The Unit Solid Angle

One of the key concepts to understand the relationship between the different photometric quantities is that of the solid angle, or steradian. A steradian is defined as the solid angle which, from the centre of a sphere, cuts off an area of the surface of that sphere equal to the square of its radius. A full sphere therefore contains 4pi steradians (as the equation for the surface area of a sphere is 4pir²).

Take as an example a unit sphere as shown in the figure below, a 1 steradian section of a 1 meter radius sphere subtends a spherical surface area of 1 square meter.

Figure 1 - The definition of a steradian showing that for a sphere of 1m radius, the area on its surface subtended by that angle is 1m².

Working backwards, the solid angle (W) in steradians is equal to the spherical surface area (A) over which it is measured divided by the square of the radius (r) at which it is measured.

Most measurements do not require a super-accurate calculation of the spherical surface area to convert between units. Flat area estimates can be substituted for spherical area when the solid angle is quite low. This can be considered to occur at a radius at least 5 times greater than the largest dimension of the light source.

Power and Intensity

Output power and output intensity are two entirely different concepts. A small battery-powered torch may have a small power output, but when closely focussed onto a tiny watch face, it may provide more light than a 100W globe hanging from the ceiling. What is really important is the distribution of that power - which depends on the characteristics of the source, the angle from the major axis of the source and the distance the light has to travel.

Figure 2 - Two light sources with vastly different power output may actually end up with the same intensity due to their output distribution.

Radiant and Luminous Flux

Radiant flux is a measure of how much radiometric power is being emitted by a source. Flux, expressed in watts, is a measure of the rate of energy flow in joules per second (J/s). Since the amount of radiometric energy contained within a photon of light is inversely proportional to its wavelength, ultraviolet photons are more 'powerful' than visible or infra-red photons.

Luminous flux, on the other hand, is a measure of the power of visible light. This measure depends on the sensitivity of the human eye and is therefore based on the CIE Luminous Efficacy Curve for photopic conditions. Thus the radiosity of any given light can be vastly different from its luminosity depending on the spectral content of its emission, meaning that we have to distinguish between the flow of energy (radiometric) and the flow of light (photometric).

Irradiance and Illuminance

Irradiance is a measure of radiometric flux per unit area, or flux density, and is typically expressed in Watts per square meter (W/m²). It is basically the amount of radiometric energy flowing in a particular direction.

Illuminance, however, is a measure of photometric flux per unit area, or visible flux density. Illuminance is typically expressed in lumens per square meter (lux) or lumens per square foot (foot-candles).

In the diagram below on the left, the light bulb is producing exactly 1 candela. As 1 steradian has a projected area of 1 square meter at a distance of 1 meter, a 1 candela (1 lm/sr) light source will similarly produce 1 lumen per square foot at a distance of 1 foot, and 1 lumen per square meter at 1 meter.

Figure 3 - The illuminance of a 1 candela light globe, showing illuminance levels at one foot and l metre.

Note that as the beam of light projects farther from the source, it expands becoming less dense. In the upper diagram above, for example, you can see that the same amount of light spread over 1m² at 1m is spread over 4m² at 2m and 16m² at 4m distance. This is known as the Inverse Square Law and will be covered in more detail in the next topic.

Radiance and Luminance

Radiance is a measure of the flux density per unit solid viewing angle, expressed in W/m²/Sr. Luminance is a measure of the photometric brightness of a surface and is given in cd/m² or asb. As such it is based on the amount of light being output or reflected off surfaces in the environment. The human eye and the camera both respond only to luminance.

It should be noted that with large area sources, radiance and luminance are independent of distance because, for a fixed solid angle, the sampled area increases with distance thus cancelling inverse square losses. This is clearly shown below, where the planar surface always has the same lumens per metre squared. If you are further away, you can see more of the surface because the cone is bigger, but you still only only see 1 lumen per metre squared. Thus, a 1 lm/m²/SR results in a luminance value of 1 apostilb (the r and 2r values simply show that in section the cone forms an equilateral triangle, i.e.: a base surface area of r²).

Figure 4 - Diagram showing that luminance does not vary with distance from a planar sources as long as it subtends a relatively large viewing angle.

Luminance levels on most architectural surfaces are related to illuminance levels by the reflectance of the surface. For example, consider a wall with a reflectance of 0.6 exposed to an illuminance level of 100 Lux. The effective luminance of that surface after reflection would therefore be 60 apostilb (0.6 * 100). Given that 1 CD/m² = p asb, the luminance is usually given as 19.09 CD/m² (60/p).

Radiant and Luminous Intensity

Radiant Intensity is a measure of radiometric power per unit solid angle, expressed in watts per steradian. Similarly, luminous intensity is a measure of visible power per solid angle, expressed in candela (lumens per steradian). Intensity (I) is related to irradiance (E) by the inverse square law, as shown below:

The diagram below encapsulates this basic photometric relationship.

Figure 5 - Diagram showing the relationship between output power and intensity.

One of the biggest sources of confusion regarding power and intensity in lighting involves the difference between Mean Spherical Candela and Beam Candela, both of which use the candela unit (lumens per steradian). Mean spherical measurements are made in an integrating sphere and represent the total output in lumens divided by the full 4p SR in a sphere. Thus, a one candela isotropic lamp produces one lumen per steradian.

Beam candela, on the other hand, samples a very narrow angle and is only representative of the lumens per steradian at the peak intensity of the beam. This measurement is usually misleading since the sampling angle need not be defined. A flashlight with a million candela beam sounds very bright, but if its beam is only as wide as a laser beam, then it won't be of much use in an architectural setting. Be wary of specifications given in beam candela because they often misrepresent the total output power of a light source.

Related Links

Lighting - the Electronic Textbook

http://www.saud.ku.edu/book/contents.htm

Radiometry vs Photometry FAQ

http://www.optics.arizona.edu/Palmer/rpfaq/rpfaq.htm