I have the relation of $ k$ and $ w$ , and now I need to plot $ k$ vs $ w$ ($ k\sim 10^7$ , $ w\sim 10^{13}$ ). The relation is given as follows:

parameters:

`\[Epsilon][2] = \[Epsilon][1] = 9.5; \[Epsilon][3] = 1;\[Mu] = 1.257*10^-6;c = 3.*10^8;\[Epsilon]0 = 8.85*10^-12;d = 20.*10^-9; \[Sigma] = Im[1/(-I* w)*Abs[(7.5*10^14*(1.6*10^-19)^2)/(0.2*0.91*10^-30)]]; `

relation:

`1 - ((\[Epsilon][1] Sqrt[-\[Epsilon][2] w^2/c^2 + k^2] - \[Epsilon][ 2] Sqrt[-\[Epsilon][1] w^2/c^2 + k^2] - \[Sigma] Sqrt[-\[Epsilon][1] w^2/c^2 + k^2] Sqrt[-\[Epsilon][2] w^2/c^2 + k^2]/(\[Epsilon]0 w ))/(\[Epsilon][ 1] Sqrt[-\[Epsilon][2] w^2/c^2 + k^2] + \[Epsilon][ 2] Sqrt[-\[Epsilon][1] w^2/c^2 + k^2] - \[Sigma] Sqrt[-\[Epsilon][1] w^2/c^2 + k^2] Sqrt[-\[Epsilon][2] w^2/c^2 + k^2]/(\[Epsilon]0 w ))) (\[Epsilon][ 3] Sqrt[-\[Epsilon][2] w^2/c^2 + k^2] - \[Epsilon][ 2] Sqrt[-\[Epsilon][3] w^2/c^2 + k^2])/(\[Epsilon][ 3] Sqrt[-\[Epsilon][2] w^2/c^2 + k^2] + \[Epsilon][ 2] Sqrt[-\[Epsilon][3] w^2/c^2 + k^2])Exp[-2 d Sqrt[-\[Epsilon][2] w^2/c^2 + k^2]]=0 `

The method `ContourPlot`

doesn’t work well, so I turn to use `FindRoot`

and `NSolve`

to find the list of solutions $ (k,w)$ so I can plot it manually. However, even `FindRoot`

doesn’t work well. For example, when I set `w=3 Pi * 10 ^12`

, the solution of k should be $ k=1.41179*10^8$ . However, I cannot use `FindRoot`

to find this solution since the LHS of the relation is a non-monotonic function: Now I have to choose the value of k0 very carefully for `FindRoot`

to start with, or Mathematica cannot give me the correct answer. However, what I need to do is to plot the relation between $ k$ and $ w$ , so I really don’t want to find k0 by looking into the plot with different $ w$ one by one, I hope to find a way to make Mathematica smartly choose k0 and find the solution.

I am grateful for any help, thank you!