Germanium is widely used for infrared lens design, especially in the 8-12 micrometer wavelength spectrum. Excellent correction of aberrations can result due to germanium’s high refractive index (4.0) and the aspheric surfaces possible using the diamond turning process.

However, germanium lenses often exhibit troublesome thermal defocus. This is mainly due to its large change in refractive index with temperature changes. The first order optics analysis shown in the following viewgraph copy provides insight into the problem.

The focal length of a plano-convex lens varies with its radius of curvature R and the refractive index N of the material as shown in the first equation shown above.

The change in focal length due to changes in radius of and refractive index are determined from the partial derivatives of the lens focal length equation.

The change in radius with temperature is calculated as the product of the thermal coefficient of expansion of the lens material, α, and the temperature difference from the temperature at start to end of the thermal excursion., dT.

Similarly, the change in refractive index is calculated as the product of the change in refractive index with temperature, dN/dt, and the temperature range, dT

These calculated changes in radius and temperature are then used to express the change in lens focal length with temperature as shown in the last equation.The two terms in brackets give the relative impacts of focus change due to the thermal expansion coefficient, α, and the index rate of change, dN/dT. Inserting approximate values for these, as shown in the red-outlined box to the right, reveals that the effect of the index temperature rate of change far exceeds that of the thermal expansion coefficient, by a factor of 44x.