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 and Computation, Ab-Initio Physics, Capasso, and Loncar groups.

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 difference of the 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 pumped) sources.

Casimir Forces: 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.]

The 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 Casimir computer.]

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] and [McCauley et. al., PRA, 2010] . 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).

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.

Publicity:
  • People in physics. American Physical Society: Physics Central, 2007 [video]
  • Nonlinearities could be strengthened by photonic crystals. PhysOrg, 2007. [http]
  • Attractive repulsion. PSC Projects in Scientific Computing, January 2009 [http]
  • Scale models can compute Casimir forces. Slashdot, March 2009. [http]
  • How to build Casimir molecules.Technology Review Physics Blog, December 2009. [http]
  • 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]
Patents:
  • Efficient terahertz sources based on difference-frequency generation in triply-resonnt resonators.                            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.