Fig. 1. Schematic of controlling nonlinear light in scattering medium via the scattering-matrix method. (a) Generation of nonlinear speckle pattern without shaping the wavefront of the input fields. (b) Nonlinear signals generation and scattering process in LN powder. (c) Intensity of nonlinear signal for a variable polarization of input field Ein(ω2). (d) Schematic of nonlinear signal focusing via WS technique. With an appropriate wavefront of Ein(ω2), a focal spot of nonlinear harmonic can be achieved.

Fig. 2. The SM measurement process. (a) Experimental setup. L1 to L5, lenses, f1−5=30, 200, 30, 200, and 100 mm; HWP, half-wave plate; SLM, spatial light modulator; M1 and M2, reflecting mirrors; DM1 and DM2, dichroic mirrors; Obj., objective; F, filter; and CCD, charge-coupled device camera. Inset: SEM image of the LN powder (scale bar: 1μm). (b) Phase pattern (Hadamard basis) displayed on the SLM, and each input mode is scanned from 0 to 3π/2 in 4 steps (0, π/2, π, and 3π/2). The pixels displayed around the phase pattern are used to generate nonlinear reference field. (c) Speckle pattern of nonlinear signal corresponding to each Hadamard basis. (d) The measured SM KNL connects the input modes (horizontal axis) and output modes (vertical axis). β and θ are the amplitude and phase of the element of KNL, respectively.

Fig. 3. Reconfigurable focusing of nonlinear signals via wavefront-shaping method based on SM. (a) Measured SM that connects the input modes (horizontal axis) and output nonlinear modes (vertical axis). Hue and brightness represent phase and amplitude, respectively. (b) Calculated phase patterns for focusing nonlinear signals on different positions of the ROI with superpixel coordinates (3, 5) (red); (9, 9) (blue); (6, 3), and (6, 7) (green, double spots). (c) Focal spots located at different subregions of the ROI with the corresponding optimized phase patterns. (d) Intensity cross section of the nonlinear focal spots located on different subregions of the ROI (red, green, and blue curves).

Fig. 4. Statistical properties of the SM: distribution of the (a), (b) real and imaginary parts of the measured SM and (c) the normalized singular value of the SM. The blue dashed line represents the singular value distribution from the quarter-circle law. Inset: normalized singular value of the matrix after removing the neighboring element. (d) Normalized amplitude profile of the focusing operator KNLconj(KNL).

Fig. 5. Nonlinear focusing along predefined trajectories. (a) Predefined scan trajectory in the shape of the letter “S” of the nonlinear focus. The scanning direction is marked by the arrow. (b) Nonlinear focus of each position in the scan path at different times. (c) Actual trajectory of the nonlinear focus in the shape of the letter “S.” (d) Predefined scan trajectories in the shape of the letters “J,” “T,” and “U.” (e) Actual trajectory of the nonlinear focus in the shape of the letters “J,” “T,” and “U.”