Wavefront shaping in multimode fibers by transmission matrix engineering

In recent years, multimode optical fibers (MMFs) are at the focus of numerous studies aiming at enhancing the capacity of optical communications and endoscopic imaging systems.^1,2 Ideally, one would like to utilize the transverse modes of the fiber to deliver information via multiple channels, simultaneously. However, inter-modal interference and coupling between the guided modes of the fiber result in scrambling between channels. One of the most promising approaches for unscrambling the transmitted information is by shaping the optical wavefront at the proximal end of the fiber in order to get a desired output at the distal end. Demonstrations include compensation of modal dispersion,^3–5 focusing at the distal end,^6–10 and delivering images^11–15 or an orthogonal set of modes^16,17 through the fiber. Similar approaches are applied to other types of random media, such as biological tissue and turbulent atmosphere.¹⁸ 

Typically, in wavefront shaping, the incident wavefront is controlled using spatial light modulators (SLMs), digital micromirror devices (DMDs), or nonlinear crystals. In all cases, the shaped wavefront sets the superposition of guided modes that is coupled into the fiber. For a fixed transmission matrix (TM) of the fiber, this superposition determines the field at the output of the fiber, as depicted in Fig. 1(a).^11,19,20 Hence, in a fiber that supports N guided modes, wavefront shaping provides at most N complex degrees of control. However, many applications require the number of degrees of control to be larger than the number of modes. For example, one of the key ingredients for spatial division multiplexing is mode converters, which require simultaneous control over the output field of multiple incident wavefronts. To this end, complex multimode transformations were previously demonstrated by applying phase modulations at multiple planes.^21–24 However, this requires free-space propagation between the modulators, thus limiting the stability of the system and increasing its footprint.

In this work, we propose and demonstrate a new method for controlling light at the output of MMF, which does not rely on shaping the incident light and that can be implemented in an all-fiber configuration. Inspired by the ongoing efforts to generate on-chip mode converters by manipulating modal interference in multimode interferometers,^25–27 we directly control the light propagation inside the fiber to manipulate its TM, allowing us to generate a desired field at its output [Fig. 1(b)]. Since the TM is determined by ∼N² complex parameters, TM-shaping provides access to much more degrees of control than shaping the incident wavefront.

Our method relies on applying controlled weak local macro bends along the fiber. Fiber macro bends and stresses change the local propagation constants of the fiber modes and induce mode coupling.²⁸ Random deformations are commonly used for scrambling the modes of the fiber and for modal filtering.^29–32 Precisely controlled bends and stresses, however, which are routinely used for programmable polarization control in fibers,³³ have only recently been proposed for controlling modal dynamics in multimode fibers.^34,35 Inspired by these recent works, here, we use an array of computer-controlled piezoelectric actuators to locally apply pressure on the fiber at multiple positions and control the fiber’s TM. Since the stress induced by the bends changes the boundary conditions of the system, it modifies the TM such that different bends yield different speckle patterns at the distal end [Fig. 1(c)]. We can therefore try to obtain a desired field at the output of the fiber by imposing a set of controlled bends, without modifying the incident wavefront. Since in this approach the input field is fixed, it does not require an SLM or any other free-space component. Such an all-fiber configuration is especially attractive for MMF-based applications that require high throughput and an efficient control over the field at the output of the fiber. As a proof-of-concept demonstration of TM-shaping, we demonstrate focusing the light at the distal end of the fiber and conversion between fiber modes.

Our method relies on applying controlled weak local bends along the fiber. To this end, we use an array of computer-controlled piezoelectric actuators to locally apply pressure on the fiber at multiple positions.^34,35 The TM of the fiber changes with the curvatures of the bends, which are determined by the travel of each actuator. To obtain a target pattern at the distal end, we compare the intensity pattern recorded at the output of the fiber with a desired target pattern. Using an iterative algorithm, we search for the optimal configuration of the actuators, i.e., the optimal travel of each actuator, which maximizes the overlap of the output and target patterns.

The experimental setup is depicted in Fig. 2. A HeNe laser (wavelength of λ = 632.8 nm) is coupled to an optical fiber, overfilling its core. We placed 37 piezoelectric actuators along the fiber. By applying a set of computer-controlled voltages to each actuator, we controlled the vertical displacement of the actuators. Each actuator bends the fiber by a three-point contact, creating a bell-shaped local deformation of the fiber, with a curvature that depends on the vertical travel of the actuator [see Figs. 2(b) and 2(c)]. For the maximal curvature we applied (R ≈ 10 mm), we measured an attenuation of a few percent per actuator due to the bending loss. The spacing between nearby actuators was set to be at least 3 cm, which is larger than $d2λ$ for d the core’s diameter, such that the spatial field distribution significantly changes upon propagation between two adjacent actuators. At the distal end, a CMOS camera records the intensity distribution of both the horizontally and vertically polarized light.

We used two types of multimode fibers: a fiber supporting few modes for demonstrating mode conversion and a fiber supporting numerous modes for demonstrating focusing. For the focusing experiment, we used a 2 m-long graded-index (GRIN) multimode optical fiber with a numerical aperture (NA) of 0.275 and core diameter of d_MMF = 62.5 μm (InfiCor OM1, Corning). The spectral correlation width of our fiber at λ is approximately 0.1 nm.^36,37 The fiber supports approximately 900 transverse modes per polarization at λ = 632.8 nm (V ≈ 85), yet we used weak focusing at the fiber’s input facet to excite only approximately 280 modes. For the experiments with the few mode fiber (FMF), we used a 5 m-long step-index (SI) fiber with an NA of 0.1 and core diameter of d_FMF = 10 μm (FG010LDA, Thorlabs). In principle, at our wavelength, the fiber supports six modes per polarization (V ≈ 5).

The curvature of the bends, set by the travel of each actuator, modifies how light propagates through the fiber and thus determines the speckle pattern that is received at the distal end. We can therefore define an optimization problem of finding the voltages that should be applied to the actuators to receive a given target pattern at the output of the fiber. The distance between the target and each measured pattern is quantified by a cost function, which the algorithm iteratively attempts to minimize.

For M actuators, the solution space is an M-dimensional sub-space, defined by the voltage range and the algorithm’s step intervals, and can be searched using an optimization algorithm. While the optical system is linear in the optical field, the response of the actuators, i.e., the modulation they pose on the complex light field, is not linear in the voltages. Moreover, since a change in the curvature of an actuator at one point along the fiber affects the interference pattern at all of the following positions of actuators, the actuators cannot be regarded as independent degrees of control. Similar nonlinear dependence between degrees of control is obtained, for example, for wave control in chaotic microwave cavities.³⁸ Out of the wide range of iterative optimization algorithms that can efficiently find a solution to such nonlinear optimization problems, we chose to use Particle Swarm Optimization (PSO)³⁹ as, on average, it achieved the best results out of the algorithms we tested (see the supplementary material for more details regarding the use of PSO).

To illustrate the concept of shaping the intensity patterns at the output of the fiber by controlling its TM, we first demonstrate focusing the light to a sharp spot at the distal end of a multimode fiber. We excite a subset of the fiber modes by weakly focusing the input light on the proximal end of the fiber. Due to inter-modal interference and mode-mixing, at the output of the fiber, the modes interfere in a random manner, exhibiting a fully developed speckle pattern [Fig. 3(a)]. Based on the number of speckle grains in the output pattern, we estimate that we excite the first 280 guided fiber modes.

To focus the light to some region of interest (ROI) in the recorded image, we run the optimization algorithm to enhance the total intensity at the target area. We define the enhancement factor η by the total intensity in the ROI after the optimization, divided by the total intensity in the ROI before the optimization. Since the intensity in the ROI before optimization is very sensitive to the actuators’ configuration, it is computed by averaging the output intensity over random configurations of the actuators. An additional azimuthal integration of the ensemble averaged intensity pattern is performed to improve the averaging, while accounting for the envelope of output intensity patterns.

We start by choosing an arbitrary spot in the output speckle pattern of one of the polarizations. We define a small ROI surrounding the chosen position, in an area that is roughly the area of a single speckle grain, and run the optimization scheme to maximize the total intensity of that area. Figure 3 depicts the output speckle pattern of the horizontal polarization before [Fig. 3(a)] and after [Fig. 3(b)] the optimization using all 37 actuators. The enhanced speckle grain is clearly visible and has a much higher intensity than its surroundings, corresponding to an enhancement factor of η = 25.

We repeat the focusing experiment described above with a varying number of actuators M. When a subset of actuators is used, the remaining are left idle throughout the optimization. Figure 3(d) summarizes the results of this set of experiments, showing that the obtained enhancement factor η grows linearly with the number of active actuators M, when M is sufficiently small compared to the number of guided modes. It is instructive to compare this linear scaling with the well-known results for focusing light through random media using SLMs or DMDs. Vellekoop and Mosk showed that when the number of degrees of control (i.e., independent SLM or DMD pixels) is small compared to the effective number of transverse modes of the sample, the enhancement scales linearly with the number of degrees of control. The slope of the linear scaling α depends on the speckle statistics and on the modulation mode.^40–42 For Rayleigh speckle statistics, as in our system (see the supplementary material), the slopes predicted by theory are α = 1 for perfect amplitude and phase modulation and $α=π4≈0.78$ for phase-only modulation.⁴² Experimentally measured slopes, however, are typically smaller, mainly due to technical limitations such as finite persistence time of the system, unequal contribution of the degrees of control, and statistical dependence between them. Interestingly, we measure a slope of α ≈ 0.71, which is close to the theoretical value for phase-only modulation for Rayleigh speckles and higher than typical experimentally measured slopes (e.g., α ≈ 0.57⁴³). Naively, one could expect a lower slope in our system since in our configuration the degrees of control are not independent. The large slope values we obtain may indicate that the bends change not only the relative phases between the guided modes (corresponding to phase modulation) but also their relative amplitudes (corresponding to amplitude modulation), via mode-mixing and polarization rotation. For focusing through multimode fibers supporting N guided modes, the optimal enhancement factor that can be achieved with a phase-only SLM that controls both polarizations is $ηoptimal=π4N$⁠.¹⁹ In our experiment, N = 280; hence, the maximal enhancement we achieve with 37 actuators is η ≈ 0.11 · η_optimal.

To further study the linear scaling, we performed a set of numerical simulations. We used a simplified scalar model for the light propagating in a GRIN fiber, in which the fiber is composed of multiple sections, where each section is made of a curved and a straight segment. The curved segments simulate the bend induced by an actuator, while the straight segments simulate the propagation between actuators and introduce mode-mixing between groups of degenerate modes. This mode-mixing has been proven to be essential for the enhancement process (see the supplementary material for more details). As in the experiment, we use the PSO optimization algorithm to focus the light at the distal end of the fiber. The numerical results exhibit a clear linear scaling, with slopes in the range of 0.57–0.64 (see Fig. 3 in the supplementary material). Simulations for fibers supporting N = 280 modes, roughly the number of modes we excite in our experiment, exhibit a slope of α ≈ 0.64, slightly lower than the experimentally measured slope.

As in experiments with SLMs, focusing is not limited to a single spot. To illustrate this, we used the optimization algorithm to simultaneously maximize the intensity at two target areas. Figure 3(c) shows a typical result, exhibiting an enhancement which is half of the enhancement obtained when focusing to a single spot, as expected by theory.⁴¹ In principle, it is possible to focus the light to several spots, yet in practice, we are limited by the number of available actuators.

In the previous section (“Focusing at the distal end of the fiber"), we demonstrate the possibility to use our system as an all-fiber SLM, i.e., to shape the output intensity by modifying the relative complex weight of the propagating modes. In the following, we show that we can go further by studying the feasibility of TM-shaping to tailor the output patterns in the few-mode regime, where the number of fiber modes is comparable with the number of actuators. Specifically, we are interested in converting an arbitrary superposition of guided modes to one of the linearly polarized (LP) modes supported by the fiber. To this end, we utilize the PSO optimization algorithm to find the configuration of actuators that maximizes the overlap between the output intensity pattern and the desired LP mode. The target LP modes of the step-index fiber were computed numerically for the parameters of our fiber and scaled to match the transverse dimensions of the fiber image. Figure 4 presents a few examples of conversions between LP modes using 33 and 12 actuators. A mixture of LP₀₁ and LP₁₁ at two different polarizations can be converted to LP₁₁ in one polarization [Fig. 4(a)]. Alternatively, a horizontally polarized LP₁₁ mode can be converted to a superposition of a horizontally polarized LP₀₁ mode and a vertically polarized LP₁₁ mode [Fig. 4(b)]. The Pearson correlation between the target and final patterns in these examples is 0.93. Similar results are obtained when we run the optimization with fewer active actuators, with a negligible reduction in the correlation between the target and final pattern. For example, with 12 actuators, we observe correlations of 0.90 for the conversion presented in Fig. 4(c). Optimization with less than 12 actuators shows poorer performance as the number of actuators becomes comparable with the number of guided modes.

Controlling the transmission matrix of a multimode fiber, rather than the wavefront that is coupled to it, opens the door for unprecedented control over the light at the output of the fiber. Since the number of degrees of control, the number of actuators in our implementation, is not limited by the number of fiber modes N, it can allow simultaneous control for linearly independent inputs and/or spectral components. In fact, if ∼N² degrees of control are available, one can expect generating arbitrary N × N transformations between the input and output modes. Over the past two decades, there is an ever-growing interest in realizing reconfigurable multimode transformations for a wide range of applications, such as quantum photonic circuits,^27,44–47 optical communications,^23,48 and nanophotonic processors.^25,49 These realizations require strong mixing of the input modes as the output modes are arbitrary superpositions of the input modes. The mixing can be achieved, for example, by diffraction in free-space propagation between carefully designed phase plates,^21–24 a mesh of Mach–Zehnder interferometers with integrated modulators,²⁷ engineered scattering elements in multimode interferometers,^25,26 or scattering by complex media.^38,50 In our implementation, we rely on the natural mode-mixing and inter-modal interference in multimode fibers, allowing implementation using standard commercially available fibers.

In this work, we used mechanical perturbations for changing the TM of the fiber, which limits the modulation rates due to the mechanical relaxation time of the fiber. The response time of the system to abrupt changes of the piezo benders is approximately 30 ms (see the supplementary material), allowing, in principle, for modulation rates as high as 30 Hz. In practice, our system works at slower rates (≈5 Hz), mainly due to the latency introduced to control the electronics. The total optimization time for the focusing experiments is 50 min and 12–15 min for the mode conversion experiments. Faster electronics and development of a stiffer and more efficient bending mechanism will allow higher modulation rates, limited by the resonance frequency of the piezo benders (≈300–500 Hz). In order to further increase the modulation rate, one should consider other mechanisms such as acoustic and electro-optical modulations.^51,52 Using electro-optical modulators would require fabricating a material with a strong electro-optical response, similar to the approach used in the “smart-fiber” technology.⁵¹ Such an implementation would allow achieving even faster rates and reaching a truly scalable solution.

In this work, we proposed a novel technique for controlling light in multimode optical fibers by modulating its TM using controlled perturbations. We presented proof-of-principle demonstrations of focusing light at the distal end of the fiber and conversion between guided modes, without utilizing any free-space components. Since our approach to modulate the TM of the fiber is general and not limited to mechanical perturbations, it could be directly transferred to other types of actuators, e.g., in-fiber electro-optical or acousto-optical modulators, to achieve all-fiber, loss-less, fast, and scalable implementations. The all-fiber configuration and the possibility to control more degrees of freedom than the number of guided modes make our method attractive for fiber-based applications that require control over multiple inputs and/or wavelengths. Moreover, the possibility to achieve high dimension complex operations opens the way to the implementation of optical neural networks. Our system can provide an important building block for linear reconfigurable transformations, which can be further used in combination with fibers and lasers that exhibit strong gain and/or nonlinearity for deep learning applications.

See the supplementary material for an in-depth description of the experimental implementation, the typical time scales of the optical setup, demonstration that the system exhibits Rayleigh statistics, a detailed explanation about the optimization technique and parameters, and details of the numerical simulation, accompanied by the simulation results for several fiber parameters.

We thank Daniel Golubchik and Yehonatan Segev for invaluable help. This work was supported by the Zuckerman STEM Leadership Program, the ISRAEL SCIENCE FOUNDATION (Grant No. 1268/16), the Israeli Ministry of Science and Technology, the French Agence Nationale pour la Recherche (Grant No. ANR-16-CE25-0008-01 MOLOTOF), and Laboratoire International Associé ImagiNano.