Universal Crosstalk of Twisted Light in Random Media

Structured light offers wider bandwidths and higher security for communication. However, propagation through complex random media, such as the Earth’s atmosphere, typically induces intermodal crosstalk. We show numerically and experimentally that coupling of photonic orbital angular momentum modes is governed by a universal function of a single parameter: the ratio between the random medium’s and the beam’s transverse correlation lengths, even in the regime of pronounced intensity fluctuations.

Figure 1

(a) Incident ${\mathrm{LG}}_{p=0}^{\ell =3}$ beam with transverse correlation length $\xi$ propagating through a random medium represented by a phase screen with correlation length $r_0$. (b) Experimental implementation of the split-step method via spatial light modulators (SLMs). (c) Qualitative illustration of crosstalk $C_{\ell ,{\ell }^{\prime }}$ of an incident ${\mathrm{LG}}_{p=0}^{\ell =3}$ mode, after propagation through a random medium to $z=L$, and of the vacuum-propagated ${\mathrm{LG}}_{p=0}^{{\ell }^{\prime }}(z=L)$ modes, for increasing distortion strength $w=w_0/r_0$ from top to bottom. Original and opposed OAM, i.e., $\pm \ell$, are indicated by the solid and dashed black lines, respectively.

Figure 2

Intensity-weighted transverse phase profiles of an incident ${\mathrm{LG}}_{p=0}^{\ell =5}$ mode, after propagation through Kolmogorov turbulence (top), or through Gaussian noise (bottom), for three different distortion strengths $w=1/2$, 1, 2, corresponding to ${\sigma }_R^2=0.24$, 0.76, 2.42 from left to right.

Figure 3

Numerically obtained average (2500 realizations) crosstalk matrices in moderate ($w=1$) Kolmogorov turbulence (a) and Gaussian noise (b). The crosstalk matrices (top) are obtained when projecting the image ${\tilde{u}}_{\ell }(L)$ of the incident mode ${\mathrm{LG}}_{p=0}^{\ell }$, onto vacuum-propagated modes $u_{{\ell }^{\prime }}(L)={\mathrm{LG}}_{p=0}^{{\ell }^{\prime }}(z=L)$ in the middle columns, according to Eq. (1). Left and right columns represent the beam’s phase and intensity profiles’ modifications quantified by substitution of $\arg [{\mathrm{LG}}_{p=0}^{{\ell }^{\prime }}(z=L)]$ and $|{\mathrm{LG}}_{p=0}^{{\ell }^{\prime }}|$ for $u_{{\ell }^{\prime }}(L)$ in Eq. (1), respectively. Rows $C_{\ell =5,{\ell }^{\prime }}$ (bottom) depict numerical (orange curve, error bands give 1 standard deviation) as well as experimental results (blue bars, 60 realizations). The crosstalk matrices are normalized such that ${\sum }_{\ell ,{\ell }^{\prime }=-10}^{10}{C}_{\ell ,{\ell }^{\prime }}=1$.

Figure 4

Ratio of average (2500 realizations) crosstalk between two OAM-opposed (i.e., $\ell$ and $-\ell$) modes propagated through numerically simulated Kolmogorov turbulence (a) and Gaussian noise (b) versus the ratio of the medium’s and beam’s transverse coherence lengths for a range of azimuthal indices $|\ell |$. The incident ${\mathrm{LG}}_{p=0}^{\ell }$ were projected onto ${\mathrm{LG}}_{p=0}^{\mp \ell }(z=L)$ at the receiver side. The emerging universal curve is fitted with a Gaussian (black curves) with parameters in [58]. Circles with error bars in (a) represent experimentally measured average (60 realizations) crosstalk in Kolmogorov turbulence. Insets illustrate rows at $\ell =5$ of corresponding crosstalk matrices (same axes as in Fig. 3). Error bands or bars give 1 standard deviation.