Low-power optical bistability in $\mathcal{PT}$-symmetric chirped Bragg gratings with four-wave mixing

The central theme of this article is the analysis of the usefulness of introducing four-wave mixing or modulation of Kerr nonlinearity in a nonuniform grating structure with gain and loss. To do so, we propose an inhomogeneous system in which the nonlinearity of the $\mathcal{PT}$-symmetric grating is modulated. It is proven that the proposed scheme can be fruitful only if the nonlinearity and its associated modulation terms are assumed to be self-defocusing ones. In order to cut down the intensity required for the steering, the sign of chirping is taken to be negative and the wavelength of operating light is considered to be higher than the Bragg wavelength. Alongside these settings, launching the light from the rear end of the device dramatically reduces the critical intensity to a value of 0.015 (approximately) which must be the lowest switching intensity ever reported in the context of nonlinear $\mathcal{PT}$-symmetric gratings. The ramp-like stable states in the broken $\mathcal{PT}$-symmetric regime prevail even in the company of modulation in the nonlinearity index for both of the light incident directions.