High-fidelity spatial mode transmission through a 1-km-long multimode fiber via vectorial time reversal

The large number of spatial modes supported by standard multimode fibers is a promising platform for boosting the channel capacity of quantum and classical communications by orders of magnitude. However, the practical use of long multimode fibers is severely hampered by modal crosstalk and polarization mixing. To overcome these challenges, we develop and experimentally demonstrate a vectorial time reversal technique, which is accomplished by digitally pre-shaping the wavefront and polarization of the forward-propagating signal beam to be the phase conjugate of an auxiliary, backward-propagating probe beam. Here, we report an average modal fidelity above 80% for 210 Laguerre-Gauss and Hermite-Gauss modes by using vectorial time reversal over an unstabilized 1-km-long fiber. We also propose a practical and scalable spatial-mode-multiplexed quantum communication protocol over long multimode fibers to illustrate potential applications that can be enabled by our technique.

Mode-group excitation 5 km 8 mode groups [15] 17 Mode-group excitation 2.6 km 4 mode groups [16] Supplementary Table 1. Summary of methods for modal crosstalk suppression in standard multimode fibers. It can be seen that transfer matrix inversion has only been applied to short fibers that are less than 5-m long. This is because the characterization of a high-dimensional transfer matrix can take as long as hours. Since it is technically challenging to stabilize long fibers for hours, short fibers are used for experimental demonstrations. It can also be seen that scalar time reversal has only been used for short fibers. In Supplementary Note 6 we show that scalar time reversal cannot be used for a 1-km-long fiber due to the inevitable polarization mixing in long fibers. Mode-group excitation can be used to transmit mode groups over a long fiber. However, a standard MMF only supports a few mode groups, and thus this method is not comparable to our approach. HG: Hermite-Gauss.
LG: Laguerre-Gauss. LP: linearly polarized. grating is used to generate complex-amplitude spatial modes in the first diffraction order [17]. The probe beam generated by Bob is denoted by red lines and the signal beams generated by Alice is denoted by blue lines in Supplementary Fig. 1. Bob uses a polarizing beam displacer (MBDA10, Karl Lambrecht) to generate a horizontally polarized spatial mode, and the polarization can be adjusted by a subsequent half-wave plate (HWP). The generated probe beam is then coupled into a 1-km-long multimode fiber (MMF) by an aspheric lens L1 (C110TMD-B, Thorlabs). The spatial mode beam waist size in the MMF used in our experiment is w 0 = 5.06 µm [18]. The MMF is comprised of two 500-m-long fibers (Clearcurve OM3, Corning) that are spliced together. The scrambled probe beam received by Alice is collimated by an aspheric lens L2 (C230TMD-B, Thorlabs), and a subsequent Sagnac interferometer is used as a customized polarizing beam displacer to coherently separate the horizontal and vertical polarization components of the scrambled probe beam to two beams that propagate along the same direction but are displaced with respect to each other [19]. The Sagnac interferometer as a customized polarizing beam displacer (PBD) provides more flexibility than the commercially available polarizing beam displacer because the transverse separation between the two displaced beams can be tuned by adjusting the mirrors in the Sagnac interferometer. A 1-km-long SMF is used to provide a coherent reference light source to interfere with the scrambled probe beam. A digital camera (Camera 1, BFS-U3-16S2M-CS, FLIR) is used by Alice to measure the interference pattern and perform the off-axis holography [20]. In this way, Alice can measure the amplitude, phase, and polarization of the scrambled probe beam via a single-shot measurement. The time-reversed signal beam is generated by the SLM and then directed back to the MMF. More details about the detection and generation of vector beam can be found in [21], and the alignment procedure for time reversal can be found in [22]. Bob measures the unscrambled signal beam by another camera (Camera 2, BFS-U3-31S4M-C, FLIR) and performs the digital spatial mode decomposition to obtain the crosstalk matrix.

Supplementary Note 2 -Experimental setup
In our experiment, both Alice and Bob perform data processing for off-axis holography in MATLAB on a desktop computer the digital signal processing for Alice can even be avoided if the camera for off-axis holography and the SLM for generating the time-reversed signal beam are exactly placed at positions that are imaging planes with respect to each other [22]. When the positions of camera and SLM are perfectly aligned, we can directly imprint the digital interference pattern recorded by the camera onto the SLM without performing any digital signal processing. Instead of carefully aligning the position of the camera,

Supplementary Note 3 -Definition of the single mode index
The Laguerre-Gauss and Hermite-Gauss mode are typically denoted by two indices, which are (p, ) for Laguerre-Gauss mode and (m,n) for Hermite-Gauss mode. For the convenience of presentation we use a single mode index j = 0, 1, 2, · · · to denote the modes. Here we refer to the convention of Zernike polynomials [26] and adopt the following definitions. For a specific mode index j, the mode group number N can be calculated as N = ceil((−3 +

Supplementary Note 4 -Crosstalk matrix and the normalized modal fidelity
The 210×210 unnormalized crosstalk matrix of the scrambled probe beams received by Alice when Bob transmits standard Laguerre-Gauss and Hermite-Gauss modes (i.e. in the absence of vectorial time reversal) are shown in Supplementary Fig. 3 and Supplementary Fig. 4. Due to the strong spatial mode scrambling, the average unnormalized modal fidelity in this case is ≈1% for both Laguerre-Gauss modes and Hermite-Gaussian modes. The 210×210 unnormalized crosstalk matrix in the presence of vectorial time reversal are presented in Supplementary Fig. 5 and Supplementary Fig. 6. The crosstalk matrix is calculated as follows. The received vectorial mode is denoted as |φ = |ψ 1 , H + |ψ 2 , V , where H and V represent the horizontal and vertical polarization state, ψ 1 and ψ 2 represent the corresponding spatial mode, and |φ is normalized such that φ |φ = 1. Each element in the crosstalk matrix is the squared inner product between the received mode |φ and a particular Laguerre-Gauss or Hermite-Gauss mode. As an example, for a horizontally polarized Laguerre-Gauss mode |LG j , H , the squared inner product can be expressed as | φ |LG j , H | 2 , where 0 ≤ j ≤ 104 is the single mode index. For a Laguerre-Gauss mode, it is normalized such that LG j , H|LG j , H = 1 and LG j , V|LG j , V = 1. Similar normalization is also applied to Hermite-Gauss modes. It should be noted that since the spatial modes with 0 ≤ j ≤ 104 do not form a complete basis set, the sum of each row in the crosstalk matrix is less than unity, i.e. ∑ 104 However, we didn't normalize the crosstalk matrix and directly present the unnormalized modal fidelity in the manuscript. After normalizing the sum of each row of the crosstalk matrix to unity, the modal fidelity can be higher as shown in Supplementary Fig. 7. It can be seen that the average of normalized modal fidelity is 91.5% for Laguerre-Gauss modes and 89.3% for Hermite-Gauss modes.
This is because the crosstalk due to coupling to higher-order modes ( j ≥ 105) is discarded. This is permissible in an experiment because the higher-order modes can in principle be separated by a mode sorter in practical applications and thus does not contribute to the crosstalk.
To aid readers for analyzing the crosstalk matrix, we also present the crosstalk distributions in the following four categories.
Here we assume the mode of interest is a horizontally polarized Laguerre-Gauss mode |LG j , H and the received time-reversed mode is |φ = |ψ 1 , H + |ψ 2 , V as an example. The four crosstalk categories are (1) crosstalk from coupling to modes inside the crosstalk matrix with degenerate polarization, which can be expressed as C 1 = ∑ k | φ |LG k , H | 2 for 0 ≤ k ≤ 104 and k = j.
(2) crosstalk from coupling to modes inside the crosstalk matrix with orthogonal polarization, which can be expressed as (3) crosstalk from coupling to modes outside the crosstalk matrix with degenerate polarization, which can be expressed as (4) crosstalk from coupling to modes outside the crosstalk matrix with orthogonal polarization, which can be expressed as These results are shown in Supplementary Fig. 8. The normalized modal fidelity is calculated by dividing the corresponding diagonal element (i.e. the unnormalized modal fidelity) of the crosstalk matrices (shown in Supplementary Fig. 5 for Laguerre-Gauss modes and Supplementary Fig. 6 for Hermite-Gauss modes) by the sum of elements in each column.

Supplementary Note 5 -Experimental generation fidelity of SLM
In the manuscript we show that the average modal fidelity without normalization is 85.6% for Laguerre-Gauss modes and 82.6% for Hermite-Gauss modes. The main reason for imperfect modal fidelity is attributed to the imperfect spatial mode generation fidelity of the SLM. In the experiment, Alice uses a SLM to generate the phase conjugate of the scrambled probe beam, and Bob uses a SLM to generate the Laguerre-Gauss or Hermite-Gauss modes. We characterize the fidelity of the generated spatial modes and horizontal polarization component of scrambled probe beams using off-axis holography, with the results shown in Supplementary Fig. 9. We also measure the complex field of the generated signal beam |ψ s , and calculate the overlap integral with the scrambled probe beam |ψ p to get the signal beam fidelity as | ψ s |ψ p | 2 . We take the product of the probe spatial modal fidelity and the signal beam fidelity as the experimental generation fidelity, which is a simple estimate of time-reversed modal fidelity. In Supplementary Fig. 9(a-d) we show the ideal and generated spatial mode and time-reversed signal beam for HG (2,5) and LG(1,4) modes. In Supplementary Fig. 2(c,d) in the manuscript we present the experimental generation fidelity for individual Laguerre-Gauss and Hermite-Gauss modes, which shows reasonable agreement with the fidelity of time-reversed modes. In general, the modal fidelity for Laguerre-Gauss modes is slightly higher than that of Hermite-Gauss modes, and the Laguerre-Gauss modes with = 0 mode have a higher fidelity than those with = 0. In addition, the signal beam generation has a slightly lower fidelity than the well-defined Laguerre-Gauss and Hermite-Gauss modes. The maximum experimental generation fidelity we measured is 90.8% for Laguerre-Gauss modes and 88.4% for Hermite-Gauss modes, and the average experimental generation fidelity is 85.8% for Laguerre-Gauss modes and 83.7% for Hermite-Gauss modes. Thus, by improving the experimental generation fidelity through using a higher-quality SLM, the performance of vectorial time reversal can be further enhanced. LG(1,0) LG (3,2) HG(1,1) HG (4,4) Supplementary Figure 10. (a, b) The measured modal fidelity for unscrambled Laguerre-Gauss modes and Hermite-Gauss modes received by Bob when Alice performs scalar time reversal (i.e. using one polarization only).

Supplementary Note 7 -Polarization crosstalk matrix
In the manuscript we show the normalized polarization crosstalk matrix within the individual spatial mode subspace (see Fig. 3a in the manuscript). Here we continue to use LG j to show how the calculation is performed. For the received timereversed mode |φ , we first calculate the unnormalized squared inner product for horizontally (H), vertically (V), diagonally (D), and anti-diagonally (A) polarized LG j mode as F u H = | φ |LG j , H | 2 , F u V = | φ |LG j , V | 2 , F u D = | φ |LG j , D | 2 , and F u A = | φ |LG j , A | 2 , where |D = (|H + |V )/ √ 2 and |A = (|H − |V )/ √ 2. Then the normalized fidelity can be calculated . These four numbers form one row of the crosstalk matrix, and the rest of the matrix can be similarly calculated. The normalized polarization crosstalk matrix for Hermite-Gauss modes is also calculated in this way. Since the calculation is performed for a (4×4)-dimensional state space, the crosstalk is small. It should be noted that such calculation is permissible because in an experiment one can use a polarization-independent spatial mode sorter and a polarizing beamsplitter to access the (4×4)-dimensional state space, and polarization crosstalk that couples to other spatial modes can be discarded.