Conoscopic Observation

Simulate conoscopic interference figures (isochromates and isogyres) for uniaxial and biaxial crystals between crossed polarizers.

Model: Textbook approximation

cos²V_z = (n_β⁻² − n_γ⁻²)/(n_α⁻² − n_γ⁻²), 2V = acute optic axial angle

each pupil point ν = NA·r: k_z = √(n(s)² − ν²) per eigenwave, Γ = k₀d(k_z,slow − k_z,fast)

crossed polars: T = sin²2(ψ − P) sin²(Γ/2); parallel: T = 1 − sin²2(ψ − P) sin²(Γ/2)

Light

Polars

CrystalWavelength (nm)Plate thickness d (µm)

Tilt from the plate normal0 °

0–60 °

Stage rotation0 °

0–360 °

Conoscope NA0.85

0.1–1.3

Sellmeier fit: G. Ghosh, Opt. Commun. 163, 95 (1999), doi:10.1016/S0030-4018(99)00091-7; 204–2172 nm, room temperature.

Tilt is that of the optic axis (uniaxial) or the acute bisectrix (biaxial); the stage rotation turns the tilt direction from x.

Optic sign

Negative (−)

n_α / n_β / n_γ = 1.4861 / 1.6584 / 1.6584

Acute optic axial angle 2V

0° (uniaxial)

the two optic axes coincide

Birefringence n_γ − n_α

0.1722

Largest retardance in the field

6.35 µm

10.8 orders at 589 nm; NA 0.85

Conoscopic figure, crossed polars, 589 nm

Isochromes are lines of equal retardance; the dark isogyres are where the slow axis is parallel to a polarizer.

Polarizer P along x (E–W); the figure spans ±NA.

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

Plane-wave eigenmodes of a uniaxial or biaxial plate per pupil point (Born & Wolf ch. 15), ideal polars, fringes averaged per pixel; white light scales R by Δn(λ). Not modelled: Fresnel and lens rotation, 2V dispersion.

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
Veiras F. E., Perez L. I., Garea M. T. (2010). Phase shift formulas in uniaxial media: an application to waveplates. Appl. Opt. 49, 2769. doi:10.1364/AO.49.002769
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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