Solid Angles

Solid angle is defined as an amount of spherical surface area divided by the squared radius:

\[\omega \equiv \frac{A}{r^2}\]

In radiometry, we consider an observer looking outward from the origin of a sphere. A is the area subtended by the observer’s field of view. For a typical optical system, this would be the area of a spherical cap. The differential area of a spherical cap is the product of the differential arc length in the zenith direction, \(rd\theta\), and the differential arc length of the spherical cross section in the azimuthal direction, \(r\sin(\theta) d\phi\).

\[\begin{split}\omega_{cap} &= \frac{A}{r^2} = \frac{1}{r^2} \int_{\phi=0}^{2\pi} \int_{\theta=0}^{\theta_H} \left(r d\theta\right) \left(r \sin(\theta) d\phi\right) = \int_{\phi=0}^{2\pi} d\phi \int_{\theta=0}^{\theta_H} \sin(\theta) d\theta \\ &= 2\pi \left[ -\cos\theta \Big|^{\theta_H}_0\right] = 2\pi \left(1-\cos\theta_H\right)\end{split}\]

where \(\theta_H\) is the half-angle of a right circular cone. When projecting onto a flat surface, e.g. a sensor, projected solid angle (Ω) is used. The projection of a spherical cap is a circle. The area of that circle is \(\pi r_{cs}^2\), where \(r_{cs}\) is the radius of the spherical cross-section, \(r\sin\theta_H\). The projected solid angle is therefore

\[\Omega_{cap} = \frac{A}{r^2} = \frac{\pi r_{cs}^2}{r^2} = \frac{\pi\left(r^2\sin^2\theta_H\right)}{r^2} = \pi\sin^2\theta_H\]