# Lambert's law

Angular dependence of the radiation intensity in a Lambert radiator. : Angle to the surface normal; I: radiant intensity; S: source or reflection surface
${\ displaystyle \ theta}$

The Lambert's law (also Lambertian cosine law ) describes formulated by Johann Heinrich Lambert , as indicated by the perspective effect decreases the radiation intensity with a flat werdendem angle. If a surface follows Lambert's law and the radiance of the surface is constant, the result is a circular distribution of the radiant intensity .

Since man with his eye only the luminance evaluated (the luminance is the photometric correspondence of the beam density), such a Lambertian material Lambert's law appears regardless of the viewing direction as equally bright. Applies to every surface element of a light source , so it is called Lambert Spotlights . In particular, an ideal black body is a Lambert radiator. If light is reflected from a surface according to Lambert's law, then one speaks of ideal diffuse reflection .

## Mathematical description

Let the angle be against the surface normal and the size of the Lambertian surface element, then the radiation intensity is proportional to the product of the cosine of the angle and the area: ${\ displaystyle \ theta}$${\ displaystyle A}$${\ displaystyle I (\ theta)}$

${\ displaystyle I (\ theta) \ sim A \ cos (\ theta)}$

The ratio of radiation intensity and reduced area (projected in the viewing direction), the proportionality factor , is the constant radiance of the area: ${\ displaystyle L}$

${\ displaystyle L = {\ frac {I (\ theta)} {A \ cos (\ theta)}}}$

The dashed line of the radiation intensity in the picture on the right complies with the equation ( -axis horizontal, -axis vertical). ${\ displaystyle x ^ {2} + y ^ {2} = yI _ {\ mathrm {max}}}$${\ displaystyle x}$${\ displaystyle y}$

## experiment

Reflection behavior of paper. Explanation in the text

The pictures above illustrate the statement of Lambert's law in an experiment. A laser beam falls from the left at the level of the red marking on the edge of the image (shown in red in the right image) and hits a strip of paper (shown in white) perpendicular to the image plane. The beam runs flat over a screen that makes the light scattered by the paper (yellow arrows) visible to the camera. In the first picture the paper is perpendicular to the beam - the distribution of the scattered light is symmetrical. In the second picture the paper is at an angle - the distribution is almost symmetrical to the perpendicular to the paper, a slight preference for the scattering in the direction of reflection can be seen. The third picture is transparent paper that lets through almost as much light as it does backscatter - there is no longer any pronounced multiple scattering, so that the deviation from Lambert's law is greater.

## Examples

In reality there is no material that exactly fulfills Lambert's law. In particular, the radiance of each surface has a directional dependence and this changes when the direction from which the surface is illuminated changes. Even standards that are used to calibrate measuring devices can only be described well by Lambert's law in certain reflection directions and wavelength ranges. With wavelengths outside the visible spectral range and with reflection or illumination directions of more than a few 10 ° to the vertical, deviations of several 100% from Lambert's law can occur even with normal.

There are a number of materials that come close to a Lambertian material at least so far that they appear almost equally bright to the human eye from all directions of observation:

• Matt paper: Small air pockets between the paper fibers form scattering centers for the visible light. If they are missing, for example after soaking the paper with water or oil, paper loses some of its reflective properties and becomes translucent (partially translucent).
• Diffuser / frosted glass : Here, too, scattering centers inside a transparent material ensure that light is diffusely scattered. While milk glass scatters back rather than letting it through, diffusers are also used transmissively.
• The emission surface of light emitting diodes (without optics such as a lens).
• Surfaces made of sintered PTFE ( Spectralon ) : Optical PTFE is often used in an integrating sphere , which in turn is supposed to simulate a Lambert radiator.
• Reflection standards made of pressed or trowelled barium sulfate

## Lommel-Seeliger Law

A better approximation for the backscattering of very dark areas is the Lommel-Seeliger law . It also takes into account a dependency on the viewing angle : ${\ displaystyle \ phi}$

${\ displaystyle I (\ theta) = {\ frac {A \ cos (\ theta)} {\ cos (\ theta) + \ cos (\ phi)}}}$

## Individual evidence

1. A. Höpe, K.-O. Hauer: Three-dimensional appearance characterization of diffuse standard reflection materials. Metrologia 47 (2010) 295-304.
2. Physikalisch Technische Bundesanstalt PTB: On the rotational invariance of reflection normals. Retrieved July 20, 2012.
3. ^ First mentioned in: Johann Heinrich Lambert, Ernst Anding: Lambert's photometry: 1. Th. Das directe Licht. - 2. Th. The weakening of light by transparent bodies, especially by glass . W. Engelmann, 1892, p. 177 ( limited preview in Google Book search).