> 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?
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
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
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
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.
So all in all you can do a whole lot with optical phase, and of course
this is far from an exhaustive list.
Dr Philip C D Hobbs
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510