# 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.

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.

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

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