To meet the demand for higher capacity fiber-optic communication, multimode fibers have gradually attracted attention, but they introduce spatial distortions. To overcome this limitation, wavefront shaping technology promises to control scattered light after it is transmitted through multimode fibers. In this work, we introduce a Hadamard encoding algorithm (HEA) to control 1550-nm light that has passed through a multimode fiber. A series of Hadamard bases is iteratively added to the current optimum phase map, and the coefficient of each order is determined through a simple four-step phase-shifting mechanism. Using a laser source at 1550-nm wavelength, we experimentally achieved an optical focus through a 2-meter-long multimode fiber. With 1024 orders, the experimental enhancement reached 690, which is 86% of the theoretical value. As far as we know, this is the best result ever reported in focusing 1550-nm light through a multimode fiber. Moreover, we note that the HEA can also be used to reduce the intensity of the targeted light, suggesting broad applications in glare suppression. These results demonstrate superior performance in controlling targeted light transport through a multimode fiber at a telecommunication wavelength. We anticipate that this work will open new possibilities in a variety of applications in fiber optics.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Interchannel crosstalk in multimode fibers generates spatial distortions that cause a scrambled intensity profile. This phenomenon significantly restricts the wide use of multimode fibers for delivering images in fiber-optic communication and capturing images in biophotonics. A novel technique, termed wavefront shaping, has been proposed to overcome this limitation [1–6]. Instead of using a planar incident wavefront, wavefront shaping modulates the incident light into a special wavefront with an optimum phase map. Unlike a planar wavefront, this special wavefront is transformed into a bright optical focus after passing through the multimode fiber. Based on the methods used to determine the optimum phase map, wavefront shaping techniques can be classified into three categories: optical time reversal/optical phase conjugation [3,4,7–21], transmission matrix (TM) measurement [5,6,22–26], and feedback-based wavefront shaping [1,2,27–31]. Among all three techniques, only optical time reversal/optical phase conjugation determines the optimum phase map in a single measurement. However, it requires a rather complicated optical system and precise alignment between the camera and the spatial light modulator (SLM) [32–34]. Controlling scattered light by measuring the TM of the scattering medium has also been demonstrated. This method poses difficulties in focusing, because the intensity at the targeted position stays at the background level during measurements, which yields a relatively low signal-to-noise ratio (SNR) and can cause measurement errors in obtaining wavefront information. Different from the TM method, in feedback-based wavefront shaping, the intensity at the targeted position increases with every iteration, providing a relatively high SNR during the whole optimization process.
Most previous wavefront shaping experiments have used the spectrum range of 400 - 1100 nm, and only a few studies have been conducted at 1550 nm [16,26,35], which is one of the most important wavelengths commonly used in fiber-optic communication. Since infrared optical detectors, such as cameras, are either lower in quantum efficiency or more vulnerable to environmental noises than visible-light detectors of roughly the same price, a wavefront shaping method with a robust optimization algorithm is desirable. Digital optical phase conjugation uses interferometry to boost signal strength and has been applied to focus 1550-nm light through a 30-cm long multimode fiber, although it uses a rather complicated design . In comparison, feedback-based wavefront shaping employs a simple and reference-free design, and certain algorithms can maintain the SNR at a relatively high level throughout the whole optimization process. For these reasons, this work uses feedback-based wavefront shaping to focus 1550-nm light from a multimode fiber.
As Fig. 1(a) shows, feedback-based wavefront shaping starts with a planar wavefront. After passing through a scattering medium, the transmitted light exhibits a speckle pattern in the output plane, as seen in Fig. 1(a). A detector monitors the light intensity at the targeted position, generating feedback information that is sent to a computer. The computer implements an optimization algorithm to calculate the desired phase map and sends it to an SLM. The SLM then shapes the incident light and creates an enhanced intensity at the targeted position. This process is repeated, and after many rounds of iterations, a sharp focus is formed, as shown in Fig. 1(b). Based on different implementations, several optimization algorithms have been proposed, and two typical examples are the stepwise sequential algorithm (SSA) and the continuous sequential algorithm (CSA) [1,2]. These two algorithms optimize only one segment at a time, which yields a very small contrast for each measurement and makes these algorithms sensitive to noise. To improve the SNR, a partitioning algorithm (PA) has been proposed that randomly selects a partition of segments simultaneously at each iteration . However, the optimization process generates a significant amount of redundant information so that the final convergence is slow. Inspired by biological evolution process, a genetic algorithm (GA) has been demonstrated to be particularly advantageous in low-SNR environments, but at the cost of a slow convergence rate . Other optimization algorithms, such as the particle swarm [36,37], four-element division , and harmony search algorithms , have also been proposed, each with its own pros and cons.
In this work, using feedback-based wavefront shaping with a Hadamard encoding algorithm (HEA), we show that targeted light transport can be controlled through a multimode fiber at 1550 nm. Hadamard matrices are widely used in a broad range of applications, including modern telecommunications, digital signal processing, and compressive sensing [40,41]. In HEA, each column of a Hadamard matrix is used as one input basis, and it starts with the patterns of low spatial frequencies. Unlike SSA and CSA, HEA modulates half of the segments simultaneously, leading to a high SNR. Compared with the randomly selected segments in PA, the columns of the Hadamard matrix are mutually orthogonal. Therefore, HEA converges faster and can form a focus that is brighter than the one obtained using PA for the same number of measurements. Using HEA, we experimentally demonstrated focusing 1550-nm light through a 2-meter long multimode fiber and two stacked ground diffusers. Since HEA is not sensitive to external noises and converges relatively fast, a bright focus was experimentally achieved, with an enhancement reaching 86% of the theoretical value. In addition to enhancing targeted light transport, we also extended HEA to reduce the light intensity at the targeted position, which is particularly important for glare suppression. The remainder of the paper is organized as follows. The principle of HEA is described and flowcharted in Section 2. In Section 3, we describe the experimental setup and show the experimental and numerical results of focusing 1550-nm light through thick scattering media, including a 2-meter long multimode fiber and two stacked ground diffusers. In Section 4, we show that HEA can also be used for glare suppression. Finally, we summarize our conclusions in Section 5.
2.1 Mathematical representation of wavefront shaping
A deterministic scattering process can be described using a transmission matrix in which each element t(j,k), j, k = 1,…, N satisfies a circular Gaussian distribution . For a given input field Ein,the corresponding output field Eout isEquation (1) indicates that every element in the output field is a summation of all the input elements multiplied by a corresponding transmission coefficient. Therefore, each element in Eout is contributed from a random summation of N phasors, which results in a speckle pattern . In order to form a focus, e.g., to increase light intensity for the j-th output element, the N phasors A(j,k) are required to constructively interfere with each other. To achieve this goal, the input field Ein(k) must be modulated with the optimum phase map so that all the A(j,k) are aligned.
2.2 Principle of the Hadamard encoding algorithm
There are a variety of optimization algorithms to determine the optimum phase map. In this section, we introduce HEA, which employs a series of Hadamard bases and a four-step phase-shifting mechanism. These Hadamard bases are naturally ordered, which are generated by taking the inverse fast Walsh-Hadamard transform of an identity matrix. To initialize the optimization process, we first generate a Hadamard matrix of order 1024 and use each column of this matrix as an input basis . Since the active area of the SLM is two dimensional (2D), is reshaped into a series of 2D matrices (32 × 32), shown in Fig. 2(a). As we can see, starts with the patterns of low spatial frequencies. In order to fully utilize the active area of the SLM (1920 × 1080 pixels), it is divided into 1024 segments, with each segment consisting of 60 × 33 pixels. A small number of pixels at the edge are unused and fixed to zero phase. Next, this series of Hadamard bases is sent to the SLM in order. For each order, a four-step phase-shifting mechanism is applied to calculate the current optimum phase map . For example, at the n-th order, four phase maps,43]. After is obtained, the current optimum phase map is updated asFig. 2(b). The above procedure is repeated until all orders are processed. A flowchart of this optimization process is depicted in Fig. 2(c).
3. Experimental setup and results
3.1 Experimental setup
The experimental setup of the system is shown in Fig. 3. A continuous 1550-nm laser (Agilent technologies N7714A) is used as the light source. A half-wave plate and a polarizing beam splitter are used to adjust the optical power in this system. Then, the beam is expanded by a pair of lenses to cover the active area of the phase-only SLM (PLUTO-2-TELCO-013, Holoeye). After being modulated and reflected by the SLM, the light is directed by a beam splitter to a multimode fiber with a core diameter of 200 μm and a numerical aperture (NA) of 0.22. Two identical objective lenses (20 × , NA = 0.5) couple light into this multimode fiber and collect the output light. An 8-bit infrared CCD camera (CMLN-13S2M-CS, infrared coating, Point Grey) captures the speckle pattern. Both the CCD camera and the SLM are controlled by a computer.
3.2 Focusing light through a multimode fiber
As an initial demonstration, we focused light through a 2-meter long multimode fiber that supported 3.2 × 103 modes at 1550 nm. Figure 4(a) shows the speckle pattern observed by the CCD before optimization. We chose the central pixel of the camera as the targeted position at which to form a focus. As shown in Fig. 4(b), after HEA was performed, a bright focus emerged at the center of the plane. Conventionally, the enhancement, η, is introduced to evaluate the quality of a focus, which is defined as the ratio between the intensity of focus after optimization Ifoc and the initial ensemble averaged intensity of target area before optimization [1,2]. Experimentally, we compute the enhancement using the following formula:Figure 4(c) plots a typical experimentally achieved enhancement as a function of the number of measurements (the number of iterations × 4). As with other feedback-based wavefront shaping algorithms , three rounds of optimization processing were performed. The enhancements are 274, 638, and 690 at the end of each round, reaching 34%, 79%, and 86% of the theoretical value, respectively. The optimum phase map that corresponds to the final focus in Fig. 4(b) is also shown in Fig. 4(d). We note that the best enhancement we achieved is still lower than the theoretical value. The main reasons might be the instability of the system, environmental and electronic noises, and imperfect modulation of the SLM. Nonetheless, to our knowledge, this is the best result that has ever been reported for focusing 1550-nm light through a multimode fiber. The excellent performance of the system is enabled by HEA, which quickly increases the enhancement at the beginning and maintains a good SNR throughout the entire optimization process. One unsatisfactory part of this system is that due to the slow refresh rate of the SLM (60 Hz) and the camera (18 Hz), each measurement takes about 200 ms including the calculation time. Thus, the total system runtime is about 40 minutes for three rounds of iterations. Although the current hardware limitations restrict our system speed, we anticipate that the system speed can be further improved by using faster cameras, SLMs, and processors. Nonetheless, we hope that this work sets a foundation for improving the data throughput for optical communication using multimode fibers. In future works, by either using multiple SLMs or splitting one SLM into several segments, multiple optical modes at the distal end of the multimode fiber can be independently controlled, enabling the improvement of data throughput.
We also performed numerical simulations to compare with the experimental results. The elements in the transmission matrix were drawn from a circular Gaussian distribution. In our simulation, we set the number of the input elements to be 1024 and targeted one element in the output plane. The focus achieved by simulation is shown in Fig. 4(e), and Fig. 4(f) plots the achieved enhancement as a function of the number of measurements. By comparing Figs. 4(c) and 4(f), we found that the experimental results are in good agreement with the simulation results. For comparison, simulated enhancements as a function of the number of measurements using GA  and PA  are also plotted in Fig. 4(f). As we can see from the figure that enhancements obtained using GA and PA exhibit much slower convergence rates compared to the one obtained using HEA.
3.3 Focusing light through two stacked ground diffusers
We also performed experiments in which the multimode fiber was replaced with two stacked 120-grit ground diffusers (DG10-120, Thorlabs) as the scattering object. To quantify the optical density of the two stacked diffusers, we measured the optical power transmitted through the diffusers at 3 meters away with a photodetector . We found that due to the existence of the diffusers, the transmitted power reduced to 6.5 × 10−4 of the transmitted power without the diffusers. Using Beer’s law, this scattering object was quantified with an optical density of 14.7 at 1550 nm, which is enough scattering to mimic a random process. Compared with the multimode fiber used in the previous experiments, this scattering object has a much larger NA, generating orders of magnitude more degrees of freedom.
Figure 5 shows the experimental results. A random speckle pattern obtained by directly transmitting light through the scattering object is shown in Fig. 5(a). Applying HEA to optimize the incident wavefront results in a sharp focus, as shown in Fig. 5(b). As before, Fig. 5(c) shows the experimentally achieved enhancement as a function of the number of measurements. After three rounds of optimization, the enhancement reaches 620, which is 77.3% of the theoretical value. The optimum phase map that corresponds to the final focus in Fig. 4(b) is shown in Fig. 5(d). These results again demonstrate that HEA is extremely well suited for phase optimization at the wavelength of 1550 nm, where external noises, i.e., thermal noise and detector noise, play important roles.
4. Glare suppression simulation results
In addition to increasing the light intensity at the targeted location, here we show that HEA is also capable of suppressing glare. Wavefront shaping methods for glare suppression have been reported in [45,46], with potential applications in transportation, astronomy, and biophotonics. As discussed in section 2, is the phase that maximizes the intensity. Therefore, becomes the phase that minimizes the intensity, which can be visualized in Fig. 6(a). To demonstrate this capability, a numerical validation was performed. Figure 6(b) shows a typical speckle pattern before optimization, with the targeted area encircled in red. The optimized intensity pattern after 1024 iterations is shown in Fig. 6(c). Clearly, the final intensity of the target position is many orders of magnitude smaller than that of other positions. Figure 6(d) plots the normalized intensity variation (normalized by the average intensity of the rest of the speckles) of the target position as a function of the number of iterations. The plot shows that the normalized intensity drops down to 10−30 at around 56 iterations. After that, the values become too small to be accurately calculated and thus have no physical meaning. Moreover, different from the situation in which the enhancement increases linearly or quadratically (polynomially) with the number of iterations, we note that the intensity decays nonlinearly (non-polynomially) with the number of iterations at a surprisingly fast rate. Therefore, suppressing glare using HEA could be particularly valuable for improving the image quality of a weak target that would otherwise be overwhelmed by background noise.
In conclusion, we have demonstrated focusing 1550-nm light through a multimode fiber and two stacked ground diffusers. Enabled by the excellent performance of HEA, the experimentally achieved enhancement through the multimode fiber is 690, which is 86% of the theoretical value. To our knowledge, this is the best result ever been reported for focusing 1550-nm light through a multimode fiber. For the ground diffusers, we experimentally achieved an enhancement of 620, which is 77% of the theoretical value. Our results demonstrate superior performance in controlling targeted light transport through a multimode fiber at a telecommunication wavelength, and we anticipate that this work will open new possibilities in fiber optics.
National Natural Science Foundation of China (61435006, 61525502, 61490710); The Science and Technology Planning Project of Guangdong Province (2017B010123005, 2018B010114002); Local Innovation and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X121).
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