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6 Theory and applications of integrating spheres

Integrating spheres are very versatile optical elements designed to achieve homogenous distribution of optical radiation through multiple Lambertian reflections at the sphere’s inner surface. The primary radiation source can either be located inside the sphere or in front of the source’s entrance port. In the latter, only the optical radiation entering the sphere is relevant for the sphere’s internal radiation distribution.

As long as we restrict ourselves to those regions that are shielded from direct irradiation by the primary source and are thus only illuminated by reflections at other of the inner surface, the theory of the ideal integrating sphere leads to two important conclusions:

  • Irradiance of the sphere’s inner surface is proportional to the total radiant power either emitted by a source inside the sphere or entering the sphere through its entrance port. Geometrical and directional distribution of the primary source’s radiation do not influence irradiance levels as long as direct illumination of the respective location is prevented. This property becomes important particularly when an integrating sphere is used as the input optical element of a detector for radiant power.
  • Radiance reflected by a region of the sphere’s inner surface shielded from direct illumination is constant in its directional distribution and independent from the specific location where the reflection occurs. Thus, the sphere’s exit port can be used as an ideal Lambertian source since optical radiation leaving the sphere is characterized by homogenous radiance and exitance distributions. This property becomes important particularly when a sphere is used as a standard calibration source.

Fig. 1: Integrating sphere used as a standard source for optical radiation. Multiple Lambertian reflections
inside the sphere result in homogenous radiance and exitance distributions at the sphere’s exit port.