Thermal Conductivity for Optics
Room-temperature thermal conductivity of window materials (Crystran) and what it means for an absorbing optic: thermal resistance, face-to-face and beam centre-to-edge temperature rise, and the dn/dT thermal lens.
Choosing a material fills its room-temperature conductivity and dn/dT from the Crystran data sheet (IR grades transmit to 3 µm.). Edit either for another material or temperature.
Thermal Resistance d/(kA)
72.46 K/W
face to face, heat flowing through the thickness
Face-to-Face ΔT
72.46 K
P d/(kA): P conducted through a slab of area A; heat absorbed at one face and removed at the other; for bulk absorption cooled at one face the drop is half
Heat Flux
1.000 W/cm²
P/A
Beam Centre-to-Edge ΔT
5.766 K
P/(4πkd): P absorbed uniformly along d inside A, radial heat flow
Thermal Lens f (dn/dT only)
21.40 m
2kA/(P dn/dT), no stress or end-face terms
Room-Temperature Data (Crystran)
| Material | k (W/(m·K)) | dn/dT (×10⁻⁶ /K) |
|---|---|---|
| Diamond | 2600 | 9.6 |
| Silicon | 163 | 160 |
| Germanium | 58.62 | 396 |
| Sapphire (Al₂O₃) | 27.21 | 13.1 |
| ZnS (multispectral) | 27.2 | 38.7 |
| MgF₂ | 21 | 1.7 |
| ZnSe | 18 | 61 |
| BaF₂ | 11.72 | -15.2 |
| LiF | 11.3 | -16 |
| CaF₂ | 9.71 | -10.6 |
| NaCl | 6.5 | -40.83 |
| Crystal quartz | 6.2 | -6.5 |
| KBr | 4.816 | -40.8 |
| Fused silica | 1.38 | 12.9 |
| N-BK7 | 1.114 | — |
| CsI | 1.1 | -99.3 |
| KRS-5 (TlBr–TlI) | 0.544 | -235 |
| AMTIR-1 (Ge₃₃As₁₂Se₅₅) | 0.25 | 71 |
Model
ΔT = P d/(kA) (Fourier's law through a slab)
T(r) = T(0) − Q r²/(4k), Q = P/(A d): ΔT₀ = P/(4πkd)
f = (kA/P)(½ dn/dT)⁻¹ = 2kA/(P dn/dT)
Steady state with a constant k at room temperature. The lens is Koechner's result for a uniformly heated rod with radial heat flow; inside a top-hat beam the curvature depends only on Q/k, so it holds in a wider window too. A slab cooled through both faces has twice the lens power. Not modelled: the stress-optic and end-bulging lens terms, a Gaussian heat profile, axial heat flow in a thin window, and k(T), which rises steeply in crystals below room temperature.
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
Steady state, constant room-temperature k (Si: Crystran's 273 K value). ΔT = Pd/(kA) through a slab absorbing at one face (half that for uniform bulk absorption); P/(4πkd) beam centre to edge for uniform heating with radial flow; dn/dT lens f = 2kA/(P dn/dT) (Koechner's rod). Stress-optic and end-bulging lens terms, Gaussian heating, axial flow and k(T) are not modelled.
- Koechner W. (2006). Solid-State Laser Engineering, 6th ed. Springer. Ch. 7: temperature profile and thermal lens of a uniformly heated rod. doi:10.1007/0-387-29338-8
- Crystran. Handbook of Infrared and Ultraviolet Optical Materials, data sheets: thermal conductivity and dn/dT. crystran.com
- Glassbrenner C. J., Slack G. A. (1964). Thermal conductivity of silicon and germanium from 3 °K to the melting point. Phys. Rev. 134, A1058–A1069. doi:10.1103/PhysRev.134.A1058