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High-fidelity multi-channel optical information transmission through scattering media

Rongjun Shao, Chunxu Ding, Linxian Liu, Qiaozhi He, Yuan Qu, and Jiamiao Yang

Open Access Open Access

Abstract

High-fidelity optical information transmission through strongly scattering media is challenging, but is crucial for the applications such as the free-space optical communication in a haze or fog. Binarizing optical information can somehow suppress the disruptions caused by light scattering. However, this method gives a compromised communication throughput. Here, we propose high-fidelity multiplexing anti-scattering transmission (MAST). MAST encodes multiple bits into a complex-valued pattern, loads the complex-valued pattern to an optical field through modulation, and finally employs a scattering matrix-assisted retrieval technique to reconstruct the original information from the speckle patterns. In our demonstration, we multiplexed three channels and MAST achieved a high-fidelity transmission of 3072 (= 1024× 3) bits data per transmission and average transmission error as small as 0.06%.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Light serves as the fundamental channel for human perception of the world and communication. In homogeneous media with consistent refractive indices, light propagation can be accurately described or predicted, enabling the effective extraction of information from light. However, in scattering media such as clouds, fog, turbid solutions, biological tissues, and multimode optical fibers, the irregular refractive index distribution inside them induces random scattering effects, which disrupt the integrity of the optical field, leading to information distortion [1,2]. Particularly in highly scattering conditions, detectors can only capture irregular speckles, and traditional optical methods have proven inadequate for the recovery of original information from the scattered optical field. Hence, achieving fast, high-fidelity, and high-throughput information transmission through scattering media is a prominent and formidable challenge in the field of optics.

In recent years, several methods for transmitting information through scattering media have been proposed, such as speckle-correlation imaging [3–5], optical phase conjugation [6–8], neural network-based approach [9–11], and the transmission matrix-based approach [12–15]. Among these, the transmission matrix-based approach characterizes the scattering process using a complex matrix, and employs it to extract the original information from the detected speckle patterns. Compared to other methods, the transmission matrix-based approach offers the advantages of robust task adaptability and a simplified optical path. It can reconstruct patterns resembling the information distribution of the incident light field and is widely used in fields such as endoscopic imaging [16–20] and non-line-of-sight imaging [21]. Currently, the advanced methods based on the transmission matrix have achieved reconstruction patterns with a correlation as high as 0.99 to the information distribution of the incident light field [22]. Unfortunately, this similarity between patterns does not guarantee precise matching between the reconstructed and original data. For example, when transmitting grayscale images directly through scattering media, the received images are affected not only by substantial noise interference but also by the loss of some details [23–25], compromising data fidelity. To address this, the current high-fidelity transmission approach converts signals into binary format and employs an appropriate threshold to reduce noise impact in the recovered patterns, achieving nearly lossless information transmission [26–28]. This binary strategy can also use a digital micromirror device (DMD) as the information input carrier, enabling high transmission rates. However, the binary strategy gives a compromised throughput for individual transmissions. While significantly increasing the number of input units may seem like a simple solution to mitigate this drawback, in practice, it substantially escalates the time and memory resources required for measuring the transmission matrix (TM) but also prolongs the signal retrieval process [29,30]. As a result, there is a practical limit to increasing the number of input units, whose compensatory effect on data throughput falls short of ideal.

Here, we propose multiplexing anti-scattering transmission (MAST) for the simultaneous, high-fidelity transmission of multiple sets of binary information through scattering media. We converted multi-channel binary information into a complex pattern by mapping the binary data from each input unit into complex numbers using Gray codes. Using a DMD-based complex amplitude modulation technique, complex patterns are loaded into the optical field within millisecond-scale timeframes and transmitted through a scattering medium. Ultimately, aided by the scattering matrix-assisted retrieval technique, we extracted the incident optical field from the output speckle pattern and decoded it to recover the original multi-channel binarized information. We built an MAST experimental system with 1024 input units. This system achieved high-fidelity transmission of 1024 × 3 bits in each transfer, with an impressively low average error rate of only 0.06%. Furthermore, it accomplished grayscale image transmission and color image transmission with Pearson correlation coefficients of 1 and 0.999, respectively. MAST has the potential to significantly increase information throughput while ensuring high transmission accuracy, and it is expected to find further applications in scenarios such as multi-mode optical fibers and non-line-of-sight communications.

2. Principle of MAST

As illustrated in Fig. 1(a), L binary images (G1, G2, …, GL) represent the information to be transmitted. At the transmitter end, these binary images are synthesized into a complex-value pattern, which is then loaded into the incident optical field ES. ES is subsequently scattered into the scattered light field Eout as it passes through the scattering medium. On the receiving end, we retrieve the incident optical field ES from the speckle pattern created by Eout using the transmission matrix of the scattering medium. Then, the originally transmitted information (G1, G2, …, GL) can be recovered.

 figure: Fig. 1.

Fig. 1. Principle of MAST. (a) The MAST transmission link. At the transmitter end, binary information from L channels is encoded into complex-valued patterns and carried by the complex amplitudes of the optical field as it propagates through the scattering medium (SM). At the receiving end, the original signal is extracted from the speckle pattern using the transmission matrix. (b) The distribution of information in the complex space when L is set to 3. (c) Input light field of the transmission matrix measurement period.

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We use the DMD-based complex amplitude modulation technique [31–33] to load the complex amplitude image into the ES. The complex-value xp of the p-th input unit in ES is encoded by the combination of the p-th pixel (gp1, gp2, …, gpL) in each binarized image (G1, G2, …, GL). The detailed encoding rule is: when gp1, gp2, …, gpL are all zero, xp is defined as 0, referred to as the zero signal. It is important to note that when xp is 0, the phase value of xp becomes irrelevant and can be any constant, such as 0. In all other cases, the values of xp are evenly distributed on the unit circle in the complex space with a phase interval of 2π/(2^L-1). The encoding rule is expressed as:

$${x_p} = \left\{ \begin{array}{ll} \begin{array}{{cc}} {0,}&{{g_{p1}},{g_{p2}}, \ldots ,{g_{pL}}\textrm{ are }0} \end{array}\\ \begin{array}{{cc}} {\exp \left[ {i \cdot 2\pi \left( {\frac{l}{{{2^L} - 1}}} \right)} \right],}&{else} \end{array} \end{array} \right., $$
where, l represents the value corresponding to the Gray code [gp1, gp2, …, gpL]. Figure 1(b) shows the distribution of xp values in complex space when L is 3.

To measure accurate transmission matrix T of the scattering process, we introduce reference light ER at the entrance of the scattering medium, allowing ER to propagate through the medium alongside the signal light ES. In practice, we simply modulate the complex amplitude of the incident light by the DMD to combine the two as Ein = ER + ES. This overcomes the challenge of obtaining a high-quality reference light field after long-distance transmission through the scattering medium, as the conventional method entails introducing ER from an additional optical path, which is less practical. In transmission matrix measurement period, the system's output light field, Eout, can be expressed as Eout = TEin = EOR + TES, where EOR represents the scattered light field generated by ER after passing through the scattering medium (i.e., EOR = TER). According to the four-step phase-shifting interferometry [34], we introduce an additional phase Δφ to ES with keeping ER constant. The intensity distributions corresponding to Eout at Δφ = 0, $\pi$/2, $\pi$, and 3$\pi$/2 are recorded as $I_{\textrm{out}}^0$, $I_{\textrm{out}}^{\pi /2}$ $I_{\textrm{out}}^\pi$, and $I_{\textrm{out}}^{3\pi /2}$, respectively. These intensity distributions are expressed as:

$$ \left\{\begin{array}{l} I_{\text {out }}^0=\left|E_{O R}\right|^2+\left|\boldsymbol{T} E_S\right|^2+2 \operatorname{Re}\left(\bar{E}_{O R} \boldsymbol{T} E_S\right) \\ I_{\text {out }}^{\pi / 2}=\left|E_{O R}\right|^2+\left|\boldsymbol{T} E_S\right|^2+2 \operatorname{Re}\left(i \bar{E}_{O R} \boldsymbol{T} E_S\right) \\ I_{\text {out }}^\pi=\left|E_{O R}\right|^2+\left|\boldsymbol{T} E_S\right|^2-2 \operatorname{Re}\left(\bar{E}_{O R} \boldsymbol{T} E_S\right) \\ I_{\text {out }}^{3 \pi / 2}=\left|E_{O R}\right|^2+\left|\boldsymbol{T} E_S\right|^2-2 \operatorname{Re}\left(i \bar{E}_{O R} \boldsymbol{T} E_S\right) \end{array},\right. $$
where, $\bar{E}$OR represents the conjugate transpose of EOR, and the operator Re(.)is used to extract the real part of a complex number. Then, we get the approximate value of the transmission matrix T as follows:
$${{\boldsymbol T}_{obs}} = {{\boldsymbol \Phi }_R}{\boldsymbol T} = \left( {\frac{{I_{out}^0 - I_{out}^\pi }}{{4|{E_{OR}}|}} + i\frac{{I_{out}^{3\pi /2} - I_{out}^{\pi /2}}}{{4|{E_{OR}}|}}} \right)E_S^{ - 1}$$
where ΦR is a complex diagonal matrix with diagonal elements having unit amplitudes and phases related to the $\bar{E}$OR phase. $E_{\mathrm{S}}^{-1}$ is the inverse matrix of ES. As ΦR does not impact intensity computations, the Tobs can accurately describe the relationship between the input light field and the intensity of the output light field.

To directly retrieve the incident optical field ES from a single recorded speckle pattern during information transmission, we construct the speckle-correlation scattering matrix [35,36] based on Tobs:

$${Z_{pq}} = \frac{1}{{{\Sigma _p}{\Sigma _q}}}\left[ {{{\left\langle {t_p^\ast {t_q}E_{out}^\ast {E_{out}}} \right\rangle }_r} - {{\left\langle {t_p^\ast {t_q}} \right\rangle }_r}{{\left\langle {E_{out}^\ast {E_{out}}} \right\rangle }_r}} \right], $$
where tp and tq are the p-th column of Tobs, ${\left\langle \cdot \right\rangle _r}$ indicates a space average, $\Sigma_p=\left\langle\left|t_p\right|^2\right\rangle_r$ is a normalization constant, and the symbol * denotes the complex conjugate of the corresponding variable. The corresponding output field can be expressed as ${E_{out}} = \mathop \sum \limits_{p = 1}^N {x_p}{t_p}$. Under the assumption that the columns of Tobs (t1, t2, …,tN) and Eout are randomly diffused fields due to high scattering, the Tobs can be considered as a Gaussian random matrix. Then, Eq. (4) is rewritten as
$${Z_{pq}} = {\alpha _p}\alpha _q^\ast{+} \frac{1}{{{\Sigma _p}{\Sigma _q}}}{\left\langle {t_p^\ast E_{out}^\ast } \right\rangle _r}{\left\langle {{t_q}{E_{out}}} \right\rangle _r}, $$
where $\alpha_p=\frac{1}{\Sigma_p}\left\langle t_p^* E_{\text {out }}\right\rangle_r$, since the columns of the Tobs are uncorrelated to each other, the general orthogonality relations $\frac{1}{\sqrt{\sum_p \sum_q}}\left\langle t_p^* t_q\right\rangle_r \approx 0$ hold. Thus, the speckle-correlation scattering matrix becomes ${Z_{pq}} = {\alpha _p}\alpha _q^\ast $, the sole eigenvector X of which is the estimate of the incident field. Following this, we utilize the double-phase retrieval algorithm [30,37] and Tobs to further process X, resulting in the recovery of the incident field ES. In the end, decoding the recovered ES to reconstruct the original binary images (G1, G2, …, GL). To be specific, when the amplitude of the recovered xp is zero, gp1, gp2, …, gpL are also zero. When the amplitude of the recovered xp is non-zero, the gp1, gp2, …, gpL are determined by the Gray code corresponding to the phase of the recovered xp.

3. Numerical simulation and parameter optimization

To suppress noise interference during the data transmission process and achieve high transmission accuracy, we optimize the parameters of the MAST through numerical simulations. Figure 2 analyzes the impact of the ratio γ, representing the number of output units (M) relative to input units (N), on the noise resilience of MAST in retrieving the phase of the incident field. We use a handbag map (from the Fashion-MNIST dataset) to represent the phase of the incident light field and fix the number of input units N to 1024. Figure 2(a) illustrates the results of a single-phase retrieval simulation, and Fig. 2(b) presents the average of 20 simulation results. Evidently, higher γ values enhance phase retrieval accuracy, contributing to improved MAST information transmission accuracy. For example, the γ should be no less than 25 to ensure sufficient phase retrieval accuracy, when the system signal-to-noise ratio (SNR) is 5. However, it's important to note that the effectiveness of increasing γ has its limits. In practice, the selection of channel count must be made judiciously, taking into account the prevailing noise conditions to ensure that information transmission accuracy meets the necessary requirements.

 figure: Fig. 2.

Fig. 2. The effects of noises on the phase retrieval of the MAST. (a) Phase retrieval results of the MAST and the Pearson correlation coefficients with the phase of the incident field are denoted below each result (b) The Pearson correlation coefficient between the retrieved phase and the phase of the incident field as a function of the value of γ at given SNRs.

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Figure 3(a) displays the results of a single simulation for MAST information transmission with L = 3, γ=25, and N = 1024 under specified noise conditions. It is evident that when the number of channels is 3, the MAST exhibits good anti-scatter transmission capability and maintains a high data transmission accuracy even at SNR = 5. Under the conditions of γ = 25 and N = 1024, we further analyzed the impact of the number of channels, L, on the transmission accuracy of MAST in the presence of noise by averaging results from 20 simulation runs (Fig. 3(b)). It can be seen that for the same transmission accuracy, MAST with a higher number of channels has more stringent requirements for system noise. Conversely, when noise levels are fixed, reducing the number of channels proves advantageous in lowering the transmission error rate. This is due to the fact that fewer channels ensure a wider dispersion in complex-valued encoding, which, in turn, reduces the demands on phase retrieval precision, thereby enhancing the system's adaptability to noise. In general, selecting 3 channels is a relatively reasonable choice for practical applications.

 figure: Fig. 3.

Fig. 3. The effects of noises on the transmission accuracy of the MAST. (a) Information transmission results for given SNRs at L = 3, N = 1024, γ=25, and the data accuracy rate are denoted on the right side of each result. (b) Transmission error rate as a function of the number of channels L for given SNRs.

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4. Results

According to the above principle, we built an MAST experimental system based on a single DMD (Fig. 4(a)). Coherent light emitted from a 532 nm laser (Verdi G2, Coherent, Inc.) was first expanded and directed onto a DMD (1024 × 768 pixels, Texas Instruments, Inc.). Behind the DMD, we incorporated an optical 4f configuration, which allowed only the first-order diffraction light from the DMD to pass through. Using superpixel modulation, a single DMD was able to high-fidelity generate complex amplitude optical fields at the back focal plane after the optical 4f configuration, allowing for rapid loading of complex-valued signals. Subsequently, the incident field carrying the complex-valued signal passed through a highly scattering medium (a stack of three diffusers, DG10-120, Thorlabs, Inc.), and the emerging scattered field was collected by lens L2 and a CMOS camera (164 fps, acA1920-155um, Basler).

 figure: Fig. 4.

Fig. 4. Experimental setup and characterization. (a) Experimental setup of the MAST. (b) Phase factor of each micromirror within a superpixel. (c) Visualization of the calibrated transmission matrix. Hue and saturation represent the phase and amplitude of the elements in the transmission matrix, respectively. (d) Speckle pattern estimation based on the calibrated transmission matrix.

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In the direction of the first-order diffraction light, each micromirror in the DMD had a phase factor associated with its position. We selected 4 × 4 micromirrors to constitute a superpixel, and the particular phase factor of every micromirror in a superpixel was shown in Fig. 4(b). In this case, a superpixel could generate 6561 distinct complex amplitude states by controlling the on/off states of the internal micromirrors. Prior to optical transmission, we calibrated the MAST system's transmission matrix (Fig. 4(c)). The number of input units in the MAST system was fixed at 1024 (32 × 32). To ensure transmission accuracy, the γ was set to 64, corresponding to 256 × 256 output units. We used the calibrated transmission matrix to estimate the output speckle intensity distribution for a given incidence optical field (Fig. 4(d)) to evaluate the precision of the transmission matrix measurement. The estimated speckle pattern closely approximated the actual pattern, with a Pearson correlation coefficient [9,22] of around 0.96. This demonstrated the calibrated transmission matrix effectively characterized the experimental system.

Considering the impact of noise, we verified MAST's data transmission performance under L = 3. In this setup, the experimental system can update 1024 × 3-bit data per transmission. Figure 5(a) shows the results of two data transmissions at γ=64, demonstrating MAST's remarkable transmission accuracy. It successfully receives lossless binary information, even with discretely random data. Subsequently, we assessed the influence of γ values on MAST's transmission accuracy. We conducted 50 independent transmission experiments at various γ values and analyzed the results, as depicted in Fig. 5(b). With γ=64, MAST achieved an average error rate of about 0.06% in data transmission. As γ decreased, there was a noticeable increase in the data transmission error rate. When γ was reduced to 25, the average transmission error rate rose to 0.78%. Notably, the adjacency of the ‘100’ and ‘001’ codes in our encoding scheme, not fully adhering to the Gray code principle of adjacent codes differing by only one binary digit, leads to mutual interference between G1 and G3, marginally increasing their error rate compared to G2. To maximize fidelity, we conducted transmission experiments for grayscale and color images under the condition of γ=64 (Fig. 5(c)). The received grayscale and color images exhibit Pearson correlation coefficients of 1 and 0.999, respectively, with the original images, effectively preserving the fine details of the original images. Therefore, MAST is capable of tripling the data throughput for practical DMD-based anti-scattering information transmission with almost no loss in accuracy.

 figure: Fig. 5.

Fig. 5. Results of data transmission experiments. (a) Results of two data transmissions with γ set to 64. (b) Mean and standard deviation of the error rate of 50 independent transmission results for different γ values. (c) Transmission results of 8-bit grayscale and color images at γ=64.

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5. Discussion

In conclusion, we proposed MAST as an effective approach to improve data throughput for high-fidelity anti-scattering information transmission without increasing the number of input units. MAST integrated multiple binary datasets in the complex domain with the assistance of Gray codes and expanded the number of optical channels through complex amplitude modulation with a DMD. Furthermore, MAST obtained a high-precision scattering transmission matrix without an additional reference light path. Based on this matrix, it constructed a speckle-correlated scattering matrix for accurately recovering input information perturbed by the scattering medium, realizing multi-channel data transmission. In the demonstration, we showed that MAST could achieve high-fidelity anti-scattering data transmission with an impressively low average error rate of only about 0.06% at a channel count (L) of 3 and an output-input unit ratio (γ) of 64. Additionally, it accomplished rapid transmission of grayscale and color images through scattering media with very high correlation. In short, MAST could multiply the transmission throughout of traditional binary data approaches while maintaining excellent noise immunity.

We believe that MAST achieves high transmission accuracy for two primary reasons when appropriately selecting the channel count and the output-input unit ratio: 1) Complex-value coding preserves the discretization of information, reducing the accuracy requirements for recovering the input field. The incorporation of Gray codes also minimizes the impact of errors in input field recovery. 2) The speckle-correlated scattering matrix has strong resilience to disturbances under high γ conditions, allowing the recovery of complex signals highly similar to the input field. Hence, MAST offers flexibility to adjust the channel count and the output-input unit ratio based on application requirements and conditions, ensuring fast, high-throughput, high-fidelity data transmission.

Looking ahead, there are several avenues for further enhancing MAST's performance. Limited by the camera's frame rate, the current experimental system hasn't yet harnessed the full potential of the DMD's high modulation rate. The integration of high-speed cameras in the future could increase the modulation rate to 22kHz. Developing multi-level amplitude encoding strategies could further tap into the amplitude dimension's potential for carrying information, thereby increasing MAST's data capacity. Exploring optical field retrieval techniques with improved recovery of high-frequency information can prevent the loss of detail in complex patterns due to noise, enhancing the transmissions accuracy of MAST. While presently suited for static scattering media, MAST's modulation efficiency drops in dynamic scattering media. In dynamic environments, the modulation rate of MAST is limited by the speed of transmission matrix measurement and the camera's frame rate. Future research into rapid measurement or compensation techniques for the transmission matrix, in conjunction with high-speed cameras, could enable MAST to excel even in dynamic scattering conditions.

We believe that MAST holds great promise in applications such as multi-mode fiber communication and non-line-of-sight information transmission, significantly improving information throughput compared to existing methods. Additionally, it also allows for secure scattering encryption of multiple unrelated pieces of information, enhancing data security.

Funding

National Natural Science Foundation of China (No. 62205189, 62375171); Shanghai Pujiang Program (22PJ1407500); Natural Science Foundation of Shanxi Province (202103021224015); Shanghai Jiao Tong University 2030 Initiative (WH510363001-10); Oceanic Interdisciplinary Program of Shanghai Jiao Tong University (SL2022ZD205); Science Foundation of Donghai Laboratory (DH-2022KF01001).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Principle of MAST. (a) The MAST transmission link. At the transmitter end, binary information from L channels is encoded into complex-valued patterns and carried by the complex amplitudes of the optical field as it propagates through the scattering medium (SM). At the receiving end, the original signal is extracted from the speckle pattern using the transmission matrix. (b) The distribution of information in the complex space when L is set to 3. (c) Input light field of the transmission matrix measurement period.

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Fig. 2.
Fig. 2. The effects of noises on the phase retrieval of the MAST. (a) Phase retrieval results of the MAST and the Pearson correlation coefficients with the phase of the incident field are denoted below each result (b) The Pearson correlation coefficient between the retrieved phase and the phase of the incident field as a function of the value of γ at given SNRs.

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Fig. 3.
Fig. 3. The effects of noises on the transmission accuracy of the MAST. (a) Information transmission results for given SNRs at L = 3, N = 1024, γ=25, and the data accuracy rate are denoted on the right side of each result. (b) Transmission error rate as a function of the number of channels L for given SNRs.

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Fig. 4.
Fig. 4. Experimental setup and characterization. (a) Experimental setup of the MAST. (b) Phase factor of each micromirror within a superpixel. (c) Visualization of the calibrated transmission matrix. Hue and saturation represent the phase and amplitude of the elements in the transmission matrix, respectively. (d) Speckle pattern estimation based on the calibrated transmission matrix.

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Fig. 5.
Fig. 5. Results of data transmission experiments. (a) Results of two data transmissions with γ set to 64. (b) Mean and standard deviation of the error rate of 50 independent transmission results for different γ values. (c) Transmission results of 8-bit grayscale and color images at γ=64.

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Equations (5)

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x p = { 0 , g p 1 , g p 2 , … , g p L  are  0 exp ⁡ [ i ⋅ 2 π ( l 2 L − 1 ) ] , e l s e ,
{ I out  0 = | E O R | 2 + | T E S | 2 + 2 Re ⁡ ( E ¯ O R T E S ) I out  π / 2 = | E O R | 2 + | T E S | 2 + 2 Re ⁡ ( i E ¯ O R T E S ) I out  π = | E O R | 2 + | T E S | 2 − 2 Re ⁡ ( E ¯ O R T E S ) I out  3 π / 2 = | E O R | 2 + | T E S | 2 − 2 Re ⁡ ( i E ¯ O R T E S ) ,
T o b s = Φ R T = ( I o u t 0 − I o u t π 4 | E O R | + i I o u t 3 π / 2 − I o u t π / 2 4 | E O R | ) E S − 1
Z p q = 1 Σ p Σ q [ ⟨ t p ∗ t q E o u t ∗ E o u t ⟩ r − ⟨ t p ∗ t q ⟩ r ⟨ E o u t ∗ E o u t ⟩ r ] ,
Z p q = α p α q ∗ + 1 Σ p Σ q ⟨ t p ∗ E o u t ∗ ⟩ r ⟨ t q E o u t ⟩ r ,
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