I am wondering in Homodyne detector structure with a beamsplitter, why phase of input beam to beamsplitter can be conjugated? I mean for example for an input beam operator b onto the beamsplitter at homodyne detector output we will have
$$ y=j \left[ b \exp(i \phi) - b \exp(-i \phi) \right] $$
I am concerned to know why either incident beam phase or phi can be conjugated?
Any comment regarding this concept would be greatly appreciated.