Learning and avoiding disorder in multimode fibers
[M.W. Matthès, arxiv, 2010.14813 (2020)]
In the past 10+ years, numerous advances were made for endoscopic imaging, micromanipulation, or telecommunication applications with multimode fiber. The main limitation to real-life applications is the sensitivity to perturbations that sometimes causes the transmission property of the fiber to change in real-time. To address this issue, the authors (we) show that, even in the presence of strong perturbations, there exists a set of channels that are almost insensitive to perturbations. Interestingly, these channels can be found using only measurements from small perturbation leveraging the so-called generalized Wigner-Smith operator. This requires the measurement of the transmission matrix, which is done thanks to a new technique based on deep learning frameworks that compensate automatically for misalignments and aberrations, allowing fast and easy acquisitions.
Studying disorder in multimode fiber is not that easy. Unlike scattering media, speckle patterns are observed even when there is no disorder behavior. In particular, the transmission matrix of the system, that links the input field to the output one, when measured in the basis of diffraction-limited spots in the input and output facets, looks random. To assess accurately the transmission properties of the fiber, one has to measure the transmission matrix in the mode basis (i.e. the basis of the propagating modes of the fiber). It is supposed to be diagonal when there is no disorder, by definition of the propagating modes that are stable during the propagation. This is what we call the no-coupling or low-coupling regime: for small perturbations, the light stays in the same mode(s) it is injected in. When light injected in one mode statistically explores all the other modes with the same probability, i.e. in the strong coupling regime, some average properties can be predicted. However, from a few centimeters to a few kilometers, typical MMF systems are neither in the no coupling nor in the strong coupling regime; disorder strongly influences light propagation but some aspects of the ordered behavior survive. This intermediate regime has been little investigated due to the difficulty to experimentally measure the mode basis transmission matrix.
In the paper, the authors:
- Introduce a technique to quickly and painlessly measure the mode basis transmission matrix;
- Demonstrate the existence of channels robust to strong perturbations;
- Give an interpretation of this effect using analogies with scattering media.
Measurement of the mode basis transmission matrix
Ideally, one can measure the transmission matrix in the pixel basis \(\mathbf{H_{pix}\), i.e. the matrix that links the field on each pixel of a modulator in input to each pixel of a camera in output (Fig.1a) and then project it on the mode basis to obtain the mode basis matrix \(\mathbf{H_{modes}}\):
\begin{equation}
\mathbf{H_{modes}} = \mathbf{M_o}^\dagger . \mathbf{H_{pix}}. \mathbf{M_i} \,\,, \tag{1}
\label{eq:proj}
\end{equation}
where \(\mathbf{M_i}\) (resp. \(\mathbf{M_o}\)) represents the change of basis matrix between the input (resp. output) pixel basis and the mode basis of the fiber. If there is no aberration or misalignment, these matrices can be calculated by computing the spatial profiles of the fiber modes. In real-life, slight aberrations or misalignments modifies dramatically the results. Using Eq \ref{eq:proj}, we obtain the matrix depicted in Fig.1b for a short 30 cm straight graded-index fiber. It looks random!
To solve this problem, and avoid having to spend weeks calibrating the setup, we developed a numerical procedure to compensate for the imperfections automatically. The model is similar to an artificial neural network, except that each layer has a physical meaning. The keys parts are layers that represent aberrations represented by Zernike polynomials. Each layer is fully differentiable and has only one parameter to be learned, the strength of the aberration. It can then be used using a standard deep learning optimization procedure. The architecture is presented in Fig1.d. We train our network to maximize the energy in the mode basis, as we know that in a perfect situation, we inject all the light in the fiber modes. We obtain after optimization a mode basis transmission matrix like the one presented in Fig1.c, it is almost diagonal!
Figure 1. Principle of the experiment. a. Schematic of the experiment. b. and c. mode basis transmission matrix before and after the compensation of the aberrations. d. Schematic of the model architecture used for the compensation of the aberrations.
Deformation robust channels
To study the effect of perturbations, we introduce a transverse deformation using an actuator (see Fig.2b). For each deformation, we measure the transmission matrix. We show in Fig.2a and c the aspect of the mode basis transmission matrix without deformation and for a strong deformation. Qualitatively, strong deformations have the effect of progressively populating the off-diagonal elements of the TM while reducing the energy on the diagonal.
Figure 2. Effect of deformations.
To look for channels robust to perturbations, we study the eigenmodes of the generalized Wigner-Smith operator introduced in [P. Ambichl, Phys. Rev. Lett., 119 (2017)]:
\begin{equation}
\mathbf{Q}_{\Delta x} = -i\mathbf{H}_\text{modes}^{-1} .
\partial_{\Delta x} \mathbf{H}_\text{modes} \tag{2}
\label{eq:WS}
\end{equation}
We estimate this operator for small deformations and compute its eigenmodes, referred to as the deformation principal modes. We show in Fig3 the stability of all those channels compared to the injection of the fundamental mode and a random wavefront. For almost all the deformation principal modes, the output stays higly correlated, even for strong deformations, much stronger than the deformations used to evaluate the Wigner-Smith operator.
Figure 3. Stability of the deformation principal modes.
Investigation of the effect of deformations
It is quite surprising that the deformation principal modes, that are estimated using only the knowledge from small deformations, are still robust for much stronger deformations. To understand this effect, we show that the deformation of the transmission matrix can be estimated using only two components:
\begin{equation}
{\mathbf{\hat{H}}_\text{modes}(\Delta x_j)} = {\mathbf{H}_\text{modes}(\Delta x = 0) \left[ \alpha_j \mathbf{U}_1 + \beta_j \mathbf{U}_2 + \mathbb{I} \right]}
\tag{3}
\label{eq:sv_estimation}
\end{equation}
It means that knowing the unperturbed transmission matrix, using the components \(\mathbf{U}_1\) and \(\mathbf{U}_2\) and only two coefficients \(\alpha\) and \(\beta\), we can estimate the transmission matrix of the deformed fiber with good accuracy, even for strong deformations. We can give a qualitative interpretation of the two significant components. \(\mathbf{U}_1\) is close to identity, traducing the loss of energy in the diagonal compared to no deformation. It is equivalent to the decay of the ballistic light in the presence of a scattering environment in free space. The second vector \(\mathbf{U}_2\) shows a well defined symmetric pattern that corresponds to an energy conversion between modes with close-by radial and angular momenta. This is analogous to the conversion between ballistic and single scattered photons in scattering media. Coupling between modes further away it is the equivalent of multiple scattered photons in scattering media. However, strong mode coupling also comes with important losses due to coupling to non-guided modes that leak out of the fiber, leading to a low energy contribution of this effect.
Figure 4. The aspect of the main deformation components of the transmission matrix.