Supplement 1.8: Lambert emitters, cosine emitters      (1/4)

 

A special type of anisotropic emitter is the cosine emitter or Lambert emitter, named after the mathematician and physicist Johann Heinrich Lambert (1728–1777). Among other things, he dealt with the subject of photometry in his publication Photometria sive de mensura et gradibus luminis, which can be read here in German translation (last access: 26.05.2025).

The radiant intensity of cosine emitters decreases with increasing angle ϑ in the range from 0 to $\pm \pi /2$ according to $I(\vartheta )=I(0)cos\vartheta$. This is plausible: as the angle ϑ increases, a small area appears smaller in perspective in the same way as the cosine, as the graphic shows. A matt glass pane (‘frosted glass’) illuminated evenly from behind fulfils the conditions of a cosine emitter. Emitters do not always behave in this way: their radiant intensity can be concentrated more towards the front or distributed more widely.

Light-emitting diodes (LEDs) with flat housings can be cosine emitters. Their data sheet shows a circle as the directional characteristic, as in the diagram on the right. Lens-like housing shapes can be used to achieve a narrower or wider emission.

This property also characterises reflective surfaces. A surface painted with matt paint can be a good cosine reflector in the spectral range in which the paint reflects: when illuminated, it also reflects as shown on the right. This is referred to as diffuse reflection.

For scientific purposes, pure Teflon (Spectralon^®) is used as a cosine reflector. Since it is also almost absorption-free from ultraviolet to near-infrared and therefore ideally white, it serves as a reflection standard, for example in the colour analysis of paints.

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