Linear mathematical model for the underlying mechanism of extended ptychographic iterative engine

•The underlying mathematics was investigated for the first time to prove the existence of the unique solution of extended ptychographic iterative engine imaging.•A linear mathematical model was set up to reconstruct the illumination and sample simultaneously from only seven frames of recorded diffraction intensities directly.•An efficient computing method without limitation of spectrum size was developed to improve computing efficiency, and the mathematical relationship between the reconstruction results and imaging system was illustrated.•Experimental factors were also considered and it was demonstrated theoretically and numerically that the proposed method had a strong robustness to detecting noise and positioning error. A linear model by writing the recorded diffractive intensities as a set of linear equations was put forwarded to determine the complex amplitude of the illumination and sample analytically for the extended ptychographic iterative engine (ePIE) imaging. An efficient computing method that can process large matrices was proposed to improve computing efficiency. The influences of detector noise and the positioning error of the sample were studied to explain why satisfactory reconstruction can be obtained experimentally when the linear equations of diffractive intensities are incorrect. This study demonstrates for the first time that ePIE technique has a unique and deterministic solution even with noisy experimental data, overcoming the main obstacle hindering the application of the ePIE algorithm in optical measurement and optical metrology, where a mathematically unique solution is crucial.