Fig. 1. Principle of constructing Lommel beams through highly scattering media with TM-based PSF engineering method. (a) When a Lommel beam is incident on a highly scattering medium (HSM), the transmitted light becomes a speckle field due to multiple scattering. (b) Flow chart of TM-based PSF engineering method. (c) With the calculated wavefront as the input field impinging on this HSM, the desired Lommel beam can be obtained at the output plane behind the HSM. OPC, optical phase conjugation.

Fig. 2. Experimental setup. L, lens; M, mirror; BS, beam splitter; DMD, digital micro-mirror device; F, filter; OBJ, objective lens; HSM, ZnO scattering layer; CMOS, complementary metal-oxide-semiconductor camera.

Fig. 3. Creation of Lommel beam with parameters n = 2, c = 0.7, ρ₁ = 240, and ρ₂ = 40 pixels of the CMOS camera through a highly scattering medium. (a), (b) The theoretical intensity and phase profiles of the Lommel beam. (c), (d) The intensity and phase profiles of the Lommel beam’s angular spectrum. (e), (f) The simulated intensity and phase profiles of the Lommel beam through a highly scattering medium. (g), (h) The measured intensity and phase profiles of the Lommel beam through ZnO scattering layer in experiment. (i), (j) The intensity profiles in (a), (e), and (g) along the white dashed line (x axis) and the blue dashed line (y axis) in (a), respectively.

Fig. 4. Construct Lommel beams through a ZnO scattering layer with different parameters c and n. (a)–(d) The intensity profiles of Lommel beams with n = 2 and different c: (a) 0.1, (b) 0.4, (c) 0.7, (d) 0.9. (e)–(h) The phase profiles corresponding to (a)–(d). (i)–(l) The intensity profiles of Lommel beams with c = 0.7 and different topological charges n = 1, 2, 3, 4, respectively. (m)–(p) The phase profiles corresponding to (i)–(l).

Fig. 5. Lommel beams with different orientations were constructed through a ZnO scattering layer experimentally. (a)–(d) The intensity patterns of Lommel beams with different parameters φ₀: (a) 0, (b) π/4, (c) π/2, (d) 3π/4, and the same c₀ = 0.6, n = 2. (e)–(h) The phase profiles corresponding to (a)–(d), respectively.

Fig. 6. Raster scanning of Lommel beams and generation of multiple Lommel beams simultaneously through the ZnO scattering layer. The parameters are c₀ = 0.7, n = 1, and φ₀ = 0. (a)–(f) The raster scanning of Lommel beam. (g)–(i) Construct multiple Lommel beams simultaneously.