Double Refraction (Birefringence)
Calculate ordinary and extraordinary ray paths, walk-off angle, lateral separation, and retardation in uniaxial crystals.
1/n(θ)² = cos²θ/no² + sin²θ/ne² (θ: wave normal to optic axis)
tan θS = (no²/ne²) tan θ, ρ = θS − θ (ρ > 0: ray turned away from the axis)
kx = sin θi conserved; e wave: k⊥²/ne² + k∥²/no² = 1
Γ = (2πd/λ)(kz,e − kz,o)
Sellmeier fit: G. Ghosh, Opt. Commun. 163, 95 (1999), doi:10.1016/S0030-4018(99)00091-7; 204–2172 nm, room temperature.
The optic axis lies in the plane of incidence, tilted by the axis angle from the surface normal toward the incoming beam's side (+x). Both beams leave the plate parallel to the incident beam.
no / ne
1.65836 / 1.48614
Δn = -0.17222 (negative uniaxial)
e wave
n = 1.56519
wave normal 45.00° from the optic axis
Walk-off (e ray − e wave normal)
-6.23°
largest: 6.27° at θ = 41.87°
Refraction angles (from the normal)
o ray 0.00°
e wave normal 0.00°
e ray -6.23°
Beam separation after the plate
1.0921 mm
Retardance (e relative to o)
-1581.906 waves
-569486.3°; negative: the e wave leads
e index and walk-off against the wave-normal angle to the axis
Separation and retardance against the angle of incidence
Retardance against wavelength (this plate and geometry)
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Plane waves into a lossless uniaxial plate, optic axis in the plane of incidence: e wave on the index ellipsoid, ray along its normal, Γ = k₀d(k_z,e − k_z,o). Not modelled: reflections, optical activity.
- 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
- 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