Adaptive spatio-temporal phase retrieval for ultra-short pulse wavefront shaping

Beam shaping of ultra-short pulses is essential for medical ultrasound, where single-cycle excitations are required to achieve high axial resolution and improve frame rate. Conventional methods, such as the Gerchberg-Saxton (GS) algorithm or more recent deep learning approaches, are generally effective for continuous-wave excitation but degrade significantly under single-cycle conditions. In diagnostic imaging, high frame rate is critical for applications demanding rapid scanning. In this context, multi-line transmission (MLT) leverages beam shaping to synthesize multiple simultaneous foci, thereby increasing frame rate. In parallel, structured illumination methods for super-resolution and acoustical holography likewise depend on actively shaping single-cycle pulses to produce controlled patterns, highlighting the need for precise short-pulse beam shaping. To address this challenge, we introduce the spatio-temporal adaptive reconstruction (STAR) algorithm, which performs active beam shaping directly in the time domain by integrating the generalized angular spectrum method (GASM) into an iterative optimization scheme. STAR enforces constraints on both the transducer and focal planes, enabling accurate control of single-cycle excitations. Simulations showed that STAR consistently outperformed GS for multi-focus patterns. For example, in a four-foci configuration, STAR achieved a correlation of 0.80 compared to 0.64 for GS, with significantly improved uniformity across focal peaks. Resolution analysis demonstrated that STAR reduced the minimum distinguishable foci spacing from 1.09 mm with GS to 0.87 mm. Experimental hydrophone measurements confirmed these improvements. Across multi-foci patterns, STAR produced more distinct and balanced foci compared to those observed with GS. These results demonstrate that STAR provides robust and efficient active beam shaping of single-cycle pulses, maintaining accuracy across different depths and frequencies for diagnostic applications.