Dye-doped zeolite microcrystals*
A new class of materials for optical applications are the
guest-host systems, in which a species
of
(sufficiently small) molecules is encapsulated into the
"hollow spaces" of a medium such as
porous silica or a
zeolite, which is therefore called the
"host" matrix. Such hybrid materials exhibit new properties
that
neither the host nor the guest molecules can produce alone. This is
because
the pores in which the guest molecules reside serve to impose order
and
stability that limits their motion and can even force the incorporated
molecules to assume a particular orientation.
This happens for example in zeolite microcrystals that are doped with
an organic dye as "guest" molecules. The dye is able to emit
light by fluorescence, and because of the imposed orientation of the
organic molecules in their host matrix, the light is polarized.
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The nanopores of the zeolite form a "molecular sieve" with a
hexagonal arrangement. If one grows crystallites out of this material,
this translates into a hexagonal facet structure. The light
generated by the fluorescent dye molecules now gets trapped inside
this
self-assembled hexagonal microresonator by reflecting off the
facets. The polarization properties of the medium favor propagation in
a
cross-sectional plane of the crystal, whose size is
below 10 micron diameter. |
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These tiny grains of hybrid material can in fact be
made into extremely small lasers. Apparently, the light
is
trapped sufficiently long to provide the necessary feedback for light
amplification. But pores in a material usually reduce
the overall refractive index n, and in fact the zeolite-dye system has
n=1.45 - this makes it easy for rays in such a resonator to escape to
the surrounding air by the law of refraction.
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This raises the question: what do the modes look
like ?
What kind of long-lived ray paths can one imagine in this cavity ?
There can only be lasing from modes corresponding to rays with angles
of incidence in the "whispering-gallery" region, i.e. travelling
close to the surface. But are
there such WG modes ? Look at these periodic ray paths:
All orbits here have the same angle of incidence, and the same
length. They constitute a family of rays, all of
which can be expected to make a contribution to
the resonator modes. There is nothing a priori
that
singles out the symmetric inscribed
hexagon (left),
and in fact
admixtures from all of these
orbits can
contribute to any
single
mode.
What about their escape behavior ?
Total internal reflection confines these modes, but the classical
Fresnel formulas break down at the corners. The
degenerate periodic orbit hitting the corners (right) is
therefore special because simple total internal reflection
doesn't work there. So the corners
are where the escape should happen in a
preferred
way.
That is in fact what is seen in experiment and also what I find in my
wave simulations. Here is an example:
This is for microcryastals with n=1.45 at (a) 4 and (b) 8 micron
diameter,
wavelength 600…700 nm.
The experiments were carried out by Uwe Vietze and Franco Laeri. This
group in Darmstadt, in collaboration with Bremen University and
the MPI
für Kohleforschung did the pioneering development leading to
zeolite
nanocomposites with lasing action. For the small cavities studied
here,
the ray picture doesn't give a correct description of the actual
emission pattern due to strong corner diffraction.
Together, we conducted a
comprehensive investigation of this
micro-optical system, published in Applied Physics B ,
see also my publication list.
The hexagonal resonator is interesting from a theoretical point of
view because its internal dynamics is pseudointegrable,
a
pathological condition. It causes the rays patterns to look
rather
neat compared with a chaotic cavity - but don't let this fool
you.
Even if we neglect the copmplication of leakage from the
resonator,
the wave fields for this system cannot be written down in any
simple
way, except for a subset of solutions whose intensity vanishes
on all
the diagonals (and hence in particular on the corners). But
the
interesting solutions are precisely those with nonzero
intensity on
the diagonals, because the ray picture tells us that those
types of
fields favor escape at the corners.
The pathological properties of pseudointegrable
systems can be studied especially well in
polygonal billiards , and in this way the importance of
diffraction as a correction to
geometric
optics is revealed.
The theoretical
interest in the
hexagonal
resonator combines
in an
ideal way with
the great
significance of this geometry in
naturally
occuring and artificially
engineered micro-optical
systems. Perhaps
the first hexagonal microstructure
that comes to mind is in fact the
snow
crystal
(that's what snowflakes
are made of) . In tropical
cirrus clouds, such
crystals are less than 50
micron small, i.e. they are
micro-optical
systems. Their scattering
properties are of great
importance in
meteorology because of
their potential climatic
effects.
Of course, people have not
been that interested
in the possibility of
long-lived resonances in
snowflakes... certainly my
kids are more interested in
the lifetime of
the snowflake itself.
Photonic
states
in
a hexagon are,
however, very
important when it
comes to
microlasers made
of Gallium Nitride
- one of the most
promising
candidates for the
elusive goal of
producing blue
light
with a
semiconductor
laser diode. This
compound grows
with a hexagonal
facet
structure as well
!
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