We have shown
in Figure 8.2
that parameters optimized for trap depth are near
optical cut-off.
It is worth gathering together the physical reasons for this.
Firstly, on general geometric grounds,
the typical available intensities in a guide scale inversely with the
effective cross-sectional area of the bound mode, which, far from cut-off
(that is, when
)
follows very closely the cross-sectional area of the guide.
This makes it favorable to shrink guiding structures to areas less than a
square wavelength, where they generally become single-mode.
Secondly, once we're in the single-mode regime, as we approach
cut-off the mode power is carried increasingly outside the guide, increasing
the ratio of surface intensity to guide center intensity.
Thirdly, the evanescent decay length in the vacuum is longer
as we approach cut-off
(recall that in the case of the slab, this is exactly expressed by
[117], where the factor of 2
arises because we are considering intensity decay length rather than
amplitude).
In the special case of
,
the decay length diverges to infinity as we approach
cut-off.
Finally,
the ratio
becomes larger as we approach cut-off, with corresponding
beneficial effects on depth and coherence (due to increasing the
`goodness factor' we will introduce in Section 8.3.1).
Unfortunately, these purely theoretical reasons for approaching cut-off
are in opposition to more practical ones.
The closer to cut-off a guide is, the more sensitive it is to manufacturing
variations in cross-section: in our case this will be predominantly a
sensitivity to .
The result is that small variations in
cause large variations in mode size, or, at worst, complete cut-off.
If the mode size change is rapid (nonadiabatic)
along the z axis, (for instance if
this change is due
to surface roughness or refractive index inhomogeneities)
the resulting mismatches will be
a source of scattering of
the guided power.
Any coherent scattering back down the guide will set up
periodic modulations of the light field over long distances. (One way to
reduce the distance over which coherent addition is possible
is to use very broad
line-width light sources, which would dramatically reduce
modulations due to both guided and stray scattered light).
The consequence for the atoms would be a z-dependent trap depth and shape,
and this could lead to partial reflection or even localization of
the matter waves.
In general we expect scattering
to
limit how close to cut-off we can reliably operate.
With regard to the substrate, further practical issues arise.
In the above cut-off calculation we chose the simplest
case of , corresponding to a guide surrounded by vacuum.
A real substrate with
has the
unfortunate effect of limiting the propagation constant
of strictly bound modes
to be larger than the freely propagating wavevector in the substrate;
in other words,
must hold or the light field
will rapidly tunnel into the
`attractive potential' of the
substrate.
This in turn limits the decay lengths and
that can be achieved.
The trap properties quoted in the abstract and in Sections
8.3.1 and 8.3.2 rely on very low
values (1.07 for the
mode, 1.18 for
)
for the reason that a low
is the only way to
create long evanescent decay lengths in the vacuum.
(This is equivalent to Ovchinnikov et al.
choosing reflection angles very close to
critical[156].)
However, in Section 8.3.3 we present preliminary results for
a substrate of sodium fluoride (the lowest-index common optical
mineral, at
), and do not believe the substrate alters
the basic feasibility of our waveguide.
For completeness, here we list some other possible
approaches to the substrate issue.
1) Use an aerogel substrate, which can have exceptionally low
refractive indices and low loss (films of several m thickness with
indices of about 1.1 can be produced[147,150]).
2) Use a dielectric multilayer substrate
with an effective index
of unity or less (very low loss multilayer mirrors[169]
with effective indices
less than unity
can be created).
3) Investigate if there exist guide shapes which have sufficiently small
tunnelling rate into a conventional substrate that the fact that the modes are
not strictly bound becomes irrelevant (for instance, a wedge shape with the
smallest face in contact with the substrate).
4) Unsupported guiding structures could
be produced over short distances[176].
Finally, it is important to note that the idea of replacing the substrate by a
metallic reflective layer
is not practical because they are too lossy.
Ultimately, the best values of and
, the best guide
cross-sectional shape,
and the substrate
choice
will depend on many of the above
factors and is an area for further research.
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