Heterodyne detection

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Heterodyne detection is a method of detecting radiation by non-linear mixing with radiation of a reference frequency. It is commonly used in telecommunications and astronomy for detecting and analysing signals.

The radiation in question is most commonly either radio waves (see superheterodyne receiver) or light (see interferometry). The reference radiation is known as the local oscillator. The signal and the local oscillator are superimposed at a mixer. The mixer, which is commonly a (photo-)diode, has a non-linear response to the amplitude, that is, at least part of the output is proportional to the square of the input.

Let the electric field of the received signal be

E_\mathrm{sig}\cos(\omega_\mathrm\mathrm{sig}t+\varphi)\,

and that of the local oscillator be

E_\mathrm{LO}\cos(\omega_\mathrm{LO}t).\,

For simplicity, assume that the output of the detector I is proportional to the square of the amplitude:

I\propto \left( E_\mathrm{sig}\cos(\omega_\mathrm{sig}t+\varphi) + E_\mathrm{LO}\cos(\omega_\mathrm{LO}t) \right)^2
=\frac{E_\mathrm{sig}^2}{2}\left( 1+\cos(2\omega_\mathrm{sig}t+2\varphi) \right)
+ \frac{E_\mathrm{LO}^2}{2}(1+\cos(2\omega_\mathrm{LO}t))
+ E_\mathrm{sig}E_\mathrm{LO} \left[ \cos((\omega_\mathrm{sig}+\omega_\mathrm{LO})t+\varphi) + \cos((\omega_\mathrm{sig}-\omega_\mathrm{LO})t+\varphi) \right]
=\underbrace{\frac{E_\mathrm{sig}^2+E_\mathrm{LO}^2}{2}}_{constant\;component}+\underbrace{\frac{E_\mathrm{sig}^2}{2}\cos(2\omega_\mathrm{sig}t+2\varphi) + \frac{E_\mathrm{LO}^2}{2}\cos(2\omega_\mathrm{LO}t) + E_\mathrm{sig}E_\mathrm{LO} \cos((\omega_\mathrm{sig}+\omega_\mathrm{LO})t+\varphi)}_{high\;frequency\;component}
+ E_\mathrm{sig}E_\mathrm{LO} \cos((\omega_\mathrm{sig}-\omega_\mathrm{LO})t+\varphi).

The output has high frequency (2ωsig and 2ωLO) and constant components. In heterodyne detection, the high frequency components and usually the constant components are filtered out, leaving the intermediate (beat) frequency at ωsig − ωLO. The amplitude of this last component is proportional to the amplitude of the signal radiation. With appropriate signal analysis the phase of the signal can be recovered as well.

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