Relativistic definition of the phase of the electromagnetic field, involving two Lorentz invariants, based on the Riemann-Silberstein vector is adopted to extend our previous study [I. Bialynicki-Birula, Z. Bialynicka-Birula and C. Sliwa, Phys. Rev. A 61, 032110 (2000)] of the motion of vortex lines embedded in the solutions of wave equations from Schroedinger wave mechanics to Maxwell theory. It is shown that time evolution of vortex lines has universal features; in Maxwell theory it is very similar to that in Schroedinger wave mechanics. Connection with some early work on geometrodynamics is established. Simple examples of solutions of Maxwell equations with embedded vortex lines are given. Vortex lines in Laguerre-Gaussian beams are treated in some detail.

In this work, we search for signatures of gravitational millilensing in Gamma-ray bursts (GRB) in which the source-lens-observer geometry produces two images that manifest in the GRB light curve as superimposed peaks with identical temporal variability (or echoes), separated by the time delay between the two images. According to the sensitivity of our detection method, we consider millilensing events due to point mass lenses in the range of $10^5 - 10^7 M_{\odot}$ at lens redshift about half that of the GRB, with a time delay in the order of $10$ seconds. Current GRB observatories are capable of resolving and constraining this lensing scenario if the above conditions are met. We investigated the Fermi/GBM GRB archive from the year 2008 to 2020 using the autocorrelation technique and we found one millilensed GRB candidate out of 2137 GRBs searched, which we use to estimate the optical depth of millilensed GRBs by performing a Monte-Carlo simulation to find the efficiency of our detection method. Considering a point-mass model for the gravitational lens, where the lens is a supermassive black hole, we show that the density parameter of black holes ($\Omega_{BH}$) with mass $\approx10^6 M_\odot$ is about $0.007 \pm 0.004$. Our result is one order of magnitude larger compared to consist with previous work in a lower mass range ($10^2 - 10^3 M_{\odot}$), which gave a density parameter $\Omega_{BH} \approx 5\times 10^{-4}$, and recent work in the mass range of $10^2 - 10^7 M_{\odot}$ that reported $\Omega_{BH} \approx 4.6\times 10^{-4}$. The mass fraction of black holes in this mass range to the total mass of the universe would be $f\approx \Omega_{BH}/\Omega_M=0.027 \pm 0.016$.