By modulating the amplitude, phase, or polarization of the optical field, scattered light modulation can compensate the distortion caused by nonuniform microscopic refractive index distributions in a scattering medium, and has been extensively applied in biomedicine, optical communications, and microscopy [1–4]. The practical effectiveness of scattered light modulation is influenced by various factors, such as the scattering dynamics, stochastic properties, and depolarization of the medium [5–8]. To analyze this influence, people often establish a physical model to approximately simulate the scattered light propagation through the medium [9]. An accurate model can provide an effective guidance for optimizing scattered light modulation, thereby improving the focus contrast or the fidelity of the data transmission in the scattering medium.

Currently, several methods have been proposed to simulate the scattering effect, including the transmission matrix [10–13], Monte Carlo and radiative transfer equation [14,15], and angular spectrum [16–18]. The transmission matrix method is commonly used to calculate the change of the optical field before and after propagating through a scattering medium [19]. The Monte Carlo and radiative transfer equation method is often employed to analyze the energy change of light [20]. Both methods, however, fail to quantify the various factors influencing the modulation effectiveness. In contrast, we have previously developed an angular spectrum model for scattered light modulation [16], which can trace and simulate the propagation of light throughout the system. This model decomposes the complex scattering process into a series of free-space diffraction and random phase shifts so that people can analyze the effects introduced by the medium on the modulation performance. This method has shown a good potential in anti-scattering, high-contrast light focusing, and high-resolution microscopy [21,22]. Furthermore, the angular spectrum model has a good scalability. It can be combined with the optical point-spread function to analyze the degradation of image quality in the scanning microscopy [17]. Combined with the transport theory [18], it can also efficiently analyze the variation of energy during scattering propagation. However, the existing angular spectrum models neglect depolarization occurring in the scattering medium [23,24]. The depolarization can cause a random crosstalk between the two orthogonally polarized channels, affecting the modulation effectiveness. For example, it hinders the high-fidelity data communications in optical fibers [25,26] and restricts the focus contrast generated in biological tissues [27,28]. Therefore, it is desirable to improve the angular spectrum model to analyze the polarization-related factors for the further development of scattered light modulation.

Here, we establish a full polarization angular spectrum model (fpASM) to describe the scattering of polarized optical fields. As an approximation, we decomposed the field propagation into a series of optical field changes and free-space diffraction. In the model, a Jones matrix simulated the random effects of the scattering particles on the amplitude, phase, and polarization of the optical field; the angular spectrum method simulated the free-space diffraction. We demonstrated the capability of fpASM in time-reversal data transmission and anti-scattering light focusing. We analyzed the influence of different depolarization effects on the modulation effectiveness, including the data fidelity and the focal contrast. Our findings demonstrated that fpASM could properly deal with various depolarization conditions and might benefit further advancements in scattered light modulation.

Figure 1 illustrates the principle of fpASM. The scattering medium is approximated as a series of scattering planes and nonscattering free spaces. Jones matrices and angular spectrum diffraction are used to calculate the layer-by-layer propagation, tracing the propagation of the scattered optical field. We describe the polarized optical field with a pair of orthogonal bases as E(x,y)=EH(x,y)e→H+EV(x,y)e→V,(1)where e→H and e→V represent, respectively, the horizontal and vertical polarization. Either EH or EV is the complex amplitude of a polarized component. (x,y) are the Cartesian coordinates. For convenience, we represent the polarized optical field E(x,y) with a vector [EH(x,y);EV(x,y)].

Specifically, we decompose the scattering medium with a thickness of l into M layers of scattering planes arranged at regular intervals [Fig. 1(a)]. Each interval between two adjacent scattering planes is a free space without a scattering effect. The scattering plane is made up of many pixels [Fig. 1(b)] with a size of Dpixel. The scattering caused by a pixel on the optical field can be described in terms of a Jones matrix as[EH′EV′]=exp(iδ2)exp(iφ)⁢[cosδ2−isinδ2cos2θ−isinδ2sin2θ−isinδ2sin2θcosδ2+isinδ2cos2θ][EHEV],(2)where [EH;EV] and [EH′;EV′] are the incident and transmitted optical fields, respectively. For convenience, we have omitted the Cartesian coordinates (x,y) in Eq. (2).

Caused by a scattering plane, the common distortion of two fields in orthogonal polarization directions is represented by a complex phase change φ=2πnΔl/λ+ik. The refractive index n for an arbitrary pixel in a scattering plane follows the Gaussian distribution with a mean nmean and standard deviation ndev. Δl=l/(M−1) is the distance between the two adjacent scattering planes. k represents the attenuation of amplitude.

The scattering effect on the polarization of light is described by θ and δ. θ represents the angle between the fast axes of the pixels and the x axis, following a uniform distribution of [−π/2,π/2]. δ represents the orthogonal polarization phase difference induced by the fast and slow axes, following the Gaussian distribution with a mean of 0 and a standard deviation of δdev∈[0,π]. All of these parameters in Eq. (2) make the distortion random, mimicking a scattering medium.

We use the angular spectrum method to calculate the nonscattering free-space diffraction between two adjacent scattering planes [Fig. 1(c)]. We assume that the optical field [Es,H(x,y);Es,V(x,y)] transmitted from the previous scattering plane (source plane) becomes [Ed,H(x,y);Ed,V(x,y)] in front of the next scattering plane (destination plane). First, a Fourier transformation is calculated for the optical field on the source plane by Es,P(kx,ky)=12π∬Es,P(x,y)exp[−i(kxx+kyy)]dxdy,(3)where Es,P represents either Es,H or Es,V. kx and ky correspond to the spatial frequencies in x and y directions, respectively. Upon the propagation from the source plane to the destination plane with a distance Δl, the source optical field Es,P(kx,ky) experiences a change and Ed,P(kx,ky)=Es,P(kx,ky)exp[in0Δl(2πλ)2−kx2−ky2],(4)where n0 is the refractive index of the free space and λ is the wavelength of light. In the end, the optical field Ed,P(x,y) that will interact with the next scattering plane can be obtained as Ed,P(x,y)=12π∬Ed,P(kx,ky)exp[i(kxx+kyy)]dkxdky.(5)

According to the principle above, by sequentially calculating the Jones matrices of the scattering planes and the diffraction in the free spaces, it becomes possible to effectively describe the scattering propagation of the polarized optical field.

To implement the model, all the parameters for the simulation were initialized. We used a collimated laser beam with a wavelength λ of 532 nm and a diameter of 1 mm as the light source. The scattering medium was divided into M=81 scattering planes, with an interval Δl=20μm. The mean and standard deviation of the refractive index were nmean=1.4 and ndev=1.6×10−3, respectively [16]. The attenuation factor was k=10−5. The refractive index of the nonscattering free space was n0=1. Each plane was discretized into 1000×1000 pixels with a pitch Dpixel=5μm.

In our model, the macroscopic depolarization property of the scattering medium was determined by the standard deviation δdev of the orthogonal polarization phase difference in Eq. (2). Therefore, we first established the relationship between this parameter and the mean depolarization distance ldp of the medium. The mean depolarization distance represents the mean distance after which the polarization distribution of the optical field becomes even [29]. Thus, we can calculate ldp by identifying the depth at which the scattered light intensity ratio Is,H/Is,V approximates to 1.

Figure 2(a) shows Is,H and Is,V, obtained from the simulation (δdev=π/3), at the depths of 0.05, 0.30, and 0.60 mm. The intensity ratio Is,H/Is,V approached 1 at the depth of ldp=0.60mm. Similarly, we found ldp to be 1.4 mm when δdev=π/5. We obtained the Is,H/Is,V as a function of depth [Fig. 2(b)]. By measuring Is,H/Is,V and matching the closest points on the curves of different δdev, the value of δdev corresponding to an actual medium can be determined. Figure 2(c) shows the relationship between ldp and δdev. Based on these facts, we could simulate various scattering media by setting different δdev in our model.

A high fidelity of data transmission is crucial for large-channel capacity and long-distance optical communications [30]. In practice, a transmission medium such as multimode fiber often induces polarization crosstalk and reduces the fidelity [19]. Data transmission techniques based on time reversal can compensate the crosstalk [31]. However, the mechanism of how the polarization crosstalk impacts the fidelity remains unclear. For this reason, we simulated the time reversal process with fpASM. In the forward step [Fig. 3(a)], a horizontally polarized image was scrambled by the scattering medium. In the playback step [Fig. 3(b)], the modulated time-reversal optical field propagated backward and recovered the incident image after interacting with the medium.

We calculated the fidelity under various depolarization conditions [Fig. 3(c)]. We used ln(l/ldp) to characterize different depolarization conditions in the medium. While ln(l/ldp) was increasing, more and more incident light was depolarized. Our results showed that a full-polarization modulation [27], simultaneously modulating the two orthogonally polarized components propagating backward, could achieve a higher fidelity than a single polarization modulation. Specifically, when the playback component polarized in the same direction as the incident polarization, the fidelity decreased under a strong depolarization due to the polarization crosstalk. Conversely, when playing back in the orthogonal direction, the fidelity exhibited an opposite trend, because the effective information in the vertically polarized component increased with the degree of depolarization. The full-polarization modulation combined the two components so that the polarization crosstalk could be suppressed. Therefore, the full-polarization modulation recovered the image with a higher fidelity than a single polarization modulation in a medium showing a strong depolarization.

Scattering depolarization can degrade the contrast of focus [32]. To understand this phenomenon, we first simulated the anti-scattering light focusing with fpASM [Fig. 4(a)]. In this simulation, the lens was modeled by a transmission function exp[iπ(x2+y2)/λf] in which f=100mm. The free-space propagation was calculated by the angular spectrum method. Two holograms IH and IV for the two orthogonally polarized components were captured by a camera. From them, we generated two binary patterns Em,H and Em,V for the two digital micromirror devices (DMDs) used in the playback modulation [33]. To focus the light against scattering, the conjugated light modulated by the DMDs propagated backward through a focusing lens. Figure 4(b) shows the simulation results under a strong depolarization condition (δdev=π/3). The three images corresponded to the foci formed by the full, horizontal, and vertical polarization modulation. Represented by the peak-to-background ratio (PBR), the contrast of focus of the full-polarization modulation was larger than that of single polarization modulation by a factor of 2.

In the experimental demonstration, we first quantified the depolarization of a scattering medium [Fig. 4(c)]. Two polarizers were placed before and after the medium. Polarizer 1 set the polarization of incident light from the 532 nm laser (Verdi V5, Coherent, Inc.) and Polarizer 2 selected a particular polarized component in the scattered light. According to the speckles captured by the camera (Basler ace acA1920-155um, Basler AG), the intensity ratio of the horizontally polarized component to the vertically polarized component was 16.99/17.38≈0.98, indicating a strong depolarization.

We focused light against this scattering medium. The number of controlled optical modes of one DMD (JUOPT-DLP 7000, JUOPT Technology Co., Ltd.) was 48×48=2304. The peak intensity and PBR of focus of full-polarization modulation were higher than those of the other two methods by factors of four and two, respectively [Fig. 4(d)]. Furthermore, we analyzed the contrast of focus under various depolarization conditions [Fig. 4(e)]. The PBR of the full-polarization modulation was not affected by the change of depolarization conditions and kept at a high level. In comparison, the PBRs were declining considerably while the degree of depolarization was increasing if we modulated only the horizontally or vertically polarized component because half of the information carried by the light was lost without an effective control. The result demonstrated that fpASM could help to improve the contrast of anti-scattering light focusing.

We have proposed fpASM, an optimized angular spectrum method modeling the polarized light propagation for scattered light modulation. This model decomposes the scattering medium into a series of scattering planes and free spaces, with the Jones matrices describing the distortion on amplitude, phase, and polarization of the scattered light and the angular spectrum method calculating the free-space propagation. We established the relationship between several model parameters and the macroscopic properties of the scattering medium. With this model, we first analyzed the influence of depolarization on the fidelity of time-reversal data transmission, and found that the full-polarization modulation provided the best fidelity among the three modulation methods. Then, we analyzed the influence of depolarization on the contrast of anti-scattering light focusing and found that the contrast of focus generated by the full polarization modulation was higher than that of single polarization modulation by a factor of 2, under the strong depolarization condition.

We introduced the Jones matrix to simulate the scattering distortion on the information of the light field, which made each pixel in the scattering plane appear to be a birefringent crystal in form. Nevertheless, this did not limit the applicability of our model to different scattering media. Instead, we just applied the clothing of the familiar optical process to approximately simulate the unknown scattering propagation inside the medium. By setting some random parameters, which were not even clearly defined in the actual scattering media but could be determined according to the scattering phenomenon of media, we made the multi degrees-of-freedom information change process no longer birefringent but an approximation of a random scattering event. As the results showed, lots of repetitions of random birefringent events and free-space diffraction could indeed effectively complete the simulation of multiple scattering.

Besides the scattered light modulation based on time reversal, other methods can benefit from fpASM. For example, by incorporating fast decorrelation components [16], our model has the potential to evaluate whether the speed of the feedback-based optimization methods [5] meets the dynamic scattering. Moreover, the fpASM can be further improved. First, we assumed that the incident light was highly coherent. Therefore, a comprehensive model for the incoherent light, based on the incoherent diffraction transmission theory [34], is warranted. Second, fpASM considered only the forward propagation of the optical field in the scattering medium and neglected reflection. Adding a weak reflection layer may produce a more accurate model [18]. Third, introducing a controllable guide star can enable focusing within a scattering medium and analyze its internal dynamics [16,35].

In addition, many applications may benefit from our model. For example, by focusing light against scattering in a tissue, a photo-therapist can optimize the dosage, minimizing the damage to normal cells [36]; by analyzing the detrimental effects of scattering on the image quality, the resolution of optical microcopy may be further improved [37]; the scattering effect of liquid crystal polarization optical elements may be simulated to support high-resolution augmented reality and virtual reality [38]; and, after quantifying the dynamics of atmosphere, our model also has the potential to mitigate the negative impacts of turbulence in free-space optical communications [39]. Therefore, we expect this approach to support advancements in biomedicine, imaging, and optical communications in the future.