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Controllable linear $π$-phase modulation in a thermal atom vapor without diffraction or absorption

A scheme is proposed to achieve substantial controllable phase modulation for a probe field propagating through a thermal atomic vapor in double-$Λ$ configuration. The phase modulation is based on the linear susceptibility of the probe field, paraxial diffraction is eliminated by exploiting the thermal motion of atoms, and residual absorption is compensated via an incoherent pump field. As a result, a strong controllable uniform phase modulation without paraxial diffraction is achieved essentially independent of the spatial profile or the intensity of the probe field. This phase shift can be controlled via the intensities of the control or the incoherent pump fields. A possible proof-of-principle experiment in alkali atoms is discussed.