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The Laser Interferometer Space Antenna (LISA) will detect gravitational waves(GWs) emitted by massive black hole binaries (MBHBs) in the low-frequency($\sim$mHz) band. Low-mass lenses, such as dark-matter (DM) subhalos, havesizes comparable to the wavelength of these GWs. Encounters with these lensesproduce wave-optics (WO) effects that alter waveform phase and amplitude. Thus,a single event with observable WO effects can be used to probe the lensproperties. In this paper, we first compute the probability of observing WOeffects in a model-agnostic way. We perform parameter estimation overapproximately 1000 MBHBs with total mass, mass ratio, and redshift spanning theranges relevant to LISA. We then calculate lensing rates using threesemi-analytical models of MBHB populations. In both cases, we use a waveformmodel that includes merger, ringdown, and higher-order modes. We use two lenspopulation models: the theory-based Press-Schechter halo mass function and anobservation-based model derived from Sloan Digital Sky Survey, called themeasured velocity function. We find that the probability of detecting WOeffects can be as large as $\sim 3\%$, $\sim1.5\%$, and $\sim 1 \%$ at$1\sigma$, $3\sigma$, and $5\sigma$ confidence levels, respectively. The mostoptimistic MBHB population model yields $\sim 8$, $\sim 4$, and $\sim 3$ eventsat the same confidence levels, while the rates drop to $\sim 0.01$ in the morepessimistic scenarios. The most likely lens masses probed by LISA are in therange $(10^3, 10^8)\, M_{\odot}$, and the most probable redshifts are in therange $(0.3, 1.7)$. Therefore, LISA observations of WO effects can probe DMsubhalos, complementing strong lensing and other observations.

Probing wave-optics effects and dark-matter subhalos with lensing of gravitational waves from massive black holes