Integrating spheres - for measurement and as homogeneous light sources

The diffusely reflecting inner surface emits according to Lambert's law: every illuminated point gives the light off equally in all directions. Together with the spherical geometry this produces a property no other shape delivers – the illuminance on the wall depends neither on where the light enters nor on which way the source radiates. That is precisely why a sphere can measure the luminous flux of a lamp whose radiation pattern is unknown.

The light is not reflected once but many times. How often is given by the sphere multiplier M:

M = ρ / [ 1 − ρ (1 − f) ]   with ρ = reflectance of the coating, f = port fraction of the inner sphere surface

Two figures show what matters. At a port fraction of 2 % and a reflectance of 0.98 the multiplier is M ≈ 25. Drop the reflectance to 0.95 and M falls to about 14.

Three percentage points of reflectance almost halve the signal. Two things follow that datasheets rarely state: the coating determines the sensitivity of the entire measurement. And a sphere ages measurably – barium sulfate yellows under UV, PTFE takes up oil and dust. Either lowers ρ by a few percentage points and the signal by a third. Integrating spheres should therefore be recalibrated at regular intervals.

The choice therefore follows the wavelength, not the price. Anyone measuring in the UV-C at 254 nm has no choice with barium sulfate – there PTFE is the only coating with a usable reflectance. Anyone needing a two-metre sphere for luminous flux measurements has no choice either, the other way round: solid PTFE cannot be built at that size.

Three quantities are linked, and none can be fixed on its own.

The port fraction f. Every opening is a spot without reflection and acts like a black patch. As a rule of thumb all openings together should stay below 5 % of the inner sphere surface; for accurate luminous flux measurements well below that. The formula above gives the reason: f enters the sphere multiplier directly.

The diameter. It follows the device under test. For the sphere to integrate the flux rather than image the source, the test object has to be small against the sphere – as a rule of thumb at most one tenth of the sphere diameter. A 30 cm LED luminaire therefore needs a sphere of around three metres, while a single 5 mm LED manages with 50 mm. The second link is the port fraction: a larger sphere tolerates a smaller fraction at the same port diameter.

The baffle. A small screen between source and detector. Without it, light falls directly from the source onto the detector and the measurement depends on the radiation direction again – exactly what the sphere is there to prevent.

Self-absorption. The test object itself swallows light: a lamp with a large base takes more from the sphere than a small LED. This is corrected with an auxiliary lamp, measured once with and once without the test object. For accurate luminous flux measurements it is not an accessory but a prerequisite.

A laser beam brings three properties that distort any direct measurement: a beam profile unevenly distributed across the detector area, an angle of incidence the responsivity reacts to, and a polarisation. The integrating sphere removes all three: after the first diffuse reflection the light is unpolarised, without direction and evenly spread across the wall. The detector sees nothing but an irradiance.

The second reason is power density. A 10 W beam of 1 mm diameter carries roughly 1.3 kW/cm² – enough to destroy any semiconductor detector instantly. The sphere spreads the same power over its entire inner surface; at 100 mm diameter that is 314 cm². What reaches the detector is thus more than three orders of magnitude lower.

Pulsed lasers are integrated over the pulse: what is measured is energy in joules rather than power in watts. The sphere changes nothing about that – here too it delivers the advantage that beam profile and alignment do not matter.

Two points remain. First, the reflectance at the laser wavelength has to be known – a sphere calibrated at 550 nm measures wrongly at 1064 nm. Second, the coating is thermally limited: at continuous-wave powers in the hundreds of watts, thermal design belongs in the conversation.