Robust transmission in topological insulators makes it possible to steer waves without attenuation along bent paths within imperfectly fabricated photonic devices. But the absence of reflection does not guarantee the fidelity of pulsed transmission which is essential for core photonic functionalities. Pulse transmission is disrupted by localized modes in the bulk of topological insulators which coexist with the continuum edge mode and are pushed deeper into the band gap with increasing disorder. Here we show in simulations of the Haldane model that pulse propagation in disordered topological insulators is robust throughout the central portion of the band gap where localized modes do not arise. Since transmission is robust in topological insulators, the essential field variable is the phase of the transmitted field, or, equivalently, its spectral derivative, which is the transmission time. Except near resonances with bulk localized modes that couple the upper and lower edges of a topological insulator, the transmission time in a topological insulator is proportional to the density of states and to the energy excited within the sample. The variance of the transmission time at the band edge for a random ensemble with moderate disorder is dominated by fluctuations at resonances with localized states, and initially scales quadratically

We calculate the time delay of a disordered topological insulator confined as quasi-one-dimensional strip based on the tight-binding model. Two approaches, the field decomposition and time delay decomposition, are tested to extract the complex energy of the discrete modes in the presence of the edge mode. The previous discussion of time delay and DOS are mainly based on the scattering matrix, we show here that the spatial matrix of Green’s function between the output and input can be directly used to calculate the DOS. This facilitates the calculation of DOS because it is easier to obtain the Green’s function, in measurements and numerical simulation such as recursive Green’s function method. When modal overlap is small, the transmitted field can be decomposed into a superposition of discrete modes and the edge channel. This decomposition is possible even though transmission is dominated by robust transmission in the edge state. When the upper edge mode is not coupled to the lower edge mode via disorder, the eigenchannel time delay is then equivalent to the integral of intensity inside the sample, whose spectrum can be decomposed into a superposition of Lorentzian lines.

The increase of phase by 2π when tuning through a resonance in a disordered TI, contrasts with the increase of π in a trivial random medium. The phase change of 2π can be understood from the complex representation of the field in which the curve of the complex field encircles the original point in the presence of the continuum.

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Kang, Y., Huang, Y. & Genack, A. Z. Dynamics of transmission in disordered topological insulators. Physical Review A 103, 033507 (2021)