1. INTRODUCTION

Holographic endomicroscopy is a powerful platform for minimally invasive bioimaging and remote sensing [1–3]. For this method, the light scrambling effects of a multimode fiber (MMF) need to be overcome, which can be achieved by wavefront shaping (WFS) [4]. MMFs support the propagation of thousands of spatial modes within a small cross-sectional area of about ${0.01}\;{\rm{mm}} \times {0.01}\;{\rm{mm}}$ and therefore allow microscopy at high spatial information density. The imaging potential of endomicroscopy has already been exploited to visualize neural activity at arbitrary depths in a living brain [5–7]. However, spatial resolution, imaging speed, contrast, and field-of-view (FOV) can be improved further.

The most popular approach of MMF-based endomicroscopy exploits raster-scan imaging [3,6,8]. By WFS, the phase front of the light incident on the fiber is modulated such that constructive interference is achieved at the desired focus location at the fiber output. In order to form an image, a sample is illuminated by a sequence of these focal spots at different locations. The performance of this raster-scan imaging mainly depends on the quality of the generated foci. Ideally, a perfect diffraction-limited focal spot should be generated at any point at the fiber output facet. Control of a large fraction of modes is essential for nonaberrated foci [9]. The highest fraction of optical power in a single spot can be achieved by oversampling (the number of controls higher than the number of fiber modes [6,10]). However, for a field-of-view of only 200 µm in diameter, this would require the control of about 34·10³ phase segments, which will lead to a long optimization time and require a spatial light modulator with a large number of pixels. Image quality can also be improved by the use of MMF with a square core, which was found to be superior to the more traditional round core since the eigenmodes of the square-core fiber fill out the core more uniformly [11]. Computational algorithms can increase image quality by modeling a spatially variant point-spread function and performing deconvolution [12].

Here, we investigate the performance of a round-core MMF-based endoscope with partial mode control, which is characteristic for a situation with a large-core MMF and a finite number of controlled pixels of a round illumination beam in its Fourier plane. We use phase-only modulation, conventional fibers, and a common Fourier plane configuration [6,13,14]. We show that, counterintuitively, coupling the input light to the center of the MMF is not always the best strategy. We show the trade-off between the shape and contrast of the generated focal spots. The center input position is beneficial for high contrast, while edge input positions help to reduce aberrations.

2. EXPERIMENTAL SETUP

The experimental setup is presented schematically in Fig. 1(a). We use a diode-pumped continuous wave (CW) laser with a wavelength of 532 nm. A half-wave plate (HWP), a polarizing beam splitter (PBS), and a beam blocker are used to modulate the beam power. Two achromatic doublets ${{\rm L}_1}$ (focal length, ${f_1} = {{50}}\;{\rm{mm}}$) and ${{\rm L}_2}$(${f_2} = {{150}}\;{\rm{mm}})$ form a telescope to expand the beam incident at the digital micromirror device (DMD) from Texas Instruments driven by the DLP V-9501 VIS module (Vialux). The DMD phase pattern is imaged on the back focal plane of Objective #1 (Olympus, ${\rm{20 \times }}$, ${\rm{NA}} = {0.4}$) by a $4f$ system consisting of lenses ${{\rm L}_3}$(${f_3} = {{150}}\;{\rm{mm}})$ and ${{\rm L}_4}$(${f_4} = {{100}}\;{\rm{mm}})$. The pinhole (${P_2}$) only allows the ${+}{{1}}$st diffraction order of the DMD to pass. Objective #1 couples light to an MMF. We use a round-core step-index multimode fiber with a diameter of 50 µm and an NA of 0.22 (Thorlabs). The fiber length is around 30 cm. Objective #2 (Olympus, $20 \times $, ${\rm{NA}} = {0.4}$) images the fiber output on Camera #1. A beam splitter (BS) between Objective #1 and lens ${{\rm L}_4}$ images the input facet of the MMF on Camera #2 with Objective #1 and lens ${{\rm L}_5}$ (${f_5} = {{70}}\;{\rm{mm}}$) placed in $4f$ configuration.

 

Fig. 1. (a) Experimental setup (HWP, half-wave plate; PBS, polarizing beam splitter; M, mirror; L, lens; P, pinhole; DMD, digital micromirror device; BS, beam splitter; Obj, objective; MMF, multimode fiber. (b) Examples of the input fields before WFS at different input positions with zero phase on the DMD. (c), (d) Output speckle patterns measured without WFS for the input beams at (c) $d_{\text{in}} = {{0.1r}}$ and (d) $d_{\text{in}} = {{0.9r}}$.

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The DMD consists of ${{1920 \times 1080}}$ tilting micromirrors, which are arranged on a rectangular grid. Each micromirror can individually rotate to ${+}12^\circ$, the “on state,” or to ${-}12^\circ$, the “off state.” We use the Lee hologram method to generate desired wavefronts [15]. Micromirrors are grouped together to form square segments consisting of $13 \times 13$ to $55 \times 55$ elements. Within each DMD segment, a binary 2D grating is created. Displacement of the “on state” and the “off state” lines of a grating within a segment provides the desired phase shift of the segment. The active area of the DMD has been restricted to a maximum of ${{550 \times 550}}$ micromirrors. This area has been experimentally chosen by measuring the contribution of each segment into the output field. With a fixed maximum of ${{550 \times 550}}$ micromirrors, the more segments we have, the smaller the size of each segment. As a result, after WFS, the light from a single segment will cover a larger portion of the fiber input facet when we use more DMD segments because of the Fourier transformation by the lens.

In the experiments, the position of the input light on the fiber input facet is controlled by the direction of beam propagation after the DMD. Changing the DMD grating’s periodicity leads to a change in the light direction. We use five input positions with the distance between the center of the core and the center of the input beam ${d_{\text{in}}}$ varied from ${0.1}r$ to ${0.9}r$, where $r$ is the fiber core radius (25 µm). Figure 1(b) shows the sum of five input fields before WFS with a flat phase pattern on the DMD. The input field has a butterfly shape instead of an ideal Gaussian shape because of the aberrations introduced by the not perfectly flat DMD surface and other optical components. The white circle indicates the fiber core. Figures 1(c) and 1(d) show the output speckles measured without WFS for the input beams at ${d_{\rm in}} = {0.1}r$ and ${d_{\text{in}}} = {0.9}r$, respectively. For each input position, we generate focal spots on the fiber output facet by the stepwise sequential WFS algorithm [16]. The phase of each segment has been stepped through three equidistant points (0, ${{2}}\pi {\rm{/3}}$, and ${{4}}\pi {\rm{/3}}$), while the intensity in the desired region is measured. We calculate the optimal phase by fitting the data with a cosine function. An example of an optimized phase pattern created by using 254 DMD segments for the input beams at ${d_{\rm in}} = {0.1}r$, $r = {{25}}\;\unicode{x00B5}{\rm m}$ is shown in Fig. 2(a). The black circle shows the fiber core.

 

Fig. 2. (a) Example of the optimized phase pattern created by using 254 DMD segments for the input beams at $d_{\text{in}} = {{0.1r}}$, ${\rm{r}} = {{25}}\;\unicode{x00B5}{\rm m}$. (b) Sum of 10 foci optimized with the same parameter. Black circle shows the fiber core. (c) Zoomed-in foci at a distance of 0 µm (left), 15 µm (center), and 22.5 µm (right) from the fiber center. The corresponding ellipticities are 0.11 (left), 0.29 (middle), and 0.59 (right). The scale bar is 2 µm.

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The most time-consuming step for a 2D wavefront optimization is recording the target intensity by a camera. The optimization time is linearly increasing with $N_{\rm seg}$. We use a camera (Basler acA2440-75µmMED) with a frame rate of 75 Hz. Therefore, 40 ms is required to optimize a single DMD segment, or ${\sim}{{1500}}$ segments can be optimized per minute.

3. RESULTS AND DISCUSSION

To investigate the influence of the light input position, we couple light to the MMF at different locations and perform the WFS experiments. Five input positions ${d_{\text{in}}} = [{0.1}r,\;{0.3}r,\;{0.5}r,\;{0.7}r,\;{0.9}r]$ are investigated. For each input position, 50 foci are generated sequentially along a single line on the fiber output facet. The controlled number of segments on the DMD in the experiment varies from ${{1}}{{{0}}^2}$ to ${{4}}{{{1}}^2}$. The square DMD pattern overfills the incident circular beam; thus, we correct the effective number of contributing segments with a factor of $\frac{\pi}{4}$. Hence, a number of DMD segments (${N_{\text{seg}}}$) varying from $\frac{\pi}{4} \cdot 10^2$ to $\frac{\pi}{4} \cdot 41^2$ has been investigated. Figure 2(b) shows 10 foci optimized on the fiber output facet with ${N_{\text{seg}}} = {{254}}$. The optimization has been done with ${d_{\rm in}} = {0.1}r$. Figure 2(c) shows three zoomed-in foci optimized at the center of the fiber (left) and at a distance of 15 µm (center) and 22.5 µm (right) from the fiber center. We see that the shape and intensity of the focal spots change with the position on the output facet. The focal spots become more elongated with increasing distance to the fiber center.

For a quantitative analysis, we fit each generated focal spot with a 2D Gaussian function and calculate the width of a focal spot along the short (${{2}}{w_{{\min}}}$) and along the long (${{2}}{w_{{\max}}}$) main axes at half maximum. The focus shape is characterized by the ellipticity $\varepsilon = {{1}}\;{ - }\;{w_{{\min}}}/{w_{{\max}}}$. The power in the focal spot is characterized by a power ratio (${\rm{PR}}) = {I_{\text{foc}}}/{I_{\text{tot}}}$, where ${I_{\text{tot}}}$ is the total power at the fiber output facet, and ${I_{\text{foc}}}$ is the power inside the elliptical area around the peak with ${w_{{\min}}}$ and ${w_{{\max}}}$ main axes of the ellipse. The focus ellipticity $\varepsilon$ and the power ratio PR are measured for a varying number of DMD segments ${N_{\text{seg}}}$ and input beam position ${d_{\text{in}}}$. The full measurement data set is shown in Appendix A.

Figures 3(a)–3(d) show the ellipticity $\varepsilon$, the power ratio PR, and the widths of the focal spot (${{2}}{w_{{\min}}}$ and ${{2}}{w_{{\max}}}$) along the main axes as functions of ${N_{\text{seg}}}$. The signal is averaged over all 50 foci generated at the fiber output facet. Different colors correspond to different coupling positions of the input beam with reference to the fiber center: ${d_{\text{in}}} = {0.1}r$ (black line), ${0.3}r$ (blue line), ${0.5}r$ (green line), ${0.7}r$ (red line), and ${0.9}r$ (magenta line). All the curves for the ellipticity $\varepsilon$ and the power ratio PR saturate when the number of segments reaches 1200. This is approximately equal to the total number of modes supported by the MMF for one polarization. However, it is not always possible to control a large fraction of modes, especially if the fiber core (and the resulting FOV) is getting larger. For example, for a fiber with a core diameter of 500 µm and ${\rm{NA}} = {0.8}$ at wavelength 532 nm, we would need to control around ${{1}}{{{0}}^7}$ modes for two polarizations, whereas the typical SLM has only around ${{1}}{{{0}}^6}$ pixels.

 

Fig. 3. (a) Experimentally measured ellipticity $\varepsilon$, (b) power ratio PR, (c) beam waists $2w_{\min}$, and (d) $2w_{\max}$ averaged over 50 foci distributed over output facet as a function of the number of segments ($N_{\rm seg}$) on the DMD. Results of semianalytical simulations: (e) $\varepsilon$, (f) PR, (g) $2w_{\min}$, and (h) $2w_{\max}$ averaged over 50 foci distributed over output facet as a function of $N_{\rm seg}$. Different colors correspond to different distances between the input beam and the fiber center: $d_{\text{in}} = {{0.1r}}$ (black line), ${{0.3r}}$ (blue line), ${{0.5r}}$ (green line), ${{0.7r}}$ (red line), and ${{0.9r}}$ (magenta line). The black lines show the diffraction limit of 1.21 µm.

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We see that important characteristics of the optimized foci such as the shape and the power ratio crucially depend on the beam position on the fiber input. The focus width along the short axis (${{2}}{w_{{\min}}}$) does not show a notable change with ${N_{\text{seg}}}$, as shown in Fig. 3(c), while the width along the long axis (${{2}}{w_{{\max}}}$) converges to the diffraction limit only for a high ${N_{\text{seg}}}$ [see Fig. 3(d)]. This effect is more pronounced for the central light coupling. To obtain a good focal shape with a relatively low number of segments and, consequently, reduce the aberrations in raster-scan imaging, through a round-core fiber, we need to couple light closer to the core’s edge [red and magenta curves in Fig. 3(a)]. On the other hand, results presented in Fig. 3(b) demonstrate that the central coupling provides more power to the focus, which yields better imaging contrast.

Our experiments demonstrate a trade-off between imaging contrast and aberration in raster-scan imaging through a round-core MMF. Additionally, we investigate this trade-off for different parameters of fiber length, diameter, and working distance through numerical modeling. We simulate light propagation in a round-core multimode fiber with the same setup configuration. The stepwise WFS algorithm in the experiment is an iterative algorithm, which changes the wavefront as set by the DMD to optimize the focus at the target plane. In the simulation, we separate the calculation of one propagation cycle into three substeps. First, we model the light propagation from DMD to the fiber assuming perfect relay optics. Second, we simulate the propagation through an ideal multimode fiber using a semianalytical mode-solver [17]. This solver calculates the fiber modes (for both polarizations) and their corresponding propagation constants for a given fiber geometry. The propagation of any input field ${E_{\text{in}}}(x,y)$ through a fiber of length $L$ to output field ${E_{\text{out}}}(x,y)$ is then given by Eq. (1):

The simulation of the stepwise WFS algorithm follows the same steps as described in Section 2. The input field ${E_{\text{in}}}(x,y)$ is assumed to be a plane wave and is divided into segments. The target intensity is recorded when each segment is swept through (0, ${{2}}\pi {\rm{/3}}$, and ${{4}}\pi {\rm{/3}}$). The optimal phase pattern for this segment is calculated by fitting the intensity as a cosine function of phase. After going through all the segments, the optimized wavefront for each focus spot is updated individually. As a result, 50 focus spots at the fiber output for one input position have been simulated for a different number of segments. Both the ellipticity $\varepsilon$ and power ratio PR are also averaged over all 50 foci generated at the fiber output facet. This process is repeated for five different positions $0.1 r$ to $0.9r$ by shifting the input field ${E_{\text{in}}}(x,y)$, where $r$ is the fiber core radius.

The simulation results with the same parameters as in the experiment are shown in Figs. 3(e)–3(h). Different colors correspond to different distances between the input beam and the fiber center: ${d_{\text{in}}} = {0.1}r$ (black line), ${0.3}r$ (blue line), ${0.5}r$ (green line), ${0.7}r$ (red line), and ${0.9}r$ (magenta line). Our semianalytical simulations also show that the shape and power ratio are determined by the position of the input beam. To obtain a good shape, we need to couple light closer to the core edge [red and magenta curves in Fig. 3(e)]. On the other hand, the central coupling provides higher power in the focus [black curve in Fig. 3(f)]. Therefore, to improve imaging contrast, input close to the fiber core center is preferable. The saturation value of the enhancement from the experiment is a bit higher than the simulation. It can be explained by the limited dynamic range of the camera and underestimated background. Another reason could be fiber bending that leads to mode coupling, which is neglected in the simulation.

We use our simulation model to investigate the trade-off between the shape and power ratio for different parameters: shape of the input beam, fiber length, fiber diameter, and working distance. The experimental and simulation results are summarized in Figs. 4(a) and 4(b), respectively; the detailed description of the simulated data is given in Appendix B. In Fig. 4, the roundness (which is calculated by $1 - \varepsilon$) and the power ratio, PR, averaged over 50 foci uniformly distributed over the fiber output and optimized by 314 segments on the DMD are plotted as a function of input position (${d_{\text{in}}}$). The solid lines show results for a 50 µm fiber core, fiber length of 30 cm, and plane-wave DMD illumination. Simulations for 1 m fiber length show exactly the same results as for fiber length of 30 cm. The dashed line in Fig. 4(b) shows the simulation results for a Gaussian beam on the DMD. Experiments and simulations demonstrate that the roundness improves and the power ratio reduces with ${d_{\text{in}}}$ closer to the edge. Simulations confirm that the trade-off between the round shape and the intensity of the foci is present for different shapes of the input beam, fiber lengths, and the core diameter, as presented in Appendix B, Fig. 6 for a MMF with 105 µm core. The simulation was also performed for imaging planes located at 100 and 200 µm away from the fiber output facet. The result is shown in Appendix B, Fig. 7. Similar to the previous results, we see that a higher intensity corresponds to a central input position, while coupling light closer to the edge helps to provide a better shape. Thus, the trade-off between aberration and intensity remains in force for imaging at a distance from the fiber facet.

 

Fig. 4. (a) Experimental and (b) simulated data for roundness, $1- \varepsilon$ (blue), and power ratio, PR (orange), averaged over 50 foci uniformly distributed on the fiber output facet and optimized by 314 DMD segments as a function of the input beam position (${d_{\text{in}}}$). The solid lines show results for the fiber core of 50 µm, fiber length of 30 cm (or 1 m) and plane wave on the DMD. Dashed lines show the results of simulation for a Gaussian beam on the DMD.

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Fig. 5. Ellipticity $\varepsilon$ (central column) and the power ratio PR (right column) of a generated focus on the fiber output facet as a function of the number of segments on the DMD ($N_{\rm seg}$) and the distance between the fiber center and the position of the optimized spot ($d_f$) for light coupled to the MMF at a distance about (a) ${{0.1r}}$, (b) ${{0.5r}},$ and (c) ${{0.9r}}$ as illustrated in the left column. White and red circles in the first column show the minimum and maximum sizes of the input beam after WFS, respectively. Yellow indicates good performance.

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To summarize, we have experimentally demonstrated the trade-off between the shape and the relative intensity of the focal spots generated on the output facet of a round-core MMF. We show that, in the case of a low number of controlled segments, coupling the input beam to the fiber center can be used to improve imaging contrast, while coupling to the fiber edge can be used to reduce aberrations. Input positions around ${{0.5 - 0.7}}r$ allow for good balance between focus contrast and shape. The aberrations can be corrected by postprocessing; however, it requires a complicated approach and an additional precalibration step since standard deconvolution algorithms work for a uniform point-spread function [12,19,20]. The semianalytical simulations confirm our experimental findings and reveal the existence of the trade-off for different experimental conditions. State-of-the-art MMF-based endoscopes have a FOV of only 50 µm in diameter. Our research shows how a larger FOV can be reached with a larger-core MMF while only controlling a subset of the fiber modes.

APPENDIX A: EXPERIMENTAL DATA OF ELLIPTICITY AND POWER RATIO

This appendix presents the measured ellipticity $\varepsilon$ and power ratio PR for three different input positions ${d_{\text{in}}}$ as a function of DMD segments ${N_{\text{seg}}}$ and distance between the fiber center and the optimized focal spot on the fiber output facet (${d_f}$).

The ellipticity and power ratio of 50 foci at output facet with ${N_{\text{seg}}}$ from $\pi /{{4}}\;{{\cdot}}\;{{1}}{{{0}}^2}$ to $\pi /{{4}}\;{{\cdot}}\;{{4}}{{{1}}^2}$ for three different input positions (${d_{\text{in}}} = {0.1}r$, ${0.5}r$, ${0.9}r$) are presented in Fig. 5. The first column illustrates the input fiber facet. The small black dots indicate the center positions of the input beam. The white and red circles show the minimum and maximum sizes of the input beam after WFS with ${N_{\text{seg}}} = \pi /{{4}}\;{{\cdot}}\;{{1}}{{{0}}^2}$ and ${N_{\text{seg}}} = \pi /{{4}}\;{{\cdot}}\;{{4}}{{{1}}^2}$ active segments on the DMD, respectively. The second column shows the ellipticity of the optimized foci, $\varepsilon$, as a function of the number of controlled segments on the DMD (${N_{\text{seg}}}$) and the distance between the fiber center and the optimized focal spot on the fiber output facet (${d_f}$). The ellipticity $\varepsilon$ is close to 0 when the generated focus has a nearly perfect shape and is about 1 when the focus is elongated. The third column represents the power ratio, PR, as a function of ${N_{\text{seg}}}$ and ${d_f}$. Note the color bars for the second and third columns are reversed. The yellow color indicates good performance: good shape and high intensity. We perform phase-only modulation and control only a single polarization, which results in the relatively low values for the PR.

When the input light is coupled to the fiber center, the shape of the optimized foci on the output facet depends on the number of controlled segments. For a low ${N_{\text{seg}}}$, the round shape can be achieved only within the ${d_f} = {{5}}\;\unicode{x00B5}{\rm m}$ region [see Fig. 5(a)]. Foci generated close to the edge have an elongated shape with $\varepsilon$ up to 0.8. The reason is that a different number of fiber modes effectively contributed to the generation of foci at different target positions. Compared with foci located at the edge of the core, a relatively smaller fraction of fiber modes contributes to the foci around the center of the output facet [8]. Thus, fewer segments should suffice to offer a diffraction-limited focal spot with high contrast at the center of the output facet. Similarly, to generate a focal spot near the edge of output facet, we need more segments. This explains the high ellipticity (blue area) on the top left corner in Fig. 5(a) middle column. However, the blue part shrinks if the input beam approaches the edge of the core. As shown in Fig. 5(c), a nearly perfect focus shape with low ellipticity can be achieved independent of the number of segments or positions of the optimized foci by an input beam closer to the edge of the MMF core. This is because the greater variety of fiber modes is addressed by such an input. As shown in Figs. 1(c) and 1(d), when ${d_{\text{in}}} = {0.1}r$, the output pattern without wavefront shaping shows elongated rings, while the unoptimized output for edge input positions shows a more complete speckle pattern with a finer and more homogeneous structure. Compared with the elongated speckles for the center input position, the more random, finer structure for the edge input position is a better initial condition for generating a focal spot. Clearly, to improve the shape of the foci for a low number of controlled segments, the edge input position should be used.

The power ratio in the optimized spot at the fiber output increases with the number of segments, as predicted by WFS theory [4], for any input position. However, in contrast with the focal shape, the maximal PR reduces when the input position approaches the fiber edge (see the third column of Fig. 5). The reason is that, for an input position closer to the fiber edge, more control power is lost, as shown in the first column of Fig. 5. When ${d_{\text{in}}} = {0.1}r$, almost all light is coupled to the fiber for different ${N_{\text{seg}}}$. When ${d_{\text{in}}} = {0.5}r$, a part of the light is lost for high ${N_{\text{seg}}}$. When ${d_{\rm in}} = {0.9}r$, we lose light even with low ${N_{\text{seg}}}$. Obviously, to improve PR for a low number of controlled segments, the center input position should be used.

The horizontal stripes in Fig. 5(b) originate from “blind spots” of WFS [1]. During WFS, the reference field is a speckle pattern. Because of the well-known exponential distribution of the intensity of fiber speckles, some positions have relatively low intensity where WFS fails. The power ratios fit to these failed foci will be lower than normal, but the ellipticities follow that of the original speckles.

APPENDIX B: SIMULATIONS OF WFS THROUGH AN MMF FOR DIFFERENT PARAMETERS

Here, we show the simulation results for different parameters: input beam shape, length, diameter, and imaging distance from the fiber facet. In the first set of simulations, we use our model to simulate WFS in an MMF with a core diameter of 50 µm and length of 30 cm. Instead of plane wave illumination on the DMD, we simulate a Gaussian beam with an FWHM of 10 cm. The results are shown in Figs. 6(a) and 6(b). There is no perceptible difference between the plane wave [Figs. 3(e)–3(h)] and Gaussian inputs.

 

Fig. 6. Semianalytical simulation of the ellipticity $\varepsilon$ and the PR averaged over 50 foci as a function of the number of segments for different parameters: Gaussian input beam, fiber with 1 m length, fiber with 105 µm diameter. Different colors correspond to different coupling distances between the input beam and the fiber center: $d_{\text{in}} = {{0.1r}}$ (black line), ${{0.3r}}$ (blue line), ${{0.5r}}$ (green line), ${{0.7r}}$ (red line), and ${{0.9r}}$ (magenta line).

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Fig. 7. Semianalytical simulation of the power ratio PR and $2w_{\min}$ averaged over 50 foci at a distance of 100 and 200 µm to the fiber facet as a function of $N_{\rm seg}$ for a fiber with 50 µm diameter and 30 cm length with a plane-wave input. Different colors correspond to different coupling distances between the input beam and the fiber center: $d_{\text{in}} = {{0.1r}}$ (black line), ${{0.3r}}$ (blue line), ${{0.5r}}$ (green line), ${{0.7r}}$ (red line), and $0.9r$ (magenta line). The horizontal black line at (b), (d) is the diffraction limit at the fiber facet.

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In the next set of simulations, we use our model to investigate the influence of fiber length and simulate WFS in an MMF with a core diameter of 50 µm, with different lengths and with a plane wave input. The results for 1 m MMF presented in Figs. 6(c) and 6(d) show the expected behavior.

In the next set of simulations, we use our model to investigate the influence of a fiber core diameter. We simulated WFS in an MMF with a length of 1 m, different core diameters, and plane wave as an input. The results for the MMF with a 105 µm core are presented in Figs. 6(e) and 6(f). Compared with 50 µm fiber diameter [presented in Figs. 3(e)–3(h)], 105 µm diameter fiber shows more pronounced dependence on the beam shape and power ratio from input beam positions (${d_{\text{in}}}$). The reason is that the MMF with a 105 µm core has four times as many guided modes as an MMF with a core of 50 µm.

In the final set of simulations, we check if the trade-off between aberration and intensity remains valid for focusing at a distance from the output fiber facet. We simulated WFS in an MMF with a core diameter of 50 µm, length of 30 cm, and plane-wave input. Foci have been optimized at a distance of 100 µm [see Figs. 7(a) and 7(b)] and 200 µm [see Figs. 7(c) and 7(d)] from the output fiber facet. The field-of-view expands according to the free-space propagation and the distance of target foci to the axis ${d_f}$ changes accordingly. The power ratio shows the same trend as before: the closer the input position to the center of fiber, the higher the intensity. The minimal beam waist (${{2}}{w_{{\min}}}$) decreases as ${d_{\text{in}}}$ goes to the fiber edge, which means the aberration caused by low ${N_{\text{seg}}}$ reduces. Therefore, we can conclude that the trade-off between aberration and intensity remains in place for focusing at a distance from the output MMF facet.

Funding

Nederlandse Organisatie voor Wetenschappelijk Onderzoek.

Acknowledgment

This work has been carried out within ARCNL, a public–private partnership among UvA, VU, NWO, and ASML, and was partially financed by “Toeslag voor Topconsortia voor Kennis en Innovatie (TKI)” from the Dutch Ministry of Economic Affairs and Climate Policy. We would like to acknowledge Marco Seynen for software support and thank Wei Li and Ksenia Abrashitova for careful reading and fruitful comments.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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