The instrument
A calcite dichroscope is a short tube with a small rectangular window at the stone end, a cleaved rhomb of calcite (Iceland spar) with a glass prism either side to straighten the light path, and a lens at the eye. Calcite is strongly doubly refractive, so it shows two images of the window side by side, made of light vibrating at right angles to each other. Hold a stone close to the window with white light behind it, and each image shows the stone’s color for one direction of vibration. The type built with two polarizing filters is called the London dichroscope; its filters tint the colors, which is why calcite is preferred.
A doubly refractive stone splits light into two rays that vibrate at right angles. In a uniaxial stone the o-ray (ordinary ray) always vibrates across the optic axis, and the e-ray (extraordinary ray) in the plane of the axis and the light. When the stone absorbs the two differently, the two windows differ in color: the stone is dichroic (pleochroic, with two colors). A singly refractive stone does not split the light, so its windows always match.
The model
The model is the brief’s version 1: one straight path of d = 5 mm through a polished slab, in unpolarized daylight (CIE illuminant D65). The shape you choose changes only the drawing; the colors are always for the same 5 mm, so a longer path cannot darken one stone against another.
- Directions
- k is the line of sight, into the screen. c is the optic axis, tilted θ from k. The o-ray vibrates along o = c × k (normalized), the e-ray along e = k × o. Looking straight down the axis (c parallel to k) there is no split.
- Absorption
- αo(λ) for light vibrating across the axis, αe(λ) along it, as natural-log coefficients in cm−1. The e-ray at angle θ takes αe′ = αo cos2θ + αe sin2θ, and each ray passes T = exp(−s α d), with s, the saturation, at 1.
- The dichroscope
- Turned by φ, window 1 passes vibration along
p1 = cos φ u + sin φ v (u, v
the screen’s axes), window 2 at right angles. With cos2ψ = (p1 · o)2:
W1(λ) = ½ S(λ) [To cos2ψ + Te′ sin2ψ]
W2(λ) = ½ S(λ) [To sin2ψ + Te′ cos2ψ]
S is the illuminant. Unpolarized light splits into two unrelated (incoherent) waves, so their powers simply add: there is no interference term, whatever the stone’s retardation. - What follows
- Turned 45° from the stone’s vibration directions, each window takes half of each ray and the two match. Turned 90°, they swap. Down the axis, both rays are ordinary and the windows match at any turn. An isotropic stone (diamond, cubic zirconia) has αe = αo.
The page shows, for each window, its L*, C* and h° (CIELAB) and its share of each ray, cos2ψ and sin2ψ; between them, the CIEDE2000 color difference ΔE00 and its class.
The spectra
Optics belong to the species, color to the variety: ruby and blue sapphire share corundum’s refractive indices, but not their absorption. The indices come from the 3D Gemstone Simulator’s gemstone data. The absorption comes from a shared spectral registry of measured, polarized spectra, each on a 5 nm grid from 380 to 780 nm. A measured spectrum is decadic absorbance A for a sample of thickness t. The lowest absorbance over the file’s whole range, which runs into the near infrared where these stones barely absorb, is taken as the baseline (reflection losses and the instrument’s offset). The rest is averaged over each 5 nm bin and converted: α = ln(10) A / t. Saturation s = 1 is the sample’s own color content.
| Gemstone | o-ray (E ⊥ c) | e-ray (E ∥ c) | Status |
|---|---|---|---|
| Ruby | GIA’s published absolute Cr3+ cross-sections in corundum, both polarizations (Dubinsky, Stone-Sundberg & Emmett 2020), at 2,164 ppma Cr: the content that gives the measured E ⊥ c absorption of a natural ruby from Chanthaburi, Thailand (GRR 1843, 0.787 mm) at 560 nm | Fitted, both rays | |
| Blue sapphire | Medium blue sapphire GRR 1020a, 4.289 mm, both polarizations measured | Fitted, both rays | |
| Emerald | Emerald GRR 3570, Afghanistan, 0.859 mm, both polarizations measured | Fitted, both rays | |
| Amethyst | Amethyst from the Anahí Mine, Bolivia: a 6.17 mm slab cut on (0001), so the light travels along the axis and vibrates across it | Derived from the o-ray: the three color-center bands fitted, then each scaled by its published E ∥ c to E ⊥ c ratio: 1.06 near 545 nm, 1.92 near 410 nm, 1.40 near 357 nm (Hassan & Cohen 1974) | o-ray fitted, e-ray provisional |
| Diamond, cubic zirconia, moissanite | Colorless: no absorption in the visible | None needed | |
The measured files are G.R. Rossman’s, on the Mineral Spectroscopy Server at Caltech. Two assignments were checked before use. Emerald’s o-ray is the yellowish green and its e-ray the bluish green, from its polarized spectra (Wood & Nassau 1968; Webster 1955); a common cutting rule of thumb says the reverse. Amethyst’s e-ray absorbs more and is the more reddish (Haidinger, in Holden 1925), which the derived spectrum follows. Real amethyst is zoned and its color centers are slightly biaxial, so its two colors vary across a stone.
From spectrum to color
Each window’s spectrum is weighted by the CIE 1931 2° color-matching functions and summed over the 81 wavelengths, with D65 as the light. The result is normalized so an empty window (no stone) has Y = 1, and that empty window is the white point for CIELAB. L*, C* and h° and the color difference come from these values. ΔE00 is CIEDE2000, checked against all 34 test pairs of Sharma, Wu & Dalal (2005).
To draw the windows, the color goes from XYZ to Display P3 on screens that show it and to sRGB (the IEC 61966-2-1 matrix) elsewhere. Deep reds and greens like ruby’s and emerald’s lie outside sRGB; they are brought in by lowering chroma at constant lightness and hue until they fit, and the page says when it has done so and by how much. The figures always come from the true color, before that step.
How strong
The class names the color difference across the optic axis, with the dichroscope lined up, through 5 mm in daylight: under 1 is none, 1 to 5 weak, 5 to 16.5 distinct, over 16.5 strong. They were set so the seven gemstones land where the trade puts them, and then frozen: sapphire 41.6 and ruby 17.9 strong, emerald 15.3 distinct, amethyst 3.3 weak, diamond, cubic zirconia and moissanite 0. Ruby and emerald sit close, so the line between distinct and strong falls halfway between them. The page also names the class for whatever tilt and turn you set, so it reads none at 45°.
What the model leaves out
- The e-ray rule assumes weak absorption and low birefringence; both hold for the colored stones here.
- Walk-off and image doubling are ignored: they move the image, not its color.
- Quartz’s optical activity is ignored; it is negligible in unpolarized light.
- Reflection losses at the surfaces are left out; they dim both windows almost equally.
- The light takes one straight path. Real faceted stones mix paths, and their face-up color mixes both rays.
- Color zoning, a stone’s own color depth and the path length are fixed: one sample per variety, 5 mm.
- Biaxial stones, which can show three colors (tanzanite or iolite, for example), are not among the seven.
- Light other than daylight. Most phone and laptop screens give out polarized light: a real dichroscope held against one shows two different windows even with no stone, so never test one against a screen.
Validation
The brief’s cases, run on the page’s own data each time the site is built. The thin-limit case (V7) asks for 1 µm; this ruby absorbs 75 cm−1 at 400 nm, so 1 µm really does tint white light slightly (ΔE00 0.19). It is run at 0.1 µm, with the check that the difference falls in proportion to the path. Checked by hand as well: turning the eyepiece 90° visibly swaps the windows, the ring stops at 0°, 45° and 90°, and the six lesson links in the brief open their exact view.
| Case | What it checks | Result | Needs | Outcome |
|---|---|---|---|---|
| V1 | Isotropic null: diamond and cubic zirconia, 1,000 random orientations and turns each | largest ΔE00 0.0e+0 | ΔE00 < 0.01 | Pass |
| V2 | Optic-axis null: every doubly refractive stone within 0.5° of its axis, any turn | largest ΔE00 0.0014 | ΔE00 < 0.05 | Pass |
| V3 | The 45° null: across the axis, turned 45°, window 1 equals window 2 | largest difference 1.4e-14 | within 1e-9 | Pass |
| V4 | Swap: window 1 at ψ equals window 2 at ψ + 90° | largest difference 1.4e-14 | within 1e-9 | Pass |
| V5 | Energy: window 1 + window 2 is the same at every turn | largest change 3.0e-16 | within 1e-9 | Pass |
| V6 | Monotonic: lined up, the spectral difference never falls as the stone tilts from 0° to 90° | holds for all four dichroic stones | all dichroic stones | Pass |
| V7 | Thin limit: 0.1 µm of each stone, both windows against the empty window, and ten times the difference at 1 µm | largest ΔE00 0.0191 at 0.1 µm, 0.19 at 1 µm | ΔE00 < 0.1 at 0.1 µm, in proportion to the path | Pass |
| V8 | Colorless and doubly refractive: moissanite, 1,000 random orientations | largest ΔE00 0.0e+0 | ΔE00 < 0.01 | Pass |
| V9 | Hue order across the axis, lined up: ruby e-ray toward orange; sapphire o-ray toward violet, e-ray toward green; emerald e-ray bluer; amethyst e-ray more reddish | ruby 353° to 14°, sapphire 266° and 139°, emerald 161° to 173°, amethyst 327° to 343° (o to e) | all hold | Pass |
| V10 | Strength classes: each stone across its axis, lined up, in its class in the trade | diamond 0.0 none; moissanite 0.0 none; cubic zirconia 0.0 none; ruby 17.9 strong; sapphire 41.6 strong; emerald 15.3 distinct; amethyst 3.3 weak | none < 1 ≤ weak < 5 ≤ distinct < 16.5 ≤ strong | Pass |
| V11 | Pipeline: the empty window in sRGB; CIEDE2000 against all 34 Sharma, Wu & Dalal (2005) pairs | white (1.0000, 1.0001, 0.9999); largest ΔE00 error 4.9e-5 | ±0.005; within 1e-4 | Pass |
| V12 | Units: every measured dataset round-trips absorbance to α and back, its thickness recorded | 5 datasets, largest error 2.2e-16 | within 1e-9 | Pass |
Sources
We are not affiliated with any of these publishers.
- G.R. Rossman, Mineral Spectroscopy Server, California Institute of Technology: polarized visible spectra of corundum (GRR 1843, GRR 1020a), beryl (GRR 3570) and quartz (amethyst, Anahí Mine).
- E.V. Dubinsky, J. Stone-Sundberg and J.L. Emmett, Gems & Gemology 56(1), 2 to 28, 2020, and its absorption cross-section data file.
- D.L. Wood and K. Nassau, “The characterization of beryl and emerald by visible and infrared absorption spectroscopy”, American Mineralogist 53, 777 to 800, 1968.
- R. Webster, Journal of Gemmology 5(4), 1955.
- F. Hassan and A.J. Cohen, American Mineralogist 59, 1974; A.J. Cohen, “Color centers in the α-quartz called amethyst”, American Mineralogist 41, 874 to 891, 1956.
- E.F. Holden, American Mineralogist 10, 1925, citing W. Haidinger, 1854.
- “An introduction to pleochroism in faceted gems”, Gems & Gemology, Fall 2014.
- J.W. Anthony and others, Handbook of Mineralogy: beryl and corundum.
- CIE, “Colour-matching functions of CIE 1931 standard colorimetric observer” (CIE 018:2019, DOI 10.25039/CIE.DS.xvudnb9b) and “CIE standard illuminant D65” (DOI 10.25039/CIE.DS.hjfjmt59).
- G. Sharma, W. Wu and E.N. Dalal, “The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations”, Color Research & Application 30(1), 21 to 30, 2005.
- IEC 61966-2-1:1999, the sRGB color space.
The Dichroscope shows what a calcite dichroscope shows when you look through one of our seven gemstones in daylight: the two windows side by side, as you tilt the stone and turn the dichroscope. The window colors are computed from measured polarized absorption spectra with CIE colorimetry. Nothing is picked by hand: every number on the page comes from an equation below or from a documented data source. It is a teaching model of the instrument. It does not measure or identify any real stone.