Topological defects and disorder counteract each other1–5. Intuitively, disorder is considered detrimental, requiring efforts to mitigate its effects in conventional topological photonics6–9. We propose a counter-intuitive approach that exploits a real–momentum topological photonic crystal that harnesses real-space disorder to generate a Pancharatnam–Berry phase10,11, without disrupting the momentum-space singularity originating from bound states in the continuum12. This methodology allows flat optical devices to encode spatial information or even extra topological charge in real space while preserving the topology of bound states in the continuum in momentum space with inherent alignment. Here, as a proof of concept, we demonstrate the simultaneous and independent generation of a real-space broadband vortex or a holographic image alongside resonant momentum-space vortex beams with a narrow bandwidth, which cannot be achieved with conventional methods. Such engineered disorder contributes to vast intrinsic freedoms without adding extra dimensions or compromising the optical flatness13,14. Our findings of real–momentum duality not only lay the foundation for disorder engineering in topological photonics but also open new avenues for optical wavefront shaping, encryption and communications.
