currently I notice a secord order PDE named transport-of-intensity equation (see page 6 of this paper).
The equation if as follow: \begin{equation} \left(\nabla_{\perp}^{2}+2 i k \frac{\partial}{\partial z}\right) U_{A}(\mathbf{x}, z)=0, \end{equation} where $\nabla_{\perp}^{2}$ is the transverse Laplacian operator defined by $\nabla_{\perp}^{2}=\left(\partial^{2} / \partial x_{1}^{2}+\partial^{2} / \partial x_{2}^{2}\right)$ and $z$ denotes the propagation direction. The varying component $U_{A}(\mathbf{x}, z)= \sqrt{I(\mathbf{x}, z)}\exp(i\phi(\mathbf{x}, z)).$
My question is about secord order Laplacian and my solution of that part is is: \begin{equation} \begin{aligned} \nabla_{\perp}^{2} I(\mathbf{x}, z)^{\frac{1}{2}}\exp(i\phi(\mathbf{x}, z)) = & \nabla_{\perp} \cdot \left[ \nabla_{\perp}( \ I(\mathbf{x}, z)^{\frac{1}{2}}\exp(i\phi(\mathbf{x}, z)) \ ) \right] \\ =& \nabla_{\perp} \cdot \left[ \frac{1}{2}I(\mathbf{x}, z)^{-\frac{1}{2}}\exp(i\phi(\mathbf{x}, z)) \nabla I(\mathbf{x}, z) +\\ i\exp(i\phi(\mathbf{x}, z))I(\mathbf{x}, z)^{\frac{1}{2}} \nabla \phi((\mathbf{x}, z)) \right] \end{aligned} \end{equation}
I dont know whether my partial dervation is right. Could anyone tell me?
Besides, could I eliminate the $\exp(i\phi(\mathbf{x}, z))$ before excuting the second order derivative (the result from $2 i k \frac{\partial}{\partial z} \sqrt{I(\mathbf{x}, z)}\exp(i\phi(\mathbf{x}, z))$ has the same term) ?
Could anyone show the hints for the following calculation? Thanks!