wavelength, 200-900nm;slit width 2nm;scanning speed medium;sampling interval 0.5s;scanning mode,single. CIE1931 XYZ Colour Matching Functions. The International Commission on Illumination proposed the CIEXYZ color system in 1931, and the tristimulus value XYZ can be obtained by matching the isoenergetic spectrum.The tristimulus value can calculate the color based on the spectrum collected from the surface of the object or the transmission through the object: ∫ S(λ)y ̅(λ) △ λ λ (16) S(λ) is the relative spectral power distribution of the observation light source.For non-luminous objects, φ(λ) is the product of the spectral transmittance T(λ) and the relative spectral power of the light sourceS(λ),expressed asT(λ)S(λ),or the product of the spectral reflectance R(λ) and the relative spectral power distribution of the light sourceS(λ), expressed asR(λ)S(λ).k is the naturalization coefficient. For non-luminous objects, the Y value of the selected standard illuminant was adjusted to 100 [31] [32].

Colour Space Conversion
In order to describe easily colors, the color tristimulus values in CIEXYZ are non-linearly converted to obtain the color parameters L*, a*, b* in the CIE1976L*a*b* color space system.The formula for conversion is as follows: * = 116( ) For D65 light source, Xn=95.04, Yn=100, Zn=108.88. For light source A, Xn=109.85, Yn=100, and Zn=35.58.Xn, Yn, Zn are the colorimetric data obtained from the CIE1931 standard colorimetric observer (2°). CIE1976 L * a * b * Colour System. The CIE1976L * a * b * color space is the most widely used in the field of colorimetry.The system consists of plane chromaticity axes a * and b * and vertical axis L * .a * stands for red, -a * stands for green; b * stands for yellow, -b * stands for blue.The chroma C * and hue angle h°can be calculated based on chromaticities a * and b * . * = √ * 2 + * 2 (20) h = arc tan b * a *

Data availability
The dataset for this study is available from the corresponding author upon reasonable request.