Initialisation

Needs["MaTeX`"]
SetOptions[MaTeX,"Preamble"->{"\\usepackage{amssymb,amsmath,latexsym,amsfonts,amsthm,xcolor,bm,mhchem}","\\definecolor{darkgreen}{RGB}{0, 180, 0}"}];

E1[t_,λ_]:=(1/2) (4 t-(Sqrt[16 (t^2)+16 (t^2) (λ^2)]))
E2[t_,λ_]:=(1/2) (4 t+(Sqrt[16 (t^2)+16 (t^2) (λ^2)]))

(1/2)(U+(Sqrt[(((4 λ t))^2)+(U^2)]))/.U->4t

(Subscript[λ, c])=I;

With[{t=1,Mag=2},
Show[Plot3D[{Re[E1[t,λx+I λy]],Re[E2[t,λx+I λy]]},{λx,-2,2},{λy,0,3},
BoxRatios->{1, 1, 1},ImageSize->420,PlotStyle->Opacity[0.6],Mesh->5,Boxed-> False,
AxesEdge->{{-1,-1},{1,-1},{-1,-1}},
AxesLabel->{
MaTeX["\\text{Re}(\\lambda)",Magnification->Mag],
MaTeX["\\text{Im}(\\lambda)",Magnification->Mag],
Rotate[MaTeX["\\text{Re}(E)/t",Magnification->Mag],(π/2)]},
AxesStyle->Directive[Thick,18,Black],ViewPoint->{1.3`,-1.6,0.7}],
ListPointPlot3D[{{0,1,2}},PlotStyle->Black],
ParametricPlot3D[{λ,0,Re[E1[t,λ]]},{λ,-2,2},PlotStyle->{Thickness[0.01],Orange}],
ParametricPlot3D[{λ,0,Re[E2[t,λ]]},{λ,-2,2},PlotStyle->{Thickness[0.01],Blue}],
ParametricPlot3D[{Re[(Subscript[λ, c])]+(1/2)Cos[θ],Im[(Subscript[λ, c])]+(1/2)Sin[θ],Re[E1[t,λ]/.λ->( (Subscript[λ, c])+(1/2)Exp[I θ])]},{θ,π/2,-π/2},PlotStyle->{Thickness[0.007],Black}],ParametricPlot3D[{Re[(Subscript[λ, c])]+(1/2)Cos[θ],Im[(Subscript[λ, c])]+(1/2)Sin[θ],Re[E2[t,λ]/.λ->( (Subscript[λ, c])+(1/2)Exp[I θ])]},{θ,π/2,3π/2},PlotStyle->{Thickness[0.007],Black}],ParametricPlot3D[{Re[(Subscript[λ, c])]+(1/2)Cos[θ],Im[(Subscript[λ, c])]+(1/2)Sin[θ],Re[E2[t,λ]/.λ->( (Subscript[λ, c])+(1/2)Exp[I θ])]},{θ,π/2,-π/2},PlotStyle->{Thickness[0.007],Black,Dashed}],ParametricPlot3D[{Re[(Subscript[λ, c])]+(1/2)Cos[θ],Im[(Subscript[λ, c])]+(1/2)Sin[θ],Re[E1[t,λ]/.λ->( (Subscript[λ, c])+(1/2)Exp[I θ])]},{θ,π/2,3π/2},PlotStyle->{Thickness[0.007],Black,Dashed}]]]