LASERS, cavity waves, coherence

[William Beaty <billb@ESKIMO.COM>]

On Sun, 2 May 1999, David W. Steyert wrote:

> William Beaty wrote:
> > I think it should be called "concentric" rather than "confocal", since
> > the mirrors have their center of curvatures in common, not their focal
> > lengths.  If the laser cavity was con-focus, then the output-beam
> > would be a parallel beam superposed on a sphere-wave beam with origin
> > at the center of focus (probably not usable for, say, holography).
>
> Which caused me to disagree:
> My recollection was that resonators of many different types have a role
> in lasers.  Without wanting to solve problems by appeal to authority, I
> will say that a check of my favorite laser book (Principles of Lasers,
> by Svelto) confirmed my recollection:
>
> "The disadvantages of a concentric resonator are that:(i) it produces a
> very small spot size at the resonator center, which can be a problem for
> high-power lasers. [The center of the ruby ball in William's post would
> suffer hot spot damage p.d.q.  (DWS)](ii) it is rather sensitive to
> mirror misalignment. . . . Confocal resonators, on the other hand,
> typically give a spot size that is too small for effective use of all
> the available cross section of the laser medium.  Plane-parallel
> resonators can make good use of the cross section.  Like concentric
> resonators, however,they are rather sensitive to mirror misalingment.
> For the various reasons discussed above, the most commonly used laser
> resonators make use of either two concave mirrors of large radius of
> curvature (say from two to ten times the resonator length) or a plane
> mirror and a concave mirror of large radius."

Hi David!  Very interesting.  I suspect that this could answer a number of
questions I've had about laser operation.  I've wondered about that
hotspot problem myself.  (And as always, I want to find out where my
conceptual mistakes are, so that *I* don't go around spreading
misconceptions.)

If the standing wave in the laser cavity is not a unique plane wave or
sphere wave, won't this have serious effects on the structure of the
exiting beam?  For a confocal cavity, I can't picture a wave pattern which
doesn't result in combinations of several sphere-waves having various
curvatures.  The resulting output beam would seem to issue from several
point-source emitters along the axis within the laser cavity.

Suppose we have a short, fat laser resonator with semi-confocal mirrors:

     100%               99.9%

      |                   \
      |                    |     LASER CAVITY
      |                     |
      |                     |
      |                     |
      |                     |
      |                     |
      |                    |
      |                   /

      <-------  F  -------->

 =========================================================================

     A (overly-crude) ray diagram of the
     standing wave and the output beam:

                                                               _____\
                                                     _____-----     /
                                          ______-----
      |                   \         ______
      |____________________|_______________________________________\
      |             _____---|                                      /
      |  _____------        |
      |--_____               |
      |       ------______  |
      |___________________==|==____________________________________\
      |                    |        -------______                  /
      |                   /                      -----_____
                                                           -----____\
                                                                    /

The output beam would contain two waves, one with a 1F radius of
curvature, and a plane-wave with infinite radius.  Perhaps the typical
standing wave is different than I have drawn?  Is there some particular
standing-wave pattern which results in a sphere-wave output beam?


> Svelto goes into a lot of treatment finding the standing waves that will
> survive in a stable cavity, and I suspect that the reason ray diagrams
> can mislead us is that we have to consider the wave nature of light, if
> not while it's interacting with the medium at least while it's bouncing
> around in our cavity.

I can see that this would be very true for a long, narrow cavity where any
sphere waves become identical to plane waves.  Perhaps my short, fat
cavity exaggerates the non-wave character of light, and creates issues
which might not exist at all if the cavity is long and narrow.

> For the various reasons discussed above, the most commonly used laser
> resonators make use of either two concave mirrors of large radius of
> curvature (say from two to ten times the resonator length) ...

If the radius of mirror curvature was a large multiple of the resonator
length, then the standing wave might become several superimposed
sphere-waves with differing radii, and if the exiting beam was focused
with a lens, there would be a series focused hotspots along the beam axis.
This sounds familiar...  somebody told me that most lasers have immense
spatial coherence, but the coherence has a periodic character:  two beams
must be within a few cm of the same length to be coherent, *OR* they must
be within a few cm of a multiple of the resonator length.  In other words,
the spatial coherence of the beam is sinusoidal along the beam path, only
decaying to zero after a distance equivalent to a large number of cavity
lengths.  Has anyone heard of this effect?  If real, where does it come
from?  Is it true of classroom HeNe lasers?

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