Wave Propagation in Linear Media | re-examined

8 Application of the quadrature routine
Knowing that we will end up with a plethora of very similar functions, we establish a consistent
naming convention to avoid confusion and to reduce the risk of programming errors. All
function names are thus composed of distinct elements to distinguish them from each other.
A severe obstacle to the legibility of the names is the fact that Mathematica permits no
underscores within a function's name because the underscore is reserved to designate patterns.
So we have no possibility toseparate the tokens that make up a function name as in other
programming languages.
We start with the de nition of a number of tokens, each of which describes a part of the
purpose the function is intended for.
Form 2 fRect, Tria, Gaussg denotes the shape of the initial wavepacket and applies
to all waveform-dependent functions.
Region 2fInc, Ref, Out, Trans, Evan, Tung distinguishes the regions a function may
be applicable to. Some modules depend on the direction of the wave propagation or
whether the evanescent or the transmitted part of the integral is regarded. Others
simply are di erent inside and outside the barrier.
FunctionType 2 fConst, Shape, Oscg denotes constants, shaping functions for the
spectra of the waves, and the oscillatory factors of the integrands.
Polarity 2 fNULL, Pos, Negg is used to distinguish integrands de ned for either the
positive or negative real axis. The empty element NULL applies to all functions that
need no distinction between the two cases.
Gradient 2fNULL, Gradg signi es whether the function is used for the computation of
the wave function or its gradient. In the latter case, the name includes the additional
token Grad.
These tokens are used to compose the various function modules. Which tokens are actually
combined and what values are meaningful depends on the class of modules under consideration.
Region-dependent factors have the form kRegionjFunctionTypek. For example,
RefCoeff is the coe cient function for the re ected wave, TunOsc means the oscillatory
factor of the integrands in the tunnel.
Integrands and waveform-dependent factors have the more complicated structure
kFormjGradientjfFunctionType, RegiongjPolarityk. Examples are GaussConst for a
constant factor, RectShapeNeg for a factor of an integrand and nally TriaGradEvan
or GaussRefNeg for the entire integrands.
Quadrature templates are rather heterogenous and have the naming structure kfNULL,
FormgjfEvan, TransgjfNULL, Exact, TruncgjTempk, as for instance TriaTransExactTemp
or GaussEvanTruncTemp. The elements Exact and Trunc refer to alternative computing
strategies for the non-Gaussian and Gaussian wave packets, respectively.
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