Solid angle calculation

General Surface (approximation)

This equation is approximate and assumes the solid angle of the surface is small, i.e. the distance from the surface to the observation point is large compared to the surface diameter or radius. $$\Omega = \frac{A\cos \theta }{d^2}$$ where $A$ is the surface area, $d$ is the surface to observation point distance, and $\theta$ is the surface tilt angle.

Circular disc

$$\Omega = \frac{2 \pi \cos \theta \left ( 1-d \right )}{\sqrt{r^2 + d^2}}$$ where $r$ is the disc radius, $d$ is the surface to observation point distance, and $\theta$ is the surface tilt angle.

f/number

$$\Omega = \frac{\pi}{4\left ( f/\# \right )^2}$$ where $f/\#$ is the f-number of the optical system.

Small Source

$$E=\frac{LA}{d^2}$$ where $L$ is the source radiance, $A$ is the lens area, and $d$ is the distance from the lens to the detector or image plane.

Extended Source

$$E=\frac{L\pi}{4\left ( f/\# \right )^2}$$ where $L$ is the source radiance, and $f/\#$ is the f-number of the optical system.

Image/detector plane flux from point source

Non-imaging (no lens)

$$\phi = \frac{IA}{d^2}$$ where $I$ is the source intensity (watts/sr or photons/sec-sr), $A$ is the detector area, and $d$ is the distance from the lens to the detector or image plane.

Imaging System

$$\phi = \frac{IA}{d^2}$$ where $I$ is the source intensity (watts/sr or photons/sec-sr), $A$ is the lens area, and $d$ is the distance from the point source to the lens.

Image/detector plane flux from finite/extended source

Non-imaging (no lens) - Finite Source

$$\phi = \frac{L A_s A_d \cos \theta_d \cos \theta_s}{d^2}$$ where $L$ is the source radiance, $A_s$ is the source area, $A_d$ is the detector area, $d$ is the distance from the source to the image plane, $\theta_s$ is the source tilt angle, and $\theta_d$ is the image plane tilt angle.

Imaging system - finite source - solid angle of source

$$\phi = \frac{L A_l A_s \cos \theta_s}{d^2}$$ where $L$ is the source radiance, $A_s$ is the source area, $A_l$ is the area of the lens, $\theta_s$ is the source tilt angle, and $d$ is the distance from the source to the lens.

Imaging system - finite source - solid angle of image/detector

$$\phi = \frac{L A_l A_d \cos \theta_d}{d^2}$$ where $L$ is the source radiance, $A_l$ is the area of the lens, $A_d$ is the detector area, $\theta_d$ is the image plane tilt angle, and $d$ is the distance from the lens to the image.

Extended source

$$\phi = \frac{\pi L A_d \cos \theta_d}{4 \left ( f/\# \right )^2}$$ where $f/\#$ is the system f-number, $A_d$ is the detector area, $L$ is the source radiance, and $\theta_d$ is the image plane tilt angle.