In the present study, the effect of the depth of the confinement potential on the nonlinear optical of a GaAs quantum well with short-range bottomless exponential potential is studied in detail. The energy eigenvalues and eigenfunctions of this structure are calculated within the framework of effective mass and envelope function approximations. Analytic formulas for the linear, third-order nonlinear and total absorption coefficients and relative refractive index changes are obtained using the compact-density matrix approach (CDMA) and iterative method. Based on this model, our obtained numerical results are reported as a function of incident photon energy for several values of the depth of the confinement potential. The results show that the linear, third order nonlinear, and total absorption coefficients and relative refractive index changes are strongly affected by the depth of the confinement potential.