Density plot for data with a wide dynamic range

Question

I'm generating some figures for a paper. One of them has a pattern like the following figure:

DensityPlot[
 Abs[Exp[-Sqrt[x^2 + y^2]/5] BesselJ[0, Sqrt[x^2 + y^2]]], {x, -20, 
  20}, {y, -20, 20}, ColorFunction -> "SunsetColors", 
 PlotPoints -> 100, PlotRange -> All, PlotRangePadding -> 0, 
 FrameTicksStyle -> 
  Directive[FontFamily -> "Arial", 12, Plain, Thick], 
 ImageSize -> {Automatic, 150}]

Since the magnitude in the middle could be much higher than that in the outer region, some rings cannot be seen if I print it. In addition, since the pattern represents magnitude, I prefer a kind of "hot" color map to render these data. How can I solve the problem such that the pattern can be seen clearly on a printed copy?

Answer 1

ColorFunction takes an actual function as input. Thus, eg,

Manipulate[
 DensityPlot[
  Abs[Exp[-Sqrt[x^2 + y^2]/5] BesselJ[0, Sqrt[x^2 + y^2]]], {x, -20, 
   20}, {y, -20, 20}, 
  ColorFunction -> Function[{f}, RGBColor[#, #, #] &@(f^alpha)], 
  PlotPoints -> 100, PlotRange -> All, PlotRangePadding -> 0, 
  FrameTicksStyle -> 
   Directive[FontFamily -> "Arial", 12, Plain, Thick], 
  ImageSize -> {Automatic, 150}],
 {{alpha, .2}, 0.01, 2}
 ]

or with SunsetColors:

Manipulate[
 DensityPlot[
  Abs[Exp[-Sqrt[x^2 + y^2]/5] BesselJ[0, Sqrt[x^2 + y^2]]], {x, -20, 
   20}, {y, -20, 20}, 
  ColorFunction -> (ColorData["SunsetColors"][#^alpha] &), 
  PlotPoints -> 100, PlotRange -> All, PlotRangePadding -> 0, 
  FrameTicksStyle -> 
   Directive[FontFamily -> "Arial", 12, Plain, Thick], 
  ImageSize -> {Automatic, 150}],
 {{alpha, .2}, 0.01, 2}
 ]

Answer 2

Two possibilities:

DensityPlot[Abs[Exp[-Sqrt[x^2 + y^2]/5] BesselJ[0, Sqrt[x^2 + y^2]]]^(
 1/4), {x, -20, 20}, {y, -20, 20},
 ColorFunction -> "SunsetColors",
 PlotPoints -> 100, PlotRange -> All, PlotRangePadding -> 0, 
 FrameTicksStyle -> 
  Directive[FontFamily -> "Arial", 12, Plain, Thick], 
 ImageSize -> {Automatic, 150}]

DensityPlot[
 20 Abs[Exp[-Sqrt[x^2 + y^2]/5] BesselJ[0, Sqrt[x^2 + y^2]]], {x, -20,
   20}, {y, -20, 20},
 ColorFunction -> "SunsetColors",
 ColorFunctionScaling -> False,
 PlotPoints -> 100, PlotRange -> All, PlotRangePadding -> 0, 
 FrameTicksStyle -> 
  Directive[FontFamily -> "Arial", 12, Plain, Thick], 
 ImageSize -> {Automatic, 150}]

In the first attempt, I raised the small values by taking the fourth root of the Abs. In the second version, I turned off ColorFunctionScaling and multiplied by 20 to raise the small values. In that case, the hight values are cut off.

You can replace the power 1/4 in the first version by 1/2 to darken the lower values.