Mesoscopic Optics
Jens U.Nöckel
A worldwide rush is on to develop integrated optical
components, aiming to increase data
transmission rates and volumes in fiber optic networks in
response to exponentially increasing demand. Of central
importance
to this 1.3 trillion US dollar market sector are
dielectric microcavity resonators for use in photonic
applications such as filters,
multiplexers and lasers. Using semiclassical
methods and tools from nonlinear dynamics, we analyze the
spectral and emission characteristics of such open
resonators.
Waveguiding and long-lived resonator modes
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Extremely small, high-quality resonators
can be fabricated by exploiting total
internal reflection at the interface between a
circularly symmetric dielectric and the surrounding
air; see Fig.1 which shows the cross-sectional
intensity distribution of a long-lived photonic mode
in a dielectric disk. Classically forbidden
tunnel leakage across the rim (black line) leads only
to minute outcoupling losses which moreover are
isotropic. Coupling to such a resonator is very inefficient
unless another optical component
is brought extremely close to the rim so as to
overlap with the evanescent near field. This is not
amenable to highly reproducible device fabrication.
An alternative and potentially much more efficient
coupling mechanism is introduced when the rotational
invariance of the resonator is broken, thus creating
preferred emission directions,
cf. Fig.2. Shown in false-color is the
intensity of a long-lived mode as obtained from
exact solution of Maxwell's equations. The green
lobes extending into the far-field indicate the
preferred directions along which coupling into or
out of the resonator is now possible with great
efficiency.
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Fig. 1
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Shown at the bottom of Fig.2 is the classical ray
picture corresponding to the wave solution depicted above.
As indicated by the arrow, rays escape according to Snell's
law after a number of total internal reflections because the
angle of incidence is not conserved in the oval cavity. The
wave solution mimicks this classical behavior and hence the
emission directions as well as the lifetime of the mode can
be predicted with the help of the ray picture. To establish
the connection between rays and waves, semiclassical methods
are required.
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Fig. 2
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Phase space structure in the emission profile
The classical ray dynamics in general deformed
resonators exhibits a coexistence of periodic,
quasiperiodic and chaotic trajectories which is
difficult to disentangle by ray-tracing. A more
structured representation of the multitude of
different trajectories can be obtained by making
phase-space portraits. In this Poincaré
section, black dots represent individual reflections of
rays at the boundary, giving the position (polar angle
) and angle of incidence .
This representation (Fig. 3)
of an orbit as a sequence of
points (,
sin)
contains more information than
is actually needed to reconstruct a given trajectory,
but it reveals the hidden structure of the phase
space. Chaotic orbits generate the grainy region
filling most of the area in this plot.
Island structure ("solid" lines) belongs to
non-chaotic, regular motion. Emission properties of
the resonator can be related to transport in
this anisotropic phase space.
Superimposed onto the Poincaré
section is a false-color density representation of a
particular resonator mode.
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Fig. 3
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The high intensity around a chain of four islands of
stability corresponds to a resonator mode associated
with the corresponding stable motion. Its real-space
intensity is shown in Fig. 4, where a
bowtie-shaped pattern is apparent. The focussing
inside and outside of the resonator is a
desirable property that is achieved here by
intentionally inducing the transition to partially
chaotic ray dynamics. Oval lasers like the one
shown here have been fabricated at Bell Labs and at
less than 100 µm diameter (5.2 µm wavelength)
delivered roughly 3 orders of magnitude more power
than a comparable circular (non-chaotic) cavity.
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Fig. 4
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