Orthoscopic Observation

Model orthoscopic observation of birefringent samples with rotating stage. Calculate intensity vs rotation angle and interference colors.

Model: Textbook approximation

1/n(θ)² = cos²θ/n_o² + sin²θ/n_e², R(λ) = |n(θ_c) − n_o|(λ)·d

section at 45°: crossed T(λ) = sin²(πR/λ), parallel 1 − sin²(πR/λ); colour = D65-weighted spectrum → CIE 1931 → sRGB

Polars

Accessory

CrystalSection thickness d (µm)

Optic axis from normal θc90 °

0–90 °

Stage rotation ρ30 °

0–360 °

Sellmeier fit: G. Ghosh, Opt. Commun. 163, 95 (1999), doi:10.1016/S0030-4018(99)00091-7; 198–2053 nm, room temperature.

θc = 90° puts the optic axis in the section plane (maximum retardance); θc = 0° looks along the axis (none). ρ is the azimuth of the section's e-wave vibration direction from the polarizer (E–W).

Interference colour, crossed polars

R = 275 nm, order R/550 nm = 0.50 (at ρ = 30°)

n_o / n_e at 550 nm

1.5459 / 1.5551

positive crystal, Δn = 0.0092

Δn_eff at 550 nm

0.00916

|n(90°) − n_o|

Retardance R at 550 nm

274.8 nm

interference order 0.50

Extinction positions

ρ = 0°, 90°, 180°, 270°

slow axis parallel to a polarizer: T = 0 at any retardance

Transmission at 550 nm against stage rotation

Four extinctions per turn, where the section's slow axis is parallel to a polarizer.

Michel-Lévy colours, crossed polars, section at 45°

0550 nm (order 1)1100 nm (order 2)1650 nm (order 3)

Marker: this section at R = 275 nm. Each strip colour comes from the full spectrum, not from a table.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Normal incidence on a uniaxial section: Δn(θc) from the index ellipsoid, Sellmeier dispersion, Jones chain with ideal polars and a 550 nm red plate; D65, CIE 1931 2°, clipped sRGB. No reflections.

Born M., Wolf E. (1999). Principles of Optics, 7th ed., ch. 15 Optics of crystals, pp. 790–852. Cambridge University Press. doi:10.1017/CBO9781139644181.024
Sørensen B. E. (2013). A revised Michel-Lévy interference colour chart based on first-principles calculations. Eur. J. Mineral. 25(1), 5–10. doi:10.1127/0935-1221/2013/0025-2252
CIE (2019). CIE 1931 colour-matching functions, 2 degree observer (CIE dataset); sRGB display colours per IEC 61966-2-1:1999. doi:10.25039/CIE.DS.xvudnb9b
Ghosh G. (1999). Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals. Opt. Commun. 163, 95–102. doi:10.1016/S0030-4018(99)00091-7
Dodge M. J. (1984). Refractive properties of magnesium fluoride. Appl. Opt. 23, 1980–1985. doi:10.1364/AO.23.001980
Malitson I. H., Dodge M. J. (1972). Refractive index and birefringence of synthetic sapphire. J. Opt. Soc. Am. 62, 1405 (abstract); formula in Dodge M. J. (1986), Handbook of Laser Science and Technology vol. IV, part 2, p. 30, CRC Press.
Lomheim T. S., DeShazer L. G. (1978). Optical-absorption intensities of trivalent neodymium in the uniaxial crystal yttrium orthovanadate. J. Appl. Phys. 49, 5517–5522: YVO₄ dispersion formula. doi:10.1063/1.324471
Zelmon D. E., Small D. L., Jundt D. (1997). Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol% magnesium oxide-doped lithium niobate. J. Opt. Soc. Am. B 14, 3319–3322. doi:10.1364/JOSAB.14.003319
DeVore J. R. (1951). Refractive indices of rutile and sphalerite. J. Opt. Soc. Am. 41, 416–419. doi:10.1364/JOSA.41.000416

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