I am new to Mathematica and trying to reproduce the following figure using `ContourPlot`:

[![enter image description here][1]][1]
(From: https://iopscience.iop.org/article/10.1088/1402-4896/abb2e0/meta)


My code: 

    n1 = 0.5
    n2 = 0.5
    sig1 = 0.1
    sig2 = 0.1
    U1 = -1
    U2 = 1
    
    ContourPlot[n1/((Omg - K*U1)^2 - 3*K^2*sig1) + n2/((Omg - K*U2)^2 - 3*K^2*sig2) - 1/K^2 == 1, 
               {K, 0, 1}, {Omg, -2, 2}, Axes -> True, Exclusions -> 
               {(Omg - K*U1)^2 - 3*K^2*sig1 ==  0, (Omg - K*U2)^2 - 3*K^2*sig2 == 0}]

The output:

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/F54yA.png
  [2]: https://i.sstatic.net/IWzSx.png

As seen in the output plot, `ContourPlot` only returns and plots the real solutions. 

Is there another Mathematica function that plots both the complex and real solutions with different line styles, and a double axis as shown in the first figure?