Some time back, I attended a seminar of a mathematician at SLAC. He discussed the information contained in phase, and the impossibility of measuring this at optical frequencies.
To illustrate, he presented some phase diagrams. He played around with those, to show the information contained - and missing.
It was misleading, as those were derived from 2-D magnitude images; i.e. sample the magnitudes, run the digital filters, extract the phase domain. Those phase diagrams weren't real sampled data.
Phase is proportional to time delay. So let's talk time domain circuitry and sampling. If you're satisfied with
90* resolution, what's the highest frequency one can sample, state of the art, using interleaved techniques and whatever cleverness?
Holograms are made with visible light, and encode the phase with good accuracy. That's maybe 600 THz. You don't get the full bandwidth up to 600 THz, just a narrow frequency band that's determined by the reference beam, but... you did say 'highest frequency'. Expanding upward through X-rays (atom-spacing wavelengths and below) will require more cleverness yet.
There are all sorts of things that folks might call "optical phase", some of which are much harder to measure than others.
_Full-bandwidth instantaneous phase of thermal light from a broad area source._ At any point on a visibly incandescent object such as the Sun or a tungsten filament, the E field has a well-defined magnitude, phase, and direction. (Otherwise it couldn't obey Maxwell's equations.)
Points more than a wavelength or two apart have independent phases, and all those independent phases have variations of order unity in times of
10**15 seconds or a bit faster, so at 8 bits per sample you'd need to measure on the order of 10**24 bytes per second per square centimetre of surface. There's no way of _storing_ all that data even if you could measure it. In any case, the instantaneous phase and polarization can be described very well statistically from first principles, so there's nothing useful to be gained by measuring it.
_Narrower-band instantaneous phase of an unresolved portion of a thermal source._ This is much easier, because we lose a factor of about
1E8 in area, times the bandwidth ratio. You can measure that phase by interfering it with a laser beam and looking at the RF. I've actually designed an instrument like that, in cooperation with an outfit in New Mexico called Mesa Photonics. It wss for a DARPA program looking for HF plumes from clandestine uranium enrichment.
_Phase differences in laser light propagating through different paths,_ as in ordinary interferometry and holography. This includes Doppler lidar and other such measurements, as well as FM detectors such as Fabry-Perots and unbalanced Mach-Zehnders used as delay discriminators.
_RF phase shifts between two laser beams with slightly different optical frequencies._ This includes laser-to-laser phase locking and heterodyne laser linewidth measurements. Beating two lasers together gives you the phase difference, so in order to infer the line shape of one laser you have to assume that the two are similar.
Using three lasers gives you three pairwise phase differences, so you can get the individual lineshapes and frequency differences uniquely. (You obviously can't get the instantaneous average frequency, but you can sometimes use a frequency-locked Ti:sapphire laser to get that too.)
_FM-to-AM measurements._ It's quite common to do FM derivative spectroscopy, where you put sinusoidal FM on a diode laser. The instantaneous optical frequency walks up and down the spectral lines, and you can show by a bit of very pretty math that the Nth harmonic interrogates the Nth derivative of the line shape.
Second-derivative spectroscopy produces the second derivative of the line shape, and second-derivative spectra are widely tabulated. The big advantage of that is that it suppresses the sloping baseline of the spectra and enhances the sharp features, which is where most of the interesting spectroscopy lives.
_"Phase of the phase"_ measurements. Back in the long ago when I was a wet-behind-the-ears postdoc, I built an atomic- and magnetic-force microscope proof-of-concept proto, which eventually became the IBM SXM ('scanned anything microscope'). It used a resonant cantilever about
100 um long, made by electro-etching a tungsten wire. The point on the end was also formed by etching and then bent mechanically into an L-shape.
The L-shaped cantilever was wiggled near its mechanical resonance using a piezo bimorph actuator, and its motion detected using a heterodyne interferometer.
The phase and amplitude of the cantilever's vibration vibration of the cantilever depend on the tuning of the cantilever's resonance, just as in every other lightly-damped second-order system. When the tip is very near the sample, the resonance gets shifted--the gradient of the tip-sample force (atomic, van der Waals, and/or magnetic) appears as a change in the spring constant of the cantilever.
The microscope works by detecting the heterodyne signal with a fast lock-in amplifier and servoing the tip-to-sample distance to keep the lock-in signal constant.
Detecting only the amplitude of the tip vibration makes it vulnerable to stiction--the normal adsorbed water layer makes the tip stick to the sample, so the vibration stops. The servo thinks the tip is way, way too close, so it pulls it back and back until it breaks loose. This of course makes it ring strongly at its free resonance, so the servo thinks the tip is way, way too far away, and sends the tip crashing into the sample again--lather rinse repeat.
Moving the excitation frequency a bit further away, so that it's outside the servo bandwidth, and detecting the phase of the response instead, allows servoing stably much closer to the sample.
Those are most of the more upmarket optical phase measurements, the ones actually associated with the phase of the electromagnetic fields in some clear way.
_Phase unwrapping._ Phase is generally measured modulo 2 pi, though PLL things can go much further in some cases. Joining a set of these 'wrapped' phases into a continuous function requires unwrapping the phase, i.e. adding judiciously chosen multiples of 2 pi to each data point to get rid of the jumps. This isn't too hard in 1D, but in higher dimensions it becomes a thorny problem in general.
_Phase retrieval._ There are also phases associated in various ways with the image intensity, e.g. the phase of the optical transfer function. There are some fairly famous "phase retrieval" algorithms that allow measuring things like topography from intensity-only images.
The original Fienup algorithm iteratively applies a positivity constraint (optical intensity is never negative) and enforces compact support in the frequency domain, because an optical system can't reproduce spatial frequencies higher than 2 lambda/NA, where NA is the numerical aperture of the received light (related to the f-number).
More recent phase retrieval algorithms use the propagation-of-intensity equation, which is based on the paraxial Helmholtz propagator.
It isn't quite right to say that either. You can measure the closure phase once you have three or more measurement points in the optical. That is how VLBI and optical aperture synthesis interferometry works.
They have to have optical bench quality mixers and incredibly thermally stable environment for all the optical components. They tend to work in the near IR to um bands rather than optical because it is easier.
COAST at MRAO Cambridge was the original proof of concept. The servos that kept the white light fringe in play caused it to be christened the "telescope that sings". They applied radio astronomy techniques in the optical band and overcame very significant engineering challenges.
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Main limitation is that it will only work for a handful of very bright stars - but they did get some impressive results for the time.
Sometimes you can back solve the 2D intensity image to obtain a self consistent 3D solution. It depends how complex the target is. They have become increasingly good at making suitable heuristic assumptions and so obtaining 3D structures that will produce a given 2D interference pattern. It just requires a lot of data at different cunningly chosen angles. Sometimes the structure remains ambiguous with multiple targets able to reproduce exactly the same set of diffraction spots.
This is the current state of the art near optical interferometer:
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It is very tough to do since they have to correlate all of the baselines at once which means a hell of a lot of pass compensators and beam splitters. It has a tiny working field of view and needs bright compact objects to stand any chance of working at all.
The ALMA Atacama desert mm wave interferometer is another closure phase aperture synthesis imaging system in the same vein.
There are attosecond lasers that produce a light pulse that's only a few wiggles long. Those can be used to sample another light source (in a nonlinear medium) or have been used to sample electrical signals using basically a very fast photoconductor.
That's mostly above my pay grade. But I'm not complaining.
In Japan, they have a tradition, persons of noteworthy accomplishment are declared a national resource. I'm going to write my congressman, to initiate that here, and add your name to the list.
Anyway, for context, at SLAC, they're analyzing molecular structures, using X ray scattering. They've dropped out of the particle biz, doing more chemistry than physics, so to speak. That's nominal, though the real raison d'etre is a PhD factory.
I'm not sure which of your recommended techniques would apply there.
"National resource" sounds bad--visions of being drained to cover up stupid policy mistakes. ;)
Phase retrieval in X-ray crystallography is an area of active research, and AFAIK the general case hasn't been solved. A quick search pulled up this math paper
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, which indicates that nobody knows how to do it ATM.
The usual method is to guess a structure based on the positions of the diffraction peaks, which give the linear separations of the scattering centres (atoms); compute the expected diffraction pattern; and iteratively refine the model structure. (That step is where the magic happens.)
That is actually a bad translation into English by whoever made it.
The actual official Japanese translation is "National Living Treasure" and it encompasses people in the arts and artisan skills who can make the likes of traditional samurai swords, pottery or phenomenally complex puzzle boxes according to the traditional ways of doing things.
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It comes with a stipend of 2MY = $20k a year to help them survive. There are just over a hundred of them in existence.
Think living history museum and you will not be too far out.
The general case is probably unsolvable. Radio astronomers worked out a long time ago (ISTR ~1980's that many patterns of sky brightness could lead to the same autocorrelation function ie. intensity).
Point scatterers is marginally more tractable but still very tough.
One trick they have up their sleeve sometimes is to substitute a heavier atom into one of the scattering centres which breaks the degeneracy.
I met a guy a couple of years back that claimed they had a more general solution for some reasonably important subclass of these problems but I never got a chance to quiz him on the details. I never really was into phaseless observations because we worked very hard to preserve phase if we could and closure phase across as many closed loops if we couldn't.
Often it was a hybrid of the two. Absolute phase to determine position on the sky and closure phase to refine what it really looks like.
A big point in favour of the crystallographers vs. the astronomers is that they can rotate their samples any way they like, so the problems aren't strictly equivalent.
And a reason SLAC gets involved, is that they can produce polarized X-rays which gives orientation info even on polycrystalline or powdered samples. A crystal's X-ray absorption by photoelectric effect depends on outgoing electrons and that outgoing electron can only go in the direction that polarization of the X-ray permits. Some of those directions are blocked by the lattice, so absorption probability is modulated accordingly.
Of course, now the problem is electron wavefunction phase, not electromagnetic.
Wow, very ..um.. illuminating. I just attended a talk in a conference last week which seems to be related. The topic was plenoptic imaging, with an added twist of utilizing (Hanbury-Twiss Brown -like) correlations somehow to determine the ray directions. The illumination had to be either chaotic light or entangled photon pairs for the scheme to work. I'll need to find time to wrap my head around the talk contents.
There are a lot of interesting things going on in quantum imaging, some of which might even be useful someday. AFAIK most have theoretical SNR advantages over the semiclassical approach, but only at very low power levels--you almost always win big by just cranking up the power.
'Quantum illumination' looks like being an exception--it apparently works even in sunlight.
I'd be interested in what you find out about the plenoptic technique.
So Quantum Illumination still rides on? I was left under impression that the field is declining, judging by the overview doi:10.1109/MAES.2019.2957870 (although I have only glanced through the paper). Interesting to hear!
Regarding the plenoptic technique, the conference organizers have not yet made the presentation slides available. I jotted down a reference during the talk however, doi:10.1103/PhysRevLett.116.223602 . You can probably see at a glance whether the technique makes sense at all, for me it's slower to get. ( I just downloaded the PRL, it actually seems to containt pretty much all that was presented in the talk).
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