Archimedean polyhedron integrating sphere

Principle of Integrating Sphere

 

I. Basic Concepts

 

Luminous flux $\varphi$：

 

The amount of light emitted by a light source per unit time is called the luminous flux of the light source, and its unit is lumen ($lm$）。

 

Luminous exitance$M$：

The luminous exitance of a light source is the luminous flux emitted per unit area of the light source into a hemispherical space (2π steradians), and its unit is lumen per square meter ($lm/m^2$）。

$M=\frac{\varphi }{S}$

 

Illuminance$E$：

The quantity that describes the degree to which a surface is illuminated is called illuminance, which refers to the amount of luminous flux received per unit area. Its unit is $lm/m^2$ or  $lux$。

$E=\frac{\varphi }{S}$

 

For a surface that becomes a surface light source after being illuminated, its luminous exitance is proportional to the illuminance.

$M=\rho E$

 

In the formula, $\rho$ is called the diffuse reflectance.

 

Luminance$L$：

 

The luminance of a light source in a certain direction is the luminous flux emitted by the unit projected area of the light source in that direction within a unit solid angle, and its unit is candela per square meter ($cd/m^2$).

$L=\frac{\varphi }{S\cos \theta ·\Omega }$

For a cosine radiator, the luminance $L$ does not change with direction, and it has the following relationship with the luminous exitance $M$:

$L=\frac{M}{\pi }$

 

II. Principle

 

In an ideal state, the following assumptions are made:
(1) The inner surface of the integrating sphere is a complete geometric spherical surface;
(2) The inner wall of the sphere is a neutral isotropic diffuse reflection surface, and the reflectance is the same everywhere;
(3) There are no objects inside the sphere, and the light source inside the sphere is also regarded as an abstract light source that only emits light without a physical entity.

As shown in the figure below, the center of the sphere is$O$, the radius is $r$, the diffuse reflectivity of the sphere wall is $\rho$, and $s$ is the light source, which can be placed anywhere inside the sphere with a luminous flux of $\varphi$

 

The illuminance produced by the light source $s$ at any point $B$ on the sphere wall is the superposition of the following components: the illuminance directly produced by $s$ shining on point $B$; the secondary illuminance produced by the light from $s$ hitting other parts of the spherical surface and then diffusely reflecting to point $B$; the tertiary illuminance produced by the light after the first diffuse reflection on the spherical surface undergoing a second diffuse reflection on the spherical surface and then reaching point $B$; and so on.

Let the illuminance produced by $s$ at any point $A$ inside the sphere be $E_a$. If point $A$is regarded as a secondary light emitter, the luminous exitance $M=\rho E_a$。Since the inner wall of the sphere is an ideal diffusing layer, the luminance near point $A$is：

 

$L=\frac{M}{\mathrm{\pi }}$

 

The secondary illuminance produced at point $B$ by the primary diffused light emitted from the tiny area $da$ near point $A$ is:

$d{E}_2=\frac{d^2\varphi }{dS}=\frac{\rho E_{a}da}{4{\mathrm{\pi r}}^2}$

 

The secondary illuminance produced at point $B$ by the primary diffused light from the entire spherical surface is:

$E_2=\frac{\rho \varphi }{4\pi r^2}$

 

Similarly, the tertiary illuminance produced at point $B$ by the secondary diffused light from any small area $da$ on the sphere wall can be obtained as:

$d{E}_3=\frac{\rho E_{2}da}{4{\mathrm{\pi r}}^2}$

 

The tertiary illuminance produced at point $B$ by the secondary diffuse reflection light from the entire spherical surface is:

$E_3=\frac{{\rho }^2\varphi }{4\pi r^2}$

 

By analogy, the illuminance of any subsequent order can be obtained.

 

 

$E=E_1+E_2+E_3+\cdots =E_1+\frac{\rho }{1-\rho }\frac{\varphi }{4\pi r^2}$

 

In the formula,$E_1$ is the illuminance directly produced by the light source $s$ on point $B$.

If a baffle is placed between the light source $s$and point $B$ to block the light directly emitted to point $B$, then $E_1=0$, and thus the illuminance at point 

$B$ is:

$E=\frac{\rho }{1-\rho }\frac{\varphi }{4\pi r^2}$

 

In this way, by measuring the illuminance $E$ at the window on the sphere wall, the luminous flux $\varphi$ of the lamp can be obtained.