Enhancement of Kerr nonlinearity completely without absorption

We investigate the enhancement of the Kerr nonlinearity in a four-level tripod-type atomic system completely without absorption. We study the effect of the incoherent pumping on the nonlinear susceptibility of the probe field and show that compared with that without the incoherent pumping, the Kerr nonlinearity can be enhanced with completely suppressed absorption at two different probe frequencies. Thus, we can get a symmetrical enhanced Kerr nonlinearity. Particularly, all of our results are in the case of Doppler broadening.