Polariscope

The Polariscope shows what a gem polariscope shows when one of our seven gemstones turns between two polarizing filters: dark or light, blinking or not, and, through the conoscope, its interference figure and optic sign. Everything is computed by Jones calculus across the visible spectrum from the stone’s refractive indices and measured absorption, and turned into color with the same CIE colorimetry as the Dichroscope. It is a teaching model of the instrument. It does not measure or identify any real stone.

The instrument

A polariscope is a light under a polarizing filter (the polarizer), a stage for the stone, and a second filter above it (the analyzer). Crossed, the two filters pass no light at all: the polarizer leaves the light vibrating one way, and the analyzer passes only the vibration at right angles to it. A singly refractive stone passes the light on unchanged, so it stays dark however it is turned. A doubly refractive stone splits the light into two rays vibrating at right angles, which the analyzer partly passes, so the stone lights up, except at four positions per turn where its vibration directions line up with the filters. With the filters parallel (the open position) the field is light instead.

A strain-free glass sphere (the conoscope) set on the stone gathers light from many directions at once, so the eyepiece shows an interference figure: each point is one direction through the stone. A first-order red plate (the compensator) slid in above the stone shows which of the two rays is the slower, the stone’s optic sign.

The model

The set-up

Daylight (CIE D65) through an ideal polarizer, vibrating up and down the view; the stone; in the conoscope, the red plate if it is in; then the analyzer, across the view when crossed and up and down when parallel. The page’s white is the open polariscope with no stone.

Directions

In the eyepiece’s frame, x to the right, y up and z toward the eye. The optic axis, tilted θ from the line of sight and turned φ clockwise, is c = (sin θ sin φ, sin θ cos φ, cos θ): at φ = 0 its shadow lies along the polarizer. For light going along k, the o-ray (ordinary) vibrates along c × k and the e-ray (extraordinary) along k × o, at an angle χ in the field.

The indices

At an angle A to the optic axis the e-ray’s index is
n_e(A) = (cos²A / n_o² + sin²A / n_e²)^−½,
so over a path L the two rays differ by (n_e(A) − n_o) L, a retardance γ = 2π (n_e(A) − n_o) L / λ. Straight down the axis (A = 0) the difference is zero. The indices are the 3D Gemstone Simulator’s, the same numbers the Refractometer shows: their difference is the birefringence (case P4).

Absorption

As in the Dichroscope: the o-ray takes α_o, the e-ray α_o cos²A + α_e sin²A, from the shared spectral registry, so a ruby glows red, not white.

Jones calculus

At each of 81 wavelengths, 380 to 780 nm, the stone is an absorber (amplitudes exp(−αL/2) along o and e) followed by a retarder
R = cos(Δ/2) I − i sin(Δ/2) (n · σ),
Δ = √(γ² + κ²) and n = (γ cos 2χ, γ sin 2χ, κ) / Δ on the Pauli matrices: a plain linear retarder when κ = 0. The red plate is a linear retarder of 550 nm with its slow direction at 45°, northeast to southwest. The light reaching the eye is |A · Plate · R · D · P|². Between crossed polarizers, without absorption or plate, this is the familiar I = sin²2φ sin²(γ/2).

Quartz

Amethyst is quartz, which also turns the light’s plane of vibration as it runs along the optic axis (optical activity), by ρ(λ) = 7.19 λ² / (λ² − 0.0926283²)² degrees per mm (λ in µm), 21.8° per mm at 589 nm (Chandrasekhar 1957). Off the axis we take ρ cos²A: the circular retardance is κ = 2ρ cos²A L. It is why a slab of quartz cut across its axis stays light, in color, between crossed polarizers, and why its figure is a bull’s-eye with no cross at the center.

High-order white

A gem a few millimeters thick puts its two rays many wavelengths apart: a ruby across its axis, 8 × 10⁻³ in birefringence over 5.8 mm, by about 46 µm, some 80 wavelengths of green light. Across a band only 5 nm wide the phase then turns several times, and sampling it at one wavelength per band would invent colors that are not there. The intensity is exactly a + b cos Δ + c sin Δ in the stone’s phase, so the model takes the band’s exact average, a + (b cos Δ + c sin Δ) sinc(w/2), with w the phase’s spread across the band (and, in the conoscope, across one pixel). The result is high-order white tinted by the stone’s own color, as a real polariscope shows; interference colors come only from small path differences, close to the optic axis or in the conoscope. Case P8 checks the bands against plain Jones matrices every 0.1 nm.

The path through the stone

The light takes one straight path, through the middle of the stone you choose, along the line of sight. Its length is measured on the simulator’s model of that shape at its real size: looking straight down the optic axis (the table’s normal, as in the simulator) it is the stone’s depth, 5.03 mm for the T73 (case P11); tilted, it is the stone’s thickness along the new line of sight. In the conoscope, a direction at an angle inside the stone runs a longer path, as through a slab of the same thickness.

The conoscope and the red plate

Each point of the figure, at a distance r from the center of the view, is a direction at sin ψ = r sin 40° in air, refracted into the stone. The 40° cone is a working choice, not a measured sphere. The figure’s center (the melatope) is the optic axis; tilted, it moves off center, and past 14° (moissanite) to 25° (amethyst) it leaves the view. The figure is drawn on the GPU (WebGL2) by the same equations as above, line for line; without WebGL2 the page computes it on the CPU at a lower resolution. The two agree to the last of 256 levels at every point we checked.

With the red plate in, the page reads the sign as a gemologist does. Close to the center, where the stone’s own path difference is about 200 nm, it compares the northeast and northwest quadrants: where the stone’s slow ray lines up with the plate’s slow direction, northeast to southwest, the two add, and the color climbs to blue; where they cross, it falls to yellow. Blue in the northeast means the slow ray is the e-ray, which vibrates out from the center: positive. Blue in the northwest means it is the o-ray: negative. The page reads this from the interference colors alone, the stone’s own color left out, because a deep color such as emerald’s can all but hide them on screen (case P5).

The readings

The bold reading describes a full turn, worked out at every 5°. Between crossed polarizers the stone “stays dark” when its lightest position stays under Y = 0.003 (the open polariscope being 1), and it “blinks” when its darkest position is under half its lightest. A singly refractive stone stays dark at every tilt; a doubly refractive one stays dark only straight down its optic axis, which is the reading that tells you to tilt it and look again. These thresholds are our own choice.

Strain, illustrative

Many diamonds grew under stress, which leaves them weakly doubly refractive in patches (anomalous double refraction): between crossed polarizers a strained diamond shows patchy gray light instead of staying dark. With Strain on, the diamond carries two smooth random fields fixed to the stone: a small path difference γ, up to 140 nm, and the direction χ of the slower ray, which wanders through every angle. Each point is a thin linear retarder, so between crossed polarizers it passes sin²2(χ + φ) sin²(πγ / λ) of the light, and when the polarizers are parallel one minus that. Each patch goes dark at its own turn, so the stone as a whole never does: over a turn its average light stays between 0.020 and 0.023 of the open polariscope. The same strain pointing one way would blink like a doubly refractive stone, from 0.002 to 0.35 (case P12). That is what the reading teaches: patchy light that never goes dark all at once is strain, not double refraction. The pattern, its size and its strength are invented to show the effect, not measured from any stone, and the page labels it so.

What the model leaves out

• A real faceted stone mixes many paths, reflected inside it, so its light between crossed polarizers is patchy; here the light takes one straight path, so the whole stone shows one color.
• Strain in any stone but diamond, and any measured strain at all: the diamond’s pattern is invented to show the effect.
• Aggregates, which stay light at every turn, and biaxial stones (two optic axes, a different figure): none of the seven is one.
• The birefringence’s own change with wavelength (its dispersion), the Fresnel losses at the surfaces, and the walk-off of the e-ray: they change the light by a few percent, not what you see.
• Twinning in amethyst, which can turn its bull’s-eye into an Airy spiral; the figure here is of one untwinned crystal.
• Real polarizers, which pass a little light when crossed and tint it slightly.
• Your screen. Most phone and laptop screens give out polarized light, so a real polariscope held against one shows the screen’s own light, not the stone’s.

Validation

These cases run on the page’s own data each time the site is built, and the build stops if one fails.

Sources

We are not affiliated with any of these publishers.

• R.C. Jones, “A new calculus for the treatment of optical systems”, Journal of the Optical Society of America 31, 488 to 493, 1941.
• S. Chandrasekhar, “The optical rotatory dispersion of quartz”, Proceedings of the Indian Academy of Sciences, 1957; T.M. Lowry and W.R.C. Coode-Adams, Philosophical Transactions of the Royal Society, 1927.
• G.R. Rossman, Mineral Spectroscopy Server, California Institute of Technology, and E.V. Dubinsky, J. Stone-Sundberg and J.L. Emmett, Gems & Gemology 56(1), 2020: the absorption spectra, as on the Dichroscope page.
• 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).
• IEC 61966-2-1:1999, the sRGB color space.