# Heterodyne detection

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.

• I don't see a question here. – Jason R May 10 '16 at 14:46
• Are you sure your transmitted signal $f(t)$ is supposed to be added to the signal from the local oscillator $c(t)$? That doesn't make sense to me. I do not have any knowledge of optical communications but in wireless communications the transmitted signal is multiplied with the signal from the local oscillator so the frequency of the message can either shift to baseband or to some intermediate frequency in the case of a heterodyne receiver. – KillaKem May 11 '16 at 13:36