  Sideband cooling enables preparation of trapped ion motion near the ground
state and is essential for many scientific and technological applications of
trapped ion devices. Here, we study the efficiency of continuous and pulsed
sideband cooling using both first- and second-order sidebands applied to an ion
where the motion starts outside the Lamb-Dicke regime. We find that after
optimizing these distinct cooling methods, pulsed and continuous cooling
achieve similar results based on simulations and experiments with a
$^{40}$Ca$^+$ ion. We consider optimization of both average phonon number
$\overline{n}$ and population in the ground state. We also demonstrate the
disparity between $\overline{n}$ as measured by the sideband ratio method of
trapped ion thermometry and the $\overline{n}$ found by averaging over the
ion's motional state distribution.
