Pancharatnam-based achromatic half-wave plates (AHWP) achieve high polarization efficiency over a broad waveband. These AWHPs generally contain a property whereby the optic axis is dependent on the electromagnetic frequency of the incident radiation. When the AHWP is used to measure incident polarized radiation with a finite detection bandwidth, this frequency dependence causes an uncertainty in the determination of the polarization angle due to the limited knowledge of the shape of the source spectrum and detection band. To mitigate this problem, we propose new designs of the AHWP that eliminate the frequency dependence of the optic axis over the bandwidth while maintaining high modulation efficiency. We carried out this optimization by tuning the relative angles among the individual half-wave plates of the five and nine layer AHWPs. The optimized set of relative angles achieves a frequency-independent optic axis over the fractional bandwidth, a bandwidth over which polarization efficiency is greater than 0.9, of 1.3 and 1.5 for the five- and nine-layer AHWPs, respectively. We also study the susceptibility of the alignment accuracy on the polarization efficiency and the frequency dependence of the optic axis, which provides a design guidance for each application.