XPost: sci.electronics.design   
   From: pcdhSpamMeSenseless@electrooptical.net   
      
   RichD wrote:   
   > 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?   
   >   
   > --   
   > Rich   
      
   There are all sorts of things that folks might call "optical phase",   
   some of which are much harder to measure than others.   
      
   1.  _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.   
      
   2. _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.   
      
   3. _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.   
      
   4. _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.)   
      
   5. _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.   
      
   6. _"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.   
      
   8. _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.   
      
   9.  _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   
      
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