Beam splitter energy conservation

[McLaren Rulez]

Hi,

If we describe a beam splitter as follows:

e^{ikx} -> \sqrt{T}e^{ikx} + \sqrt{R}e^{i\theta}e^{iky}
e^{iky} -> \sqrt{T}e^{iky} + \sqrt{R}e^{i\theta'}e^{ikx}

then \theta+\theta'=\pi is a condition to ensure conservation of energy according to my text.

I tried working this out by taking Ae^{ikx}+Be^{iky} incident on a beam splitter. The incident energy is A^{2}+B^{2}.

The output is

A\sqrt{T}e^{ikx} + A\sqrt{R}e^{i\theta}e^{iky} +B\sqrt{T}e^{iky} + B\sqrt{R}e^{i\theta'}e^{ikx}

Its energy is A^{2} + B^{2} + AB\sqrt{TR}(e^{i\theta}+e^{-i\theta})+AB\sqrt{TR}(e^{i\theta'}+e^{-i\theta'})

So, to preserve conservation, we must have 2cos(\theta)+2cos(\theta')=0

That gives \theta+\theta'=\pi or \theta -\theta'=\pi. But I never see this second result anywhere. Why is it there and how is it eliminated?

Thank you

 

[McLaren Rulez]

Anyone? Strange thing is that I have the answer but an extra solution which should be invalid. Thank you