Prove that with appropriate choices of the phase $\varphi$, one can recover the quadratures of $m(t).$
Attempt
The context of this question is weak optical signals combined with a strong local oscillator at a beamsplitter. And $m(t)$ refers to the complex modulating signal.
The signal field is given by: $f(t)=Re \Big[ m(t) \exp (j2 \pi \nu_c) \Big],$ with $m(t)=m_1 (t) + j m_2 (t).$
This is linearly combined with the stronger coherent local oscillator which has the form:
$$c(t) = Re \{ A \exp \Big[ j(2 \pi \nu_{het} t - \varphi) \Big] \},$$
where $\nu_{het} = \nu_c + \Delta,$ has been shifted from the carrier frequency by $\Delta.$ So the output field would be:
$$g_1 (t) = Re \{ R \ A \exp \Big[ j(2 \pi \nu_{het} t - \varphi) \Big] + T \ m(t) \exp (j 2 \pi \nu_c) \},$$
where $R$ and $T$ are reflection and transmission coefficients for the beamsplitter (I was told that for this problem we can assume for simplicity that $R$ is real and $T$ is imaginary).
I am unsure how to proceed with the proof so any explanation would be appreciated.
