Transport of intensity equation: Validity limits of the usually accepted solution

The transport of intensity equation (TIE) is the basis of a powerful non-interferometric technique for phase retrieval. When intensity gradients are present, the generalized belief is that the solution of the TIE (i.e. the phase distribution) can be found by solving two Poisson equations. In the present paper we will demonstrate that this expression for the TIE solution holds when intensity and phase gradients are parallel (or null), but in general the usually accepted solution is not correct. We perform simulations with arbitrary intensity and phase functions in order to show the phase errors derived from the use of this expression.