Peridot Optics Lab (2D): Refractive Index, Dispersion and Brilliance of a Cut Gem

Presets

Live readouts

Graphs

Light returned and leaked against pavilion angle (all other settings fixed)

Refractive index against wavelength (Cauchy fit)

Where the returned light goes: exit-angle histogram

Numerical checks

running...

How to use

Choose a preset or move the sliders and watch the coloured rays. White light is traced as seven wavelengths; the violet rays bend most. Rays that make it back up through the crown are drawn in colour, rays that leak out of the pavilion are drawn thin. Try: (1) Too shallow: light escapes through the bottom, the green "returned" curve drops; (2) Diamond: the critical angle shrinks and the colours fan out more; (3) tilt the beam and see which facets take over; (4) sweep the pavilion angle in the graph and find the best compromise between returned light and the 35 degree viewing cone; (5) set the dispersion to zero and the fire spread falls to zero. Space pauses, R resets.

The science

What this lab shows. Why a cut gem sparkles. Light enters the top of the stone and is refracted; if it meets the pavilion facets beyond the critical angle it is totally internally reflected, crosses the stone a second time and leaves through the crown towards the eye. Different colours bend by slightly different amounts (dispersion), which splits white light into the coloured "fire". The lab traces rays in the central cross-section of a round brilliant, with every ray splitting at each surface into a reflected and a transmitted part.

1. Snell's law and the critical angle.

n₁ sin θ₁ = n₂ sin θ₂, θ_c = asin(1/n)

For peridot (gem olivine, n_α about 1.654 to n_γ about 1.690, mean about 1.672) θ_c is about 36.7 degrees, and for quartz-like amethyst (n about 1.55) it would be 40.2 degrees; for diamond (n = 2.417) it is only 24.4 degrees, which is why diamond traps light so well. A lower critical angle means more rays are totally reflected for the same cut.

2. Fresnel reflection. Even when a ray is not totally reflected, part of it reflects. For unpolarised light with c_i = cosθ_i and c_t = cosθ_t:

R = (1/2) [ ((n₁c_i - n₂c_t)/(n₁c_i + n₂c_t))² + ((n₁c_t - n₂c_i)/(n₁c_t + n₂c_i))² ], R(0) = ((n-1)/(n+1))²

Rays are split and followed until their weight falls below 0.4 % or they exit. Absorption inside the stone is ignored (peridot's olive green comes from Fe²⁺ in the olivine lattice and would add a transmission loss, strongest in the blue).

3. Dispersion (Cauchy). n(λ) = A + B/λ², with B fitted so that n(430 nm) - n(687 nm) equals the dispersion slider (about 0.020 for peridot; 0.044 for diamond by the usual B-G convention) and A so that n(589 nm) is the chosen index. Seven wavelengths from violet to red are traced separately. The "fire" figure is the spread of the mean exit angle of upward-leaving light across the spectrum.

4. Cut proportions. The profile is built from the table width, the crown angle, the girdle thickness and the pavilion angle; crown height = (1/2 - table/2) tan(crown) and pavilion depth = (1/2) tan(pavilion). The graph sweeps the pavilion angle at fixed other settings: too shallow and light leaks out of the bottom (a "window"), too deep and the light comes back but leaves the crown at steep angles away from the viewer. Around 40 to 41 degrees is a sensible compromise for a stone of this index (peridot's higher index tolerates slightly shallower pavilions than quartz) (the standard round brilliant uses 40.75 degrees with a 34.5 degree crown).

Checks (run in the page). (1) a ray entering the stone at 35 degrees refracts to asin(sin35/n) exactly; (2) at the critical angle +0.2 degrees nothing escapes through a flat face and at -0.2 degrees most of the light does; (3) the normal-incidence Fresnel reflectance equals ((n-1)/(n+1))²; (4) a ray through a 60 degree prism at symmetric incidence is deviated by 2 asin(n sin(A/2)) - A.

Simplifications. A real brilliant has 57 facets and a three-dimensional ray path; this lab uses the 2D meridian section (skew rays are not included), so brilliance values are a teaching proxy and not a grading. Peridot is biaxial and strongly birefringent (n_α 1.654, n_β 1.670, n_γ 1.690; doubling of the back facets is visible in real stones) and is represented here by a single index, dispersion by the Cauchy two-term form, and polarisation only through the averaged Fresnel formula. The 3D page rotates the same profile into an eight-fold gem for appearance; its light paths are those of the section.