Silicon Photonics: Fundamentals and Devices

11.4 Distributed Bragg Reflector

We now consider the second device, the distributed feedback or distributed Bragg reflector structure, in which coupling may exist between the forward- and backward-propagating modes [1–3]. The basic structure is shown in Figure 11.4. A periodic undulation has been introduced over the region 0 < z < L. Such kind of undulation may be accomplished by etching the layer and then growing another layer on top of it. The DBR structure is introduced in narrow-width single-mode lasers used for telecommunication and the reader may consult the literature for more information.

Figure 11.4 Schematic structure of a Bragg grating. The periodic perturbation in RI is introduced by changing the structure of width d_p along the x-direction.

11.4.1 Phase-Matched Grating

We first consider the phase-matched case. As stated already, the periodicity of perturbation along the z-direction allows coupling of modes propagating in opposite directions. The two modes of interest in this configuration have propagation constants +β for the forward-traveling (+z) mode (mode 1) and −β for the backward-traveling (−z) mode (mode 2). The field profiles f(x)'s remain the same for both the forward- and backward-propagating modes.

Assume that the structure is lossless. From energy conservation, it is required that the decrease of power along the positive z-direction in mode 1 must be exactly ...