Supplement 1.7: Radiation quantities and radiometry (3/5)

The radiant intensity

The radiant intensity is determined by the directional dependence of the radiant power in space or by the directional characteristics of a radiation source. It indicates the radiant power present in a unit solid angle of 1 sr.

Formula symbol: $I$
Unit of measurement: Watts per steradian, $[I] = \frac{W}{sr}$

The relationship with radiant power is:

I = \frac{Φ}{Ω}

The concept of the solid angle $Ω$ is illustrated in the following diagram.

A solid angle Ω in a hemisphere. It is equal to the ratio of the spherical segment area a to the square of the sphere radius R: Ω = a/R². The spherical segment area is represented by a circle as its boundary line; however, there are no restrictions on the shape of solid angles.

Solid angles are measured in the dimensionless unit steradian, denoted by the symbol sr – not to be confused with plane angles, which are measured in radians and denoted by the symbol rad. An explanation of plane angles and solid angles can be found in the tutorial Remote Sensing using Lasers, from which the illustrations on this page are taken.

Since the surface area of a sphere with radius R is $4 π R^{2}$ , the total space around the centre of the sphere corresponds to the solid angle $Ω = 4 π sr$ . The above-mentioned unit solid angle $Ω = 1 sr$ is therefore quite large; however, solid angles can be scaled down or up.

The following therefore applies to the relationship between radiant intensity and radiant power of an emitter that is equally bright in all directions (isotropic):

I = \frac{Φ}{4 π}

Φ = 4 π I

Example: Efficiency of a lens collimator ↓

The light from a very small lamp (‘point light source’) that emits light equally brightly in all directions (isotropic) is to be projected into as parallel a beam as possible using a plano-convex lens with a focal length of f = 100 mm and a diameter of D = 50 mm. To do this, the lens is placed at a distance equal to the focal length from the source. What proportion of the total radiation is found in the parallel beam of light?

The solid angle covered by the lens is

Ω = \frac{a}{R^{2}} = \frac{π {(D / 2)}^{2}}{f^{2}} = 0.196 sr

The lamp illuminates the entire room with a solid angle of $4 π \approx 12.6 sr$ . The yield is therefore only 15.9·10^-3 or 1.59 per cent.

However, there is an error in the calculation: the area $a$ is assumed to be the area of a circle. However, it is a curved spherical segment area (or: a spherical section) whose surface exceeds that of a flat circular area. The situation is illustrated in the following diagram.

A collimator consisting of a lamp and a lens. The lamp does not actually emit light isotropically in all directions, but that is irrelevant here. The plano-convex lens with diameter D is shown in blue. The convex side faces upwards, and the apex of the convex side is at a distance R=f from the lamp.

The flat side of the lens with the area $π {(D / 2)}^{2}$ faces downwards. The convex side of the lens faces upwards, h being the thickness of the lens. From an optical point of view, this orientation is correct, as the refraction is distributed across both glass surfaces. Otherwise – if the flat side of the lens faced upwards – the convex side of the lens would have to perform all of the refraction of the lamp light, which would cause greater imaging errors. The diagram further assumes that the convex surface of the lens coincides with the spherical segment surface. This is not actually the case: such a plano-convex lens would have a radius of curvature of approximately 52 mm, i.e. only about half the radius of the sphere. The surface area M of the shell of the spherical segment that we use instead is

M = 2 π R h

see, for example, Bronstein-Semendjajew: Handbook of Mathematics, chapter on geometry, section on stereometry. The following applies to the right-angled triangle spanned by the opening angle ϕ:

{(R - h)}^{2} + {(\frac{D}{2})}^{2} = R^{2}

i.e.:

(R - h) = \sqrt{R^{2} - {(D / 2)}^{2}}

On the other hand, $h = R - (R - h) = R - \sqrt{R^{2} - {(D / 2)}^{2}}$ ,

which eliminates the unknown variable h. This results in the following for the spherical segment area:

M = 2 π R (R - \sqrt{R^{2} - {(D / 2)}^{2}})

and for the corrected solid angle:

Ω = \frac{a}{R^{2}} = \frac{M}{f^{2}} = \frac{2 π}{f} (f - \sqrt{f^{2} - {(D / 2)}^{2}}) = 0.200 sr

The difference to the result of the simplified calculation remains negligible.

There is another error that should at least be mentioned: the focal length f and the radius of curvature R must not be equated. A real lens has two principal points from which the focal lengths are measured on both sides. In the orientation chosen here, with the focal point at the location of the lamp, the corresponding principal point is not at the apex of the convex lens side, but inside the lens.

Task 1: The intensity of the sun's rays ↓

Equations ↓

How can the beam intensity be represented in a solid angle with any orientation in space? This requires a suitable coordinate system. Cartesian (x,y,z) coordinates would be rather impractical here. Spatial polar coordinates (spherical coordinates) are ideal:

Distance r from the origin (or radius R of a sphere),
angle φ to the x-axis (azimuth angle), and
angle ϑ to the z-axis (zenith angle).

The relationship with Cartesian coordinates and the transformation from one coordinate system to another can be found in the tutorial Remote Sensing using Lasers.

If the radiant power varies in different directions, a differential representation must be used. This is calculated from the differential radiant power divided by the differential solid angle around the orientation under consideration:

I = \frac{d Φ}{d Ω}

The following graphic shows a differential solid angle formed from the area da at a distance R from the coordinate origin

d Ω = \frac{d a}{R^{2}}

A differential surface element da on the surface of a sphere. The surface da does not have to be rectangular, as the edge of infinitesimal surfaces has no defined shape. The x-axis mentioned above points in the direction of φ=0, the z-axis in the direction of ϑ=0; neither are explicitly labelled in the diagram.

The sides of $d a$ are $R sin ϑ d φ$ in the azimuth direction and $R d ϑ$ in the zenith direction. Their product divided by $R^{2}$ gives the differential solid angle