Propagation of phase noise from the input to the nulled intensities for different kernel-nulling architectures. Values are normalized by the peak transmission of an off-axis signal. The raw nulls (dashed lines) are compared with their corresponding kernel-null observable (solid lines), showing the suppression of second-order phase noise (by a few orders of magnitude) for small input phase error. This effect decreases as the input phase errors depart from the small phase approximation. The behavior of the Bracewell nuller is shown for reference (dashed blue lines).