Integrating sphere

An integrating sphere (also known as an Ulbricht sphere) is an optical component consisting of a hollow spherical cavity with its interior covered with a diffuse white reflective coating, with small holes for entrance and exit ports. Its relevant property is a uniform scattering or diffusing effect. Light rays incident on any point on the inner surface are, by multiple scattering reflections, distributed equally to all other points. The effects of the original direction of light are minimized. An integrating sphere may be thought of as a diffuser which preserves power but destroys spatial information. It is typically used with some light source and a detector for optical power measurement. A similar device is the focusing or Coblentz sphere, which differs in that it has a mirror-like (specular) inner surface rather than a diffuse inner surface. In 1892, W. E. Sumpner published an expression for the throughput of a spherical enclosure with diffusely reflecting walls. R. Ulbricht developed a practical realization of the integrating sphere, the topic of a publication in 1900. It has become a standard instrument in photometry and radiometry and has the advantage over a goniophotometer that the total power produced by a source can be obtained in a single measurement. Other shapes, such as a cubical box, have also been theoretically analyzed. Even small commercial integrating spheres cost many thousands of dollars, as a result their use is often limited to industry and large academic institutions. However, 3D printing and homemade coatings have seen the production of experimentally accurate DIY spheres for very low cost.

Theory The theory of integrating spheres is based on these assumptions:

Light hitting the sides of the sphere is scattered in a diffuse way i.e. Lambertian reflectance Only light that has been diffused in the sphere hits the ports or detectors used for probing the light Using these assumptions the sphere multiplier can be calculated. This number is the average number of times a photon is scattered in the sphere, before it is absorbed in the coating or escapes through a port. This number increases with the reflectivity of the sphere coating and decreases with the ratio between the total area of ports and other absorbing objects and the sphere inner area. To get a high homogeneity a recommended sphere multiplier is 10-25. The theory further states that if the above criteria are fulfilled then the irradiance on any area element on the sphere will be proportional to the total radiant flux input to the sphere. Absolute measurements of instance luminous flux can then be done by measuring a known light source and determining the transfer function or calibration curve.

Total exit irradiance For a sphere with radius r {\displaystyle r} , reflection coefficient ρ {\displaystyle \rho } , and source flux Φ {\displaystyle \Phi } , the initial reflected irradiance is equal to:

E = ρ Φ 4 π r 2 {\displaystyle E=\rho {\frac {\Phi }{4\pi r^{2}}}\,}

Every time the irradiance is reflected, the reflection coefficient exponentially grows. The resulting equation is

E = Φ 4 π r 2 ρ ( 1 + ρ + ρ 2 + . . . ) {\displaystyle E={\frac {\Phi }{4\pi r^{2}}}\,\rho (1+\rho +\rho ^{2}+...)}

Since ρ < 1 {\displaystyle \rho <1} , the geometric series converges and the total exit irradiance is:

E = Φ 4 π r 2 ρ 1 − ρ {\displaystyle E={\frac {\Phi }{4\pi r^{2}}}\,{\frac {\rho }{1-\rho }}\,}

Applications

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