Gravitational lensing observables, including anomalies in image positions, flux ratios, and time delays, serve as usual probes of dark matter substructure. When dark matter substructure possesses sufficient perturbations, it may lead to the formation of extra images in otherwise canonical doubly or quadruply imaged systems. With the advent of increasingly precise observational instruments, previously undetectable images may become measurable and image number anomalies therefore could be an increasingly viable method. In this paper, we utilize the gravitational lensing phenomenon of image number anomaly to derive constraints on dark matter substructure. We present the extra images induced by distinct forms of dark matter substructure, specifically primordial black holes (PBHs) and fuzzy dark matter and show that higher angular resolution observations increase the probability of detecting additional lensed images. Based on a null detection of image number anomalies in a sample of 3500 lens systems generated from the strong lensing halo model-based mock catalogs, we derive upper limits on the abundance of PBHs. At the 95% confidence level, the PBH abundance is constrained to $\lesssim 0.125\%$, 0.08%, and 0.04% for PBH masses in the range $\sim {10}^7–{10}^9{M}_{\odot }$, corresponding to angular resolutions of ${0.1}^{\prime \prime }$, $0.05^{\prime \prime }$, and $0.01^{\prime \prime }$, respectively. Similarly, we exclude particle masses below 0.4, 0.6, and $3.5\times {10}^{-22}\mathrm{eV}$ for fuzzy dark matter at the same confidence level for the respective resolutions. Furthermore, the abundance of $\mathrm{PBHs}\lesssim 0.9\%$ could be constrained at an angular resolution of ${0.5}^{\prime \prime }$ for the Legacy Survey of Space and Time observations. Finally, we discuss methodologies for identifying image number anomalies in special cases and demonstrate feasibility using a fitting procedure.

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• Dark matter
• Gravitational lenses