On the exploration of structured light transmission through a multimode fiber in a reference-less system

Optical fibers, particularly single-mode fibers, serve as the bedrock of modern telecommunications, facilitating swift data transfer across significant distances.^1,2 However, recent strides in optical wavefront shaping have sparked a resurgence of interest in the potential of multimode fibers (MMFs) for both long-distance communication and applications in minimally invasive medical procedures, such as endoscopy.^3–7 

Multimode fibers (MMFs) are notable for their distinct advantage of offering higher data rates.^8,9 As we enter a new era of data-intensive applications, such as ultra-high-definition video streaming, cloud computing, and Internet of Things (IoT), we confront the impending saturation of fiber-based networks. Multimode fibers present a potential solution to this bottleneck. Their capacity for handling multiple light paths simultaneously allows for an increased volume of data transmission, potentially mitigating network congestion and accommodating the increasing global demand for high-speed data communications. Moreover, these fibers exhibit compatibility with single fiber endoscope procedures, providing an exciting opportunity to advance the state-of-the-art in this critical field of medicine.^10,11 Single fiber endoscopy is a minimally invasive technique that permits clinicians to examine areas within the body that are usually inaccessible, such as the gastrointestinal tract, lungs, and blood vessels. Incorporating multimode fibers into this procedure could significantly enhance the information content and resolution of images obtained during the endoscopic examination, thereby potentially improving diagnostic accuracy and aiding in real-time decision making during medical procedures. However, despite these promising benefits, the transmission of structured light—a light beam with a well sculpted intensity and phase distribution—through multimode fibers presents significant challenges.^12,13 The primary complication arises from modal dispersion, a phenomenon where light traveling through different paths or modes within the fiber arrives at the end of the fiber at different times. This temporal misalignment can distort the transmitted data and degrade the overall performance of the fiber-based system. Moreover, the process of retrieving a sizable transmission matrix (TM)—a matrix representing how an input light field is transformed into an output light field—in a reference-less system adds a layer of complexity to this problem. The difficulty of this task is compounded when the medium significantly scatters the light, as is the case with biological tissues or certain types of glass. Scattering disrupts the propagation of light and randomizes the phases of the light modes, complicating the extraction of a reliable TM and hampering the transmission of structured light. These challenges underscore the need for innovative approaches to fully leverage the potential of multimode fibers in high-capacity data transmission and biomedical imaging applications. Also, the challenge of imaging through opaque materials has traditionally been seen as insurmountable, given the incoherence of scattered transmitted light.^14,15 The ability to measure the phase of a light beam with a camera is limited due to the oscillating nature of the optical electric field, and the phase problem of light optics compounding the difficulty further.^16,17 However, elastic scattering has been shown to be deterministic for a solid static or slowly moving target.¹⁸ Innovative research since 2007 has demonstrated the potential to cancel complex scattering effects by shaping the wavefront of the incident beam directed toward the scattering media.^{3,6,7,19–22}

This paper presents a method for the complete retrieval of the TM in a single step, thus enabling coherent light propagation through a strongly scattering multimode fiber.¹⁷ We explore the fusion of iterative phase retrieval algorithms²³ and correlations between experimentally established focal points, a step that could potentially allow for arbitrary interference manipulation and effective phase correction.²⁴ Furthermore, we leverage technological advancements, such as spatial light modulators (SLMs) and digital mirror devices (DMDs), to quantify and measure the optical response function of the scatterer to the propagation of the incident coherent light.^25–27 

This discussion also encompasses the generation of structured beams through fibers,^28–30 a nascent field that holds immense promise in advanced communication and imaging systems. In this context, we tackle the limitations of conventional phase recovery techniques, such as off-axis holography and the Gerchberg–Saxton method, underscoring the need for innovative solutions like the one proposed here. Our work is pioneering in its use of the Generalized Gerchberg–Saxton (GGS)³¹ algorithm in tandem with the Scanning Focused Output (SFO) method to perform full TM retrieval in a single process.³² Significantly, we introduce a method to apply phase correction by employing direct correlations between rows in the TM. The measured and corrected TM is then utilized to execute various complex applications.²⁴ 

Our methodology’s validity is substantiated by successfully transmitting a variety of structured light beams through the multimode fiber. We explore the performance with two prominent examples, Laguerre–Gauss (LG) and Hermite–Gaussian (HG) types.^28,33,34 These beam types were selected due to their distinct characteristics and their critical role in optics. For instance, the LG beams, with their orbital angular momentum, are a cornerstone in high-capacity data transmission^35,36 and quantum communication. On the other hand, the HG beams with their rectangular symmetry and zero orbital angular momentum find application in areas, such as laser cutting,^37,38 optical trapping,^39,40 to name a few. Both of these beams provide a stringent testing ground for the phase recovery capability of our method. The phase profiles of these beams are measured using off-axis holography.⁴¹ 

We further demonstrate its accuracy by projecting different alphabetical patterns through multimode fibers and compare the results before and after correction. This transmission of such intricate patterns accentuates the robustness and high-resolution capabilities of our method.

In addition to validating the method, we take a thorough examination of its inherent limitations. Recognizing these restrictions is pivotal, as it not only leads to the development of more refined techniques but also aids users in determining the most suitable method for their specific application. We juxtapose our approach with other cutting-edge methods, considering several critical aspects, such as accuracy, complexity, computing time, and the need for external references. This comprehensive comparison provides a clear perspective of the strengths and weaknesses of our method relative to other existing techniques.

Finally, we expand the discourse to a broader context, highlighting the potential implications of our approach on various applications involving multimode fibers. These include areas such as biological endoscopy,^42,43 telecommunications,⁴⁴ image transmission,^33,45–48 image reconstruction,^42,49,50 and quantum information science.⁵¹ For instance, the potential of achieving higher resolution in biological endoscopy can aid in more precise medical diagnoses, while the improved data rate can mitigate the looming data bottleneck in telecommunications.

In order to recover the complete transmission matrix (TM) of the optical system including multimode fiber (MMF), we commence with an established reference-less TM retrieval method, employing the Generalized Gerchberg–Saxton (GGS) algorithm. This involves the collection of speckle images stemming from a multitude of random phase inputs. The retrieved TM is subsequently employed to create distinct foci as demonstrated by a plethora of well-established methods such as Semi-Definite Programming (SDP),¹⁴ Extended Kalman Filter-Modified Speckle-correlation Scatter Matrix (EKF-MSSM),⁵² and phase recovery-Variational Bayes Expectation-Maximization (prVBEM).^53–55 Here, we prefer the GGS algorithm due to its computational efficiency; the details of our retrieval process and illustrations are shown in  Appendix A.

Inherent to the retrieval process in a reference-less system, however, is the omission of inter-point information, leading to an overall phase discrepancy across rows in the TM. This phase difference results in non-ideal output when projecting continuous patterns through the MMF, or patterns with complex phase profiles.

In order to rectify the phase discrepancy between points, phase data spanning across all points must be collected. This is conventionally accomplished through the use of a reference arm. When a reference beam is present, the phase at each point is gauged relative to the reference phase, ensuring phase consistency throughout. However, in a reference-less system, points are spatially isolated and lack mutual correlation.

One feasible method to recover the phase is through capturing speckle images at varied planes. Light propagating in free space results in interference, and an additional phase retrieval procedure can be initiated to compute the phase. However, this mandates either the displacement of the fiber coupler or the camera. An alternative approach that capitalizes on the unique characteristics of the speckles is to perform phase shifts between neighboring points and use their interference values to calculate the phase differences. However, this method requires at least three extra measurements per point, and much higher stability is to be required.

Figure 2 presents a meticulously devised schematic diagram of our experimental procedure for comprehensive TM recovery.

At the outset of the process, an expanded continuous wave (CW) laser beam, operating at a wavelength of 632.8 nm, is subjected to modulation by a phase-only Spatial Light Modulator (SLM)—specifically, the HOLOEYE PLUTO 2.1 model. This modulated light is subsequently channeled into a multimode fiber. The MMF is a 1 m long step-index with a numerical aperture (NA) of 0.22 and a core diameter of 50 μm (Thorlabs M14L01). The MMF supports about 1500 modes at 632.8 nm.

The MMF serves as a medium for high-capacity signal transmission, causing random interference within the light beam as it propagates through the fiber, leading to the generation of speckle patterns. The light emerging from the distal end of the MMF is collected by another coupler and a camera (Thorlabs CS2100M-USB), which records the resultant speckle pattern. This configuration enables a comprehensive investigation of light propagation through the MMF and the challenges associated with structured light transmission in a reference-less system.

To complement our phase retrieval algorithm, we develop a direct correlation for phase correction (DCPC) method to extract the phase difference between any pair of neighboring points.

Figure 3 portrays the complex interference between adjacent foci experimentally generated by our preferred GGS algorithm. We strategically project two proximate foci onto the imaging plane.

To one of these foci, we apply an incremental phase shift ϕ, valued at 0, π/2, π, and 3π/2. Figures 3(a)–3(d) illustrate the corresponding interference patterns captured during this phase modulation. The focus undergoing the phase shift is outlined within a dotted circle, with the location of the resultant interference indicated by an arrow. The experiment extends to include three neighboring foci, as depicted in Figs. 3(e)–3(h). Here, it is shown that when two foci are out of phase (approximately π phase difference), there is no interaction between them, resulting in a dark gap. When they are in phase, however, their interference is at maximum, resulting in an elliptical spatial distribution.

To achieve full retrieval of the TM, all foci need to be corrected with respect to a single focus. A proper scanning procedure is needed to recover the full TM. Figure 4 delineates two potential scanning orders, represented by an exemplary small array of foci. The real experimental array is significantly larger.

Figure 4(a) demonstrates a scanning progression from top-left to bottom-right, with the order represented by numbered black arrows. Foci in the first column are corrected first by the correlation between itself and its upper neighbor. Foci in subsequent columns are corrected with their left neighbors. Direct phase references between points, marked by blue arrows, reveal the potential for error propagation in the opposite direction. Accumulated errors may diverge between rows, increasing with lattice size.

Figure 4(b) presents a modified scanning approach, with a central focus serving as the reference point rather than a corner focus, effectively reducing the steps of direct reference and subsequent phase errors at the far edge. It should be noted that while there are scanning orders that require fewer steps to achieve the correction of the whole basis, they are not presented in this work considering that direct correlation only takes ∼30 ms to complete correcting a 15 × 15 grid.

Phase recovery is inherently an iterative procedure, and as such, can be halted at any given juncture. Essentially, this allows us to focus on recovering a specific region if that is the primary area of interest. For example, the transmission of a first order Hermite–Gauss beam (HG₀₁) only requires two neighboring foci with an exact phase difference of π. In this particular scenario, a single phase correction is all that is necessary.

The phase profiles of HG beams are divided into sections depending on the indices n and m, each having a π-phase shift relative to its neighbor.

In Fig. 5, we present the transmitted HG beams through the multimode fiber. We clearly show that many different HG modes can be propagated with little to no distortion in their spatial distribution: HG₀₁, HG₁₁, HG₂₀, and HG₂₂ are fully transmitted through the MMF.

Their retrieved phase profiles are displayed in the bottom row of Fig. 5. Each Gaussian component that forms the HG beams has a π-phase difference with respect to its immediate neighbor. Therefore, we only need a single phase correction for each component to effectively transmit these modes.

To verify that the phases of the HG modes are transmitted accurately, we added a reference arm to our experimental setup (not shown) to perform phase retrieval using the off-axis holographic method. These phase profiles are shown in Fig. 5(b), demonstrating remarkable agreements with theoretical phase patterns in Fig. 5(c), thus validating our method. Further discussion is shown in  Appendix B.

In Fig. 6, we show the propagation of various LG beams through the MMF with high fidelity. The intensity profiles of LG₀₁, LG₁₁, LG₋₂₀, and LG₃₀ are well preserved at the output of the fiber as demonstrated in Figs. 6(a) and 6(b). In Fig. 6(c), we also show that the spiral phase of each beam is carried through the fiber. We verify the accuracy of the phase profile using two different methods:

• Experimentally, we use the same off-axis holography method to show the accuracy of the HG mode. The results of these experiments are presented in Fig. 6(c).

• Computationally, we performed an iterative phase retrieval calculation based on Error Reduction (ER) and relaxed averaged alternating reflections (RAAR) algorithm to converge to a stable phase to obtain a global minimum in the error of less than 1%.^58,59 The calculation results are presented in Fig. 6(d).

It is noteworthy that the phase profile retrieved experimentally and computationally shows very good agreements. The colors red and blue are chosen to represent physically equivalent phases π and −π, respectively. This choice is made to easily determine the output beams’ topological charge. More importantly, phase profiles in Fig. 6(c) show high correlation values with the corresponding desired output, thus validating our method of generating a complete TM for an MMF. Additional phase profile comparisons and the limitations of these methods are discussed further in  Appendixes B and  C.

In this section, we investigate the projection of handwritten alphabetical characters through the multimode fiber (MMF) and highlight the importance of using the Direct correlation for Phase Correction (DCPC) algorithm to improve the quality of the projections. The handwritten patterns used in our experiment are derived from the EMNIST dataset,⁶⁰ commonly employed for machine learning training purposes. These characters serve as desired targets for our projections.

Figures 7(a), 7(d), and 7(g) display three different characters from the EMNIST dataset, representing the desired output. To evaluate the performance of the transmission matrix (TM) prior to correction, we project the characters using the uncorrected TM. The results obtained using solely the combination of the Gerchberg–Saxton (GGS) algorithm and the Single Focal Point Optimization (SFO) technique are shown in Figs. 7(b), 7(e), and 7(h). Although these patterns vaguely resemble the desired output, they suffer from noticeable intensity distribution discontinuities. This discrepancy arises due to the random out-of-phase relationship between the columns of the TM, which cannot be recovered using reference-less phase retrieval techniques alone.

In contrast, Figs. 7(c), 7(f), and 7(i) demonstrate significantly improved projection outputs achieved after the application of the DCPC algorithm. These corrected results exhibit a continuous intensity distribution and, more importantly, bear a stronger resemblance to the desired output presented in the left column.

To quantitatively assess the difference in quality, we perform a cross-correlation analysis between each projection output and its corresponding theoretical output. The results, shown in the top right corner of each figure, reveal that the data obtained with the corrected TM yields much higher correlation values compared to the data obtained without correction. This confirms the effectiveness of the DCPC algorithm in improving the TM and highlights the accuracy of the phase retrieval process.

Experimentally, the cross-correlation coefficients of the projected characters are limited by the number of modes within the fiber and the decrease in quality of the foci at the edge of the fiber output.

Overall, the successful projection of handwritten patterns through the MMF, along with the significant enhancement achieved through DCPC, underscores the potential of our approach for accurate transmission of structured light information and its application in various fields such as telecommunications and healthcare.

In this study, we highlight the effective utility of the Gerchberg–Saxton–Fienup Optimization (GSFO) method as a versatile and streamlined approach for the retrieval of the transmission matrix of a scattering medium, even in a system devoid of reference points. The application of the GSFO method, combined with the Sequential Focal Optimization (SFO) technique, enables the highly accurate and computationally efficient recovery of the complete transmission matrix.

The acquired matrix allows us to perform a variety of important tasks, such as image transmission via multi-focusing and, notably, pattern projection with a high correlation value in a single attempt. These tasks underscore the significance of the combined GSFO and SFO approach in generating a transmission matrix. In addition to the accurate retrieval of the reference-less TM, we have devised a novel method DCPC to fully recovers the phase discrepancies between all points of the TM by making use of the correlation coefficients denoted μ between adjacent foci. Specifically, the accurate retrieval of the entire matrix is a crucial step in these processes, demonstrating the method’s broader utility and effectiveness.

Furthermore, our study also explores the transmission of Hermite–Gaussian and Laguerre–Gaussian beams through a Multimode Fiber (MMF). We successfully propagate various patterns of these structured light beams, including the projection of handwritten characters. Our findings emphasize the potential of direct correlation for phase correction in enhancing the projection results obtained through MMF measurements.

Looking ahead, we envision that the GSFO and SFO tandem method will facilitate more complex fiber-based applications, such as endoscopy imaging or the projection of structured light beams through an MMF in reference-less optical systems. We anticipate that these advancements will open up new possibilities in the field of optical imaging and projection, adding another layer of versatility and functionality to MMF operations.

M.N. acknowledges the support from the National Geospatial Intelligent Agency Grant No. #HM04762010012, and P.B. and M.N. acknowledge the support from the American Society for Cell Biology and PAIR-UP Award No. #PIC-03/RPI/N’Gom.

E.F. and M.N. acknowledge the support from the U.S. Department of Defense, DoD Air Force Research Laboratory under Grant No. #FA9550-23-1-0325.

N.P.N. and E.F. acknowledge the support from the U.S. Department of Energy, (DOE) Office of Science under Grant No. DE-SC0023148.

We are grateful to Yuecheng Shen for sharing the scripts of the Gerchberg–Saxton (GGS) and Extended Kalman Filter-Multiple Scattering Self-Consistent Method (EKF-MSSM) algorithms used in this article.

The authors have no conflicts to disclose.

V.T. and T.W. contributed equally to this work.

Viet Tran: Formal analysis (equal); Investigation (lead); Methodology (supporting); Visualization (equal); Writing – original draft (equal). Tianhong Wang: Formal analysis (equal); Investigation (supporting); Methodology (lead); Visualization (equal); Writing – original draft (equal). Nimish P. Nazirkar: Methodology (supporting); Software (lead); Visualization (equal). Pascal Bassène: Conceptualization (lead); Data curation (lead); Formal analysis (supporting); Validation (equal); Writing – review & editing (supporting). Edwin Fohtung: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Moussa N’Gom: Conceptualization (Lead); Funding acquisition (equal); Investigation (equal); Project administration (Lead); Resources (Lead); Supervision (Lead), Writing – review (equal).

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

The iteration process continues until the correlation between T_f and T_f−1 reaches 99.999%. However, performing the calculation of the whole TM causes the correlation function to be trapped in local maxima. Therefore, we use the GGS algorithm to retrieve the TM one row at a time. The detailed process is described in Algorithm 1, where n indicates the nth row of the matrix.

To ensure the accuracy of each individual TM row, we input we input 6N random phases (where N is the number of superpixels). Other algorithms, such as feedback-assisted GGS (fGGS), require fewer measurements if only a few rows of the TM are needed.⁶¹ 

In SFO, the light is sequentially focused along a grid-like pattern at the distal end of an MMF, effectively scanning through the speckle pattern. It has been previously used to perform fiber-based imaging.²⁰ It was then extended to address aberrations-free fluorescence imaging through a scattering medium.⁶² This progression has made SFO an ideal method to effectively measure the TM of a scattering medium, as each point at the output speckle represents a row in the TM. Thus, by using the GGS to focus a single point through the MMF column by column, from left to right, an entire TM can be recovered. The TM calculations are performed by choosing a basis of 15 × 15 in the speckle pattern and sequentially reconstructing its elements by performing the GGS algorithm. We repeat this process for the entire structure based on the SFO principle. In this way, we effectively scan through every pixel in the output pattern.

The collective outputs from this process are shown in Fig. 8(a). The intensity distribution of the different foci has a profile that corresponds well to the theoretical profile of speckles from an MMF.

Figures 8(b) and 8(c) show the reconstruction results of both amplitude and phase of the normalized TM, respectively. The TM calculation process takes on average 29.6 s to complete. The inverse TM is then used in the phase correction process using direction correlation to adjust phase difference.

To obtain the accurate phase of the propagated fields through the MMF, we compare the phase profile of the experimental outputs with their corresponding theoretical values. The results are shown in Fig. 9.

LG beams with no radial mode (p = 0) and small topological charge, e.g., LG_±10, LG_±20, have excellent correlation values (above 90%) when compared to their theoretical profile. LG_±11 with 1 radial mode, have more intricate phase profiles, which has a direct effect on the correlation values; lowered to $∼75%$⁠. LG beam with topological charge ℓ > 3, we observe a sharp decrease in the quality of the output phase pattern, especially with LG₅₀ as evidenced in Fig. 9.

HG beams have a similar trend to that of LG beams. They have high correlation values for low modes and low correlation values for modes m ≥ 3 as shown in Fig. 9.

There are several factors that contribute to the limitation of our experiments, one of which is the number of spatial modes $(∼1500)$ that can be carried by the model fiber used in this experiment. In Fig. 8, one can observe that the intensity of the foci generated diminishes as the focus is further away from the center of the fiber due to the Gaussian profile of the input light.

Furthermore, phase retrieval is based on intensity measurements, which effectively discard all phase information contained in the signal’s entries. Therefore, the accuracy of our method suffers from the fundamental requirements for intensity measurements to be an indicative of its accuracy. Hence, the quality of the light transmission is doubly affected, first by the GGS algorithm to obtain the preliminary TM, then by the SFO method to generate the foci.

An additional limitation is in the size of the input field: LG beam with a high topological charge (ℓ = 5 for example) have a larger diameter and will therefore have most of their signals propagate at the outer edge of the MMF. For these reasons LG₅₀ will not be transported as accurately in the MMF as LG₁₀ as demonstrated in Fig. 10. These limitations can be solved by using an MMF with more supported modes, more measurements, and/or by using a faster wavefront shaping device.