Birefringent Crystals
nₒ(λ) and nₑ(λ) from each crystal's published Sellmeier formula; Δn = nₑ − nₒ, and a plate of thickness d with the optic axis in its face retards by |Δn|·d/λ waves.
Sellmeier fit: G. Ghosh, Opt. Commun. 163, 95 (1999), doi:10.1016/S0030-4018(99)00091-7; 198–2053 nm, room temperature.
nₒ (ordinary)
1.54422
nₑ (extraordinary)
1.55332
Δn = nₑ − nₒ
0.00910
positive uniaxial
Retardance of 1 mm
15.451 waves
162.2° modulo one wave; optic axis in the plate face, normal incidence
Zero-order half-wave thickness
32.36 µm
λ/(2|Δn|); a quarter-wave plate is half as thick
Crystal quartz: nₒ and nₑ
Birefringence Δn, all crystals
All crystals at 589 nm
| Crystal | nₒ | nₑ | Δn | Fit range (nm) |
|---|---|---|---|---|
| Crystal quartz (α-SiO₂) | 1.5442 | 1.5533 | 0.0091 | 198–2053 |
| Calcite (CaCO₃) | 1.6584 | 1.4861 | -0.1722 | 204–2172 |
| Magnesium fluoride (MgF₂) | 1.3777 | 1.3895 | 0.0118 | 200–7000 |
| Sapphire (α-Al₂O₃) | 1.7681 | 1.7600 | -0.0081 | 200–5000 |
| Yttrium orthovanadate (YVO₄) | 1.9997 | 2.2270 | 0.2272 | 488–3390 |
| Lithium niobate (LiNbO₃, congruent) | 2.2998 | 2.2136 | -0.0862 | 400–5000 |
| Rutile (TiO₂) | 2.6131 | 2.9089 | 0.2958 | 430–1530 |
| β-barium borate (BBO) | 1.6706 | 1.5525 | -0.1182 | 220–1060 |
| Potassium dihydrogen phosphate (KDP) | 1.5094 | 1.4682 | -0.0411 | 214–1529 |
“—”: the wavelength is outside that crystal's fit range.
Model
Δn = nₑ − nₒ, retardance = |Δn|·d/λ waves, d(λ/2) = λ/(2|Δn|)
nₒ(λ) and nₑ(λ) come from each crystal's published Sellmeier formula (the same data sets as the polarization calculators), only inside the wavelength range it was fitted over and at the temperature of its measurements. Δn > 0 marks a positive uniaxial crystal (quartz, MgF₂, YVO₄, rutile), Δn < 0 a negative one. The retardance is for light at normal incidence on a plate whose optic axis lies in its face; off-axis cuts and tilted beams are on the Waveplate and Birefringent Polarizer calculators.
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
Exact for each published Sellmeier fit inside its wavelength range (outside it the page shows no value); the fits match measurements to about 1e-4 in n. Retardance assumes normal incidence with the optic axis in the plate face, at the fit's temperature.
- refractiveindex.info database (CC0). Sellmeier data sets, each with its original reference shown on the page. refractiveindex.info
- 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
- Eimerl D., Davis L., Velsko S., Graham E. K., Zalkin A. (1987). Optical, mechanical, and thermal properties of barium borate. J. Appl. Phys. 62, 1968–1983. doi:10.1063/1.339536
- Zernike F. (1964). Refractive indices of ammonium dihydrogen phosphate and potassium dihydrogen phosphate between 2000 Å and 1.5 µ. J. Opt. Soc. Am. 54, 1215–1220. doi:10.1364/JOSA.54.001215