Integrating sphere

An integrating sphere is a hollow sphere, typically 25 mm to 2 m in diameter, whose inner surface is coated with a diffuse, near-Lambertian reflector: sintered or pressed PTFE, which reflects about 99% of visible light, barium sulfate paint, or roughened gold for the infrared. Light entering through a port strikes the wall, is scattered in all directions, and after many further reflections the radiance of the wall becomes nearly uniform and independent of the direction, position, polarization and beam shape of the input. A detector looking at the wall through a second port then measures a signal proportional to the total power entering the sphere. The idea goes back to Sumpner (1892) and Ulbricht (1900), who used it to measure the total output of lamps, and it remains the standard tool for total flux, reflectance and transmittance measurements.

Sphere radiance equation

For an input flux $\Phi$ into a sphere of internal surface area $A_s$, wall reflectance $\rho$ and total port fraction $f$ (port area divided by $A_s$), the wall radiance is

$$L = \frac{\Phi}{\pi A_s}\,M,$$

with the sphere multiplier

$$M = \frac{\rho}{1-\rho\,(1-f)}.$$

$M$ is the wall radiance in units of $\Phi/(\pi A_s)$, the radiance a single bounce off a perfect reflector would give. For $\rho = 0.98$ and $f = 0.02$, $M = 24.7$; at $\rho = 0.99$ it rises to 33.2, and at $\rho = 0.95$ it falls to 13.8. Opening more ports reduces it sharply: with $\rho = 0.98$ and $f = 0.05$, $M = 14.2$. The multiplier is therefore sensitive to coating reflectance, which ages and changes with wavelength.

As a worked example, a sphere 101.6 mm (4 in) in diameter has $A_s = \pi d^2 = 0.0324$ m². With 1 W entering and $M = 24.7$, the wall radiance is 243 W/(m²·sr). A detector filling a 1 cm² port receives the flux crossing that port, $\pi L$ times its area, or 76 mW: 7.6% of the input. Larger spheres give more uniform output and tolerate larger samples, but deliver less flux to a given detector, since $A_s$ grows as $d^2$.

Uses in the laboratory

An integrating sphere in front of a photodiode makes an optical power meter head that is insensitive to beam position, divergence and polarization, and it attenuates a high-power beam before it reaches the detector; this is the usual choice for measuring divergent sources such as laser diodes, LEDs and bare fiber ends. Calibrated with a spectroradiometer, the same arrangement measures total luminous flux in lumens and the spectral radiant flux of lamps, the photometric side of which is set out under radiometry vs photometry. With the sample placed at a port, the sphere collects all light reflected or transmitted into the hemisphere, specular and diffuse together, which gives total reflectance and total transmittance including scattering. With an internal lamp or a laser input, the exit port becomes a uniform radiance source for calibrating cameras and radiometers, typically uniform to about 1% or better over the port.

Time response

Multiple reflections delay the light. The mean path between reflections is $2d/3$, so the exponential decay time of the sphere is

$$\tau = -\frac{2d}{3c\,\ln[\rho(1-f)]}.$$

For the 101.6 mm sphere with $\rho = 0.98$ and $f = 0.02$, $\tau = 5.6$ ns. This broadens short pulses and corresponds to a 3 dB bandwidth $1/(2\pi\tau)$ of about 28 MHz, which matters when measuring pulse energy or modulated sources.

Pitfalls

The detector must not see the point where the input beam first strikes the wall, or any directly illuminated area; baffles between the input spot and the detector port are placed for this reason. A sample, detector or lamp holder inside the sphere absorbs light and changes $M$, so substitution or auxiliary-lamp methods are used to correct for self-absorption when comparing samples. Coatings fluoresce under ultraviolet excitation, absorb near water and organic contaminant bands, and lose reflectance when dirty or damp, which shifts calibration. High-power laser beams can damage the coating at the first-strike spot. The spectral dependence of $M$ is magnified relative to that of $\rho$, so a drop in reflectance from 0.98 to 0.97 at some wavelength (with $f = 0.02$) lowers $M$ from 24.7 to 19.6, a 21% loss of throughput there.

Common questions

What is an integrating sphere used for?

To measure the total power or luminous flux of a source regardless of its beam shape, to measure total reflectance and transmittance of materials including diffuse components, and to provide a uniform radiance source for calibrating cameras and radiometers.

Why is the inside of an integrating sphere white?

The coating must reflect nearly all incident light diffusely. High reflectance gives a large sphere multiplier and a uniform wall radiance; diffuse, Lambertian scattering removes any memory of the input direction after a few bounces.

How does port size affect an integrating sphere?

Every port is a hole through which light escapes, lowering the multiplier. The total port fraction is usually kept below about 5% of the sphere surface so that uniformity and throughput stay close to those of a closed sphere.

References: D. G. Goebel, "Generalized integrating-sphere theory," Appl. Opt. 6, 125 (1967); W. R. McCluney, Introduction to Radiometry and Photometry, 2nd ed. (Artech House, 2014).