Euclid Quick Data Release (Q1). The shape of cluster haloes revealed by intracluster light
Intracluster light (ICL) offers a luminous route for measuring the shapes and orientations of cluster haloes. We used Euclid Quick Data Release (Q1) images to quantify ICL shapes for 205 clusters at $0.1 \leq z \leq 0.8$ selected from the eROSITA All-Sky Survey and the Dark Energy Survey. We measured the ICL properties across five bands ($I_\mathrm{E}$, $Y_\mathrm{E}$, $J_\mathrm{E}$, $H_\mathrm{E}$ and the coadded $YJH_\mathrm{E}$), and find that the shape parameters are consistent across the filters, with the $H_\mathrm{E}$ band probing the profiles to the largest cluster-centric distances ($0.1R_{200\mathrm{c}}\sim100$ kpc). The ICL ellipticity ($e=1-b/a$) distribution peaks at 0.5, consistent with halo ellipticities inferred from strong and weak lensing studies, suggesting that the ICL traces the large-scale structure of the clusters. Cluster cores, dominated by brightest cluster galaxy (BCG) light, are round ($e\sim0.2$), while the ICL becomes progressively more elongated at large radii, reaching an ellipticity of $\sim0.5$ by $0.1R_{200\mathrm{c}}$. We find that the ICL is closely aligned with the BCG (87% lie within 30$^\circ$), but shows a weaker alignment with the satellite galaxy distribution (61% within 30$^\circ$). Robust position angles could be measured using the ICL for 63% of the sample, whereas position angles could be measured for only 30% of the sample using the satellite galaxies. Stacking clusters in redshift bins allows measurements of the ICL surface brightness to $0.6$-$0.8R_{200\mathrm{c}}$ and ellipticity beyond $0.3$-$0.4R_{200\mathrm{c}}$ (300-400 kpc), but we observed a plateau in the ICL ellipticity beyond $0.1R_{200\mathrm{c}}$. These results demonstrate the capability of Euclid to characterise the ICL morphology with high statistical precision, allowing new constraints on cluster halo structure and its evolution with redshift.