I invite you to fall asleep to some of
my publications (you can view
abstracts and PDF versions of the papers). In addition,
while I have yet to invent a perpetual motion machine, you
can also view some of the patents
resulting from this work. These works were performed in
collaboration with other members of
the Nanostructures
and Computation
and Ab-Initio
Physics groups.
Below are short descriptions of some of
my research interests. You can
also view a number of selected works I have presented
at talks and conferences, along with
some press material.
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Research Interests:
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Photonic Crystals: are periodic
dielectric structures that have a band gap which
forbids the propagation of light corresponding to a
certain frequency range. This property enables one
to control light with amazing facility and produce
effects that are impossible with conventional
optics. Photonic crystals are described exactly by
Maxwell's Equations, which we can (and do) solve by
the application of massive computational power. In
particular, my
advisor Steven
G. Johnson and colleagues have developed a
parallel finite-difference time-domain (FDTD) code
called
Meep,
that can be used to compute a large number of
photonic properties from an ab-initio
perspective. Much of our research, however, is
directed at achieving a higher level of
understanding of these systems, so that we can
predict and explain their behavior without resorting
to brute force calculation.
[Figure (left): Two-dimensional,
complete-gap PhC cavity with a large amount of
fabrication-induced (positional, roughness,
dielectric) disorder, showing that the cavity mode
remains immune (confined). Figure (right):
Calculation (statistically averaged) of photonic
bandgap for the system above: the insensitivity of
the modal properties on the disorder comes from
the the fact that the mode remains protected by
the gap up to large amounts of disorder,
afterwhich Anderson localization effects take
over.]
My first project as an undergraduae student involved
the study of disorder in complete band-gap systems:
in particular, borrowing well-known ideas from the
theory of electronic solids, colleagues and I
directly showed that systems with complete gaps
(i.e. with photonic gaps in every direction) are
immune to small amounts of disorder
[
Rodriguez et. al., OL 2009]
. Specifically, we demonstrated that the band
gap, and therefore the modes of a three-dimensional
cavity are protected from disorder (the only effect
of disorder is the reduction of the modal
lifetimes). This is especially important to the
fabrication community, and is one among many
compelling arguments for the design of
photonic-crystal-based devices.
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Nonlinear Frequency Conversion: when
the intensity of light interacting with a
polarizable material exceeds a certain
(material-dependent) threshold, its dielectric
response can no longer be described by a linear
permittivity, and the corresponding Maxwells
equations become nonlinear, giving rise to a wide
range of interesting effects. One such effect is
known as nonlinear harmonic generation: when
light of a particular frequency enters a nonlinear
medium, the nonlinearity can generate higher-order
harmonics of the input frequency. This phenomenon is
the basis of many important practical devices, such
as lasers and light-emitting diodes. Our interest in
this field is motivated by the possibility of
achieving interesting effects associated with
harmonic generation in PhC cavities, where both
temporal and spatial localization can enhance (and
at times even generate novel) nonlinear
interactions. Our recent understanding of these
processes (studied in the framework of coupled-mode
theory, described in
[Rodriguez et. al., OE 2007]
) has already bore fruit to a number of
peculiar phenomena arising from, e.g., the phase and
cross-phase modulation (nonlinear frequency shifts)
of the cavity frequencies involved in the
interaction
[Hashemi et. al., PRA 2008].
[Figure: Schematic of a nonlinear
third-harmonic process taking place inside a
Kerr nonlinear cavity coupled to a
waveguide. The bottom figure shows a
one-dimensional realization (calculation of
third-harmonic field profile) of this
system. ]
More recently, and in collaboration with
experimental colleagues at Harvard University, our
group has begun a study of sum-frequency
generation (SFG) and difference-frequency
generation (DFG) in PhC cavities. SMG and DFG
are nonlinear processes in which input light at two
different frequencies (w1
and w2) generates output light
given by either the sum
(w1+w2) or
difference
(w1-w2) of the
two input frequencies, respectively. By employing
the same tools described above, we have recently
demonstrated that the spatio-temporal confinement
provided by PhC cavities can be exploited to enable
efficient conversion of GHz to THz light. Because
THz sources are scarce and inefficient, we hope that
this (purely optical) scheme can become a promising
complement to more conventional (e.g. electrically
pumped) sources.
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Casimir Forces: the quantum vacuum is
a dynamical origin for a dramatic macroscopic
manifestation of quantum mechanics. In particular,
at small separations, neutral objects experience a
force that arises due to quantum vacuum fluctuations
of the electromagnetic field. Recent interest in the
Casimir force, discovered by Hendrick Casimir in
1948, has been fueled by a number of experiments
which indicate that the force may play a crucial
role in the stiction of microelectronic mechanics
systems (MEMS). Until recently, and due to a lack of
theoretical tools capable of handling arbitrary
settings (materials and geometries), the Casimir
force had been studied in simple geometries
consisting of parallel plates or approximations
thereof. Unfortunately, and stemming from the rather
young age of the field, most work on Casimir forces
is currently being carried out by specialists
(physicists and mathematicians). Thus, in order to
open this field to other scientists and engineers,
we believe it is fruitful to design theoretical
tools that are more accessible to a broader
audience.
[Figure: (Inset) shows a two-dimensional
geometry consisting of perfectly-metallic square
(blocks) adjacent to two perfectly-metallic
infinite plates. (Plot) shows the normalized
Casimir force between the two blocks as a function
of the plate separation, the dependence of which
is non-monotonic.]
A starting point of our study of these forces is an
analytical stress-tensor calculation developed by
E. Lifshitz and L. Pitaevskii in the 1960s. Our
first numerical implementation of this method was a
finite-difference frequency-domain (FDFD) algorithm,
described in
[
Rodriguez et. al., PRA, 2007],
in which the the force integrand of a
desired spatially discretized geometry is computed
by numerically solving for the Wick-rotated
(imaginary frequency) Green's function, at each
frequency (this requires the repeated inversion of a
positive-definite matrix). Using a proof-of-concept
implementation, we performed the first calculations
of Casimir forces in a geometry consisting of
multiple bodies, and demonstrated a surprising
non-monotonic dependence of the force between two
objects
[
Rodriguez et. al., PRL, 2007].
As of late, we have become interested on the
search of qualitatively exotic phenomena arising
from the strong interplay between geometric and
material dispersion, e.g. formation of stable
equilibria.
[Figure: Schematic of correspondence
between the Casimir force in the piston-like
geometry above at micrometer scales, and the
(equivalent) force for a transformed geometry at
centimeter scales, in which vacuum is exchanged
with a conductive (dissipative) fluid. This
exact equivalence points to a possible analog
Casimir computer.]
An alternative theoretical framework for computing
Casimir forces lies in the finite-difference time
domain (FDTD), a formulation that is interesting to
us due to the availability and generality of FDTD
codes. Toward this end, we are currently exploring a
recently proposed correspondence
[
Rodriguez et. al., submitted, 2009]
between the Casimir force as computed in imaginary
time and the force as computed in a transformed,
conductive (dissipative) medium, in real
time. This correspondence not only also allows
us to readily compute Casimir forces via table-top
experiments at the centimeter lengthscale, but also
serves as an important starting point of a purely
FDTD (time-domain) algorithm, described in
[
Rodriguez et. al., PRA, 2009]
and
[
McCauley et. al., in preparation]
. Our time-domain algorithm has been
implemented as a new and easy-to-use feature
in
Meep, which can now perform calculations of
Casimir forces in arbitrary geometries (two- and
three-dimensional structures with either
perfectly-conducting, absorbing or periodic boundary
conditions) and for arbitrary materials (dispersive
and/or anisotropic).
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[Figure: (left): Two-dimensional periodic
(square lattice) geometry, whose cut-and-project
(irrational slice) yields the one-dimensional
Fibonacci quasicrystal. Figure (right): Spectrum
of the quasicrystal as a function of the
"fictitious" wavevector (along the irrational
slice), from which it is possible to obtain the
photonic density of states.]
Photonic Quasicrystals: are
quasiperiodic dielectrics which possess long-range
order. Mathematically, they represent structures
whose Fourier transform span a finite number of
reciprocal lattice vectors. These systems are
interesting because they promise a number of unique
characteristics not found in periodic structures,
especially in two and three dimensions, where they
can have greater rotational symmetry. Because these
systems are aperiodic, studying their properties
(frequency spectrum) in two and three dimensions has
been challenging, i.e. they require large
computational cells that ultimately only capture a
portion of the aperiodic lattice. Recently, we
proposed a computational method to solve for the
spectra and eigenstates of photonic quasicrystals
that captures the entire (infinite aperiodic)
structure, by applying Bloch's theorem to a
higher-dimensional space whose irrational slice
yields the original quasiperiodic structure
[
Rodriguez et. al., PRB, 2007].
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Talks, Conferences and Press:
Talks & Conferences:
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Computing Casimir forces via tabletop
experiments: from FDFD to
FDTD [pdf]
Casimir theory @ MIT (May 2009)
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Fluctuation-induced interactions in photonic
media [pdf]
Capasso / Pitaevskii meeting @ Harvard
Invited talk @ Stanford (Shanhui Fan Group)
SIAM 2009
NSBP/NSHP Anual Meeting
Press:
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People in physics. American Physical
Society: Physics Central, 2007
[video]
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Nonlinearities could be strengthened by
photonic crystals. PhysOrg, 2007.
[http]
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Scale models can compute Casimir forces.
Slashdot, 2008.
[http]
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Patents:
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Jorge Bravo-Abad, Alejandro W. Rodriguez,
J. D. Joannopoulos, Steven G. Johnson, and Marin
Soljacic. Efficient terahertz sources based on
difference-frequency generation in triply-resonnt
resonators. (provisional filed)
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Alejandro W. Rodriguez, Marin Soljacic,
J. D. Joannopoulos, and Steven G. Johnson. Efficient
harmonic generation and frequency conversion in
nonlinear multimode cavities. (provisional filed)
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Peter Bermel, Alejandro W. Rodriguez,
J. D. Joannopoulos, and Marin Soljacic. Enhancement and
inhibition of optical nonlinearities via the Purcell
effect. (provisional filed)
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Hila Hashemi, Alejandro Rodriguez, J. D. Joannopoulos,
Marin Soljacic, and Steven G. Johnson. Nonlinear
harmonic generation and devices in mutlir-resonant
cavities. (provisional filed)
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