The ideal integrating sphere, a theoretical construction which allows the explanation of the sphere’s basic principle of operation, is characterized by the following properties:

- Its entrance and exit ports are infinitesimally small.

- All objects inside the sphere, light sources and baffles, are also infinitesimally small and their influence on optical radiation after its first reflection at the sphere’s inner surface can be neglected.

- Its inner surface is a perfectly homogenous Lambertian reflector and its reflectance ρ is independent from wavelength. For a more detailed discussion of reflective materials largely fulfilling these properties, see Example 3: The Lambertian surface (chapter “Calculation of radiometric quantities – Examples”) and Integrating spheres used with integral detectors (chapter “The detector’s input optics and its directional sensitivity”)

During the following considerations, the symbol index describes the order of reflection. So, E_{0} denotes the irradiance caused directly by the light source, whereas E_{1}, E_{2}, … denote the irradiance caused by light from the source after one, two, … reflections. Total irradiance is then given by the infinite sum

E

_{total}= E_{0}+ E_{1}+ E_{2}+ …

For convenience, the index “e”, denoting radiometric quantities is omitted. However, if the reflectance ρ of the sphere’s coating material is independent from wavelength, the derived relations also hold true for photometric quantities, which would be denoted by the index “v”.

*Fig. 1: Geometry of an ideal integrating sphere of radius R*

Let’s consider an ideal integrating sphere of radius R, consisting of a hollow perfect Lambertian reflector with infinitesimally small entrance and exit ports. An inhomogeneous radiation source produces direct irradiance levels E_{0} (the term “direct irradiance” refers to the fact that E_{0} is directly caused by the source without any reflections) which depend on the respective location at the sphere’s inner surface (Fig. 1). As a first step, we want to calculate the irradiance E_{1} of the sphere’s inner surface produced by the radiance L_{1} after the first reflection. Due to the Lambertian reflection property of the sphere’s material, the radiation reflected by a certain area element dA is characterized by a constant directional radiance distribution L. According to the paragraph “Example 3: The Lambertian surface”, the area element’s exitance M_{1} is related to the reflected radiance L_{1} by

M

_{1}= L_{1}π

and is further related to the element’s direct irradiance E_{0} by

M

_{1}= ρ E_{0}

whereby ρ denotes the total reflectance of the sphere’s inner surface.

As a consequence,

L = ρ E _{0}π

Although L does not depend on the direction relative to the surface element dA, it still depends on the location at the sphere’s inside, which is a consequence of the generally irregular direct illumination by the light source.

If we want to calculate the radiant power emitted by the area element dA and impinging upon another area element dA', we have to calculate the solid angle of dA', as seen from dA (Fig. 1). As dA' is tilted by an angle ε relative to the line of sight between the two area elements, dA' occupies the solid angle dΩ', as seen from dA:

dΩ' = cos(ε) dA' d ^{2}

with d denoting the distance between dA and dA'.

According to Equ. 2 in “Basic radiometric quantities”, the radiant power emitted by dA into the solid angle dΩ' and thus impinging upon dA' is given by

L cos(ε) dA dΩ' = L dA × cos ^{2}(ε) × dA'd ^{2}

and dividing this expression results in the (infinitesimal) irradiance dE_{1} of the sphere’s inner surface at the location of dA’ which is caused by a single reflection of direct radiation from the source at the area element dA:

dE _{1}= L ×cos ^{2}(ε)× dA = E _{0}ρ× 1 × dA d ^{2}π 4 R ^{2}

and the relation d = 2 R cos(ε), which can be easily seen from Fig. 1.

In order to obtain the irradiance E_{1} at the location of dA', which is caused by a single reflection of the source’s radiation at the whole inner surface of the sphere, the above expression for dE_{1} has to be integrated over the sphere’s inner surface:

E _{1}=∫ E _{0}ρ× 1 dA = ρ ∫ E _{0}dA =ρ Φ _{0}inner surface π 4 R ^{2}4 π R ^{2}inner surface 4 π R ^{2}

Here, Φ_{0} is the total radiant power emitted by the source and impinging upon the sphere’s inner surface.

Note that the irradiance of the inner surface after the first reflection is independent from the actual location on the sphere, which is still the case despite the inhomogeneous direct irradiance caused by direct illumination from the source.

Deriving the irradiance E_{2} caused by the source’s radiation after two reflections in the same way, we get

E _{2}=∫ E _{1}ρ× 1 dA = ρ ∫ E _{1}dA =ρ E _{1}× 4 π R^{2}= ρ E_{1}=ρ ^{2}Φ_{0}inner surface π 4 R ^{2}4 π R ^{2}inner surface 4 π R ^{2}4 π R ^{2}

Generally, the irradiance of the sphere’s inner surface caused by the source’s radiation after k reflections is given by

E _{k}=ρ ^{k}Φ_{0}4 π R ^{2}

and the total irradiance is thus given by

E _{total}= E_{0}+ E_{1}+ E_{2}+ … = E_{0}+∞ ∑ ρ ^{k}Φ_{0}= E _{0}+Φ _{0}∞ ∑ ρ ^{k}= E_{0}+Φ _{0}× ρ k=0 4 π R ^{2}A _{sphere}k=0 A _{sphere}1 - ρ

In this expression, only E_{0} actually depends on the respective location on the sphere’s inner surface. As a consequence, Etotal is independent of the actual location of the sphere’s inner surface as long as we ensure that E_{0} = 0 at this location. This means that no direct radiation from the source reaches the location, which can be achieved by using baffles. In this case, total irradiance is proportional to the total amount of radiant power Φ_{0} reaching the sphere’s inner surface directly from the source:

E _{total}=Φ _{0}× ρ = Φ _{0}× K A _{sphere}1 - ρ A _{sphere}

Since the constant K describes the enhancement of irradiance relative to the average irradiance of a non-reflecting sphere, it is called “sphere multiplier” and, for an ideal sphere, solely depends on the coating material’s reflectance ρ.