## Theoretical and numerical investigations of sub-wavelength diffractive optical structures

Research output: Book/Report › Ph.D. thesis – Annual report year: 2000 › Research

The work in this thesis concerns theoretical and numerical investigations of sub-wavelength diffractive optical structures, relying on advanced two-dimensional vectorial numerical models that have applications in Optics and Electromagnetics. Integrated Optics is predicted to play a major role in future technologies. For this to come true, more advanced optical signal processing must be achieved in miniaturized multifunctional components which should enable optimal light control and light localization. These components have complex subwavelength geometries and material distributions, such as in dielectric waveguides with gratings and periodic media or photonic crystal structures. The vectorial electromagnetic nature of light is therefore taken into account in the modeling of these diffractive structures.
An electromagnetic vector-field model for optical components design based on the classical finite-difference time domain method and exact radiation integrals is implemented for the polarization where the electric field vector is perpendicular to the two dimentional plane of symmetry. The computational model solves the full vectorial time domain Maxwell equations with general sources of illumination. Maxwell's equations are solved numerically in complex geometries and radiation integrals are applied in homogeneous regions, thus minimizing the computational time. Analysis of finte length surface relief structures embedded in polymer dielectric waveguides are presented. The importance of several geomtric parameters is indicated as far field power distributions are rearranged between diffraction orders, and as diffraction efficiencies vary. The influences of the variation in grating period, modulation depth, length, and relief profile are investigated. It is shown that the farfield power distribution pattern exhibits lobes of specific angular widths around each diffraction order. The widths of these lobes depend on the grating length. The diffraction physics of sub wavelength waveguide focusing grating couplers are studied. Two designs of focusing grating couplers integrated in multilayer planar waveguides are analyzed numerically, and the physics are discussed, as classical methods based on scalar geometrical ray optics fail to predict the nearfield characteristics when local modulation changes in the suruface profile are significatnt, altering the propagation of the incident fundamental mode. One original formula is derived for the design of focusing grating couplers. The influences of amplitude modulation, modulation depth, grating period, and device length are discussed. The focusing ability of these components is demonstrated mumerically. It is shown that for modulation depths above a certain threshold the surface perturbation changes the propagation and the effective refractive index of the incident mode significantly and the devices focusing efficiency drops. A significant fraction of the incident power is lost in unwanted directions. This lack of control is due to the highty dispersive nature of the subwavelenth diffractive gratings. Theoretical and numerical investigations of engery flow in photonic crystal waveguides made of line defects and branching points are presented. It is shown that vortice of energy flow may occur, and the net energy flow along the line defect is described via the effective propagation puted for different crystal lattices and waveguide widths. Both strong positive, strong negative and zero disperions are possible. It is shown that geometric parameters such as the nature of the lattice, the line defect orientation, the defect width, and the branching point geometry have a significant influence on the electrodynamics. These are important issues for the fabrication of photonic crsytal structures.
A novel stable Cartesian grid multi space staircase free fintie difference time domain method of sparial second order accuracy is developed. Contrary to the classical Yee scheme, it allows the correct application of electromagnetic boundary conditions. moreover, it does not represent the physical structure using a staircase approximation, but takes into account the exact geometry, and thus significantly reduces the required number of points per wavelength to accurately resolve the wave and the geometry. The new formulation is applicable to general geometries with arbitrary distributions of material in Cartesian coordinate systems. Numerical simulations show that it exhibits error levels that are orders of magnitude lower than what was achieved up until now with the classical Yee scheme for general physical problems involving dielectrics as well as perfect electric conductors. The method is stable and retains the spatial second-order accuracy. The new scheme offers a fairly straigthforward way of improving existing models based on the Yee scheme, and it eliminates the two most severe sources of errors in the finite difference time domain method based on Yee's recipe, namely the staircasing and the lack of application of the electromagnetic boundary conditions at material interfaces. GEneralizations of the method to solve three dimensional modeling problems based on the multi space formulation are possible. Non homogeneous grid layout is also possible as the local spatial resolution in each sub space can be varied as long as the discrete time step is varied accordingly with respect to the stability criterion. Also, a fourth order accurate scheme using similar principles has been investigated numerically in one and tow dimensional space. The one dimensional model shows excellent behavior and spatial fourth order convergence. However, it is observed that the two dimensional version is only well-behaved when the values of extrapolation coefficients are restricted to a certain interval.

Original language | English |
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Number of pages | 111 |
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ISBN (Print) | 87-90-97409-3 |

Publication status | Published - May 2000 |

### Projects

- Vektoriel modelleding af diffraktive strukturer
Project: PhD

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