Amplitudes in a Michelson interferometer

[Decimal]

Hello,

I am having a hard time understanding a result relating to a michelson interferometer. I always assumed that when the beam hits the wave splitter both resulting waves will have half the amplitude of the original wave. However using this assumption does not give the correct irradiance for fringes on a michelson interferometer. $$ I = 4 I_0 * cos^2(\frac {2{\pi}d} {\lambda} + \frac {\pi} {2}) $$ Here $d$ is the difference in length between the two arms of the interferometer. I can only arrive at this expression by assuming that the amplitude of both beams after the beam amplitude is still the same as the source amplitude. Is this true, and if so, why is this? Wouldn't you be creating energy in this way?

Thanks!

 

[Orodruin]

$I_0$ is the intensity of each of the beams after passing the entire system, not the source intensity.