In some practical applications, the general speckle that obeys the Rayleigh distribution cannot meet the application requirements. Therefore, it is necessary to customize speckles with a specific distribution. Recent studies on speckle customization are mainly generated by the illumination of the active laser light source. We explore a method for customizing speckles in a passive detection mode. However, it is difficult for existing customized speckle modulation methods to obtain the speckle with the same statistical distribution in a specific wide spectrum wavelength range and a specific axial distance range. Maintaining the speckle with the same statistical distribution in broadband is vital for multi-color imaging. To this end, we propose a method to customize broadband speckle modulation based on multi-wavelength inverse propagation theory and iterative algorithms.

We put forward a broadband speckle customization method for the passive detection mode. The multi-wavelength inverse propagation theory based on Fresnel diffraction and iterative algorithms is adopted to optimize the phase of phase modulators. Firstly, the incident fields of all modulated wavelengths at the source plane propagate a distance to the modulator plane, and the modulated fields which are phases modulated by the phase modulator (randomly initialization) propagate a distance to the detection plane. Secondly, we update the amplitude of detected fields with the target modulation patterns (retaining its phase) and the updated fields of all modulated wavelengths propagate inversely to the modulator plane. Thirdly, the effect of initial incident fields is eliminated and the fields at the modulator plane over all wavelengths are averaged. Finally, the phase of modulators is updated and the iteration is performed until the modulation patterns of all wavelengths are target modulation patterns. The customized broadband speckles are generated at a certain axial distance by the illumination of the optimized phase modulator with an incoherent source in experiments.

Customizing speckles with different statistical distributions including sub-Rayleigh and super-Rayleigh in the broadband is realized by simulations (Fig. 2). The modulation ability of the proposed method for customizing broadband speckle modulation is quantitatively researched (Fig. 3). Both the simulation and experiment verify the feasibility of the proposed method in customizing multi-wavelength super-Rayleigh speckles (Fig. 4). The proposed broadband speckle modulation method is applied to single-shot multi-color fluorescence super-resolution microscopic ghost imaging for improving the imaging performance. In the simulation, adopting super-Rayleigh speckle modulation exhibits better reconstruction results than that of traditional Rayleigh speckle modulation, especially under low photon numbers or low detection signal-to-noise ratios (Fig. 7).

We propose a method for customization of broadband or multi-wavelength speckle modulation and apply it to single-shot multi-color fluorescence super-resolution microscopic ghost imaging. The customization of multi-wavelength super-Rayleigh speckle modulation is realized by simulations and experiments. Additionally, the simulation verifies that compared with traditional Rayleigh speckle modulation, the super-Rayleigh speckle modulation has advantages in multi-color object reconstruction under low photon numbers. This imaging method is suitable for existing microscopic imaging systems and can combine with other fluorescence super-resolution microscopic imaging techniques to further improve spatial resolution and multi-color imaging speed. Thus, it has broad application prospect in low-dose, fast and multi-color fluorescence super-resolution microscopy imaging.