How to convert a polar plot in surface Plot

Question

I want to display PolarPlot in the form of surface plot. I don't know how to achieve this task. for example I have chossen a very simple problem

                     a=4;
                     PolarPlot[Cos[x*a],{x,0,Pi}]

which returns the above pattern. I want to display the same pattern in Surface Plot form.
for reference please see the below Pic

Answer 1

The PolarPlot for your given function is actually

a = 2;
PolarPlot[Cos[x*a], {x, 0, Pi}]

The PolarPlot that you show corresponds to

a = 4;
PolarPlot[Sin[x*a], {x, 0, 2 Pi}]

To display radius as a function of angle on a cartesian Plot

Plot[Abs[Sin[x*a]], {x, -Pi, Pi}]

Or as a LogPlot

The far-field radiation patterns that you show are not related to your function or your PolarPlot

EDIT: Using RegionPlot

RegionPlot[
 -Sqrt[x^2 + y^2] <= Sin[a*ArcTan[y/x]] <=
  Sqrt[x^2 + y^2],
 {x, -1, 1}, {y, -1, 1},
 MaxRecursion -> 5]

EDIT 2: Using DensityPlot

DensityPlot[Sin[a*ArcTan[y/x]],
 {x, -1, 1}, {y, -1, 1},
 RegionFunction -> Function[{x, y, f},
   -Sqrt[x^2 + y^2] >= f || f >= Sqrt[x^2 + y^2]],
 PlotPoints -> 100,
 MaxRecursion -> 3]