Solid angle

A solid angle measures how large a cone of directions is, seen from its apex. If the cone cuts an area $A$ out of a sphere of radius $r$ centered on the apex, the solid angle is

$$\Omega = \frac{A}{r^2},$$

in steradians (sr), a dimensionless unit. The full sphere of directions is $4\pi \approx 12.57$ sr and a hemisphere is $2\pi \approx 6.28$ sr. One steradian is about 3283 square degrees, so the whole sky is about 41,250 square degrees. The Sun's disk, 0.533° across, subtends about $6.8 \times 10^{-5}$ sr, the figure used in the radiance entry. For a small flat area $A$ at distance $r$ whose normal is tilted by $\alpha$ from the line of sight, $\Omega \approx A\cos\alpha / r^2$: a 1 mm² detector face-on at 10 cm subtends $1.0 \times 10^{-4}$ sr.

Cones and numerical aperture

Optical systems usually accept or emit a circular cone. Integrating over the spherical cap of half-angle $\theta$ gives

$$\Omega = 2\pi\,(1 - \cos\theta).$$

For small angles this tends to $\Omega \approx \pi\theta^2$, the area of a disk of radius $\theta$ (in radians). The approximation overestimates by 0.25% at 10° and by 2.3% at 30°, and at 90° it gives 7.75 sr against the exact $2\pi$. Worked values:

Since a lens or fiber is specified by its numerical aperture $\text{NA} = n\sin\theta$, the half-angle in air is $\theta = \arcsin(\text{NA})$. A multimode fiber with NA 0.22 accepts a half-angle of 12.7°, a solid angle of 0.154 sr; standard single-mode fiber at NA 0.14 accepts 8.0°, or 0.062 sr.

Use in radiometry

Solid angle is the denominator in two radiometric quantities. Radiant intensity is power per unit solid angle, in W/sr; an isotropic 1 W source has a radiant intensity of $1/(4\pi) = 0.0796$ W/sr in every direction. Radiance is power per unit projected area per unit solid angle, in W/(m²·sr). The photometric counterparts, covered under radiometry and photometry, are the candela (lumens per steradian) and the candela per square meter. Laser physics uses "intensity" for power per unit area, which is irradiance in radiometric language, so "intensity" in W/sr and in W/cm² should be read from the units.

The product of beam area and solid angle is the étendue, which no lossless passive optic can reduce. A source that fills a wide solid angle from a large area cannot be squeezed into a small fiber core with a small acceptance cone, whatever the lenses.

Light collection from an isotropic emitter

For a point emitter that radiates equally in all directions, such as a fluorescent molecule averaged over orientation, a lens of numerical aperture NA collects the fraction

$$\eta = \frac{\Omega}{4\pi} = \frac{1 - \cos\theta}{2}.$$

An NA 0.5 objective in air ($\theta$ = 30°) collects 6.7%; an NA 0.9 dry objective ($\theta$ = 64.2°) collects 3.54 sr, or 28.2%. An oil-immersion objective with NA 1.4 viewing an emitter in a medium of index 1.515 accepts a half-angle of 67.5° in that medium and collects 30.9%. These fractions set the photon budget in fluorescence microscopy and single-molecule detection, and the reason high-NA objectives are used even when resolution is not the goal. The same arithmetic estimates the power a detector intercepts: a 1 mW isotropic source seen by a 1 mm² detector at 10 cm delivers $10^{-3} \times 10^{-4}/(4\pi) \approx 8.0$ nW.

Measurement and pitfalls

Angle-resolved power is measured with a goniometer: a detector of known aperture area at a fixed distance is swept around the source, and the power divided by the detector's solid angle gives radiant intensity in each direction. Total power from a source with an unknown or very wide pattern, such as a light-emitting diode, is measured instead with an integrating sphere, which collects nearly all $4\pi$ sr at once.

Common errors are using $\pi\theta^2$ for wide cones, using the full cone angle where the half-angle belongs, and treating a Lambertian emitter as isotropic. A Lambertian surface emits with a $\cos\theta$ weighting, so a cone of half-angle $\theta$ about its normal carries the fraction $\sin^2\theta$ of its power: 25% within 30° against 6.7% of 4π for an isotropic source. Refraction at a flat window also changes the solid angle, since a cone in glass widens on exit into air.

Common questions

How many steradians are in a sphere?

$4\pi$, about 12.566 sr. A hemisphere is $2\pi$ sr.

What is the solid angle of a cone?

$\Omega = 2\pi(1 - \cos\theta)$, where $\theta$ is the half-angle. For a half-angle of 10° this is 0.0955 sr, close to the small-angle value $\pi\theta^2$.

Is the steradian an SI unit?

Yes. It is a coherent derived unit with the special name steradian, equal to m²/m², so it is dimensionless; writing "sr" in the units keeps track of which quantities are per solid angle.

References: R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); E. Hecht, Optics, 5th ed. (Pearson, 2017).