  We investigate exact and approximate techniques to calculate the emission of
gravitational radiation from cosmic string loops in order to generate beam
models covering the entire celestial sphere for a wide range of modes $m$. One
approach entails summing over contributions of stationary and nearly-stationary
points of individual, factorized, left- and right-moving modes. This
"multipoint method" generalizes traditional methods that rely on expansions
around exact stationary points of mode products. A second complementary
approach extends the method of steepest descent to generate an asymptotic
description of the beam as $m \to \infty$. We present example calculations of
the emission of power from cusp-containing loops and compare the results with
those obtained by numerically exact techniques as well as by previous
approaches. The multipoint method achieves its best results at an intermediate
range of modes, improving over previous methods in terms of accuracy. It
handles the emission from all regions of the loop not just those near cusps. We
demonstrate this capability by making a detailed study of the "pseudocusp"
phenomenon.
