  Parity-time (PT) symmetric optical sensors operating around exceptional
points have recently gained much attraction, offering an unparalleled high
sensitivity in measuring small perturbations. In the past, most of the
PT-symmetric sensors have been based on tracking the mode-splitting that arises
due to a perturbation-induced change in coupling strength between two
subcavities of the PT-symmetric system. We design a linear fiber Fabry-P\'erot
and coupled cavities sensor, tailored to operate in the broken PT-symmetric
region. We explore a new sensing metric-that is, the mode's linewidth change as
a function of perturbation-induced changes in the loss within one of the
subcavities of the PT-symmetric system. The coupling strength between the two
subcavities remains unchanged in our proposed sensor. Supported by a
mathematical formulation, we find that the full-width-half-maximum (FWHM) of
the cavity resonances exhibits a square root dependence on the refractive index
(RI) change in one of the subcavities. The proposed fiber cavity refractive
index sensor has a maximum sensitivity of $2.26\times10^7$ GHz/RIU and a lowest
detection limit of $10^{-9}$ RIU, widely outperforming the comparable cavity
sensors subject to the same refractive index change, gain, and loss settings.
