Super- and Anti-Principal-Modes in Multimode Waveguides

We introduce special states for light in multimode waveguides featuring strongly enhanced or reduced spectral correlations in the presence of strong mode coupling. Based on the experimentally measured multispectral transmission matrix of a multimode fiber, we generate a set of states that outperform the established “principal modes” in terms of the spectral stability of their output spatial field profiles. Inverting this concept also allows us to create states with a minimal spectral correlation width, whose output profiles are considerably more sensitive to a frequency change than typical input wave fronts. The resulting “super-principal-modes” and “anti-principal-modes” are made orthogonal to each other even in the presence of mode-dependent loss. By decomposing them in the principal-mode basis, we show that the super-principal-modes are formed via interference of principal modes with close delay times, whereas the anti-principal-modes are a superposition of principal modes with the most-different delay times available in the fiber. Such novel states are expected to have broad applications in fiber communication, imaging, and spectroscopy.

Popular Summary

Optical fibers, which transmit information as pulses of light through flexible transparent cables, are key components not only for long-distance communication but also for medical diagnostics, sensing applications, and emerging quantum technologies. One of the most vexing problems affecting all of these fields is that the multiple transverse modes—overlapping but distinct patterns of light—supported by “multimode fibers” get mixed during transmission. When using such a fiber for a phone call, for example, you would also be listening to all other calls going through the same fiber. Researchers predicted that multimode fibers could support special channels that are well protected from this crosstalk. Over the past two years, these “principal modes” have been realized in experiments. We present the discovery of light states that were not considered possible before—their protection against crosstalk exceeds the fundamental limit imposed by principal modes.

To generate these states, we first measure the complex transmission matrix of the multimode fiber and then apply an optimization algorithm on this data set. This algorithm is designed to maximize the frequency stability of states at the output end of the fiber as well as the states’ mutual orthogonality to prevent crosstalk.

The breakthrough we achieve in this way not only holds promise for numerous practical applications, but also the design principle we introduce for these “super-principal-modes” can be applied to synthesize other novel light states. As an example, we demonstrate that our optimization algorithm can be easily adapted to generate “anti-principal-modes” that have a minimal, rather than a maximal, frequency stability.

Figure 1

(a) Schematics of the experimental setup. The output from a frequency-tunable laser, injected by a SMF, is collimated by a lens ($C1$) and linearly polarized by a polarization beam splitter (PBS) before being split by a beam splitter (BS) into two paths. (i) The signal is spatially modulated by the SLM and then imaged onto the fiber facet of the MMF by a lens ($L$) and a micro-objective ($O$). (ii) The reference arm matches the signal arm’s path length by a pair of mirrors $M_1$ and $M_2$. The output from the MMF is collimated ($C2$) and polarized (PBS) before combining with the reference signal at the BS with a relative angle. The interference fringes are recorded by a CCD camera. (b) Experimentally measured autocorrelation function $C(\omega ,{\omega }_0)$ for the most stable PM (blue solid line). For comparison, we also show the measured average correlation of 20 random inputs (gray dashed line) with a narrower bandwidth. A super-PM eigenvector of $\rho (\mathrm{\Delta }\omega )$ with $\mathrm{\Delta }\omega =0.28\mathrm{THz}$ is evaluated using the measured transmission matrix and its correlation function (green dotted curve). The horizontal solid gray line indicates the threshold value $C(\omega ,{\omega }_0)=0.90$ that sets the correlation width. In panel (c), we plot the fluctuations of the measured correlation function of the PM shown in panel (b).

Figure 2

(a) Measured spectral correlation function $C(\omega ,{\omega }_0)$ for the output signals of the widest PM (blue solid line) and for the widest super-PM obtained from our bandwidth optimization (green dotted line). The weighting function $W(\omega )$ used in the cost function of Eq. (3) is shown as a purple dash-dotted line. The gray dashed line shows the spectral correlation of a random input. The horizontal solid gray line indicates the threshold value $C(\omega ,{\omega }_0)=0.90$ that sets the correlation width. (b) Spectral correlation width of super-PMs and PMs normalized by that of a random input. Our optimization scheme found 20 super-PMs that have a larger bandwidth than the widest PM. The orthogonality of the output-field patterns for the 20 widest PMs (c) and the 20 super-PMs (d). In the presence of loss, the PMs are not orthogonal but the super-PMs are.

Figure 3

(a) Decomposition of the widest super-PM in the basis of PMs, which are numbered from short to long delay times. The super-PM bundles together PMs with similar delay times. (b) Spectral correlation functions of the widest super-PM (green dotted line), the widest PM (blue solid line), and the widest super-PM with phases of constituent PMs randomized (green dashed line). The bandwidth enhancement of super-PMs is sensitive to the phases of constituent PMs, indicating that super-PMs are formed by interference of PMs.

Figure 4

Numerically determined anti-PM in a fiber without MDL. (a) Spectral correlation function of the narrowest anti-PM (red dotted line), which is significantly narrower than that of the random input (gray dashed line). The red dashed line is obtained by randomizing the phases of constituent PMs in the anti-PM. The phase randomization does not change the main peak, but it enhances the two side peaks. (b) The decomposition of this anti-PM in the PM basis shows two pronounced maxima at the extreme values of the involved PMs, indicating that anti-PMs are efficiently formed by combining the fastest and the slowest PMs. The data shown here result from the numerical simulation of a waveguide with $200\mu \mathrm{m}$ width, 0.22 numerical aperture, 1 meter length, and no loss.

Figure 5

Experimentally measured anti-PM in the same fiber as in Fig. 1. (a) The spectral correlation function of the narrowest anti-PM (red dotted line), which is notably narrower than that for a random input (gray dashed line). (b) Because of mode-dependent loss, the decomposition of this anti-PM in the PM basis does not give the bimodal distribution as in Fig. 4; instead, it gives a broad distribution over all available PMs. This result agrees with the numerical simulation that includes the MDL (see Ref. [75]).