A Quaternion Analysis of Time Symmetry in Spin-1 Excitation

The role of time symmetry in composite-pulse design is examined by considering a phase-alternating composite pulse pair {π(I= $\text{1}\text{2}$), π/2(I= 1)}, where the spin-1 excitation pulse has been derived from its spin-$\text{1}\text{2}$ progenitor by halving the pulse durations. The quaternion calculus is used to define the quaternion elements (Euler–Rodrigues parameters) of each composite pulse. In this manner, it is shown how an Euler–Rodrigues (ER) parametrization of the consecutive rotations implicit in each composite pulse can be used to derive simple phase and amplitude relationships between the members of such a {π(I= $\text{1}\text{2}$), π/2(I= 1)} pulse pair. The simplicity and compactness of the ER parametrization is then used to identify optimal time-symmetric sequences for spin-1 excitation by using the Lagrange multiplier method.