  We report the spectral features of a phase-shifted parity and time
($\mathcal{PT}$)-symmetric fiber Bragg grating (PPTFBG) and demonstrate its
functionality as a demultiplexer in the unbroken $\mathcal{PT}$-symmetric
regime. The length of the proposed system is of the order of millimeters and
the lasing spectra in the broken $\mathcal{PT}$-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 $\mathcal{PT}$-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 $\mathcal{PT}$-symmetric regime, the presence of multiple phase
shifts aids in controlling the number of reflectivity peaks besides controlling
their magnitude.

Optics;Pattern Formation and Solitons;Computational Physics