Re: Beats: quiz question

["John S. Denker" <jsd@MONMOUTH.COM>]

Mark Sylvester wrote:

> I recall hearing (on this list, I believe) that one can detect
> optical beats by illuminating a photodiode with lasers of slightly
> different frequency. The beat signal is detected electronically.
> Could someone recall the details?

In the general case, there's not much to add.

But let's look at a particular case that shows
why this is not just possible but important.

Illuminate the diode with two beams.  One is the
small, time-varying "input signal" beam we are
trying to measure.  The other is a strong "local
oscillator" beam, which we make sure has a
constant amplitude.

The diode output voltage is a nonlinear function of
the optical input voltage.  Expand the nonlinearity
in a Taylor series.  Drop everything beyond the
second-order term because we assume the diode is
only weakly nonlinear.  Drop the zeroth order
term by conservation of energy;  the diode doesn't
put out energy in the dark.  Drop the first-order
term because by Floquet's theorem, for a linear
system, the input frequency equals the output
frequency, and we will apply a low-pass filter
to the output of the diode that excludes multi-
hundred-terahertz signals (which is reeeeally easy
to do).  So all that remains is the voltage-squared
term.

So what we have is
  Sif = (Asig sin(wsig t) + Alo sin(wlo t))^2
where
  Asig := smallish input signal voltage
  Alo  := large local oscillator voltage
  wsig := signal frequency
  wlo  := local oscillator frequency
  Sif  := intermediate-frequency voltage

Carry out the square.  Throw away everything except
the cross term;  the discarded terms are either DC
or multi-hundred-terahertz or of order small squared.

The remaining term is
  Sif = Alo Asig sin(wsig t) sin(wlo t)
By the usual trig identity, rewrite this in terms of
sines at the sum and difference frequencies.  Discard
the sum frequency;  too high.  All that remains is
  Sif = .5 Alo Asig sin((wsig - wlo) t)

Run this IF signal right into your IF amplifier and
high-speed digitizer, and store it on disk.

This is a wonderful result.  This might be a jillion
times larger signal than you would get if you tried
to observe Asig directly (since then you would only
be sensitive to Asig^2, i.e. small squared).

You can get pretty amazingly stable lasers:
  http://www.google.com/search?q=laser+stability+sub-khz

======

For extra credit, split the LO to produce two beams,
and delay one of them by a quarter wave to produce
cos(wlo t) as well as the aforementioned sin(wlo t).
Use two diodes and two digitizers.  You now know
everything there is to know about the original input
signal.  In particular, this inphase+quadrature
detector can tell the difference between positive
and negative IF frequencies (i.e. wsig above and/or
below wlo) which a single-detector system cannot.
This can be verrrry handy in a research setting,
where you don't know exactly what you're looking
for;  you just know it's gonna be small, and you
have a rough idea of the wsig frequency.

  http://www.google.com/search?q=optical+single-sideband+heterodyne