Stationary-soliton fractals in a mode-locked laser

Stationary optical solitons were considered unrelated to fractals, which are instead associated with dynamic nonlinear processes such as soliton splitting or breathing dynamics. Here, we demonstrate the direct engraving of fractal patterns onto stationary solitons in a mode-locked fiber laser using a programmable spectral waveshaper. By applying a Weierstrass function—continuous everywhere but differentiable nowhere—as the spectral filter, we encode a self-similar fractal structure onto the soliton spectrum. The resulting fractal spectra exhibit a clear self-similarity, illustrating a high-fidelity copy to the applied fractal structure, and their fractal dimension is continuously tunable by varying the function parameter. Experimental measurements agree well with numerical simulations. Our work extends the concept of optical fractals to stationary solitons, opening a new route for designing ultrafast lasers with controllable spectral complexity.