How can I improve my code for visualizing a complex map?

Question

This code visualize a complex map, but it looks rather cumbersome:

F[z_] := z^2;
t1 = 0; 
t2 = Pi/3;
r1 = 1;
r2 = 3;
dt = (t2 - t1)/10;
dr = (r2 - r1)/10;

p1 = Show[
  Table[ParametricPlot[ReIm[r Exp[I t]], {r, r1, r2}], {t, t1, t2, 
    dt}],
  Table[ParametricPlot[ReIm[r Exp[I t]], {t, t1, t2}], {r, r1, r2, 
    dr}],
  ParametricPlot[ReIm[r Exp[I 4 dt]], {r, r1 + 5 dr, r1 + 6 dr}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[ReIm[r Exp[I 5 dt]], {r, r1 + 5 dr, r1 + 6 dr}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[ReIm[r Exp[I 6 dt]], {r, r1 + 6 dr, r1 + 7 dr}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[ReIm[r Exp[I 3 dt]], {r, r1 + 6 dr, r1 + 7 dr}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[
   ReIm[(r1 + 7 dr) Exp[I t]], {t, t1 + 3 dt, t1 + 6 dt}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[
   ReIm[(r1 + 6 dr) Exp[I t]], {t, t1 + 3 dt, t1 + 4 dt}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[
   ReIm[(r1 + 6 dr) Exp[I t]], {t, t1 + 5 dt, t1 + 6 dt}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[
   ReIm[(r1 + 5 dr) Exp[I t]], {t, t1 + 4 dt, t1 + 5 dt}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  Axes -> None, PlotRange -> All]

p2 = Show[
  Table[ParametricPlot[ReIm[F[r Exp[I t]]], {t, t1, t2}], {r, r1, r2, 
    dr}],
  Table[ParametricPlot[ReIm[F[r Exp[I t]]], {r, r1, r2}], {t, t1, t2, 
    dt}],
  ParametricPlot[ReIm[(r Exp[I 4 dt])^2], {r, r1 + 5 dr, r1 + 6 dr}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[ReIm[(r Exp[I 5 dt])^2], {r, r1 + 5 dr, r1 + 6 dr}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[ReIm[(r Exp[I 6 dt])^2], {r, r1 + 6 dr, r1 + 7 dr}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[ReIm[(r Exp[I 3 dt])^2], {r, r1 + 6 dr, r1 + 7 dr}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[
   ReIm[((r1 + 7 dr) Exp[I t])^2], {t, t1 + 3 dt, t1 + 6 dt}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[
   ReIm[((r1 + 6 dr) Exp[I t])^2], {t, t1 + 3 dt, t1 + 4 dt}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[
   ReIm[((r1 + 6 dr) Exp[I t])^2], {t, t1 + 5 dt, t1 + 6 dt}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ParametricPlot[
   ReIm[((r1 + 5 dr) Exp[I t])^2], {t, t1 + 4 dt, t1 + 5 dt}, 
   PlotStyle -> {Black, Thickness[0.01]}],
  ImageSize -> 200,
  Axes -> None,
  PlotRange -> All]

I'm wondering if there is a way to make it significantly shorter since there are lots of repeating functions used.

Answer 1

You can get filled in grid sectors fairly easily with MeshShading:

GraphicsRow[
 With[{z = r Exp[I t], col = Black},
    ParametricPlot[ReIm@#[z], {r, r1, r2}, {t, t1, t2}, Mesh -> 9,
     MeshShading -> 
      ArrayPad[{{None, col}, {col, col}, {None, col}}, {{3, 4}, {5, 
         3}}, None], Frame -> False, AxesOrigin -> {0, 0}
     ]
    ] & /@ {Identity, F}
 ]

Answer 2

At least, the generation of the mesh can be shortened to

t1 = 0;
t2 = Pi/3;
r1 = 1;
r2 = 3;
ParametricPlot[ReIm[r Exp[I t]], {r, r1, r2}, {t, t1, t2}, 
 Mesh -> {10, 10},
 PlotStyle -> {FaceForm[]},
 MeshStyle -> {{Opacity[.5], ColorData[97][1]}},
 BoundaryStyle -> {{Opacity[.5], ColorData[97][1]}}
 ]