## Abstract

Electro-optic modulators (EOMs) are components which convert electric signals to optical ones. They are needed, e.g., at the transmitter end of fiber-optic communication systems and in time-stretch analog-to-digital converters. The development of new EOM designs to handle electric signals with higher frequencies is driven by the demand for ever-increasing bandwidths in telecommunications. As the process of building prototypes is time-consuming and expensive, it is highly desirable to increase the use of numerical simulations in the design process.

The topic of this thesis is numerical methods for simulation of high-speed EOMs. Two main challenges arise in this context: first, the device is optically very large; a wave propagation problem must be solved over tens of thousands of wavelengths. Second, due to the high frequencies of the modulating signal, the problem must be solved in time domain, rather than in frequency domain, as is otherwise common for waveguiding problems. A suitable method for this type of problem is the time-domain beam-propagation method (TD-BPM), which is particularly efficient for propagation over long distances, occurring mainly along a specified direction. If further the geometry varies slowly along that direction, a simplified formulation, called the paraxial TD-BPM, has previously been employed. Based on an analysis of the equations, and comparison to the related analysis of (linear) optical fibers, a modified paraxial formulation is suggested in the current work. We show that the modification, while adding neither complexity nor computational effort to the method, increases its accuracy significantly, especially for short pulses.

The TD-BPM has previously been discretized using finite differences. In this work we derive a weak formulation and a novel discretization based on tensor product finite elements. By using finite elements, discontinuities in material parameters at material interfaces can be represented in an exact manner, which is not possible with finite differences. Furthermore, non-uniform meshes with higher resolution where the data varies rapidly can be readily used with finite elements, and function spaces can be chosen flexibly. Full-vector and scalar versions of the weak and discrete formulations are derived. Numerical results are presented for the scalar TD-BPM. The implementation is validated against analytic data for the case of pulse propagation in a straight waveguide. In addition, the results are compared to those of the finite-difference time-domain method, and the TD-BPM is shown to have higher accuracy for short pulses (pulse widths 10 – 50fs), when the modified paraxial approximation, suggested in this work, is employed. Finally, as a proof-of-concept case, the method is applied to an electro-optic modulator with simplified geometry but realistic modulating signal.