We study the performance of achromatic half-wave plates (AHWPs) as a function of the detection bandwidth of a power detector operating in the millimeter wave band and the spectral shape of the incident radiation. We focus particular attention on the extraction of the degree of incident polarization and its orientation angle from the intensity measured as a function of AHWP rotation angle, which we call the IVA (intensity versus angle). We describe the formalism to extract the two incident polarization parameters. We use this formalism to quantify the phase offset of the IVA and point to potential systematic errors in the extraction of this offset in cases where the incident spectrum is not sufficiently well known. We quantify the phase offset and modulation efficiency as a function of the relative angles between the plates in the stack and find that high modulation efficiency can be achieved with alignment accuracy of a few degrees. We present measurements of the spectral response of an AHWP made with five plates. The measurements predict a modulation efficiency that is higher than 98% for three bands centered at 150, 250, and 410 GHz.