Matlab - rootfinder for nonlinear function

I have written a function (file calculationdeterminant) which calculates a determinant that is function of k and omega. This function is complex. Now I need for each omega from 0 to 80 all the positive real roots (k's) of this function. For each omega there will be several roots. I need the values of the real roots where the function (real and imaginary part) is zero. I started by searching for the k's where the real part of the function is zero and then looked whether the imaginary part of the function is smaller than a certain tolerance, but I did not quite succeed in getting good results. Next to the values of the roots, I need a figure in which the roots are plotted with omega on the vertical axis and k on the horiontal axis, where k should go from 0 to 2 . This can in first instance be plotted with dots, but ideally, next to the plot with dots I would need a figure (with omega on the vertical axis and k on the horizontal axis again) with smooth lines, comparable to the plot in attachment. Attached you find the Model data, the determinant function and an example of the plot I would need.