Fig. 2: Ultra-wide hidden spectral correlation enables efficient dispersion compensation.

a The curves correspond to the cosine similarity of independently measured TMs with the reference TM at \(\omega _0\) (black), with the first-order estimation using the closed-form reconstruction (orange), and subsequent optimization with K = 1 (yellow) or K = 2 (purple) using gradient descent optimization, with spectral widths of 0.26, 33, 33, and 115 nm, respectively. Additionally, the correlations of individual PMs computed at \(\omega _0\) and applied to all ω are shown (gray). The color-coded frames show corresponding synthetic focusing through the MMF at varying wavelengths, where the second order enables excellent focus contrast across the entire spectrum (purple frame). b Focus contrast η at a spatial channel (gray solid circle) for varying wavelengths using the dispersion model with K = 2. The fiber facet is masked from the background (gray dashed circular support). c Spatial maps of maximal focus contrast and bandwidth obtained by focusing through all available channels at the fiber core in two polarization states. The scale bars are 20 μm