# Special solution of Helmholtz equation separated Phase

I'm searching for some special solutions for the Helmholtz-equation in 2D:

$(\partial_x^2 + \partial_y^2 + a) f(x,y) = 0$

where $f(x,y) \in \mathbb{C}$ (boundary condition: $\lim_{x,y\to \infty} f(x,y)=0$)

Now let's separate the Amplitude and Phase:

$f(x,y)=A(x,y) \cdot \exp(i\cdot g(x,y))$

with $A(x,y),g(x,y) \in \mathbb{R}$

I want now that $g(x,y)$ itself fulfills some special PDEs: $\hat{L} g(x,y)=h(x,y)$.

with $h(x,y) \in \mathbb{R}$

So the question:

When can I find a solution $f(x,y)$ fulfilling the 2D Helmholtz equation (with natural boundary conditions) such that its phase fulfills itself a PDE? With what method can I find such solutions for a given $\hat{L}$ and $h(x,y)$?