Integrating Sphere Calculator – Radiance & Luminance Simulation

How the integrating sphere simulation works

An integrating sphere mixes the injected light through multiple reflections on its diffusely reflecting, ideally Lambertian inner coating. After just a few reflections the sphere wall is illuminated almost uniformly, and the output port forms a homogeneous luminous area. The simulation models this behaviour with classical sphere theory: the radiant flux of the halogen lamps is modelled as a Planckian radiator with a typical colour temperature and amplified per wavelength by the sphere multiplier:

M(λ) = ρ(λ) / (1 − ρ(λ) · (1 − f))

Here ρ(λ) is the spectral reflectance of the coating and f the fraction of the sphere surface taken up by port openings. The calculation runs in 1 nm steps from 250 to 2500 nm using reflectance spectra measured on real Opsytec coatings.

Calculating the radiance

The spectral radiance at the output port follows from the total flux Φ(λ) of all lamps, the sphere surface A and the sphere multiplier:

L(λ) = Φ(λ) · ρ(λ) · M(λ) / (π · A)

The additional factor ρ(λ) accounts for one reflection at the baffle that blocks the direct view between lamp and output port. The radiance L in W/(m²·sr) results from integrating over 250–2500 nm. Because the sphere wall emits in a Lambertian way, the radiance is independent of the viewing angle — the integrating sphere acts as a homogeneous, angle-independent radiation source.

Calculating the luminance

For the luminance, the spectral radiance is weighted with the luminous efficiency function V(λ) of the human eye (CIE 1924, photopic):

Lv = 683 lm/W · ∫ L(λ) · V(λ) dλ

The result in cd/m² describes how bright the output opening appears to the eye or to a photometrically weighted camera. The luminous flux of each lamp is determined from nominal power and typical efficacy (lm/W); when dimmed, it drops over-proportionally with (P/P_nom)^2.2 — matching the real dimming behaviour of halogen lamps.

Influence of sphere diameter and port areas

The radiance scales inversely with the sphere surface and thus with 1/D²: a sphere twice as large delivers only a quarter of the radiance with the same lamps, but mixes the light better and offers space for more ports. Every port opening drains light from the sphere: as the port fraction f grows, the sphere multiplier drops markedly. As a rule of thumb f should stay below 5 % — beyond that, sphere theory becomes increasingly inaccurate and the simulation shows a warning. The lamp ports are arranged rotationally symmetrically on a 35° cone around the output port.

Influence of the reflectance

Because the light undergoes many reflections, the reflectance acts as a powerful lever: close to ρ = 1, a single percentage point changes the result considerably. At f = 2 %, ρ = 98 % yields a multiplier of about 25, while ρ = 94 % gives only around 12 — almost a factor of two. Optical PTFE reflects around 98 % across the entire UV/VIS/NIR range, barium sulfate around 90–93 %, gold only about 25 % below 500 nm but more than 95 % above 800 nm. The simulation uses measured spectra of these three coatings and a 2 % practical derating for ageing and application.

Assumptions and limitations of the model

• ideally Lambertian, spatially homogeneous coating; ports as fully absorbing openings; exactly one additional reflection at the baffle
• halogen lamps as Planckian radiators with constant colour temperature and typical efficacy; dimming curve (P/P_nom)^2.2
• no ray tracing — lamp filaments, sockets and baffle geometry are not resolved spatially; temperature and ageing effects only via the flat derating

The results are therefore orientation data for design purposes — without guarantee and no substitute for a measurement. On request, Opsytec performs calibrated measurements of the real sphere in its own light laboratory.

Selecting a suitable integrating sphere

Rules of thumb for the selection: optical PTFE is the first choice from UV to NIR for small and medium spheres — highest reflectance and long-term stable. Barium sulfate is particularly suitable for large spheres, as the paint can be applied economically to large areas. Gold is the coating for NIR/IR applications above roughly 800 nm. The diameter follows from the trade-off between high radiance (small sphere) and good uniformity or many ports (large sphere). Opsytec Dr. Gröbel manufactures custom integrating spheres from 25 mm to 2.5 m diameter — with baffles, arbitrary port configurations and matching light sources: luminance and radiance standards by Opsytec

Frequently asked questions

Which coating is suitable for NIR and IR applications?

Gold — above roughly 800 nm it reflects more than 95 % and is chemically stable long-term. Below 500 nm its reflectance drops to about 25 %; for UV and VIS applications, optical PTFE or barium sulfate are the right choice.

How accurate are the calculated values?

The simulation provides orientation data based on classical sphere theory with measured reflectance spectra and is calibrated against experience values from real Opsytec spheres. Manufacturing tolerances, baffle geometry and lamp scattering are modelled in simplified form — only a measurement of the real sphere is binding.

Why does the radiance decrease with larger sphere diameters?

The injected luminous flux spreads over the sphere surface, which grows with the square of the diameter. Doubling the diameter means roughly a quarter of the radiance — in return for better uniformity and more space for ports.

Is the simulated integrating sphere available as a calibrated luminance standard?

Yes. Opsytec manufactures custom integrating spheres from 25 mm to 2.5 m diameter and supplies them on request with traceable calibration from its own light laboratory.