Similarities and differences between optical and radio interferometry
David Buscher
21st NRAO Synthesis Imaging Summer School
Venus @ 19GHz (Perley 2016)
Vega @ 136 THz (Aufdenberg+ 2006)
We will order the comparison from the fundamental to the contingent
- Physics: coherence, quantum optics, thermodynamics
- Environment: atmospheric "seeing"
- Technology: telescopes, waveguides, delay lines, detectors, data analysis
Measurement equation
Optical fringe patterns measure the correlation of fields from the telescopes
\begin{aligned}
i(x)\propto &\left\langle\left|E_1+E_2 e^{2\pi i k x}\right|^2\right\rangle \\
=&\left\langle\left|E_1\right|^2\right\rangle+\left\langle\left|E_2\right|^2\right\rangle\\
&+2\Re\left\{\left\langle E_1 E_2^*\right\rangle e^{2\pi i k x}\right\}
\end{aligned}
The measurement equation scales straightforwardly with wavelength
V_\nu(u, v)=\iint I_\nu(l, m) e^{-2 \pi i \left(u l+v m\right)}\,\text{d} l \text{d} m
(u,v, w)=\boldsymbol{b}/\lambda
In the optical we often normalise to give the "Michelson visibility":
V_\nu(u,v)/V_\nu(0,0)
Coherent
Amplifiers
Quantum effects limit the performance of coherent amplifiers (Cave 1982)
Astronomical sources emit a limited number of photons per mode
\lambda = 1\,\text{cm, }T=3\,\text{K}\\
\Rightarrow \delta \approx 1.64
\lambda = 0.5\,\mu\text{m, }T=6000\,\text{K}\\
\Rightarrow\delta \approx 0.008
Interfering unamplified light gives the best sensitivity at optical wavelengths
Atmospheric
seeing
Radio wavelengths: water vapour
Optical wavelengths: temperature
Atmospheric "seeing" is caused by turbulent mixing of refractive index inhomogeneities
The wavefront corrugations have a fractal spatial structure
The Fried parameter defines a spatial scale over which the perturbations are ~1 radian
\begin{aligned}
D_{\phi}(\boldsymbol r,\boldsymbol r^{\prime})\equiv&\left\langle \left| \phi(\boldsymbol r^{\prime}+\boldsymbol r) - \phi(\boldsymbol r^{\prime}) \right|^{2}
\right\rangle\\
D_{\phi}(\boldsymbol r)=&6.88(r/r_{0})^{5/3}
\end{aligned}
The Fried parameter increases with increasing wavelength
r_0\propto \lambda^{6/5}
r_0 \sim 100\,\text{m -- 1\,km at radio wavelengths}
r_0 \sim 10\,\text{cm at optical wavelengths}
Wavefront perturbations across a single aperture are >>1 radian in the optical
Wind transport of the refractive index perturbations leads to temporal fluctuations
\begin{aligned}
D_{\phi}(t)\equiv&\left\langle \left|\phi(\boldsymbol r,t^{\prime}+t)-\phi(\boldsymbol r,t^{\prime})\right|^{2} \right\rangle=(t/t_{0})^{
5/3}\\
t_{0}\propto&\frac{r_{0}}{v}
\end{aligned}
The coherence time is wavelength-dependent
4 seconds
The seeing parameters themselves fluctuate
Technologies
An optical interferometer is mostly free-space optics
The primary beam gives a sub-arcsecond field of view
Wavefront distortions across a telescope reduce the coherent fringe signal
Adaptive optics systems at each telescope correct the wavefront
Increasing the telescope size can reduce the sensitivity
Beam transport using waveguides is yet to see mainstream adoption
Electronic delays are replaced by path delay
Correlators just overlap beams
Small apertures and short exposures mean that we have many, noisy measurements
We average observables which are insensitive to "antenna phase" errors
\text{ Power spectrum } = \left\langle | V |^2\right\rangle
\text{ Bispectrum } = \left\langle V_{12} V_{23} V_{31} \right\rangle
How will this change in the future?
