Research Interests, Publicity, and Patents
Below are short descriptions of some of
my research interests. You can
also view a number of the patents, popular articles and press materials resulting from my work in collaboration with members of the Nanostructures
Selected Research Interests:
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. My interest in this
field is motivated by the possibility of achieving interesting effects
associated with harmonic generation in PhC cavities supporting resonant
modes at all interacting frequencies, where both temporal and spatial
localization can enhance and even generate novel nonlinear
interactions. To understand these nonlinear processes, I exploited a
universal framework known as coupled-mode theory, described in [Rodriguez et. al., OE 2007], that makes it possible to describe the temporal and steady-state dynamics
of the system using only a few fundamental cavity parameters, such as the frequencies, lifetimes of the modes, and the
so-called nonlinear coupling coefficients between the various modes (which can be derived from a perturbative treatment
of Maxwell's equations in the presence of the nonlinearities, and which reduce to spatial or overlap integrals between
the various mode profiles). Our recent understanding of these
processes has bore fruit to a number of
interesting predictions, involving peculiar phenomena arising from the phase and
cross-phase modulation (nonlinear frequency shifts)
of the cavity frequencies, and this is described in
[Hashemi et. al., PRA 2008].
[Fig. 1: 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 (electric-field profile of the third-harmonic mode).]
More recently, and in collaboration with
experimental colleagues at Harvard University, our
group has begun a study of sum-frequency
generation and difference-frequency
generation in PhC cavities: two nonlinear processes in which input light at two
different frequencies generates output light
given by either the sum or
two input frequencies, respectively. By employing
the same universal descriptions mentioned 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, which lends hope that
this (purely optical) scheme can become a promising
complement to more conventional (e.g. electrically
the quantum vacuum is the theatre of a dramatic, macroscopic
manifestation of quantum mechanics. In particular, charge fluctuations
inside otherwise neutral bodies can give rise to fluctuating
electromagnetic fields, whose interactions with the bodies lead to the
so-called Casimir effect. Although this quantum pressure is tiny at
everyday lengthscales, it can reach magnitudes of up to atmospheric
pressure for objects whose size and separation is in the microns. Not
surprisingly, this phenomenon is of importance to the fabrication and
operation of small micro-devices, such as new generations of
microelectromchanical systems (MEMS)—the force is usually attractive
and therefore causes these devices to fail in an undesirable process
known as stiction.
However, because the fluctuating electromagnetic fields must obey
Maxwell's equations, the Casimir force is sensitive to both material
and geometry, begging the question: is it possible to engineer this
force by microstructuring the shape and/or surface of the interacting
objects? Until recently, and due to lack of theoretical tools capable
of calculating the force in arbitrary geoemtries, the Casimir force had
only been studied in simple geometries consisting of parallel plates or
simple approximations thereof. My work in this field involves the
design and exploitation of computational tools based on standard
techniques from classical numerical electromagnetism, to investigate
the limits and possible incarnations of this phenomenon.
[Fig. 2: Normalized
Casimir force between two metal squares as a function
of their separation h from two adjacent metal plates, the dependence of which
is non-monotonic. Inset shows a schematic of the geometry.]
first numerical method I helped develop involves the calculation of the
Minkowskii stress-tensor via the finite-difference frequency-domain
method, and is described in
Rodriguez et. al., PRA, 2007]: the force integrand of a discretized geometry is computed
and integrated over all frequencies by numerically solving the Wick-rotated
(imaginary frequency) Green's function at each
frequency and position along a surface surrounding the body of interest (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 four bodies, and demonstrated a surprising
non-monotonic force dependence between two
of the objects
[Rodriguez et. al., PRL, 2007].
[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
An alternative theoretical framework for computing
Casimir forces lies in the finite-difference time
domain (FDTD) method, a formulation that is interesting due to the availability and generality of FDTD
codes. Toward this end, we are currently exploring a
recently proposed correspondence
[Rodriguez et. al., PNAS, 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]
[McCauley et. al., PRA, 2010]
. Our time-domain algorithm has been
implemented as a new and easy-to-use feature
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
Using a combination of numerical methods, we have embarked on a journey in search of qualitatively exotic phenomena arising
from the strong interplay between geometric and
material dispersion, and this has already led to a number of interesting predictions.
|Radiative heat transfer: Coming soon.
People in physics. American Physical
Society: Physics Central, 2007
Nonlinearities could be strengthened by
photonic crystals. PhysOrg, 2007.
Attractive repulsion. PSC Projects in Scientific Computing, January 2009
Scale models can compute Casimir forces.
Slashdot, March 2009.
How to build Casimir molecules.Technology Review Physics Blog, December 2009.
- Forcefull thinking. Deixis Magazine, June 2010. [http]
- New way to calculate Casimir force. PhysOrg, May 2010. [http]
- WD-40 for micromachines. Technology Review, August 2010. [http]
- Mysterious quantum forces unraveled. MIT News, May 2010. [http]
- New way to calculate Casimir effects. Science Daily, May 2010. [http]
- Efficient terahertz sources based on
difference-frequency generation in triply-resonnt
U.S. patent #7768694. Jorge Bravo-Abad,
Alejandro W. Rodriguez, John D. Joannopoulos, Steven
G. Johnson, and Marin Soljacic.
- Efficient harmonic generation and frequency
conversion in nonlinear multimode cavities. Provisional filed. Alejandro W. Rodriguez, Marin Soljacic, J. D.
Joannopoulos, and Steven G. Johnson.
- Enhancement and inhibition of optical
nonlinearities via the Purcell effect.
Provisional filed. Peter Bermel, Alejandro W. Rodriguez, J. D.
Joannopoulos, and Marin Soljacic.
- Nonlinear harmonic generation and devices in multi-resonant cavities.
Provisional filed. Hila Hashemi, Alejandro Rodriguez, J. D.
Joannopoulos, Marin Soljacic, and Steven G. Johnson.