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- xp2[xi_] :=
- Module[{g, num, den}, g = Sqrt[4*xi^2 + (4*xi^2*(xi^2 - 1))^(2/3)];
- num = 2*xi^2*Sqrt[g];
- den = Sqrt[8*xi^2*(xi^2 + 1) + 12*g*xi^2 - g^3] - Sqrt[g^3];
- num/den];
- xz2[xi_] := xi^2/xp2[xi];
- t[xi_] := Sqrt[1 - 1/xi^2];
- (*Use particular values for low-order functions*)
- r1[xi_, x_] := x;
- r2[xi_, x_] := ((t[xi] + 1)*x^2 - 1)/((t[xi] - 1)*x^2 + 1);
- r3[xi_, x_] :=
- x*((1 - xp2[xi])*(x^2 - xz2[xi]))/((1 - xz2[xi])*(x^2 - xp2[xi]));
- (*Use nesting property for higher-degree functions*)
- r4[xi_, x_] := r2[r2[xi, xi], r2[xi, x]];
- r8[xi_, x_] := r4[r2[xi, xi], r2[xi, x]];
- ellgain[xi_, w_, w0_, ep_] := 1/Sqrt[1 + ep^2*r8[xi, w/w0]^2];
- DensityPlot[
- w0 = 1;
- ep = 0.5;
- xi = 1.05;
- min = 0.0001;
- max = 10;
- Log[Abs[
- ellgain[xi, sig + I*w, w0*I, ep]
- ]],
- {sig, -4, 4},
- {w, -4, 4},
- PlotRange -> {Log[min], Log[max]},
- PlotPoints -> 100,
- ColorFunction -> GrayLevel,
- ClippingStyle -> {Black, White}
- ]
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