This work demonstrates how the crystal symmetry of photonic crystal defect waveguides interacts with simple, experimentally realizable parity-time ($ {\cal P}{\cal T} $) symmetric regions of chip-scale absorption and amplification to control the existence and location of exceptional points in the first Brillouin zone. Our analysis is based on Heesh–Shubnikov group theory and is generalizable to a large class of devices for which the symmetry groups can be identified. Transverse, longitudinal, and transverse–longitudinal hybrid $ {\cal P}{\cal T} $ symmetries are considered, and for each, a triangular lattice photonic crystal waveguide with lattice-aligned and lattice-shifted cladding orientations is analyzed. We find that various symmetry combinations produce either strictly real-valued or strictly complex-valued eigenfrequencies at the Brillouin zone boundary. We also show how symmetry can be used to predict $ {\cal P}{\cal T} $ transitions at accidental degeneracies in the waveguide bands. It is shown how symmetry can be used to design single-mode waveguides, and we discovered exceptional points whose propagation constants are highly sensitive to the non-Hermiticity factor.

中文翻译：

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对称设计的波导色散 $ {\ cal P} {\ cal T} $ 对称光子晶体波导

这项工作表明用简单的光子晶体缺陷波导相互作用的晶体的对称性，通过实验可实现的奇偶时间（如何$ {\ CAL P} {\ CAL T】$ ）芯片尺寸的吸收和扩增的对称区域，以控制的存在和位置第一个布里渊区的特殊点。我们的分析基于Heesh–Shubnikov组理论，并且可以推广到可以识别对称组的一大类设备。横向，纵向和横向-纵向混合动力$ {\ cal P} {\ cal T} $考虑对称性，并且对于每个对称性，分析具有晶格对准和晶格移位的包层取向的三角形晶格光子晶体波导。我们发现，各种对称组合在布里渊区边界产生严格的实值或严格的复数本征频率。我们还展示了对称性如何可用于预测波导带中意外退化时的$ {\ cal P} {\ cal T} $跃迁。它显示了如何将对称性用于设计单模波导，并且我们发现了其传播常数对非赫米特因子高度敏感的特殊点。