Paradox or error in design?

Question

Currently I'm writing a homework for my school. I've made an experiment built this way:

There is a laser pointed at a half reflecting mirror which reflects 50% at a wall. The other half cross the half mirror to a full mirror this is reflecting 100% back to the middle. There the laser is reflected by a half to a second mirror and by a other half back into the laser.

The question I ask myself was how much energy is reaching the wall and how much is reflected back into the laser. Based on the idea that one half of the laser beam is circling through the mirror-halfmirror-mirror system which outputs every roundtrip a part of the energy to the wall or back into the laser, I designed two formulas (output of the Laser is 100%).

The first to describe the total amount of energy reaching the wall: $$ E_{wall}= \sum_{k=1}^\infty \frac{1}{2^k}=\lim\limits_{n \rightarrow \infty}{\sum_{k=1}^n \frac{1}{2^k}} =1 (100\%) $$

And the second to describe the amount of energy which is mirrored back into the laser: $$ E_{Laser}=\sum_{k=2}^\infty \frac{1}{2^k}=\lim\limits_{n \rightarrow \infty}{\sum_{k=2}^n \frac{1}{2^k}} =0.5(50\%) $$

My problem is now that there are 150% of total energy output. Is there any problem with my understanding of the mathematical problem behind it or is it a problem related to the set-up of the experiment or the formulas?

Answer 1

Let:
$P_L$ = a power emited by the laser,
$P_1$ = a power reflecting on the first miror (in front of the laser),
$P_2$ = a power reflecting on the side mirror (in front of the wall),
$P_W$ = a power reaching the wall,
$P_R$ = a power returning to the laser (and dissipating there).

Then: $$ \begin{align} P_1 & = \tfrac 12 P_L + \tfrac 12 P_2 \\ P_W & = \tfrac 12 P_L + \tfrac 12 P_2 \\ P_2 & = \tfrac 12 P_1 \\ P_R & = \tfrac 12 P_1 \end{align} $$ From the first and the third: $$P_1=\tfrac 12 P_L + \tfrac 14 P_1 = \tfrac 23 P_L$$ so from the third: $$P_2 = \tfrac 13 P_L$$ and from the fourth $$P_R = \tfrac 13 P_L$$ finally from the second $$P_W = \tfrac 23 P_L$$

Now $P_W + P_R = P_L$ that is your $100\%$, as there is obviously no power loss on ideal mirrors and no gain, either.

Answer 2

If I understand this correctly, half the energy goes to the wall directly after the first meeting of the beam with the half-mirror. The other half of the energy reaches the wall by , with probability 1, reach the wall by going back into the laser, coming back out of the laser, and off the half-meter to the wall. 1/2 +1/2 =1. If I am wrong and the energy going back into the laser does not come back out,then only half the energy reaches the wall