# What does it mean to measure a complex electric field?

One of the Event Horizon Telescope papers says the following:

Every antenna $$i$$ in an interferometric array records the incoming complex electric field as a function of time, frequency, and polarization: $$E_i(t, \nu, P)$$. Every pair of sites $$(i, j)$$ then gives an interferometric visibility, defined as the complex cross-correlation between their recorded electric fields,

$$V_{ij}(t, \nu, P_1, P_2) = \langle E_i(t, \nu, P_1) E_j^*(t, \nu, P_2) \rangle.$$

I understand how complex exponentials are used to represent real oscillations, and how the Fourier transform of a real function generically gives a complex function, and so on. But I always thought of complex time-dependent fields as a theoretical construct, while all measurements should give a real number, the real part of the complex number. What does it mean, then, to say that an antenna measures a complex electric field?

In practice that an antenna measures the "complex electric field" means that it is sensitive to its polarization dependent response of the instantaneous phase $$\phi(t)$$ and amplitude $$a(t)$$ of the incoming EM wave, say $$\mathbf E = \hat x E_0 a(t) \cos(\omega_0t + \phi(t))$$.

When speaking of an array of such antennas each equipped with its own receiver/transmitter it also means that there is a common time base to which these phases and amplitudes can be coherently referenced. The "complexness" of this system is manifested in this coherency.

Let there be a common reference, say, whose phase and amplitude are adjusted to be the same at the antenna phase centers $$r_c(t) = r_0(t) \cos(\omega_0t + \psi_0)$$ and along with its $$90^0$$ shifted version $$r_c(t) = r_0(t) \sin(\omega_0t + \psi_0)$$.

When both used in the (I/Q) demodulator against the received signal $$a_k(t) \cos(\omega_0t + \phi_k(t)$$ in the $$k^{th}$$ antenna you get a pair of voltages $$x_k(t)=a_k(t)\cos(\phi_k(t)-\psi_0)$$ and $$y_k(t) = a_k(t)\sin(\phi_k(t)-\psi_0)$$. This pair can be viewed as the real and imaginary part of a complex number $$z_k(t)=x_k(t)+\mathfrak j y_k(t)$$. This complex time function can be filtered, detected, filtered again, compared against thresholds, etc., a coherent baseband representation of the complex wavefields as seen by the phased array.

• Thanks for your answer, though unfortunately it goes a bit over my head, not being an experimental physicist. What's the point of having the reference wave? Can we not measure the field $E(t)$ directly and work with that? I assume we can't, but I'm not sure why. May 7, 2023 at 15:06
• For observing a remote wave with a single antenna the reason to have a frequency coherent "reference" oscillator is to remove the incoming wave's dependence on the carrier frequency, that is to convert $a(t)\cos(\omega_0t+\phi(t)) = A(t)\cos(\omega_0t)+B(t)\sin(\omega_0t)$ where the rates $\frac{|\dot a|}{|a|}\ll \omega_0$ and $\dot \phi \ll \omega_0$ to a pair of baseband signals $A(t), B(t)$ for detailed processing. That requires precise knowledge of $\omega_0$. If you have several such receivers you also have to distribute their individual phases so that .... May 7, 2023 at 15:18
• ...they can be combined into a single pair of numbers otherwise the various $A_k, B_k$ values will depend on the phases $\psi_k$ of the distributed reference oscillators at the $k^{th}$ receiver. Specifically, a down-converter mixer is a "multiplier" and a low-pass filter. E.g., multiplication: $a\cos(\omega_0t+\phi)\cos(\omega_0t+\psi)=\frac{1}{2}a(\cos (\phi-\psi)-\cos (2\omega_0t+ \phi+\psi))$. Follow this with an LPF to remove the double frequency term $2\omega_0$ and you get the baseband term $a\cos (\phi-\psi)$. Now do the same mixing with $a\sin (\omega_0t+\psi)$ and you get... May 7, 2023 at 15:28
• ... a term like $a\sin(\phi-\psi)$. If you have several antennas/receivers then you get for each receiver a pair of time functions $a\cos(\phi-\psi_k), a\sin(\phi-\psi_k)$. To minimize error, i.e., maximize Signal to Noise Ratio you want to be able to add them up, but you can only do so if all $\psi_k$ are the same; this is coherent processing. Having done that you will have the most you can get out of your receivers that is limited by normal thermal noise using linear processing. May 7, 2023 at 15:33
• The purpose of the frequency conversion is to move the received signal from the RF/IR/optical etc. region to a lower frequency where it can be measured or processed. It is possible to make low noise amplifiers even in the THz range but even then ultimately all decisions are made at dc and the earlier one moves from the higher to a lower frequency the easier the signal processing becomes. It is possible to measure the optical wave intensity incoherently (e.g., bolometer) but if you wish to combine several such measurements from the same source you lose at least half the available signal power. May 7, 2023 at 15:45

This is called I/Q detection for "in-phase" and "quadrature". The transmit is synched to a STALO (stable local oscillator), and the receive signal is mixed with the STALO in-phase, and 180 degree out of phase (quadrature). Basically one is $$\cos\omega t$$, and the other is $$\sin\omega t$$.

Wikipedia provides the following picture: