High-precision phase retrieval in mode-division multiplexing systems using extended Nijboer-Zernike polynomials

Mode-division multiplexing (MDM) systems require accurate optical field characterization, but conventional phase measurement techniques, such as off-axis holography, suffer from bulky setups and limited robustness for complex spatial modes. Here, we propose a reference-free optical field measurement method based on extended Nijboer-Zernike (ENZ) polynomials. An ENZ-based phase retrieval system operating at 1550 nm reconstructs complex pupil-plane fields from multi-focal plane intensity measurements without interferometric reference beams. Validation on LP₀₁, LP₁₁, LP₂₁, and orbital angular momentum (OAM) modes shows that the ENZ method reduces the phase root mean square error (Phase RMSE) by 52.7%, 52.2%, and 44.1% for LP modes, and 74.0%, 66.6%, and 60.2% for OAM modes, relative to off-axis holography. These results correspond to overall improvements of roughly 44%-75%, with the strongest gain consistently observed for OAM modes. Further analyses under different noise levels and experimental parameter variations confirm that the ENZ method maintains stable and reliable phase reconstruction performance. Overall, the proposed approach enhances phase reconstruction accuracy, reduces reconstruction fluctuations, and provides a robust and scalable solution for high-precision optical field measurement in multimode optical communication systems.