Wave Propagation in Linear Media | re-examined

A.2 Solutions for the step potential
(*-- Functions independent of the waveform --*)
TunCoeff[xi_] := 2 xi/(xi + Sqrt[xi^2-1]);
TunOsc[a_,b_,c_,d_,xi_] := Exp[I (a xi^2 + b xi + c Sqrt[xi^2 - 1] + d)];
RefCoeff[xi_] := (xi - Sqrt[xi^2-1])/(xi + Sqrt[xi^2-1]);
OutOsc[a_,b_,c_,d_,xi_] := Exp[I (a xi^2 + (b+c) xi + d)];
PhiArg[a_,b_,c_,d_][xi_] := a xi^2 + b xi + c Sqrt[xi^2 - 1] + d;
GaussTruncLims[w_,k_,n_,wp_] :=
{N[Sqrt[w] (1-n/(Pi k) *
Sqrt[Log[Sqrt[2 Pi] k/Sqrt[n w Sqrt[2 Pi]]/10^-(wp+1)]]), wp+2],
N[Sqrt[w] (1+n/(Pi k) *
Sqrt[Log[Sqrt[2 Pi] k/Sqrt[n w Sqrt[2 Pi]]/10^-(wp+1)]]), wp+2]};
(*-- Triangular Shape inside the tunnel --*)
TriaConst[w_,k_] := Sqrt[3/w] / (2 k Pi^2);
TriaShapePos[w_,xi_] := 1/(1-xi/Sqrt[w])^2;
TriaShapeNeg[w_,xi_] := 1/(1+xi/Sqrt[w])^2;
TriaTransPos[a_,b_,c_,d_,w_,k_][xi_] :=
TunCoeff[xi] * TriaShapePos[w,xi] * TunOsc[a,b,c,d,xi];
TriaTransNeg[a_,b_,c_,d_,w_,k_][xi_] :=
TunCoeff[xi] * TriaShapeNeg[w,xi] * TunOsc[a,b,c,d,xi];
TriaGradTransPos[a_,b_,c_,d_,w_,k_][xi_] :=
TunCoeff[xi] * TriaShapePos[w,xi] * TunOsc[a,b,c,d,xi] *
I Sqrt[xi^2 - 1];
TriaGradTransNeg[a_,b_,c_,d_,w_,k_][xi_] :=
TunCoeff[xi] * TriaShapeNeg[w,xi] * TunOsc[a,b,c,d,xi] *
(-I Sqrt[xi^2 - 1]);
TriaEvan[a_,b_,c_,d_,w_,k_][xi_] :=
TriaTransPos[a,b,c,d,w,k][xi] *
(-1 + 2 Exp[I (Pi k/Sqrt[w] xi - Pi k)] -
Exp[2 I (Pi k/Sqrt[w] xi - Pi k)]);
TriaGradEvan[a_,b_,c_,d_,w_,k_][xi_] :=
TriaGradTransPos[a,b,c,d,w,k][xi] *
(-1 + 2 Exp[I (Pi k/Sqrt[w] xi - Pi k)] -
Exp[2 I (Pi k/Sqrt[w] xi - Pi k)]);
(*-- Triangular Shape outside the tunnel --*)
TriaRefPos[a_,b_,c_,d_,w_,k_][xi_] :=
RefCoeff[xi] * TriaShapePos[w,xi] * OutOsc[a,b,c,d,xi];
TriaRefNeg[a_,b_,c_,d_,w_,k_][xi_] :=
RefCoeff[xi] * TriaShapeNeg[w,xi] * OutOsc[a,b,c,d,xi];
TriaRefEvan[a_,b_,c_,d_,w_,k_][xi_] :=
TriaRefPos[a,b,c,d,w,k][xi] *
(-1 + 2 Exp[I (Pi k/Sqrt[w] xi - Pi k)] -
Exp[2 I (Pi k/Sqrt[w] xi - Pi k)]);
TriaIncPos[a_,b_,c_,d_,w_,k_][xi_] :=
TriaShapePos[w,xi] * OutOsc[a,b,c,d,xi];
TriaIncNeg[a_,b_,c_,d_,w_,k_][xi_] :=
TriaShapeNeg[w,xi] * OutOsc[a,b,c,d,xi];
TriaIncEvan[a_,b_,c_,d_,w_,k_][xi_] :=
TriaIncPos[a,b,c,d,w,k][xi] *
(-1 + 2 Exp[I (Pi k/Sqrt[w] xi - Pi k)] -
Exp[2 I (Pi k/Sqrt[w] xi - Pi k)]);
(*-- Rectangular Shape inside the tunnel --*)
RectConst[w_,k_] := I / (Sqrt[w] 2 Pi);
RectShapePos[w_,xi_] := 1/(1-xi/Sqrt[w]);
RectShapeNeg[w_,xi_] := 1/(1+xi/Sqrt[w]);
RectTransPos[a_,b_,c_,d_,w_,k_][xi_] :=
TunCoeff[xi] * RectShapePos[w,xi] * TunOsc[a,b,c,d,xi];
RectTransNeg[a_,b_,c_,d_,w_,k_][xi_] :=
TunCoeff[xi] * RectShapeNeg[w,xi] * TunOsc[a,b,c,d,xi];
RectEvan[a_,b_,c_,d_,w_,k_][xi_] :=
RectTransPos[a,b,c,d,w,k][xi] *
(-1 + Exp[I 2 (Pi k/Sqrt[w] xi - Pi k)]);
(*-- Rectangular Shape outside the tunnel --*)
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