Re: Phosphor conversion of photons in LEDs
I was looking over
this reference TheDriver supplied. In the post, SMA is describing his experiment where he determines the luminance gain arising from a large wavien collar centered on a dedomed XP-G2. I'll describe what I think is going on here with this configuration, and I hope someone of experience would jump in to guide and correct my thinking.
It's hard to say exactly whether the XP-G2
SMA@Taschenlampen Forums pulled out of his scrap box was bare phosphor, had a sealant over the bare phosphor or if the dedome was a slice job with some silicone remaining over the bare LED. If someone here knows
SMA, that information would be helpful in my understanding of photon recycling. This fact is critical to understanding the behavior of reflected photons returning to the emitter surface.
Assuming the XP-G2 has a seal and the sealing material has a refractive index of 1.5 like silicone, the critical angle between air and silicone would be about 42 degrees. (measured from the vertical axis
) If the seal is perfectly smooth (assume for the moment), I would expect all photons approaching the phosphor at 42 degrees or more would simply bounce off the barrier and head to the opposite side of the collar for another 99% reflection and 1% absorption. It would take a number of reflections back and forth like a PONG game to be completely absorbed by the mirror.
Photons approaching at less than 42 degrees from vertical axis (regardless of wavelength) would enter the seal, then the phosphor, and some with enough energy (shorter wavelength) would be converted to lower energy photons. Those surviving, converted and unconverted photons would take another stab at a favorable exit angle that allows escaping the dome or get stuck in a reflective loop until completely absorbed by the mirror.
In essence, only the region of the mirror from 30-42 degrees supports increasing the luminance of the LED. The region from 42-90 degrees would eventually be absorbed by the mirror.
Of course this is not a perfect world with perfect surfaces, so I would expect the line to be less tightly defined (blurred/transitioned). Roughness of the LED surface would change angles of incidence to favor more re-absorption and emission, so I would anticipate the useful area of the mirror might extend beyond 42 degrees a bit.
If I use the following formula:
GAIN = K* sin2γ / sin2θ
(Ratio of flux through solid angles times a correction factor: constant * [critical angle of the die's top layer / half angle of the collar port].)
Where
γ=42 degrees (gamma) is the critical angle of the first layer of the LED and
θ=30 degrees (theta) is the half-angle of the collar's port (both measured from the vertical axis), and GAIN is about 2.20 in the posted experiment from Taschenlampen Forums, K becomes 1.23. If I set K to 1 and predict the angle that divides the useful from non-useful regions of the mirror, I get 48 degrees, which is close to the critical angle, deviated only by 6 degrees. I would therefore suspect surface roughness is responsible for the 6 degree offset in prediction of that line.
If the phosphor were bare, I would expect the critical angle to be around 34 degrees from vertical. The phosphor surface would probably be rougher than a sealed surface so I would expect the line to be lower than 40 degrees but probably not beyond 48 degrees.
This is all just conjecture from a photon noob, and I'm not trying to assert that that is what is actually going on here. But at the moment, it is what I'm thinking is happening as I'm trying to understand this physics.
Anyone here with some expertise that can give some knowledgeable insight, guidance or correction? Thanks.
EDIT: Conceptual error: I re-examined my calculations for photons approaching the die at shallow angles.
Snells law can't describe an angle of approach that prevents photons from entering a slower medium (air to phosphor) like it can in reverse (phosphor to air) to describe TIR.
If photons in air coming from 70 degrees (from vertical axis) impact the phosphor surface, refraction into the die takes place 31 degrees from the vertical axis. Even 90 degrees has a refractive angle of 34 degrees from normal, as weird as that seems to my intuition (due to particle wave duality). So refraction is always taking place from all angles of approach coming from the collar to the die.