Frequency-domain versus time-domain analysis of slow-light mesophotonic waveguides: Theoretical insights for practically realizable devices

Slow-light mesophotonic waveguides have gained increasing interest in the recent years owing to their potential for time-delay control of optical signals, crucial for all-optical circuitry, as well as EM energy storage and field enhancement which is important for driving a stronger light-matter interaction. Yet, the design of such systems has hitherto been predominantly directed by modal-type of analyses in frequency domain. We discuss here why frequency domain analysis fails for real systems, leading to mistaken predictions for the effective propagation speed of the EM signal. In particular, by utilizing a time-domain analysis we demonstrate a slow-light paradigm waveguide system which contradicts the widespread notion that the group-index is generally a good measure of the light's speed effective slow-down factor within the waveguide. In particular, our counterexample shows that it is entirely possible when two modes are compared that it would be the one with the lower group index to yield effectively the larger light slow-down factor. We analyze the salient characteristics of the modes with the higher light slow-down-factor merit to understand how to enable slow light in practical platforms. Our results suggest that in order for a large group index to practically effect a large slowdown factor it should be accompanied by a large modal index bandwidth. This can occur when both the group velocity and the group-velocity dispersion are simultaneously near-zero.

I. Introduction

Electromagnetic (EM) modeling methods fall within two large classes: frequency-domain methods and time-domain methods. In the former methods, the EM wave solutions are sought for a fixed frequency, ω. This can be done numerically, for example with the Finite Difference Frequency Domain (FDFD) method [1]–[3], or analytically for simple geometries [4]–[7]. These frequency-domain methods generally provide important information for a system's response to incoming EM waves. However, they fail to accurately predict the time-dependent response to realistic source excitations that are not necessarily perfect monochromatic plane waves. Indeed, it is possible to construct the response of the system in time-domain by utilizing frequency-domain methods together with rigorous-modal matching analysis [8]. However, this can get increasingly tedious or difficult as structures or EM-wave launch geometries get more complicated. A brute-force method For time-domain analysis is certainly attractive. The most popular method for time-domain analysis is the Finite-Difference-Time-Domain (FDTD) [9]. Free from any assumptions for simplifications, it can model actual experimental set-ups. We discuss here why for slow-light-waveguide [10] problems the FDTD is uniquely suited for designing structures that practically demonstrate a large light propagation slow-down factor. In particular, we present in the following a counter-example where a frequency-domain analysis alone clearly fails to accurately quantify the effective slow-down factor for the propagating slow waveguide mode. Fig. 1.

(A) The bi-waveguide with the size parameters indicated for the two different designs. (b) The wave dispersion for the two bi-waveguide designs, A and B. The frequency is expressed in dimensionless units, by multiplying the inverse of the free-space wavelength, with the average of the width of the two waveguide designs. The wavevector along the waveguide direction, , is also scaled with the free-space planewave wavevector, The ratio of the two gives the dimensionless parameter, which represents the modal index of the guided mode [11]. (c) Magnitude of the group index, |ng|, versus the modal index, for the two waveguide modes seen in (b) in logarithmic scale. Note: Dashed lines have been used where the group index becomes negative.