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 apparent size of an area a decreases with increasing angle ϑ by cos ϑ.

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.

The length of an arrow extending from the emitter in the direction ϑ changes with cos ϑ. The arrowheads then lie on a circle.

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.

Further examples

Clouds

In the visible spectral range, clouds are cosine reflectors, provided they are ‘optically dense’. In physical optics, this term means that there is no way for photons to pass through the cloud without being scattered multiple times by water droplets. ‘Optically dense’ does not therefore mean opaque. Rather, the photons diffuse through the cloud and leave it again after many scattering events. The direction of the solar radiation can no longer be detected. The photons scattered back into space are also spatially diffuse.

Earth from space in the visible spectral range, captured by the Meteosat Third Generation Imager-1 (MTG-I1) on 18 March 2023. The image shows optically dense white cloud fields that intensely scatter sunlight, but also partially transparent haze over the tropical East Atlantic and Indian Ocean.
Source: ESA Applications.

An evening sky with clouds. Location and time of photograph: Oldenburg, Germany, 5 April 2008, 7 p.m.

Dark cumulonimbus clouds can also be cosine scatterers: their droplet density is so high that a great deal of light is lost through absorption from the irradiated upper surface to the lower surface. The photo also shows white cumulus clouds, which are cosine scatterers. Higher, partially transparent clouds are coloured reddish by the evening sunlight and do not exhibit good cosine characteristics.