Wave Propagation in Linear Media | re-examined

8.3 Implementation of the quadrature modules
8 Application of the quadrature routine
Now that we have formulated all integrals in a way that suits our needs, we can write the
quadrature modules. Since they are all based on the same structure, we shall discuss here
only the example of the rectangular wave packet and a few functions that are related to it. All
other modules are very similar in their structure and can be understood without di culties
from the listing of the package in the appendix.
The atomic function modules we use to de ne the integrands are the transmission and reection
coe cients and the oscillatory factors of the integrands within (TunOsc) and outside
(OutOsc) the tunnel. In addition, we need the argument function of the oscillation inside
the tunnel for the optional exact determination of the integrand's zeros. These functions are
independent of the waveform.
TunCoeff[xi_] := 2 xi/(xi + Sqrt[xi^2-1]);
RefCoeff[xi_] := (xi - Sqrt[xi^2-1])/(xi + Sqrt[xi^2-1]);
TunOsc[a_,b_,c_,d_,xi_] := Exp[I (a xi^2 + b xi + c Sqrt[xi^2 - 1] + d)];
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;
Other basic functions like those describing the shape of the spectrum are di erent for each
waveform. Note that in the implementation we use the variable w for the parameter .
RectConst[w_,k_] := I / (Sqrt[w] 2 Pi);
RectShapePos[w_,xi_] := 1/(1-xi/Sqrt[w]);
RectShapeNeg[w_,xi_] := 1/(1+xi/Sqrt[w]);
With these building blocks we de ne the actual integrands that are to be used inside the
quadrature routines. As we saw while formulating the respective equations, we need two
distinct functions for the positive and negative inde nite integrals. The integrand for the
evanescent region is augmented by the coe cients and oscillatory factors that stem from the
shape of the spectrum. Therefore we can compute the integral with only one single quadrature
call. Owing to the particular structure of the oscillatory factors, we need to pass only values
for the parameters a and b to the quadrature routine and must set b and d to zero in this
case.
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)]);
The constant RectConst is not part of the integrand because it is useless to compute it
with each function evaluation. Instead, the constant is multiplied to the result of the entire
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