Find function which minize variance of intensity on target plane

Physics Asked on January 12, 2021

I am trying to find continuous function $sigma(x)$ for uniform intensity.

I set two parallel horizontal lines with length $W$ and with distance each other $H$.

The upper line is a target plane section, below one is a LED plane.

enter image description here

Intuitively, the $sigma(x)$ will have next propeties

  1. $sigma(x) = sigma(-x)$ on $x in [-frac{W}{2} , frac{W}{2}]$
  2. $ frac{partial sigma}{partial x} geq 0, x>0$ and $ frac{partial sigma}{partial x} = 0, x=0$
  3. $N = int_{-frac{W}{2}}^{frac{W}{2}} sigma(x) dx$ where $N$ is a number of LED.

I calculated intensity using $I(r,theta) = I_0 A_0frac{1}{r^2} cos^m(theta) = I_0 A_0 H^m((x-t)^2 +H^2)^{-{(m/2 +1)}}$.
On one point of $t$, the intensity can be calculated as

$$I(t, sigma) = I_0 A_0 H^m int_{-frac{W}{2}}^{frac{W}{2}} sigma(x) ((x-t)^2 +H^2)^{-{(m/2 +1)}} dx$$

and the average value is

$$I_{avg}(sigma) =I_0 A_0 H^m int_{-frac{W}{2}}^{frac{W}{2}} int_{-frac{W}{2}}^{frac{W}{2}} sigma(x) ((x-t)^2 +H^2)^{-{(m/2 +1)}} dx dt $$

Therefore, the variance of intensity will be

$$V(sigma) =frac{1}{W} int_{-frac{W}{2}}^{frac{W}{2}} (I(t,sigma) – I_{avg}(sigma))^2 dt $$

The optimized function $sigma_{opt}$ will make it a minimum.
$$V(sigma_{opt}) = V_{min}$$

If I use variation method, I think it will be

$$frac{partial V}{partial sigma} = int_{-frac{W}{2}}^{frac{W}{2}} 2 (I-I_{avg})frac{partial }{partial sigma} (I-I_{avg}) dt$$

It is work that I did and I stuck. I think the calculus of variations methods will be helpful. However, I am not familiar to work with such a multi-integrated function. (I am a sophomore undergraduate student in physics major.)

On last equation,

$$frac{partial }{partial sigma} (I-I_{avg})$$

The $I-I_{avg}$ is a functional with integral, so I think I cannot just calculate it as usual Lagrangian.

So my question is " Is it possible to find such function $sigma $ by derivative of functional? "

Additionally, are there other approachs to find such $sigma$?

One thing I worried about is it is impossible to solve, actually, I searched some papers in optics about arrange LED sources and they using computational algorithms approaches not the work I did.

I am considering using numerical solution with assumption $sigma(x) = sum a_n x^{2n}$

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