SetDirectory["~/KappaLib/"];
<<kappaLib-1.1.m
<<helper.m

vp={1.6385600128874258`,-2.79035110757227`,-0.9894755078507828`};vv={0.9919624506450033`,-0.11938437307145532`,-0.04192693617109755`};

sub={a3->a2,b3->b2};

vars={x0,x1,x2,x3};
kappa=emMatrixToKappa[(({{a1, 0, 0, -b1, 0, 0}, {0, a2, 0, 0, -b2, 0}, {0, 0, a3, 0, 0, -b3}, {b1, 0, 0, a1, 0, 0}, {0, b2, 0, 0, a2, 0}, {0, 0, b3, 0, 0, a3}}))];
fr=emKappaToFresnel[kappa,vars]/.sub;

FullSimplify[fr]

AA= (({{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, (1/2) (-D3+(Sqrt[-4+(D3^2)])), 0}, {0, 0, 0, (1/2) (-D3+(Sqrt[-4+(D3^2)]))}}));
BB = (({{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, (1/2) (-D3-(Sqrt[-4+(D3^2)])), 0}, {0, 0, 0, (1/2) (-D3-(Sqrt[-4+(D3^2)]))}}));
ss={D3->((((((a1-a2))^2)+(b1^2)+(b2^2)))/(b1 b2))};

f1=Simplify[vars.AA.vars];
f2=Simplify[vars.BB.vars];
verify= b1  b2^2 f1 f2;
FullSimplify[( fr- verify)/.sub/.ss]

Fresnel polynomial depends on x0, x1^2+x2^2, x3

f1
f2

g1[x0_,x1_,x2_,x3_,D3_]:=(x0^2)+(1/2) (-2 (x1^2)+(-D3+(Sqrt[-4+(D3^2)])) ((x2^2)+(x3^2)))
g2[x0_,x1_,x2_,x3_,D3_]:=(x0^2)+(1/2) (-2 (x1^2)-(D3+(Sqrt[-4+(D3^2)])) ((x2^2)+(x3^2)))

draw[d3_, xx_, yy_,zz_]:=Module[
{p1,p2,grayLevel},
grayLevel=0.2; (* 0 = black *)
p1=ContourPlot3D[
{g1[x0,x1,0,x3,d3]==0},
{x0,-xx,xx},{x1,-yy,yy},{x3,-zz,zz},
Axes->False,
Boxed->False,
Lighting->"Neutral",
ViewPoint->vp, 
Mesh->5,
PlotPoints->40,
ColorFunctionScaling->0.1,
MeshStyle->{Directive[GrayLevel[grayLevel],Opacity[0.5]]},
ViewVertical->vv];
p2=ContourPlot3D[
{g2[x0,x1,0,x3,d3]==0},
{x0,0,xx},{x1,-yy,yy},{x3,-zz,zz},
MeshStyle->{Directive[GrayLevel[grayLevel],Opacity[0.5]]},
Axes->False,
Boxed->False,
Mesh->{2,2,3},
MaxRecursion->10,
Lighting->"Neutral",
PlotPoints->40,
ViewPoint->vp, 
ViewVertical->vv];
Show[{p1,p2},PlotRange->{All,All,All}]
];

plot1=draw[2.05,1,1,2.165];
plot2=draw[2.3,1,1,2.2];
plot3=draw[3,1,1,2.9];
ptot=Show[GraphicsGrid[{{plot1,plot2,plot3}}]];

Export["PlotI.pdf",ptot,ImageResolution->1600]

Note running the above notebook is relatively time consuming. Images have been removed from the above due their size.