I'm calculating the received optical power by an photodiode over a certain distance $d$.
Basically, the received power is inversely proportional to the squared distance $d^{2}$ between the light and the photodiode.
However, I want to split this distance $d$ into two segments $d_{1}$ and $d_{2}$ and obtain the same power by combining them.
Formulating to simplify it to a most mathematical problem:
I have a coefficient $H$ which is inversely proportional to another number, squared:
$H = \frac{C}{d^2}$
where $C$ is a number which is function of many factors, but it is constant in this case, so it doesn't matter.
Now, I want to break this distance into two parts:
$d = d_{1}+d_{2}$
For each of these distances I can obtain a coefficient, say
$a = \frac{C}{d_{1}^2}$ and $b = \frac{C}{d_{2}^2}$
And now, from $a$ and $b$ (and/or $d_{1}$ and $d_{2}$), I want to obtain $H$ again. Like:
$H = f(d_{1}, d_{2})$
$H = f(a, b)$
How can I make it so?
I have a feeling that this should be very simple, but I'm having a hard time with it.
$H=f(d_1,d_2)$ is easy because $d=d_1+d_2$. So since $H=\frac{C}{d^2}$, you just have $H=\frac{C}{(d_1+d_2)^2}$.
To write $H=g(a,b)$, solve the equations $a=\frac{C}{d_1^2}$ and $b=\frac{C}{d_2^2}$ for $d_1$ and $d_2$, respectively. Then plug these solutions in for $d_1$ and $d_2$ in the expression $H=\frac{C}{(d_1+d_2)^2}$.