How do I light thee? Let me count the rays...

I am the first to admit having trouble envisioning photons fluttering through space, let alone remembering how beam candle power relates to footcandles. So I use a crutch. Physicists will cringe, but this model serves for the needs of human-scale lighting situations. It casually glosses over many details of physics and geometry but should provide a basic understanding which the reader can refine with further study.

My crutch is light rays. I imagine that a light source, a candle for example, emits actual countable rays of light. Not a sea of photons or waves of electromagnetic flux, but thousands of individual rays of light. My rays are straight lines extending in every direction from a light source. They may be refracted through lenses, reflected by surfaces, and absorbed.

Raw light is measured in lumens. Think of a lumen as a ray. A light source, like a flame or a filament, produces a certain number of lumens. Imagine the source having some number of rays going out in all directions. A candle has a few rays. A tungsten lamp has many rays.

The Luminous Intensity of a source is the number of lumens or rays it emits. Luminous intensity, candlepower, is expressed in units of candles or candelas. Candles are really a measure of lumens per steradian, which is a cone shaped sector of the total space around the light source. Since steradians are geometry and not lighting, I don't worry about them and simply think of candlepower as a measure of the number of rays emanating from the source.

And how exactly do we count the rays? Imagine placing a card in the path of these rays and counting how many lumens, or rays, hit the card. In a dark room there are not very many. In a bright room there are a lot. The density of the rays in a field of light, as measured by how many strike a card of a fixed size, is the illuminance, or amount of illumination.

Imaginary illumination measuring cards come in three standard sizes: 1 meter square, 1 foot square, and 1 centimeter square. When measuring the density of a field of evenly distributed light rays, obviously more light rays will fall on a bigger card, so different unit names are used to identify which size card is used. Footcandles are the measure of how many rays fall on a 1 foot square card. Lux is the measure of how many rays fall on a one meter square card, and phots use a tiny card 1 centimeter square. These three units; lux, phot, and footcandle; all measure exactly the same thing: illumination, but at different scales. Just like feet and meters both measure distance.

Using our measuring card we can count how many rays are coming from a light source. But there is a problem. As we move our card farther away from the source, the rays diverge and we count fewer and fewer actually falling on the card. Since light travels in straight lines, when the card is moved twice as far from the source the rays will have diverged so far that we would need a card twice as wide and twice as tall to catch all of the same rays: four times the area of the original card. Moving farther out to three times the distance and we will need a card three times as wide and three times as tall, or nine times the area of the original card to catch all of the same rays at the greater distance.

Mathematicians describe this relationship by stating that the size of the card increases with the square of the distance, meaning the size card needed at each distance is the square of the distance from the source. Two times as far requires a card having four times as much area. Three times as far requires nine times the area. Distance squared equals the required card area. If instead of looking at the size of the card at each distance, we look at how many rays fall on a card of the original size we see that at two times the distance we get 1/4 the number of rays, and at three times the distance we get 1/9 the number of rays (since it would take a card nine times as big to collect them all.) The amount of light falling on the card is the reciprocal, or inverse, of the square of the distance. This is the famous inverse square law.

In stage lighting the inverse square law is most often applied to determine in advance how much illumination a fixture will deliver from a particular distance. If we know the beam candlepower of the fixture (from the manufacturer's data) and the distance for which we want to calculate the illumination, we can simply divide the candlepower in lumens by the distance squared. If we measured the distance in feet, the answer will be in footcandles. If we measured in meters, it will be in lux. If we measured in centimeters it wil be in phots. And that's why there are different illumination units.

For extra credit we'll tackle the measurement of light reflected by a surface such as a painted wall or a costume or a face. This is called the luminance, or brightness, of the surface. Note that a large wall and a small face can have the same brightness, therefore we do not want to measure the total amount of light reflected from the wall, but the amount of light reflected from a standard measuring area so that we can compare the brighness of different surfaces regardless of size. Note also that this is an inside-out measuring situation. Previously we were using a card of a standard area to count rays diverging from a point source. When measuring brightness we will count rays arriving at one point (our eye) from a surface area of a standard size.

We sit in the auditorium and look at the stage. Rays fall on a uniformly painted flat and are scattered by it. Some go one way, some another, and from each square foot of the flat, some number of rays come to our eye and are counted. This is the brightness of the flat in foot-lamberts. If we count the number of rays from one square meter we are measuring the brightness in meter-lamberts, also known as nits. Don't ask. The number of rays reflected from an area one square centimeter is lamberts.

Of course this is a simplified case. Depending on how the flat is painted and lighted we will see different brightnesses across its surface. The point is that the way those different brightnesses are measured or expressed is in units of lamberts (or foot-lamberts or meter-lamberts,) which are defined by the simpler example of measuring a uniform surface illuminated by a uniform field of light.

Final confusing detail for those who care: brightness is measured in candles per area, not lumens per area, because the receptor is a point (the eye) rather than an area (the card). Basically, if you could measure area on a point it would be in steradians. So in the inside-out measurement of brightness, we use candles (lumens per steradian, you'll recall) instead of straight lumens.

Lastly, keep in mind that these are technical terms, but some have less precise common usage. Forgive the director who asks for brighter lighting when it is technically the scene that needs to be made brighter, perhaps by increasing the intensity of the illumination.
The nuts and bolts...
  • Basic measuring unit:
    lumen, which is an energy of 1/683 joules per second, measured at a wavelength of 555 nanometers.
  • Luminous intensity of a point source:
    candles or candelas = lumens per steradian. These two units differ very slightly, but are close enough to be equivalent for stage lighting purposes. What is a steradian? Why, a solid angle subtending an area of R squared at distance R from the apex, of course.
  • Illumination of a surface by a point source:
    lux = lumens per square meter
    footcandles = lumens per square foot
    phots = lumens per square centimeter
  • Brightness reflected or emitted from a surface:
    lamberts = candles per square centimeter
    foot-lamberts = candles per square foot
    meter-lamberts or nits = candles per square meter
-Eric Cornwell