Spatiotemporal Control of Light Transmission through a Multimode Fiber with Strong Mode Coupling

We experimentally generate and characterize eigenstates of the Wigner-Smith time-delay matrix, called principal modes, in a multimode fiber with strong mode coupling. The unique spectral and temporal properties of principal modes enable global control of temporal dynamics of optical pulses transmitted through the fiber, despite random mode mixing. Our analysis reveals that well-defined delay times of the eigenstates are formed by multipath interference, which can be effectively manipulated by spatial degrees of freedom of input wave fronts. This study is essential to controlling dynamics of wave scattering, paving the way for coherent control of pulse propagation through complex media.

Figure 1

(a) Schematic of the experimental setup. The continuous-wave output of a tunable laser is collimated at C1, and split into two arms by a beam splitter (BS1). Light in one arm is modulated by the SLM and imaged to the fiber facet by a lens (L1) and an objective (O). The output field from the fiber is collected by a lens (C2) that one focal length away from the fiber facet and combined with light in the other arm at a second beam splitter (BS2). By offsetting BS2 to introduce a phase tilt between the two wave fronts, interference fringes are formed. From the interferogram recorded by the camera (CCD) at the back focal plane of C2, the output field is extracted in momentum space. The mirrors (M1, M2) are used to match the path length of the two arms. The MMF has $50\mu \mathrm{m}$ core diameter and 0.22 numerical aperture. (b) Amplitude of the measured transmission matrix at $\lambda =1550\mathrm{nm}$ ($\omega =1219\mathrm{THz}$). (c) Amplitude profile of a PM at the output. (d) Decomposition of the PM in (c) by linearly polarized modes.

Figure 2

(a) Output field amplitude for a PM input with short delay time at ${\omega }_0=1219\mathrm{THz}$ (top row), or a random superposition of linearly polarized modes (bottom row). The input frequency is $\omega -{\omega }_0=-157\mathrm{GHz}$ (left), 0 (middle), and 157 GHz (right). The output fields for the PM input are similar while those for random input are totally different. (b) Spectral correlation function $C(\mathrm{\Delta }\omega )$ of the output field, experimentally measured for a PM (red solid), and calculated from the transmission matrix and input wave front of the same PM (green dotted). For comparison, $C(\mathrm{\Delta }\omega )$ for a random input is also shown (blue dashed). $C(\mathrm{\Delta }\omega )$ is normalized to one at $\mathrm{\Delta }\omega =0$. Its value decreases to 0.9 at $\mathrm{\Delta }\omega =338\mathrm{GHz}$ for the PM and 173 GHz for the random input. The agreement between the red and green curves illustrates the accuracy of the measurement.

Figure 3

(a),(b) Temporal variation of the output field amplitude in three spatial channels (three speckles grains) when an optical pulse is spatially launched into a random superposition of modes (a) or a PM at ${\omega }_0=194\mathrm{THz}$ (b). The output field is recorded in a frequency range of 400 GHz with a step size of 2.5 GHz. The Fourier transform is performed to obtain the temporal evolution of the field. The horizontal axis is the relative delay time, obtained by subtracting the average delay time for random inputs. The temporal traces of individual spatial channels are totally different for the random input, but nearly identical for the PM input. (c) Spatially integrated intensity of the input (black dotted) and the output pulses when a Gaussian pulse is injected to the MMF with random spatial profile (blue dashed) or with the profile of a PM (red solid).

Figure 4

(a) Measured spectral correlation width $\mathrm{\Delta }{\omega }_c$ of PMs with different delay times. $\mathrm{\Delta }{\omega }_c$, given by $|C(\mathrm{\Delta }{\omega }_c)|=0.9|C(0)|$, is normalized by the average bandwidth of random inputs. The shortest delay time is set to 0. (b) Intensity distribution over the path-length $U(l)$ for three measured PMs with the delay $\text{time}=0$, 0.06, 0.12 ns. The relative path-length $l$ is obtained by subtracting the average path-length of random inputs. $U(l)$ is normalized: $\int U(l)dl=1$. (c) Calculated spectral correlation width of the PMs with (red circles) and without (black crosses) mode-dependent loss. (d) Intensity distribution over the path-length for two PMs with the delay $\text{time}=0$ (blue), 0.12 ns (black), in the presence (dashed) or absence (solid) of mode-dependent loss. The mode-dependent loss narrows (broadens) the path-length distribution for the fast (slow) principle mode, thereby increasing (reducing) the spectral correlation width.