### Principle of Integrating Spheres for Laser Measurements

In our Integrating Spheres the detector faces the ** centre ** of the sphere (** not ** the point where the beam strikes the back of the sphere). The light bounces around inside the sphere and at each reflection a small fraction reaches the detector. To analyse this, we start by considering light which has been reflected from the diffuser just once.

Imagine that the detector signal is proportional to Cos(D) and the surface emission proportional to Cos(B)- called a"Lambertian Surface". An input power, W, then gives a power, P_{1}, at the detector after just one reflection of :
`P`

but, because r· Cos(D) = y = r· Cos(B), we get :
_{1} = R· W· d· Cos(D)· Cos(B) / 4 π y^{2}`P`

where S is the sphere surface area, 4 π r_{1} = R· W· d / 4 π r^{2} = R· W· d/S^{2}.

Note that the signal no longer depends on the angles: thus P_{1} is the same for any point of incidence and any angle of incidence, even for only one 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 = P`

_{1} + P_{2} + P_{3} = (R· W· d / S) / [(1− R (S−A−a) / S]

The term 1 / [(1− R (S−A−a) / S] is the** Bounce Round Factor** or the **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 paint over most of the sphere surface but to have matt black paint on the section 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 all sounds very convincing but, in practice, nothing is ever quite that easy!

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

So to get a sphere 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 lower 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:

- It spreads the light uniformly over the detector which is very important in getting the maximum possible linear range.
- It provides a very useful attenuation. This is nearly always needed to get the detector into its linear range.
- It allows a much larger input aperture and range of input angles than other methods of attenuation and diffusion. For example you can measure the power coming from the end of an optical fibre (or an optical
**fiber**for those in USA)

As an example, we use integrating spheres in all our Power Meters for CW lasers. A miniature thermopile is positioned behind a ground flashed opal diffuser. The sphere surface is formed from a special sprayed paint which, after high temperature curing, is even water scrubable. We get a sphere gain of about x5 and the paint reflectivity is within 5% over the range 450 - 1000nm. 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 4% in the first year and under 1.5% per year after that.