Fig. 1. (a) Concept of correcting aberrated states by using light to correct light. The product of an input beam (middle mode) with another containing the same phase aberration (exponential term) cancels the identical distortion present in the structure carried by a second input beam (left mode) to restore the unaberrated state (right mode) in the difference-frequency beam generated from nonlinear wave mixing. Beams are shown in the far field for conceptual clarity. (b) Experimental setup used to apply and correct distortions on structured modes with DFG. SLM, spatial light modulator; HWP, half-wave plate; I, aperture; DM, dichroic mirror; NLC, nonlinear crystal; F1, short-pass and F2 long-pass wavelength filters; and L1 (18 mm), L2 (200 mm), L3 (300 mm), L4 (75 mm), L5 (500 mm), L6 (100 mm), L7 (750 mm), and L8 (100 mm) are lenses.

Fig. 2. Three azimuthal aberrations (top insets) were applied to a Gaussian signal beam, resulting in measured intensity distortions in the far field (aberrated row). Application of the nonlinear correction process with a probe beam results in the recovery of the initial Gaussian beam, as evident in the measured far-field intensities in the bottom row (corrected). All intensities are normalized to 1.

Fig. 3. Experimental correction of astigmatism and trefoil aberrations for five different spatial states (column-wise), where ℓ={1,2,3}, and petal modes with LG superpositions (12(LGℓ+LG−ℓ)), where ℓ={2,3}. The correction has been applied for both vertical and oblique combinations of aberrations. Further, every such combination has been corrected for both positive and negative strength coefficients. The applied phase distortion has been shown in the left panel. Every experimental picture shows results for corrected (Cor.) mode with corresponding aberrated (Ab.) and not aberrated (NA) modes as insets. The expected simulated intensity and phase profiles have been shown in the top row.

Fig. 4. Correction for superpositions of astigmatism and trefoil with arbitrarily chosen strengths. Left column insets show the aberrating phases acting on spatial modes (top insets). Experimentally aberrated (Ab.) and corrected (Cor.) far-field intensities for LG beams increasing columnwise in OAM from ℓ∈[1,6] are given for aberrations with (a) three- and (b) four-mode superpositions. (c) Experimental results with the same OAM range for the LG superpositions (12(LGℓ+LG−ℓ)) are given for the same four-mode aberration.

Fig. 5. Our approach can also be used in dispersive media. For instance, (a) by resizing the input beams of different wavelengths with experimental results shown in Fig. 6 and (b) by using the same initial wavelength in the dispersive media along with a second NLC for wavelength conversion prior to the DFG stage.

Fig. 6. Experimental results: resizing one beam changes the relative strength of the aberrations for (a) ℓ=1 with astigmatism (m=2) and coefficient of 10 (measured β=1.443), (b) ℓ=1 with trefoil (m=3) and coefficient of 15 (measured β=1.431), and (c) ℓ=2 with astigmatism (m=2) and coefficient of 10 (measured β=1.466). Leftmost insets show the (i) unaberrated downconverted mode, (ii) aberrated downconverted mode (no correction), and (iii) aberrating Zernike mode phase distribution.

Fig. 7. Probe field is used as a detector for OAM modes of ℓ∈[−7,7] in the cases where (a)–(c) the beam size expands as dictated by the OAM value and (d)–(f) a size adjustment of w(ℓ+1) is included to mitigate the OAM-dependent expansion in the generated modes. Detection cross talk matrices of the system are shown in (a) and (d) without applied aberrations; (b) and (e) with the four-mode Zernike aberration; and (c) and (f) with the aberrations corrected. Each row is normalized with the maximum value.