  We present the constructon and detailed analysis of highly-optimized
self-refocusing pulse shapes for several rotation angles. We characterize the
constructed pulses by the coefficients appearing in the Magnus expansion up to
second order. This allows a semi-analytical analysis of the performance of the
constructed shapes in sequences and composite pulses by computing the
corresponding leading-order error operators. Higher orders can be analyzed with
the numerical technique suggested by us previously. We illustrate the technique
by analysing several composite pulses designed to protect against pulse
amplitude errors, and on decoupling sequences for potentially long chains of
qubits with on-site and nearest-neighbor couplings.
