<< NDSolve`FEM`

r = 0.8;
outerCirclePoints =
  With[{r = 2.},
   Table[{r Cos[[Theta]], r Sin[[Theta]]}, {[Theta],
     Range[0, 2 [Pi], 0.05 [Pi]] // Most}]];
innerCirclePoints =
  With[{r = r},
   Table[{r Cos[[Theta]], r Sin[[Theta]]}, {[Theta],
     Range[0, 2 [Pi], 0.08 [Pi]] // Most}]];

bmesh = ToBoundaryMesh[
   "Coordinates" -> Join[outerCirclePoints, innerCirclePoints],
   "BoundaryElements" -> {LineElement[
      Riffle[Range[Length@outerCirclePoints],
        RotateLeft[Range[Length@outerCirclePoints], 1]] //
       Partition[#, 2] &],
     LineElement[
      Riffle[Range[Length@outerCirclePoints + 1,
         Length@Join[outerCirclePoints, innerCirclePoints]],
        RotateLeft[
         Range[Length@outerCirclePoints + 1,
          Length@Join[outerCirclePoints, innerCirclePoints]], 1]] //
       Partition[#, 2] &]}];

mesh = ToElementMesh[bmesh];
{bmesh["Wireframe"], mesh["Wireframe"]}

glass = 1.45; air = 1.; k0 = (2 [Pi])/1.55;
n[x_, y_] := If[x^2 + y^2 <= r^2, glass, air]

helm = !(
*SubsuperscriptBox[([Del]), ({x, y}), (2)](u[x, y])) +
   n[x, y]^2*k0^2*u[x, y];
boundary = DirichletCondition[u[x, y] == 0., True];

(*region=ImplicitRegion[x^2+y^2[LessEqual]2.^2,{x,y}];*)

{vals, funs} =
  NDEigensystem[{helm, boundary}, u[x, y], {x, y} [Element] mesh, 1,
   Method -> {"Eigensystem" -> {"FEAST",
       "Interval" -> {k0^2, glass^2 k0^2}}}];
vals

 Table[Plot3D[funs[[i]], {x, y} [Element] mesh, PlotRange -> All,
  PlotLabel -> vals[[i]]], {i, Length[vals]}]