Structured light beams, such as Hermite-Gaussian beams, vortex beams and vector beams, which may possess spatially variant states of polarization in addition to amplitude and phase, have attracted increasing interest in super-resolution microscopy, laser processing, optical manipulation and communication [[1], [2], [3], [4]]. Nonetheless, when the structured light travels through an anisotropic multiple-scattering medium (AMSM), the multiple scattering would scramble its wavefront and states of polarization. The scattering poses a fundamental limitation for information transfer and imaging.

However, the scattering is a complex and deterministic coherent process, even in multiple scattering regime. The multiple scattering can couple all the spatial degrees of freedom (DOFs, including amplitude, phase, and polarization) of light before and after the scattering media. These characteristics have been employed to control the transmitted light through the scattering media by modulating the DOFs of the incident light in spatial light modulators (SLM) [[5], [6], [7], [8], [9], [10], [11], [12], [13], [14]], digital micro-mirror devices [[15], [16], [17], [18], [19], [20]], and liquid crystal displays [21,22], which have several million DOFs. This technique, often called wavefront shaping, includes transmission matrix [7,8] or vector transmission matrix (VTM) [13,15], digital optical phase conjugation (DOPC) [[23], [24], [25], [26]], and iterative optimization [9,10,27,28]. Wavefront shaping has been exploited routinely to focus light as well as to image or to transmit information even across the scattering media.

It has recently been demonstrated that structured light beams can focus through the scattering media [8,[29], [30], [31]]. For instance, Boniface et al. used the transmission matrix-based point-spread-function engineering technique to provide a way for focusing the structured light beams in phase and amplitude after a complex medium [8]. This approach generalizes the Fourier optics for wavefront shaping through the scattering media. However, the control of the generated structure light's polarization state through the scattering media has not been taken into consideration, and the point-spread-function engineering-based wavefront shaping requires a significant number of Fourier transforms, making the required input field calculation time-consuming. Gong et al. (2018) introduced a comprehensive framework that utilizes the VTM-based DOPC to customize the polarization states of focused structure light without requiring Fourier transforms. However, this framework was validated only through simulations [32]. Qi et al. have recently presented an experimental method utilizing the VTM-based DOPC to achieve vertical manipulation through the scattering media [33]. However, the scattering medium used in their work is isotropic, and the VTM of the scattering medium only contains two components. It is crucial to factor in the full VTM of an anisotropic scattering medium, since this may be utilized to precisely and completely characterize the vector scattering properties between the input and output fields. Furthermore, their approach to wavefront shaping with VTM-based DOPC does not rely on amplitude degrees of freedom; rather, it merely modifies the phase and polarization of the input field.

In our previous work, we proposed a method for shaping all the spatial DOFs of the input field and focusing vector beams through the AMSM [34]. This approach uses a two-channel angular-multiplexing holographic polarization recording geometry in conjunction with a vector SLM (VSLM) to measure the full VTM of the AMSM. The VSLM, which consists of a small angle birefringent beam splitter (BBS) and a conventional transmittance SLM, can be utilized to quantitatively modulate the input field's amplitude and phase as well as the polarization state. Nevertheless, this work only demonstrates a single spatial position and scanning focusing. Here, we present a framework to tailor structured light beams after the AMSM in one step with the full VTM and multi-focus-based DOPC. For simplicity, our method is referred to as MF-DOPC in the following text. We experimentally demonstrate that the vector beams, array vector beams, and vortex beams with modulation across all spatial DOFs can be generated through the AMSM. A method to predict the theoretical peak-to-background ratio (PBR) of the generated multifocal structured light through the scattering media is determined. We also demonstrate the advantage of this framework for generating the structured light with a clear optical ring and high PBR through the AMSM, which is verified by simulations and experiments.