Impact of slow-light enhancement on optical propagation in active semiconductor photonic-crystal waveguides

We derive and validate a set of coupled Bloch wave equations for analyzing the reflection and transmission properties of active semiconductor photonic-crystal waveguides. In such devices, slow-light propagation can be used to enhance the material gain per unit length, enabling, for example, the realization of short optical amplifiers compatible with photonic integration. The coupled-wave analysis is compared to numerical approaches based on the Fourier modal method and a frequency domain finite-element technique. The presence of material gain leads to the buildup of a backscattered field, which is interpreted as distributed feedback effects or reflection at passive-active interfaces, depending on the approach taken. For very large material gain values, the band structure of the waveguide is perturbed, and deviations from the simple coupled Bloch wave model are found.

Figure 1

Illustration of defect photonic-crystal waveguide. The red part of the structure is active, i.e., this part of the membrane structure contains embedded layers of quantum wells or quantum dots that may provide gain upon optical or electrical pumping. The lattice constant is $a$ and the length of the active region is $L$.

Figure 2

Illustration of coupled-wave analysis of PhC waveguide. (a) Schematic of surface used in the derivation of coupled mode equations, with arrows indicating outward unit normal vectors to the boundary surface. (b) Schematic diagram of coupling between the amplitudes of forward- and backward-propagating Bloch waves induced by the active material, which is represented by an imaginary perturbation of the refractive index.

Figure 3

Power reflection (dash-dotted blue line) and transmission (solid red line) as functions of the imaginary part of the refractive index for the PhC structure in Fig. 1, with an active section of length $L/a=100$. A slow-light mode at $\omega a/\left(2\pi c\right)=0.2064$ with a group index $n_{gz}\simeq 25$ [see Figs. 4 and 4] is considered. Curves are results from the 1D coupled-wave analysis, with $|r_1{|}^2={\left|r_2\right|}^2=0$, and markers are 2D FMM simulation results.

Figure 4

Propagation characteristics of a line-defect photonic-crystal waveguide with a gain section of finite length, $L=100a$. (a) Dispersion curve and (b) group index, both of passive PhC waveguide Bloch mode. (c) Power transmission and (d) power reflection, both as functions of the imaginary part of the refractive index and frequency. The results in (c) and (d) are obtained by numerically solving the 1D coupled Bloch wave equations.

Figure 5

Power transmission and reflection in (a) decibel scale and (b) linear scale as functions of the imaginary part of the refractive index for the PhC structure in Fig. 1, with an active section of length $L/a=20$. The frequency is $\omega a/\left(2\pi c\right)=0.2075$ with a corresponding group index of $n_{gz}\simeq 18.7$. Dash-dotted blue (reflection) and solid red (transmission) curves are 1D coupled-wave analysis results with zero passive-active interface reflections $\left(r_1=r_2=0\right)$, while markers are 2D FEM simulation results.