Gold’s red shift: colorimetry of multiple reflections in grooves

Abstract

The color of fine gold shows paradoxical variations that have tentatively been explained by metallurgic factors. Measurements and digital photographs show a significantly redder color than predicted by theory. A novel purely optical explanation based on the multiple reflection in grooves is suggested. The analysis in the colorimetric space RGB of the photographs of several fine gold samples and coins confirms that the gold red shift comes from minute grooves that seem black but that in fact have an extremely dark orange/red color.

Appendix

Fresnel coefficient and V-grooves

The Fresnel coefficients are given by the formulas:

$$ {F}_{\lambda}^s={\left|\frac{\cos \theta -\sqrt{\epsilon_1\left(\lambda \right)-{\sin}^2\theta +i{\varepsilon}_2\left(\lambda \right)}}{\cos \theta +\sqrt{\epsilon_1\left(\lambda \right)-{\sin}^2\theta +i{\varepsilon}_2\left(\lambda \right)}}\right|}^2\mathrm{and}\ {F}_{\lambda}^p={\left|\frac{-\left({\upvarepsilon}_1\left(\uplambda \right)+\mathrm{i}{\varepsilon}_2\left(\uplambda \right)\right)\cos \theta +\sqrt{\epsilon_1\left(\lambda \right)-{\sin}^2\theta +i{\varepsilon}_2\left(\lambda \right)}}{\left({\upvarepsilon}_1\left(\uplambda \right)+\mathrm{i}{\varepsilon}_2\left(\uplambda \right)\right)\cos \theta +\sqrt{\epsilon_1\left(\lambda \right)-{\sin}^2\theta +i{\varepsilon}_2\left(\lambda \right)}}\right|}^2 $$

The ray trajectory in a parallel V-groove with angle ω is given by its successive azimuths v_i (0 = north, counted positive clockwise) after each reflection: First, the ray enters the groove at angle θ which corresponds to azimuth v₁ = π + θ. To enter the groove, the angle must satisfy θ < α and θ > α − ω. Then it hits the first facet at an angle f₁ with the facet’s normal (n₁) and is then reflected at the same angle according to the law of reflection. According to its new azimuth v_2, it can either emerge from the groove or hit the other facet and so on. The ray’s azimuth after the ith reflexion is v_i + 1 and the normal n_i depends on which facet is hit: ⁹${n}_i=\pi -\alpha +\frac{\omega }{2}+{\left(-1\right)}^i\left(\frac{\pi }{2}+\frac{\omega }{2}\right)$. v_i is the incoming ray, the outgoing ray’s azimuth is v_i + 1 = π + 2 n_i − v_i. It makes an angle with the facet’s normal f_i = n_i − v_i + π. It emerges from the groove if 2π − α ≤ v_i + 1 < ω − α, otherwise it bounces against a facet again.

$$ \left\{\begin{array}{c}{v}_{k+1}=\frac{\omega }{2}-\alpha -{\left(-1\right)}^k\left(\frac{\omega }{2}-\alpha - k\omega -{v}_1\right)\\ {}{f}_{k+1}=\left|\pi -{\left(-1\right)}^k\left( k\omega +\alpha +{v}_1-\frac{\pi }{2}\right)\right|\kern1.5em \end{array}\right. $$

The ray always ends up emerging after a certain number of reflections that depend on the initial conditions (θ) and the groove geometry (ω and α). Its final azimuth is v_m + 1 if m is the total number of reflections. When considering the groove as an optical system, it does not generally follow the law of reflection. A ray in a groove follows the law of reflection if and only if the condition v₁ + v_m + 1 = π is satisfied. This condition is important for two reasons: on the one hand it maximizes the light received and on the other hand it allows a comparison to be made with the traditional case of a single reflection on a flat surface.

Two cases must be distinguished according to the parity of the number of reflections (m is odd or even): In the first case, when there is an even number of reflections (m = 2 k), the condition is $\omega =\frac{\pi +2}{m}$ with α − ω < θ < α. The groove angle can take only discrete values. For normal incidence (θ = 0), these authorized angles are $\frac{\pi }{2},\frac{\pi }{4},\frac{\pi }{6},\frac{\pi }{8},\dots$ For such discrete values, vertical rays are retroreflected and bounce back to their source. For other values, the ray is lost and the groove appears as black. When the the groove angle condition is satisfied: the color is modulated by the groove orientation (α). In the second case, when there is an odd number of reflections (m = 2 k + 1) the condition of emergence is independent of θ but depends on α: $\omega =\frac{\pi -2\alpha }{m-1}$ which implies $\frac{\pi }{m+1}<\omega <\frac{\pi }{m-1}$. For symmetrical grooves ($\alpha =\frac{\omega }{2}$), the condition becomes $\omega =\frac{\pi }{m}$. The color hence depends on θ and α as each configuration produces a specific color.

For retroreflecting rays, the intensity of the reflected light decreases exponentially with the number of reflections. Using the multiple-angle formula $\sin (nx)={2}^{n-1}{\prod}_{k=0}^{n-1}\sin \left(\frac{k\pi}{n}+x\right)$ the intensity becomes:¹⁰

If m even m = 2p

Hence $\omega =\frac{\pi }{m}$.

$$ {\displaystyle \begin{array}{c}{I}_m={I}_0\prod \limits_{i=1}^m\cos {f}_i={I}_0\prod \limits_{i=0}^{2p-1}\cos {f}_{i+1}={I}_0\prod \limits_{j=0}^{p-1}\cos {f}_{2j+1}\prod \limits_{j=0}^{p-1}\cos {f}_{2j+2}\\ {}={I}_0\prod \limits_{j=0}^{p-1}\sin \left(\alpha +{v}_1+\frac{j\pi}{p}\right)\prod \limits_{j=0}^{p-1}\sin \left(\omega +\alpha +{v}_1+\frac{j\pi}{p}\right)\kern3.5em \\ {}={I}_0\frac{1}{2^{m-2}}\sin \left(\alpha +\theta \right)\sin \left(\alpha +\theta +\frac{\pi }{m}\right)\kern10.75em \end{array}} $$

If m is odd m = 2p + 1 and symmetrical groove ($\alpha =\frac{\omega }{2}$): $\omega =\frac{\pi }{m}$ implying ${I}_m={I}_0\frac{1}{2^{m-2}}\sin \left(\frac{\pi }{m}+2\theta \right)\sin \left(\frac{3\pi }{2m}+\theta \right)$

The geometric factor is equal to 1 for symmetrical retroreflecting grooves (${H}_m^{\star }=1$) when $\omega =\frac{\pi }{m}$ with $\alpha =\frac{\omega }{2}$ otherwise

$$ {H}_m=\left\{\begin{array}{c}\frac{\sin \alpha +\sin \left(\frac{\pi }{2}+\alpha -2\omega \right)}{\sin \alpha +\sin \left(\omega -\alpha \right)},\kern0.5em m\ \mathrm{even}\\ {}\frac{\sin \alpha }{\sin \alpha +\sin \left(\omega -\alpha \right)}\kern1em ,\kern0.5em m\ \mathrm{odd}\end{array}\right. $$

Hence

$$ \left\{\begin{array}{c}\begin{array}{c}{f}_i^{\star }=\left|\left(k+1\right)\frac{\pi }{m}+\pi \left(\frac{1}{2}-{\left(-1\right)}^k\right)\right|\kern0.75em \\ {}{R}_i^{\star}\left(\lambda \right)=\frac{1}{2}{\prod}_{i=1}^m{F}_{\lambda}^s\left({f}_i^{\star}\right)+\frac{1}{2}{\prod}_{i=1}^m{F}_{\lambda}^p\left({f}_i^{\star}\right)\end{array}\\ {}\begin{array}{c}{H}_m^{\star }=1\kern14.25em \\ {}{I}_m^{\star }={I}_0\frac{1}{2^{m-2}}\sin \left(\frac{\pi }{m}\right)\sin \left(\frac{3\pi }{2m}\right)\kern2.5em \end{array}\end{array}\right. $$

Color space conversion

From sRGB to RGB¹¹:

$$ {\left(\begin{array}{c}R\\ {}G\\ {}B\end{array}\right)}_{D65}={\left(\begin{array}{c}\Gamma \left(\frac{R}{255}\right)\\ {}\Gamma \left(\frac{G}{255}\right)\\ {}\Gamma \left(\frac{RB}{255}\right)\end{array}\right)}_{D65}\kern-.5em \mathrm{with}\ \Gamma (C)=\left\{\begin{array}{c}\frac{C}{12.92}\kern2.5em ,\kern0.5em C\le 0.03928\\ {}{\left(\frac{C+0.055}{1.055}\right)}^{2.4},\kern0.5em C>0.03928\end{array}\right. $$

From RGB to XYZ

$$ {\left(\begin{array}{c}X\\ {}Y\\ {}Z\end{array}\right)}_{D65}=\left(\begin{array}{ccc}0.4124& 0.3576& 0.1805\\ {}0.2126& 0.7152& 0.0722\\ {}0.0193& 0.1192& 0.9505\end{array}\right)\times {\left(\begin{array}{c}R\\ {}G\\ {}B\end{array}\right)}_{D65} $$

From XYZ to xyY

$$ {\left(\begin{array}{c}X\\ {}Y\\ {}Z\end{array}\right)}_{D65}={\left(\begin{array}{c}\frac{X}{X+Y+Z}\\ {}\frac{Y}{X+Y+Z}\\ {}Y\end{array}\right)}_{D65} $$