Modelling and Simulation of Passive Optical Devices Joo Geraldo P. T. dos Reis and Henrique J. A. da Silva Introduction Integrated Optics is a field of increasing interest and there has been considerable progress in the design and development of high performance devices based on optical waveguiding structures, particularly those built by the diffusion of Titanium strips in a Lithium Niobate substrate, due to the extremely good electrooptical properties of the latter. This has called for an accurate knowledge of the components characteristics, essential for a correct device design and simulation. However, the dielectric structures usually employed in this kind of devices forbid the use of traditional analytical methods. Several numerical methods have been proposed. Among these, the finite differences and the beam propagation methods are the most popular for their accuracy and efficiency. These algorithms were combined with several others to develop modelling and simulation tools with the Matlab software package, making possible to accurately predict the dynamic and static behaviour of several passive optical devices, such as directional couplers and Mach-Zehnder modulators, from their individual models and to simulate the complete structure of an optical integrated circuit. To obtain a general basic device model, which reflects reality as much as possible, it is necessary to understand not only the diffusion of Titanium in Lithium Niobate, employed in the construction of diffused optical waveguides, but also their propagation characteristics and the electrooptic control of optical waves. Refractive Index Profile The diffusion of a Ti strip in a LiNbO 3 substrate produces a distribution that is strongly dependent on the diffusion process characteristics (temperature, duration, strip width and thickness) and is characterised by a combination of error and exponential functions The induced refractive index change is dependent on the wavelength and polarisation of the incoming light beam, due to the anisotropic nature of LiNbO 3 and is related to the Titanium concentration by After defining the dielectric structure of the device, its optical characteristics must be determined in order to simulate the propagation of light through its waveguides. However, since the complex shape of the index profile does not lend itself to an analytical analysis, it was necessary to develop numerical tools that can be used with any type of profile. The transverse dielectric domain is divided by a non-uniform grid into small cells in which the electric field is defined. To increase the accuracy, the density of cells is higher in the areas where the index change is more significant. Substituting the partial derivatives in the wave equation by five-point finite differences and applying the adequate boundary conditions between the cells, the full partitioning procedure produces a matrix problem of finding the eigen values and functions of which is solved by conventional matrix algorithms to obtain the propagation constants, te, and mode profiles, E te, of the optical waveguide. D S and D P variation with t [hours] and T [C]Extraordinary index variation profile Non-uniform grid and corresponding fundamental optical mode After the characterisation of each optical device by a three dimensional refractive index distribution, and the determination of the transverse propagating modes, the response of a waveguiding device to an arbitrary optical excitation can be found by applying the beam propagation method. This method, derived from the wave equation and based on the Fast Fourier Transform algorithm, takes the optical input and propagates it step by step throughout the device. The step propagation is based on the expression where the operator H models the half-step propagation through a homogeneous medium and N introduces a correction due to the index variation. Using some properties of the Fourier Transform, Optical Propagation Analysis Normalised field amplitude distributions for a dir. coupler and a Mach-Zehnder interferometer Travelling-Wave Electrode Analysis The control of an electrooptic device is done through metallic electrodes that are deposited on the substrate surface and produce an electric field that induces a variation in the waveguides refractive index. The analysis is based on the assumption that a TEM wave propagates through the electrodes and this has been shown to be an excellent approximation up to frequencies of several dozens of GHz. The calculation of the external field distribution and the transmission line characteristics are done using a conformal mapping approach. The propagation of the modulating wave in the electrodes is completely characterised and the interaction between the optical and modulating fields is readily determined. Electrooptic Control Analysis The electrooptic effect is a mechanism through which it is possible to change the refractive index of a devices waveguides and induce a phase variation in the propagating beam. Nevertheless, the optical and modulating fields propagate at different velocities, and the beam will experience different phase variations as it propagates through the device. The overall effect is determined by This expression can be directly applied to calculate the response of a phase or interferometric modulator. For directional coupler based intensity modulators, some changes in the analysis are required and the complete response is computed by segmenting the device into infinitesimal directional couplers. Simulations and Results The results were obtained for interferometric and directional coupler based intensity modulators. The devices were designed to have the same characteristic impedance (25 ) and the same active length, equal to the coupling length of the directional coupler modu- lator (5.63mm). Copper electrodes, with 2 m thickness, were positioned in a complementary coplanar arrangement, producing a field overlap of =0.911 and =0.552 for the interferometric and directional coupler cases. The corresponding calculated velocity mismatch factors were =0.429 and = Overlap profile between the optical and modulating fields for both devices Responses for the integrated phase shift and modulation depth of the interferometric modulator Responses for the integrated phase shift and modulation depth of the Dir. Coupler based modulator Optical mode profiles for the Directional Coupler Modulator Optical mode profiles for the Interferometric Modulator Transverse Field Profile f 3dB = 7.1 GHz f 3dB = 23.3 GHz f 3dB = 8.5 GHz f 3dB = 19.0 GHz