Optical speckles arise from the random interference of light after its path is disrupted by scattering media. Similarly, in multimode fibers (MMFs), mode crosstalk—where theoretically orthogonal spatial modes interfere due to imperfections, bending, and environmental factors—produces speckles at the output. In both cases, these random and disordered speckles complicate the retrieval of the original information, posing a long-standing challenge in optical information transmission^[1,2]. Recent breakthroughs in wavefront shaping offer a powerful solution by transforming what was once a chaotic obstacle into a controllable process. By treating the scattering medium as a deterministic transmission matrix, wavefront shaping enables precise manipulation of scattered light, restoring a one-to-one mapping between input and output modes^[3–6]. Phase conjugation, which reverses the scattering process by retracing the light’s original path, has demonstrated remarkable accuracy in refocusing scattered light^[7–11]. This advancement is particularly valuable in biomedical imaging, where tissue scattering limits imaging depth. Moreover, wavefront shaping mitigates mode crosstalk in MMFs^[12–19], improving signal integrity and transmission efficiency in communication, sensing, and endoscopy applications^[20–23].

However, real-world applications introduce additional complexities, as dynamic environments—such as moving scatterers in tissue or changing conditions in MMFs—cause the transmission matrix to evolve rapidly and unpredictably, complicating stable control^[23–26]. This challenge is particularly pronounced in freely positioned MMFs and live biological tissues, where internal fluctuations and external disturbances affect the deterministic nature of the scattering process. In such cases, adaptive, real-time wavefront shaping systems are essential^[27]. System efficiency is governed by the average mode time—the time required to measure and control each independent mode or element^[26,28,29]. It is defined as the ratio of the total system runtime to the product of the number of independent modes and the modulation efficiency. By dividing the optical speckles’ correlation time (the time scale of the dynamic process) by the average mode time, one can determine the number of modes that can be effectively controlled in a given dynamic scattering environment. The shorter the average mode time, the more modes that can be controlled before the speckle patterns change. As a result, fast wavefront shaping systems are essential for maintaining optimal performance in dynamic environments.

Researchers have been actively developing fast wavefront shaping systems to reduce average mode time. As most time is spent on wavefront measurement, these techniques can be broadly categorized into those using external reference light and those that do not. Systems employing interferometric holography are generally faster but more susceptible to environmental disturbances. For example, utilizing the time-reversal symmetry of wave equations can retrieve a single row of the transmission matrix in tens of nanoseconds^[29–32], and acousto-optic frequency encoding has achieved average mode time of a few microseconds^[33,34]. However, these systems rely on external reference light and require complex alignment, making them vulnerable to environmental factors. In contrast, referenceless wavefront shaping systems offer greater robustness. Feedback-based systems, which use optimization algorithms, eliminate the need for reference light but are less efficient due to repeated communication between wavefront measurement and control, resulting in average mode time in the millisecond range^[35–38]. Neural networks can also be employed to determine the transmission matrix, but the time-intensive processes of generating training datasets and training the networks limit their real-time performance^[39–45]. A simpler alternative is direct transmission matrix measurement through coded wavefronts using coaxial interferometry^[4,46–48] or non-interferometric phase retrieval^[49–54]. With digital micromirror devices (DMDs) operating at tens of kHz, state-of-the-art systems have achieved average mode time of 185 µs using coaxial interferometry^[55] and 95 µs with non-interferometric phase retrieval^[26]. However, coaxial interferometry sacrifices a fraction of controllable elements to serve as static reference lights, while non-interferometric phase retrieval demands significant computational resources, limiting scalability for large transmission matrices^[56,57].

To address the aforementioned challenges, we propose a coded self-referencing method that overcomes the limitations of both coaxial interferometry and non-interferometric phase retrieval, enabling fast measurement and control of dynamic scattering. This method eliminates the need for static references in coaxial interferometry, thereby preserving the number of controllable elements and avoiding the computational complexity associated with non-interferometric phase retrieval. By employing superpixel encoding with a DMD^[58] (as detailed in Appendix A), we directly synthesize interference patterns of orthogonal bases and a planar reference to probe the complex medium, enabling efficient three-step phase-shifting to extract the complex coefficients of orthogonal basis modes (explained in Appendix B). This approach allows for rapid, large-scale transmission matrix measurement and full-field wavefront control with minimal computational overhead, significantly enhancing wavefront shaping performance. Specifically, we achieved millisecond-level system runtime of 21.90 and 76.26 ms for 256 and 1024 controllable elements with full-field modulation, corresponding to average mode time of 85.54 and 74.47 µs, respectively, reaching the system’s hardware limit. As a proof of concept, we demonstrated optical focusing through unfixed MMFs and moving diffusers, showcasing the system’s ability to effectively overcome dynamic scattering and mode crosstalk.

Figure 1 illustrates the operational principle for measuring the transmission matrix of a complex medium. A series of orthogonal bases, such as Hadamard bases^[4,47,59,60], are generated to interact with the medium, producing speckle patterns [Fig. 1(a)]. The transmission matrix is then directly calculated by summing the Hadamard bases (reshaped from two-dimensional matrices to one-dimensional rows) with the associated optical fields as complex coefficients. In setups where reference light is introduced after the complex medium, as in a typical interferometric holographic configuration [Fig. 1(a)]^[61], the true transmission matrix is determined using a direct three-step phase-shifting method. Alternatively, when the reference light is introduced before the medium, the system operates as a coaxial interferometric setup [Fig. 1(b)]^[4,46]. If no reference light is used, the system functions as a non-interferometric phase retrieval setup [Fig. 1(c)]^{[26,49,50,56]}. It should be noted that in the latter two configurations, each row of the transmission matrix contains unknown complex coefficients.

Figure 1(d) illustrates the proposed coded self-referencing method, which synthesizes the combined optical field of probing light (Hadamard bases) and reference light (three-step phase-shifted) to interact with the complex medium (as detailed in Appendix C). A similar encoding scheme has recently been applied for holographic imaging in clear media, primarily under static conditions^[57]. This method retains the stability advantages of a referenceless setup while employing field encoding to reduce the computational complexity involved in reconstructing the transmission matrix. As with coaxial interferometry and non-interferometric phase retrieval, each row of the resulting transmission matrix contains unknown complex coefficients. However, these unknown coefficients do not adversely affect system performance when focusing light at specific spatial positions. In contrast to feedback-based methods^[59,62–64], our coded self-referencing approach allows for efficient transmission matrix reconstruction without the need for persistent feedback. Additionally, this method eliminates the need for dedicated reference fields, optimizing the use of available modes and reducing computational complexity, which is crucial for applications involving large and dynamically changing transmission matrices.

We constructed an experimental setup to demonstrate the coded self-referencing method and validate its operational principle. As shown in Fig. 2, a continuous-wave laser (MSL-FN-532-100mw, CNI) was used as the light source, with the beam expanded to a 1-inch diameter to fully illuminate the DMD (V-7001, Vialux). A superpixel encoding scheme was employed, grouping every $4\text{pixel}\times 4$ pixel into one superpixel^[58]. The light diffracted from the DMD was spatially filtered to pass only the 1st diffraction order, enabling the selective generation of beams with varying amplitude and phase at the imaging plane of a 4f system. To achieve fast DMD refresh rates, a central active zone of $256\text{pixel}\times 1024$ pixel was selected, providing a refresh time of 23 µs, corresponding to a refresh rate of approximately 43.48 kHz. The selected active zone facilitates higher optical energy coupling through the MMF, supported by a higher refresh rate and compatible with complex-field modulation. Opting for a 1:1 aspect ratio, however, reduces the refresh rate to 22.7 kHz. The modulated light was then coupled into a 1-m-long MMF (FC/PC-FC/PC-50/125-900 µm, Shenzhen Optics-Forest Inc.). Additionally, stacked ground glass diffusers (DG10-1500, Thorlabs) were used as the scattering media. A polarizer (LPVISA100-MP2, Thorlabs) was placed after the complex media to detect light with a specific polarization state. The output intensity distribution was captured by an avalanche photodiode (APD) (APD130A2, Thorlabs) and digitized using a data acquisition device (USB-6251, National Instruments) with a sampling rate of 1.25 MS/s. Simultaneously, a complementary metal-oxide-semiconductor (CMOS) camera (BFLY-U3-23S6M-C, FLIR) positioned at the conjugate plane acted as a supervisory module, monitoring and recording the experimental results. The system’s performance is influenced by the APD’s dynamic range and noise characteristics, which are carefully managed through the optimization of gain settings and the application of signal processing techniques.

To meet different application requirements, we designed the system to operate in two distinct configurations, referred to as Configuration 1 and Configuration 2. In Configuration 1, every $32\text{pixel}\times 32$ pixel ($8\text{superpixel}\times 8$ superpixel) were grouped into one block, resulting in 256 controllable elements. In Configuration 2, every $16\text{pixel}\times 16$ pixel ($4\text{superpixel}\times 4$ superpixel) were grouped into one block, resulting in 1024 controllable elements. Configuration 1 offers a higher signal-to-noise ratio (SNR) under the same illumination intensity but with fewer controllable elements, while Configuration 2 provides more controllable elements but with a lower SNR. Higher resolution patterns distribute input light energy across more modes, thereby reducing the power per mode and exacerbating mode-dependent losses. This leads to a lower SNR and can negatively affect the accuracy of transmission matrix measurements. The complete wavefront shaping process involves projecting coded patterns, collecting the resulting speckle intensities, computing the transmission matrix elements, and modulating the wavefront. Table 1 summarizes the time required at each stage. Due to the increased number of controllable elements, Configuration 2 requires projecting four times as many coded patterns, making the projection time four times longer—70.66 ms compared to 17.66 ms. Data collection can be performed simultaneously with pattern projection, so it adds almost no additional time. However, the computational complexity increases significantly, resulting in non-linear growth in calculation time (detailed in Appendix D). Despite the differences in computation, the overhead—including hardware initialization, communication, and DMD display of the modulated wavefront—remains nearly identical for both configurations, at approximately 4 ms. Ultimately, the total system runtime for Configurations 1 and 2 are 21.90 and 76.26 ms, respectively (detailed in Appendix E). Therefore, the average mode time are 85.54 µs for Configuration 1 and 74.47 µs for Configuration 2. Once the transmission matrix is measured, loading the optimal binary amplitude mask takes just a few tens of microseconds, limited only by the speed of the DMD.

We first experimentally demonstrated optical focusing through the MMF using 256 controllable elements. The success of the coded self-referencing method relies heavily on superpixel encoding, which enables the synthesis of complex fields via binary amplitude modulation. Figure 3(a) illustrates an example where a binary-amplitude mask (with only on/off states) generates a synthesized field by combining a Hadamard basis of order 128 with reference light at a phase of $2/3\pi$. Using this approach, 768 pre-designed complex fields were coupled into the MMF, resulting in a series of speckle patterns at the output. The intensity at the target position was captured by an APD. After retrieving a row of the transmission matrix, the DMD synthesized the conjugate field, which was used as the input to the MMF, producing a bright focus at the target position. Figure 3(b) shows the intensity captured by the APD over time, demonstrating the transformation of random speckle patterns into a bright focus within 22 ms. For visualization purposes, the intensity distributions captured by the camera are also displayed. Quantitatively, the enhancement—defined as the ratio between the peak intensity of the focused light and the ensemble-averaged intensity of random inputs—reached 180, which corresponds to 70.3% of the theoretical value of 256.

The advantages of the coded self-referencing method become more evident when handling a larger number of modes. To demonstrate this, we controlled 1024 elements and focused light on multiple points through complex media, including both the MMF and stacked ground glass diffusers. As a proof of concept, we projected three letters—“S”, “Y”, and “U”—through multiple foci. First, we measured 225 rows of the transmission matrix using the proposed method, and then applied phase conjugation to compute the input complex fields [Fig. 4(a)]. The experimental results of projecting light through the MMF and stacked ground glass diffusers are shown in Figs. 4(b) and 4(c). For MMFs, the reduced number of effectively excited modes is primarily due to the limitations of the input beam profile and coupling efficiency, which predominantly excite lower-order modes. In contrast, the diffuser’s high numerical aperture facilitates the excitation of more scattering modes, leading to smaller speckle sizes. Although the focal spot sizes varied due to differences in speckle size, all three letters were successfully projected. The average enhancements for the letters in Fig. 4(b) were 25, 38, and 29, respectively achieving over 25.3% of the theoretical value (1024/number of focal spots). Similarly, the average enhancements for the letters in Fig. 4(c) were 25, 27, and 28, respectively achieving over 22.5% of the theoretical value. These results confirm the accuracy of the proposed coded self-referencing method, particularly in managing large-scale transmission matrices.

We further demonstrated the performance of the proposed method in compensating for dynamic scattering with controllable correlation time. Following established protocols, we used moving stacked ground glass diffusers at controlled speeds, as shown in Fig. 5(a). The faster the diffuser velocity, the shorter the correlation time of the scattering process. To quantify this relationship, we captured a series of speckle patterns using a camera while translating the diffusers at various velocities, controlled by a one-dimensional motorized stage (MLJ050, Thorlabs). For each translational velocity, the correlation time was defined as the time interval during which the intensity correlation dropped to 1/e. A linear fit was applied to plot correlation time as a function of inverse velocity, establishing a direct mapping between correlation time and velocity.

To evaluate the method’s performance under dynamic scattering, we achieved optical focusing with 256 controllable elements at diffuser velocities of 0.001, 0.01, and 0.1 mm/s, as shown in Fig. 5(c). Based on the linear fit, these velocities correspond to correlation time of 2100, 210, and 21 ms, respectively. Given that the wavefront shaping system’s operational time for 256 controllable elements is 21.90 ms, the first two cases represent nearly static scattering conditions, resulting in a bright optical focus with enhancements around 159, achieving 62% of the theoretical value. In the third case, where the system’s operational time is comparable to the scattering correlation time, focus fidelity decreased, with the enhancement dropping to 62, approximately 1/e of the original value, consistent with theoretical predictions. Despite this, a bright optical focus was still achieved, demonstrating the system’s robustness in dynamic scattering environments.

We compared the time consumption of the proposed coded self-referencing method with existing non-interferometric phase retrieval techniques. For consistency, all algorithms were executed with 256 controllable elements on a computer equipped with an Intel® Core™ i9-10900X X-series processor, using Visual Studio 2022 for implementation. To ensure sufficient accuracy, different oversampling ratios were introduced, defined as the ratio between the number of intensity measurements and the number of controllable elements, with no noise added. In this comparison, we focused on the coded self-referencing method alongside the generalized Gerchberg-Saxton (GGS) method^[53] and the amplitude flow (AF) method^[56], which represent mainstream approaches in phase retrieval: iterative Fourier-based algorithms (such as the GGS algorithm) and gradient descent methods (such as the AF method). As shown in Table 2, the proposed method significantly outperforms the other phase retrieval techniques in terms of time consumption. This advantage becomes even more pronounced with an increased number of controllable elements, such as 1024, where the difference becomes substantial. Some phase retrieval methods from other studies were not included in our comparison due to the lack of available C++ code implementations^[49,50,52]. However, based on the known time complexity of various phase retrieval algorithms, it is unlikely that these methods would outperform the two techniques (GGS and AF) presented here.

In conclusion, we developed a coded self-referencing method for fast wavefront shaping to address the challenges posed by dynamic scattering. Utilizing superpixel encoding and phase conjugation, we successfully achieved optical focusing through MMFs and stacked ground glass diffusers, using both 256 and 1024 controllable elements. This system eliminates the need for reference light, avoiding the sacrifice of controllable elements required by coaxial interferometry, while also reducing the computational burden and long processing time typically encountered in non-holographic phase retrieval methods. Consequently, our method achieves the fastest average mode time among current referenceless approaches. Specifically, the average mode time for the two configurations were 85.54 and 74.47 µs, approaching the DMD’s hardware limit of 69 µs for three-step phase-shifting ($23\mathrm{\mu s}\times 3=69\mathrm{\mu s}$ for three-step phase-shifting). In applications where measurements on the other side of MMFs are prohibitive, our encoded self-referencing method can be directly integrated with guide stars, such as fluorescent particles^[26,65]. This integration facilitates measurements from the same side as the illumination beam. These results showcase the method’s potential for real-time applications in optical imaging, communication, and sensing in complex and dynamic scattering environments.