How do I model the optical coupling between two optoelectronic devices?
The key to figuring out how much of the LED's light falls onto the photodiode at a given distance lies in the unit of a steradian (sr). This is the 3D version of the 2D radian.

But if you just used the radian casually, you probably don't even know what a radian physically is. You probably were thinking in degrees first, and radians second. This won't work for steradians. It's actually the other way around and degrees is the made-up unit.



In 2D you probably have a mental shortcut of 360 degrees to a circle that you can use to bypass noticing what a radian actually is. In 3D this makes no sense since the angle isn't contsrained to a single plane.

First, a radian is the angle (whatever it may be) where the arc drawn is equal to the radius of the circle. Let that sink in for a bit. Focus less on the angle part, and more about the arc length traced out by that angle. The fraction of that arc length over the circumference is the actual definition of a radian.

The thing I want to emphasize is to think of a radian as less of an angle in and of itself, but as a specific fraction of the circle's circumference. Focus less on the angle part and more on the arc length part (technically not the arc length since circles come in many sizes so it's the arc length's fraction of the circumference).

Now a sterdian is the very similar but in 3D. A steradian is the "angle" of a cone whose area ends up being the squared of the radius on the surface of the sphere. So a steradian is actually the cone angle that traces out a particular fraction of the total surface area of a sphere.



That means that if the LED produces 1W/sr, then at 1 meter (i.e. a sphere with a radius of 1 meter) the area traced out by a steradian is 1 meter squared. That means that at 1 meter, one steradian is equal to 1 meter squared. So 1W/sr is the same as 1W/m^2 at that distance. Easy!

So what it comes down to is simply finding the area that corresponds to one steradian for an imaginary sphere around the emitter (where the detector sits at) and then using that number with the datasheet power levels.

This is the procedure:


Decide your photoemitter and photodetector distance. This is the
radius of the sphere around. The emitter sits at the center, the
detector sits at the surface.
Decide your LED current.
Look up the LED current on the curve that plots mW/sr vs drive current.
Use the math in this post to convert mW/sr to mW/area. This is the raw optical power.
Look at the peak wavelength of the emitter and look at what
detector's responsivity is at that wavelength on the curve in its
datasheet. This is a multiplier applied to the optical power    to
account for mismatch in wavelength.
Do the same thing with the field of view graphs on both the emitter
and detector datasheets. Apply both multipliers to the optical power
to account for off-axis alignment.
THe number you have now is the optical power the detector will be
exposed to. So just go to detector's datasheet and find the current
vs optical power curve and look up the output current.


It's actually embarassingly easy. I just figured this out a few weeks ago and most of that time was spent running in circles with the definition of a steradian until I realized it did not make sense because I did not understand what a radian was. It took two minutes to figure that out and from there it took another 5 minutes to figure out what a steradian was and how to use it.

[All images from Wikipedia]