Birefringence Bench: Double Refraction, Walk-off and Interference Colours

What is modelled

A uniaxial crystal (calcite, quartz or rutile) has two refractive indices: n_o for light polarised perpendicular to the optic axis and n_e for polarisation along it. A beam entering a plate is split into an ordinary wave, which obeys Snell's law, and an extraordinary wave, whose index depends on the direction of travel. The bench shows both beams in three dimensions: the turntable sets the angle of incidence, the yellow arrow is the optic axis, and the screen shows the two spots (laser) or the interference colour of the overlap (white light) between polariser P1 and analyser P2.

Equations

• Ordinary wave: sin θi = no sin φo. Extraordinary wave: 1/n(γ)2 = cos2γ/no2 + sin2γ/ne2 with γ the angle between the wave normal and the optic axis; the tangential part of n k is continuous, so n(γ) sin φe = sin θi is solved numerically (bisection) in full vector form, so any axis direction works.
• Walk-off: the energy flows along the Poynting vector s, perpendicular to E = D⊥/no2 + D∥/ne2, not along the wave normal. Measured from the axis, tan θs = (no/ne)2 tan θk, and the largest angle between them is atan[(no2 − ne2)/(2 no ne)], 6.26 deg for calcite at an axis tilt of 41.8 deg.
• Both beams leave a parallel-faced plate parallel to the incident beam. Their phase difference is Γ = (2π/λ) d (kz,e − kz,o) with kz = n cos φ the normal component of n k; at normal incidence with the axis in the face this is 2π d (ne − no)/λ.
• Between a polariser along p and an analyser along q: I = Ao2 + Ae2 + 2 Ao Ae cos Γ, Ao = (p·eo)(q·eo), Ae = (p·ee)(q·ee). Crossed at 45 deg to the axis this is sin2(Γ/2). In white light the intensity is summed over 400 to 700 nm with an approximate spectral colour table.

Simplifications

• Fresnel reflection and absorption are ignored (all light enters the plate); refractive indices are fixed numbers for sodium light with no dispersion, so the colour sequence is only qualitatively that of a real Michel-Levy chart.
• The plate is treated as infinite and parallel-faced; the Blender rhomb is a visual stand-in whose large faces are the physical entry and exit faces. Beams are treated as thin rays; in the laser mode the two spots do not interfere because they are displaced, in white mode the overlap patch (wide beam) is shown.
• The polarisation vectors of the two waves are rotated back into the plane transverse to the incident beam and renormalised, which is exact at normal incidence and a small approximation otherwise.
• The wavelength-independent retardation of a thick plate is millions of nanometres, so for plates thicker than a few micrometres the white-light colour washes out to near white, as in a real thick crystal; the colour graph therefore stops at 3000 nm.