Abstracts 3.5

Abstract: Waveguides created in planar photonic crystals offer many unique opportunities for manipulating optical pulses that cannot be realized in conventional rib waveguides. In particular, the speed of light can be dramatically reduced [1-3]. By making the light slower it becomes possible to introduce delays for signal synchronization. Additionally, in the regime of slow light the photon-matter interactions are dramatically enhanced, allowing for all-optical control of light. In particular, nonlinear self-action can be used to dynamically tune the pulse delay.

In this work, we present a novel and general approach to design directional couplers in photonic crystals where the spatial control of slow-light pulses may be realized [4]. Conventional directional couplers consist of two parallel identical waveguides, and in such structures the light launched into one waveguide is switched to another waveguide at the characteristic coupling length, defined by the beating period between the odd and even super-modes. We perform a general symmetry analysis and show that tunneling of slow-light pulses between the waveguides is possible in anti-symmetric photonic crystal couplers, whereas this is not possible in symmetrically coupled waveguides. The anti-symmetry allows co-existence of two co-propagating modes whose group velocity and band-edge dispersion are exactly matched in the slow-light regime. As a result, the beating of these modes provides slow light switching between the waveguides. The coupling distance, after which the pulses are fully switched between the parallel waveguides, can be made remarkably short (a few unit cells) and it remains constant and independent on the group velocity. This latter property enables dispersionless tunneling of slow light pulses, where the pulse dynamics is preserved even under the variation of the speed of light by several orders of magnitude, as confirmed through the finite-difference time-domain simulations.

• "Fourier Modal Method for the Analysis of Two- and Three-Dimensional PBGs"

      L. Pajewski, Cristina Ponti, G. Schentini -  Applied Electronics Department, “Roma Tre” University

Abstract: In this work we resume our work on developing Fourier Modal Methods for the analysis of two- and three-dimensional Photonic Band-Gap (PBG) structures.

A two-dimensional finite-thickness PBG may be considered as a stack of periodic grids of rods, eventually separated by homogeneous layers, i.e., as a stack of one-dimensional diffractive optical elements. It can therefore be characterized by using a rigorous diffraction theory for multilevel gratings. The general approach for exactly solving the electromagnetic problem associated with a multilevel grating involves the solution of Maxwell’s equations in each of the following regions: the grating layers, and the two homogeneous regions outside the periodic structure. In the Fourier Modal Method (FMM), a Fourier decomposition of the permittivity function of each layer of the grating, together with a plane-wave expansion of the electromagnetic fields (Rayleigh expansions outside the grating, modal expansions in periodic layers), leads to an eigenvalue problem which has to be solved in each grating layer. Then, the tangential electric and magnetic field components have to be matched at all the boundary surfaces. The resulting equation system is to be solved for the reflected and transmitted field amplitudes, so that the diffraction efficiencies can be determined. To obtain a high convergence rate, we use an improved formulation of the eigenvalue problem. To overcome numerical problems due to ill-conditioned matrices obtained on imposing the boundary conditions, and to improve numerical stability and efficiency of the implemented codes, we apply appropriate numerical techniques.

The characterization of two-dimensional PBGs through our home-made FMM simulator is very accurate and versatile, since we can study crystals with rods having an arbitrary cross-section, forming whatever kind of lattice. Of course, also PBGs made of holes in a host medium, instead of rods, may be studied.

The FMM is considered the most efficient and prevailing rigorous technique for the analysis of multilevel crossed gratings, and so for the characterization of three-dimensional finite-thickness PBGs. We implemented the method by using the correct Fourier factorization rules for discontinuous functions, obtaining a much faster convergence than with the customary FMM. For a numerically stable treatment of the evanescent waves, at the interfaces between different layers stacked to simulate the PBG, we use the S-matrix algorithm in the solution of the boundary problem.

Our home-made FMM simulator for three-dimensional PBGs permits to analyze, in an accurate and stable way, photonic crystals with implants having an arbitrary shape and arranged in whatever kind of lattice. It is possible to consider any polarization and incidence angle, therefore the off-plane stop-band structures of a PBG can be calculated: this is a fundamental potentiality of our tool, since achieving an absolute, omni-directional, band-gap is a fundamental target in the design of three-dimensional photonic crystals.

The presence of periodicity interruptions can be taken into account by both our tools for two- and three-dimensional PBGs, and this is another very important advantage of our simulators. Defects may be present due to fabrication errors, however PBGs with periodicity interruptions are often designed on purpose to act as filters or switches, since the occurrence of a sharp transmission peak in the band-gap results from defect creation.

The involved materials can be isotropic or anisotropic dielectrics. The extension to metallic and lossy media should be quite straightforward, however the convergence of the numerical implementation would be less satisfactory.

Finally, our theory is normalized to the operation wavelength, therefore it can be applied at any frequency, from microwave region to the optical one, when the use of fabrication nanotechnologies is essential in order to realize correctly working devices.

References

[1] F. Frezza, L. Pajewski, and G. Schettini, "Characterization and design of two-dimensional electromagnetic band-gap structures by use of a full-wave method for diffraction gratings", IEEE Transactions on Microwave Theory and Techniques, Vol. 51(3), pp. 941-951, March 2003

[2] F. Frezza, L. Pajewski, and G. Schettini, "Periodic defects in 2D-PBG materials: full-wave analysis and design", IEEE Transactions on Nanotechnology, Vol. 2(3), pp. 126-134, Sept. 2003

[3] F. Frezza, L. Pajewski, and G. Schettini, "Fractal two-dimensional electromagnetic band-gap structures", IEEE Transactions on Microwave Theory and Techniques, Vol. 52(1), pp. 220-227, Jan. 2004

[4] F. Frezza, L. Pajewski, and G. Schettini, "Numerical investigation on the filtering behavior of 2D-PBGs with multiple periodic defects", IEEE Transactions on Nanotechnology, vol. 4(6), pp. 730-739, Nov. 2005

[5] F. Frezza, L. Pajewski, and G. Schettini, "Full-wave characterization of three-dimensional photonic bandgap structures", IEEE Transactions on Nanotechnology, vol. 5(5), pp. 545-553, Sept. 2006