Upper bounds on focusing light through multimode fibers

TL;DR

This paper derives a general theory connecting phase-only wavefront shaping performance in multimode fibers to a participation ratio $R$ and a phase-error coefficient $\Phi$, predicting an enhancement bound $\eta_m=\alpha R_m \Phi_m (N-1)+1$. It shows that phase-only modulation in the Fourier basis yields $R\approx\pi/4$, leading to $\eta \approx (\pi/4)(N-1)+1$, while phase errors degrade performance via $\Phi$, with Hadamard-based TM measurements maintaining $\Phi\approx1$ even at low photon budgets. Experimentally, using noise-tolerant Hadamard TM measurements, the authors achieve enhancement factors near the theoretical limit (e.g., $\eta\approx5{,}000$ for $N\approx8{,}000$), validating the framework and its predictive power. The results provide a practical benchmark for phase-only wavefront shaping in MMFs and have broad implications for fiber-based imaging, communications, and high-power applications, offering a path to approaching fundamental limits in real-world systems.

Abstract

Wavefront shaping enables precise control of light propagation through multimode fibers (MMFs), facilitating diffraction-limited focusing for applications such as high-resolution single-fiber imaging and high-power fiber amplifiers. While the theoretical intensity enhancement at the focal point is dictated by the number of input degrees of freedom, practical constraints-such as phase-only modulation and experimental noise-impose significant limitations. Despite its importance, the upper bounds of enhancement under these constraints remain largely unexplored. In this work, we establish a theoretical framework to predict the fundamental limits of intensity enhancement with phase-only modulation in the presence of noise-induced phase errors, and we experimentally demonstrate wavefront shaping that approaches these limits. Our experimental results confirm an enhancement factor of 5000 in a large-core MMF, approaching the theoretical upper bound, enabled by noise-tolerant wavefront shaping. These findings provide key insights into the limits of phase-only control in MMFs, with profound implications for single-fiber imaging, optical communication, high-power broad-area fiber amplification, and beyond.

Figures (14)

• Figure 1: Wavefront shaping setup and the results are shown. (a) Experimental setup: The spatial light modulator (SLM) modulates the laser beam on the multimode fiber's proximal end and focuses on the distal end. P: linear polarizer; HWP: half-wave plate; BS: beam splitter; M: mirror; ${\rm MO}_1$ and ${\rm MO}_2$: microscope objectives; MMF: multimode fiber; NA: numerical aperture; CCD: charge-coupled device. (b) An experimental image of the speckle formation at the distal end of the MMF is shown when a random wavefront is incident on the proximal end. The interference of the waves propagating through various optical modes in the MMF results in random intensity fluctuations, giving rise to the granular appearance of speckle patterns. Here, the fiber radius is $a = 100$ µ m. (c) Experimental image of the distal end of the fiber when light is focused by wavefront shaping on the canonical (SLM pixel) basis and (d) on the Hadamard basis with $N = 8,000$. The scale bar indicates the intensity across the distal end as observed on the CCD camera and is normalized to the highest count on the image. (e) The mean enhancement factor $\eta$ averaged over azimuthal $\theta$ positions versus the normalized radial distance $r/a$ at the fiber distal end for the number of degrees of freedom $N = 8,000$. The blue solid line represents the upper limits of the enhancement factor with full-field modulation at the input equal to $\eta = N$. The violet solid line represents the upper bounds of the enhancement factor with perfect phase-only modulation when the SLM is placed on the Fourier plane of the fiber proximal end. The black and red solid lines represent the experimental enhancement factors with wavefront shaping on the Hadamard and canonical (SLM pixel) basis. The enhancement factor is higher with wavefront shaping on the Hadamard basis.
• Figure 2: (a) The mean participation ratio $R$, averaged over the core radius $a$ and the number of degrees of freedom $N$, versus the normalized radial distance $r/a$ is illustrated. The experimental $R$ (solid red line) is consistent with the numerical $R$ (solid blue line) in the Fourier basis, both following a trend close to $\pi/4$ (dashed black line) and showing no dependence on the radial distance. However, we observe a strong dependence of $R$ on the radial distance when $R$ is computed with phase-only modulation on the MMF fiber mode basis (solid black line). (b) The mean participation ratio $R$, averaged over the core radius $a$ and the radial distance $r$ at the fiber distal end, is shown with respect to the normalized number of degrees of freedom $N/M$. The experimental $R$ (red line) agrees with the numerical $R$ (solid blue line) and remains invariant with $N/M$, maintaining a value near $\pi/4$ (dashed black line). (c) The mean enhancement factor $\eta$, averaged over the azimuthal position $\theta$ and radial distances $r$ at the fiber distal end, is shown with respect to the number of degrees of freedom $N$ = 172, 484, 952, and 2,032. The experimental enhancement factor $\eta$ in the Hadamard basis (red solid line) closely follows the theoretical prediction (black solid line) and is notably higher than the experimental $\eta$ in the canonical basis (blue solid line).
• Figure 3: Phase-error measurements in wavefront shaping experiments are shown. (a) A conceptual sketch of two independently measured phase maps, $\phi_{mn}^{(1)}$ and $\phi_{mn}^{(2)}$ for the same input $n$. The phase difference, $\delta \phi_{mn}' = \arg \left(e^{i(\phi_{mn}^{(2)} - \phi_{mn}^{(1)})} \right)$, represents the measured phase error. (b) Standard deviation $\sigma\mathrm{_{\delta \phi^\prime}}$ of the Gaussian-fitted phase-error distributions as a function of the number of degrees of freedom $N$. In the canonical basis, $\sigma\mathrm{_{\delta \phi^\prime}}$ increases with $N$, indicating higher phase errors. In contrast, the Hadamard basis maintains a consistently low $\sigma\mathrm{_{\delta \phi^\prime}}$, suggesting greater robustness to phase errors. (c, d) Probability density functions of the phase errors $P(\delta \phi_{mn}')$ for the canonical and Hadamard bases, respectively, as a function of $N$. The Gaussian-fitted curves (red) show that in the canonical basis, the phase-error distribution broadens significantly with increasing $N$. In contrast, (d) shows that the Hadamard basis maintains a sharply peaked distribution around zero, indicating minimal phase errors.
• Figure 4: The impact of phase errors on the enhancement factor is illustrated. (a) Schematic representation of the phase-error coefficient $\Phi$ for the Hadamard (red) and canonical (blue) bases, where $\delta\phi$ represents phase errors. The dashed lines indicate how phase errors accumulate differently in the two bases, with the Hadamard basis maintaining $\Phi \approx 1$ due to a reduced cumulative effect of phase errors. (b) The phase-error coefficient $\Phi$ as a function of the number of degrees of freedom $N$ is shown. On the canonical basis, $\Phi$ decreases with $N$ due to a decreasing signal-to-noise ratio (SNR), which reduces interferometric visibility. In contrast, the Hadamard basis maintains $\Phi \approx 1$ due to a balanced signal-to-reference ratio. (c) The phase-error coefficient $\Phi$ for $N = 8000$, plotted as a function of the normalized photon budget. The Hadamard basis maintains a consistently high $\Phi$ compared to the canonical basis, reflecting its improved SNR under the same noise conditions. The normalized photon budget (ranging from 0 to 1) is defined so that a value of 1 corresponds to a maximum photon budget of approximately 214 mean counts on the CCD.
• Figure 5: The mean enhancement factor $\eta$, averaged over the azimuthal $\theta$ position and radial distance $r$ at the fiber's distal end, as a function of the number of degrees of freedom $N$ is shown. Solid blue and red lines show the theoretical enhancement factors, while dashed lines represent experimental measurements for wavefront shaping on the canonical and Hadamard bases, respectively. Error bars indicate the standard deviation of enhancement factors measured at different focal positions on the fiber’s distal end.