Lambert's cosine law

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

As the man with his eye only rated luminance ( luminance is the photometric equivalent of radiance ), such lambert netic material appears independent of the viewing direction as equally bright Applies Lambert's law for each surface element of a light source, so it is called a Lambert spotlight. In particular, an ideal black body is a Lambert radiator. When light is reflected from a surface according to Lambert's law, then one speaks of ideal diffuse reflection.

Mathematical Description

Is the angle from the surface normal and the size of the A Lambertian surface element, the radiation intensity is proportional to the product of the cosine of the angle and the surface:

The ratio of intensity of radiation and reduced area ( projected in the viewing direction ), the proportionality factor here is the constant luminance L of the surface:

The dashed line in the radiation intensity in the image on the right satisfies this equation ( x - axis horizontal, y is vertical ).

Experiment

The images above illustrate the testimony of Lambert 's law in an experiment. From left each a laser beam is incident in the amount of the red marking on the edge of the image (in the right picture shown in red ) and strikes a plane perpendicular to the image plane of paper (white shown). The beam is flat over a screen which makes the scattered light from the paper (yellow arrows ) of the camera is visible. In the first image, the paper is perpendicular to the beam; the distribution of the scattered light is symmetrical. The second picture shows the paper is tilted; the distribution is almost symmetrical to the perpendicular to the paper; a slight preference for the spread in the reflection direction is seen. In the third image is transparent paper, which passes almost as much light as back- scattered; there is no pronounced multiple scattering more, so that the deviation from Lambert 's law is greater.

Examples

There is in reality no material satisfying the Lambert's law accurately. Even Normal, which are used for calibration of measuring instruments, show deviations of 0.1 % to 2.5%. However, there are a number of materials that come near a Lambert material:

  • Matte Paper: Small air pockets between the paper fibers form scattering centers for the visible light. The lack of, for example after impregnation of the paper with water or oil, paper loses a part of its reflection properties, and is translucent.
  • Milk Glass: Here too, scattering centers inside the glass that light is diffusely scattered. Milk glass scatters it back more than it lets through.
  • The emission area of ​​light emitting diodes ( without plastic lens )
  • Surfaces of sintered PTFE ( Spectralon ): Optical PTFE is often used in an integrating sphere, which in turn should track Lambert radiator.
  • Reflecting standards of pressed or a filled 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 dependence on the angle of observation:

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