Etendue, Exposure, and Equivalence

There are many explanations of how exposure and light/aperture/focal length(FL) works, but most make a basic error. Particularly in respect to aperture diameter relative to FL (for the purposes of this article I am considering f-ratio to be equivalent to t-ratio).

The basic fact is that all lenses of a given f-ratio transmit the same amount of light relative to what they receive (same percentage). This is what makes f-ratio a constant of exposure (i.e. the sunny 16 rule, handheld meters, etc).


When a longer FL lens is used the f-ratio increases if the aperture diameter remains constant, and the amount of light transmitted is reduced. This is what causes a variable aperture zoom lens to be "variable," the aperture diameter (entrance pupil) remains constant. Using a longer FL lens is an increase in magnification, which is essentially just enlargement of the image circle. And when you increase the size of the image circle you spread the light out over a larger area resulting in a loss of light (known as "bellows factor"). In order to prevent this loss of light the aperture diameter must also be increased in order to maintain the same f-ratio. This is what happens with a constant aperture zoom lens. A larger entrance pupil for a longer FL does not, in itself, increase the amount of light; it compensates for what would otherwise be a loss of light (reduced exposure).
For a given shutter speed and ISO, a subject with a brightness of 10 lux and correct exposure at f/1 would require a brightness of 20 lux for correct exposure at f/1.4, and 40 lux at f/2. And this is irrespective of the focal length or sensor used. 
But when a source is recorded larger it *should* contain more light... because it is larger. And it does. Where this "more light" comes from is explained by etendue.

Etendue is the quantity (volume) of light in phase-space and characterizes how concentrated/spread out the light is. In this next diagram I have drawn (to scale) the source field of view (FOV) which is the spread of light over distance, and two different lens FOV's.


In the upper left I have calculated the etendue (total light) contained in the two source areas (defined by the lens' FOV bases). And as you can see the total light contained in both areas is the same, and it should be if the light from the source is constant.

However, I have also calculated the amount of light received by both FL's in two dimensions and in three dimensions at the bottom of the image. And as you can see, the longer FL lens (small/blue area) receives a bit over 2x as much. 

We can also calculate the amounts as areas/volumes using simpler math. In this drawing I have calculated the area of the lens's FOV's in two dimensions, and the volume of them in three dimension.

In this case the wider FOV contains more area/light in total. However, when that amount of light is flattened onto a sensor or a piece of film the smaller FOV again receives more light from the source. It is only 50% more in 2D, but in 3D it is again 2x more light (i.e. the source is 2x the size on the sensor).
Additionally I have included an example of how the amount/area of light collected can be increased by increasing the size of the objective lens (typically the objective lens diameter is the "aperture stop" of a lens and limits the maximum entrance pupil size). The same percentage of transmission (f/ratio) when there is more light/light per area equals more light transmitted. Because the speed of light is not a limiting factor, the amount of light traveling within a lens' FOV total area is the amount of light that will be recorded with a given shutter speed.

Basically, what all of this shows is that a longer FL lens transmits more light because *there is more light*. And that does mean that a larger sensor receives more light; because it requires a longer FL lens (for the same recorded FOV).


We can then correlate all of this information into the following understanding of how everything works together. 

The size of the source, and the amount of light from the source is dependent on how far from the source the light is gathered (the etendue).
The change in aperture relative to FL required to maintain the same light transmission (%) only compensates for the change in FL itself (making f-ratio a constant).

However, the intensity (exposure) of the light per point is controlled by the real distance between the camera and the source.
This is not exactly sensor illuminance/exposure... it *is* but, when the f-ratio is held constant;
-If the source is smaller than the lens FOV the exposure will change as the size and amount of light changes (matrix metering).
-If the size of the source equals or exceeds the FOV the exposure will not change as more or less of the source is included (matrix metering).
-In both cases the exposure per point will remain constant (spot metering), because the physical camera distance is constant.




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