# Unidirectional Slow Light Transmission in Heterostructure Photonic Crystal Waveguide
## 1 Introduction  
The velocity of light in vacuum is approximately \(2.99\times 10^{8}\) m/s, fast enough to circle the Earth once in 0.3 s. However, faster is not always better, such an ultrahigh speed makes it very difficult to control the optical signals in time domain [1, 2]. Slow light with a much slower group velocity makes it possible to delay and temporarily store light in all-optical systems [3, 4, 5, 6]. Among all the approaches to realize slow light, photonic crystal waveguide (PCW) has drawn much attention as it supports room temperature operation and is compatible with on-chip integration [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. However, the performance of the conventional photonic crystal (PC) systems is very vulnerable to the backscattering loss induced by disorders or defects [17, 18]. The backscattering loss is amplified as the group velocity slows down in the waveguide [19, 20].  
The development of the photonic chiral edge state (CES) [21, 22, 23, 24] may provide an approach to overcome the limitations related to backscattering loss in slow light systems, as it is robust against scattering from interfacial defects. The existence of the photonic CES at the interface of a two-dimensional (2D) magneto-optical photonic crystal (MOPC) was first theoretically proposed byHaldane [25; 26] and was then experimentally verified by [PERSON] et al. [27; 28] in microwave regime. Since the CESs are leaky in the free space [21; 29], a regular PC with an aligned band gap has to be bounded to the interface of the MOPC to prevent the radiation leakage. The constructed waveguide only supports unidirectional modes as it has only positive (negative) group velocities and the backscattering is completely suppressed. This enabled the light transmission to be immune to fabrication imperfections. The unidirectional transmission feature is of great significance for slow light systems and may provide robust designs for optical delay and light storage applications [30].  
There are various approaches to tailor the dispersion curves to realize slow light in PC systems: modification the radii of air holes [31], shifting the air holes to specific directions [32], infiltrating optical fluid into the air holes [33], infiltrating microfluidic selectively [34; 35], using asymmetric engineering structure [36], merging coupled cavities [37] and, introducing alternative rows of ellipse-holes [38]. In all the above mentioned ways, adjusting the rod radius is relatively easy to control and have less fabrication issues.  
In this paper, we propose a unidirectional backscattering-immune slow light waveguide. The waveguide is constructed by a square lattice of MOPC with the rod radii of the first two rows adjacent to the waveguide core are precisely tailored and a titled 45\({}^{\circ}\) square lattice of Alumina PC as an ancillary cladding. A flat band for slow light is obtained and the slope (group velocities) of the waveguide dispersion curves process only positive (or only negative) values, which is determined by the sign of the externally applied magnetic field. In the waveguide, the backscattering is completely suppressed for the lack of the backward transmission channel. This unique feature ensures that the design can avoid performance degradation induced by impurities or defects, which is difficult to realize in conventional slow light systems.  
The remaining content of this work is arranged as follows: to better understand the content of the subsequent sections, we briefly introduce the theoretical background of slow light in Section 2. In Section 3, we firstly introduce the design of the unidirectional slow light waveguide and then exhibit the slow light characteristics and the backscattering-immune properties of the waveguide in detail. The followed simulation results are based on the finite element method (FEM).