I need to find the roots of a non linear function, namely:

F[z_] = 2 E^(2 I b z)
z ((-1 + E^(2 I a Sqrt[z^2 - Subscript[V, 0]])) z + (1 + E^(
    2 I a Sqrt[z^2 - Subscript[V, 0]])) Sqrt[
  z^2 - Subscript[V, 0]]) + (E^(2 I a z) - E^(2 I b z)) (-1 + E^(
 2 I a Sqrt[z^2 - Subscript[V, 0]])) Subscript[V, 0]   

where:

$\omega$ is complex valued and $a,V_{0}, b$ are real parameters.

I've been trying to solve numerically this problem by defining a new function

G[x_, y_] := Module[{Z}, Z = x + I*y; Abs[N[1/F[Z]]]]

making a contour plot of this and then approximating the roots by Findroot

ContourPlot[G[x, y], {x, -10, 10}, {y, -4, 4},PlotLegends -> Automatic, 
FrameLabel -> {"x", "y"}, Mesh -> False, MaxRecursion -> 5]

but it's not giving me the better graphical quality that I need. Is there any better way to evaluate this complex roots?

Thanks in advance!

I need to find the roots of a nonlinear function, namely:

F[z_] = 2 E^(2 I b z)
z ((-1 + E^(2 I a Sqrt[z^2 - Subscript[V, 0]])) z + (1 + E^(
    2 I a Sqrt[z^2 - Subscript[V, 0]])) Sqrt[
  z^2 - Subscript[V, 0]]) + (E^(2 I a z) - E^(2 I b z)) (-1 + E^(
 2 I a Sqrt[z^2 - Subscript[V, 0]])) Subscript[V, 0]   

where:

$\omega$ is complex valued and $a,V_{0}, b$ are real parameters.

I've been trying to solve numerically this problem by defining a new function

G[x_, y_] := Module[{Z}, Z = x + I*y; Abs[N[1/F[Z]]]]

making a contour plot of this and then approximating the roots by FindRoot

ContourPlot[G[x, y], {x, -10, 10}, {y, -4, 4},PlotLegends -> Automatic, 
FrameLabel -> {"x", "y"}, Mesh -> False, MaxRecursion -> 5]

but it's not giving me the better graphical quality that I need. Is there any better way to evaluate these complex roots?