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I'm trying find graphically the solutions of the intersection of the following functions.

d = 1;
κ = ΩH/(ΩH^2 - Ω^2);
ΩH = 1;
Ω = 1.05;

ContourPlot[
  {Re[Tan[ki d] - 2 (ko ki)/(ki^2 - ko^2)] == 0, ko == (I ki)/Sqrt[1 + κ]},
  {ki, -10, 10}, {ko, -10, 10},
  LabelStyle -> 
    Directive[Black, {FontFamily -> "Latin Modern Roman", FontSize -> 20}],
  FrameLabel -> {Subscript[k, i], Subscript[k, o]},
  PlotLabel -> "1+κ<0",
  PlotTheme -> "Scientific",
  PlotPoints -> 200, 
  WorkingPrecision -> 100,
  PerformanceGoal -> "Quality",
  ImageSize -> 400,
  Exclusions -> {ki == ko, ki == -ko, Cos[ki d] == 0}]

However, I don't know why there are a discontinuity of the orange function.

enter image description here

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The discontinuities occur at the exclusions:

Exclusions -> {ki == ko, ki == -ko, Cos[ki d] == 0}
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