# Theory of Integrating Spheres

### Principles of Integrating Spheres for use in Laser Measurements

Integrating Spheres are used in many areas in optics.   When they are used for laser power measurements the internal arrangement may have to be different from usual.   The objective is to ensure that the response is essentially constant as the beam scans across the input aperture or the beam angle is changed.   At the same time the detector has to be kept within its linear range.

In our Integrating Spheres the detector is exactly on the sphere surface and faces the centre of the sphere (rather than the point where the beam strikes the back of the sphere).   The light bounces around inside the sphere and at each diffuse reflection a small fraction reaches the detector.

#### Geometry of an Integrating Sphere

The detector is facing the centre of the sphere

An area around the input aperture may be darkened to reduce the Gain

To analyse these multiple reflections in a sphere whose diffuse reflectivity is R, we start by considering light which has been scattered from the diffuser just once.   Imagine that the detector signal is proportional to Cos(D) and that the scattering emission from the surface is proportional to Cos(B) - called a"Lambertian Surface".   An input power, W, then gives a power, P1, at the detector after just one reflection of:- `P1 = R· W· d· Cos(D)· Cos(B) / 4 π y2` but, because r· Cos(D) = y = r· Cos(B), we get:- `P1 = R· W· d / 4 π r2 = R· W· d/S` where S is the sphere surface area, 4 π r2.

Note that the signal no longer depends on the angles: thus P1 is the same for any point of incidence and any angle of incidence, even for only one diffuse reflection of the light.   So, getting a signal independent of the beam position does not depend on multiple reflections of the light.   However, multiple reflections do give more signal and can help to smooth out the effects of beam position if either the diffusing paint or the detector do not exactly follow the Cosine of the angle.

The same analysis applies to light scattered twice, three times etc. The total power at the detector is then : `P = P1 + P2 + P3 = (R· W· d / S) / [(1− R (S−A−a) / S]`

We call the term 1 / [(1− R (S−A−a) / S] the Bounce Round Factor or Gain of the sphere and is typically between x1.5 and x20.

In principle, it is possible to control the sphere gain by darkening the white diffusing paint, but it is difficult to do this reproducibly.  Our method is to use a high reflectance white paint over most of the sphere surface but to have matt black paint on a section of the sphere surface just around the entrance port. This reduces the sensitivity by effectively increasing the value of ‘A’ in the term for the Gain 1 / [(1− R (S−A−a) / S].   With this trick we can get close to the optimum sensitivity for each application.

The analysis above might all sound very convincing but, in practice, nothing is ever quite that easy!

• Even the best diffusing paints have polar diagrams that fall off with angle a little faster than Cos(B).
• Nearly all diffusing coatings show a noticeable retro-reflection due to the spherical nature of the diffusing particles.   Typically you get a few % of the light in a 5 degree cone reflected back around the original laser direction (see, for example,  Edwards and Smith, 1981 ).
• Very few detectors have polar diagrams that go exactly as Cos(D).

So to get a system which has only a very slight dependence of its signal on the beam position or angle we use a thick transmitting diffuser such as Opal glass in front of the detector to get nearer to the ideal response proportional to Cos(D).   Of course, the Opal reduces the sensitivity but this may actually be quite useful for high powers.   For the lowest powers we can recover some of the signal by having a higher sphere gain.   However, the long term stability is now compromised because if, for example, we have a sphere gain of x10 then the reflectivity of the diffusing paint has only to drop by 1% to give us a 10% reduction in overall sensitivity.   Using an Opal diffuser in front of the detector also gives a slight wavelength dependence (the sensitivity usually increases with wavelength because then the opal scatters less).

Despite these difficulties, an integrating sphere can still be useful in the measurement of laser power or energy:

• The light is spread uniformly over the whole active area of the detector which is very important in getting the maximum possible linear current.
• It provides a very useful attenuation.   This is nearly always needed so that the detector is well within its linear range.
• It allows a much larger input aperture and range of input angles than other methods of attenuation and diffusion.   So, with an Integrating Sphere you can measure the rapidly diverging power coming from the end of an optical fibre.

We use integrating spheres in nearly all of our systems for both pulsed and CW lasers (see pages CW Laser Power Meters and Laser Peak Power Meters  ).
In one example to measure CW power, we have a miniature thermopile behind a ground opal diffuser.   The sphere has a special sprayed paint which, after high temperature curing, is even water scrubable.   The sphere gain is about x5 and the paint reflectivity is constant to within 1% over the range 450‑1100nm.   Experimentally we find that as the input beam is swung around over the maximum possible input angle of ±22° the sensitivity changes by only 2% in one plane and 3% in the other.   In a normal Laboratory environment long term sensitivity changes are typically less than 3% in the first year and under 1.5% per year after that.