LASERS, cavity waves, coherence

["David W. Steyert" <steyert_dw@MERCER.EDU>]

William wrote a nice, illustrated, reply to my post on cavity waves. One
part I'd like to focus on is his illustration of the ray patterns in the
cavity:
> 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?

It seems to me that the parallel rays emerging near the center of the
diagram could nearly equally well be seen as part of the center of the
emerging spherical wave.   (Remember, Chemist talking here...Never took
a course in geometrical optics, just lasers.)

The book I'm looking at works much more in terms of the wave fronts
(equiphase surfaces) One bottom line of the math is that the mirrors are
equiphase surfaces, "as expected."  It's not at all obvious that that's
what I should have expected, but I was able to rationalize it this way.
If they weren't eqiphase surfaces, then there would be a node in the
output beam, and presumably in the cavity itself, thus leading to
inefficient use of the inverted medium.  (These are the higher-order
modes, that laser types work so hard to avoid.)

Thus, the wave front (seen perhaps as a combination of Huyghens waves
from all points in the active medium) progresses from planar at the
planar mirror to spherical at the spherical mirror.  That it has to be
planar at the planar mirror can be seen from symmetry considerations of
the corresponding confocal resonator, of which we're just looking at the
front half.

William also commented on the use of resonators other than the "named"
cases we've been talking about:  spherical, confocal, planar,
hemiconfocal, etc.

It turns out that there's a stability condition, such that these "named"
resonators are at the border of stability, with the risk that slight
misalignments take you out of stability.
If the cavity is of length L, and the mirror curvatures R1 and R2, and
if you define g1 and g2 by
g1 = 1 - (L/R1)
g2 = 1 - (L/R2)
Then the cavity will be stable if  0 < g1 * g2 < 1

The confocal resonator has g1*g2 = 0, while the the plane and concentric
(spherical) resonators are different cases of g1*g2 = 1

If you change the radii slightly, then the cavity becomes more robust,
without significant risk of mode competition.  This, as I understand it,
is the case for cavities described as "nearly confocal" "nearly planar"
etc.

Hope this helps.
David.


--
___________
Dr. David W. Steyert                     steyert_dw@mercer.edu
Department of Chemistry                  (912)-752-4173
Mercer University
Macon, GA 31207
___________