Calculating Flux for non-lambertian emitter

[chrisewolf]

Hey everyone!

I am struggling with the following problem:
I am calculating the ratio of photons emitted to electrons generated in an LED. Due to the architecture the device shows a strong cavity effect and as such the emission profile is not lambertian but very "flat".
Unfortunately I can not use an integrating sphere with my apparatus. So I tried to measure the luminance (cd/m^2) for angles from the surface normal down to 80°. Indeed this measurements show a strong deviation from the lambertian cosine law.

Now I need to calculate the luminous flux (for an lambertian this is [itex]\Phi=\pi L[\latex]) but I am not sure how to properly do so for my case (though this seems like very basic optics).

The main confusion stems from the inclusion of the solid angle. I do not know how to properly discretize the solid angle element to somehow "sum up" over the half sphere I effectively measure (that is the luminance shows no dependence on the azimuth).


I am very thankful for every tip!

Have a nice sunday!
Chris

 

[eljose79]

Hi Chris,

I understand your struggle with calculating the luminous flux for your LED with a non-lambertian emission profile. In this case, the solid angle element you need to consider is the differential solid angle, dΩ, which is defined as the area of a spherical cap divided by the square of the radius of the sphere.

To properly calculate the luminous flux, you will need to integrate the luminance (cd/m^2) over the differential solid angle, dΩ, from 0 to 2π in azimuth and from 0 to 80° in elevation. This will give you the total luminous flux emitted by your LED in all directions. The equation for this integration is:

Φ = ∫∫L(θ,φ)cosθsinθdθdφ

Where L(θ,φ) is the luminance at a given angle θ and azimuth φ. This integral can be simplified by using the fact that your luminance does not depend on the azimuth, so you can integrate only over the elevation angle θ, from 0 to 80°. The equation then becomes:

Φ = 2π∫L(θ)cosθsinθdθ

Once you have calculated the total luminous flux, you can then divide it by the number of electrons generated to find the ratio of photons emitted to electrons generated.

I hope this helps and have a nice Sunday!