Calcite, Quartz & Sapphire Properties
Walk-off tan ρ = (nₒ² − nₑ²) sin 2θ / (2(nₑ² cos²θ + nₒ² sin²θ)), with nₒ and nₑ from the crystal's Sellmeier fit; quartz rotation ρ = kλ²/(λ² − λ₀²)².
Sellmeier fit: G. Ghosh, Opt. Commun. 163, 95 (1999), doi:10.1016/S0030-4018(99)00091-7; 204–2172 nm, room temperature.
nₒ
1.65836
nₑ
1.48614
Δn = nₑ − nₒ
-0.17222
nₑ(θ)
1.56519
index of the e wave at θ
Walk-off angle ρ
6.233°
e ray tilted away from the optic axis
o–e separation after 10 mm
1.0921 mm
L·tan ρ, normal incidence
Largest walk-off
6.270°
at θ = 41.87° = atan(nₑ/nₒ)
Dispersion
Walk-off at 589 nm
Calcite (CaCO₃): data sheet
| Transmission | 0.3–2.3 µm |
| dn/dT | dn_o/dT = 3, dn_e/dT = 13 ×10⁻⁶/K at 500 nm |
| Thermal conductivity | 5.526 ∥ c, 4.646 ⊥ c W/(m·K) |
| Thermal expansion | 25 ∥ c, -5.8 ⊥ c ×10⁻⁶/K |
| Knoop hardness | 155 (Mohs 3) kg/mm² |
| Young's modulus | 88.19 ∥ c, 72.35 ⊥ c GPa |
| Density | 2.71 g/cm³ |
| Specific heat | 852 J/(kg·K) |
| Melting point | 825 °C (decomposes) |
Crystran data sheets, room temperature; ∥ and ⊥ are relative to the c (optic) axis as Crystran labels them.
Transmission bands
Model
1/nₑ(θ)² = cos²θ/nₒ² + sin²θ/nₑ², tan ρ = (nₒ² − nₑ²) sin 2θ / (2(nₑ² cos²θ + nₒ² sin²θ))
nₒ and nₑ come from the published Sellmeier fits (Ghosh for calcite and quartz, Malitson & Dodge for sapphire), inside their fitted ranges. θ is the angle between the wave normal and the optic axis; ρ is the angle between the extraordinary ray (energy flow) and its wave normal (Born & Wolf, ch. 15). For a beam at normal incidence on a plate cut at θ the o and e beams leave parallel, separated by L·tan ρ: the principle of a calcite beam displacer.
Quartz optical rotation ρ = kλ²/(λ² − λ₀²)², k = 7.186 °·µm²/mm, λ₀ = 0.0926 µm (Chandrasekhar 1952), applies to light along the optic axis. It gives 21.75 °/mm at 589.3 nm; Lowry & Coode-Adams measured 21.72. Off the axis the linear birefringence soon dominates.
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
Indices from published Sellmeier fits (exact for the fit, inside its range); walk-off from the uniaxial index ellipsoid (exact). Quartz rotation is Chandrasekhar's one-oscillator fit to Lowry & Coode-Adams' measurements, 0.2 % high at 589 nm. Thermal and mechanical values are single room-temperature data-sheet values.
- 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
- 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., Handbook of Laser Science and Technology, vol. IV, part 2 (CRC Press, 1986).
- Chandrasekhar S. (1952). The optical rotatory power of quartz and its variation with temperature. Proc. Indian Acad. Sci. A 35, 103–113. doi:10.1007/BF03172225
- Lowry T. M., Coode-Adams W. R. C. (1927). Optical rotatory dispersion. Part III: the rotatory dispersion of quartz in the infra-red, visible and ultra-violet regions of the spectrum. Phil. Trans. R. Soc. A 226, 391–466. doi:10.1098/rsta.1927.0010
- Born M., Wolf E. (1999). Principles of Optics, 7th ed. Cambridge University Press. Ch. 15, optics of crystals.
- Crystran. Handbook of Infrared and Ultraviolet Optical Materials, data sheets: transmission range, dn/dT, thermal conductivity and expansion, hardness, density, specific heat, Young's modulus. crystran.com