We report the spectral features of a phase-shifted parity and time ($𝒫𝒯$)-symmetric fiber Bragg grating (PPTFBG) and demonstrate its functionality as a demultiplexer in the unbroken $𝒫𝒯$-symmetric regime. The length of the proposed system is of the order of millimeters and the lasing spectra in the broken $𝒫𝒯$-symmetric regime can be easily tuned in terms of intensity, bandwidth and wavelength by varying the magnitude of the phase shift in the middle of the structure. Surprisingly, the multi-modal lasing spectra are suppressed by virtue of judiciously selected phase and the gain-loss value. Also, it is possible to obtain sidelobe-less spectra in the broken $𝒫𝒯$-symmetric regime, without a need for an apodization profile, which is a traditional way to tame the unwanted sidelobes. The system is found to show narrow band single-mode lasing behavior for a wide range of phase shift values for given values of gain and loss. Moreover, we report the intensity tunable reflection and transmission characteristics in the unbroken regime via variation in gain and loss. At the exceptional point, the system shows unidirectional wave transport phenomenon independent of the presence of phase shift in the middle of the grating. For the right light incidence direction, the system exhibits zero reflection wavelengths within the stopband at the exceptional point. We also investigate the role of multiple phase shifts placed at fixed locations along the length of the FBG and the variations in the spectra when the phase shift and gain-loss values are tuned. In the broken $𝒫𝒯$-symmetric regime, the presence of multiple phase shifts aids in controlling the number of reflectivity peaks besides controlling their magnitude. The advantage of the proposed model is that it exhibits multi-functional capabilities like demultiplexing, filtering and lasing in a short length of the grating depending on the different operating regimes.

中文翻译：

---

相移𝒫𝒯对称周期结构

我们报告了相移的奇偶校验和时间的频谱特征（$𝒫𝒯$）对称光纤布拉格光栅（PPTFBG）并在不间断的情况下展示其作为多路分解器的功能 $𝒫𝒯$对称制度。所提出的系统的长度为毫米量级，并且在断裂处的激光光谱$𝒫𝒯$通过改变结构中间相移的大小，可以很容易地在强度，带宽和波长方面调整对称状态。出乎意料的是，由于明智地选择了相位和增益损耗值，多模激光光谱被抑制了。同样，有可能获得断层中无旁瓣的频谱$𝒫𝒯$对称的方式，不需要切趾轮廓，这是驯服不需要的旁瓣的传统方法。对于给定的增益和损耗值，发现该系统对于宽范围的相移值显示出窄带单模激射行为。此外，我们通过增益和损耗的变化报告了在不间断状态下强度可调的反射和透射特性。在特殊情况下，该系统显示出单向波传输现象，而与光栅中间的相移无关。对于正确的光入射方向，系统在特殊点的阻带内显示零反射波长。我们还研究了沿FBG长度固定位置放置的多个相移的作用，以及当对相移和增益损耗值进行调整时频谱的变化。在破碎$𝒫𝒯$在对称的情况下，多个相移的存在除了控制幅度的大小外，还有助于控制反射峰的数量。所提出的模型的优点在于，它根据不同的工作方式在短长度的光栅中展现了多种功能，例如多路分解，滤波和激射。