The concept of singularity is always illusive. When a photon moves through a spiral path, the central axis region of the cylinder becomes an undefined phase region. Moment a go, before starting a spiral rotation, the photon was travelling through the central axis. Now that it has left the linear path, we cannot tell the exact value of phase along this axis anymore, but we are certain that intensity is zero here. Phase singularity does not mean phase zero, it does not mean infinity, when an entity loses a value and we cannot tell what it is, then, it is singularity. Here, who carries phase singularity? Our reply would be “wavefront”. So wavefront carries a phase singularity. Phase singularity region has a shape, its linear here. The poynting vector precesses around the phase singularity. —a line of undefined phase (and zero intensity) along the optical path.
The pointing vector looks like a cone, and its circular diameter could be increased manifolds. The helicity (vorticity) of the vortex beam is characterized by its topological charge value ℓ, which signifies the number of 2π phase twists acquired by the wavefront’s azimuth along a path equal to one wavelength. Certainly the distance variation between light source and lens would show that in the wall, the light has turned from a ring to a dot and then turn to a ring again, as the separation increases. Increasing and decreasing of ℓ, or topological charge changes the diameter of the light ring.
For the vector vortex beam, there are two pointing vectors moving simultaneously. Since the ring acquires an additional twist, there are two poynting vectors, one that move along with the optical axis and another along the spiral path. For one angular momentum, the shape of singularity region is a line, for the two angular momenta, the shape of singularity region is a spiral.
For the tensor vortex beam, there is an additional twist in the spiral path of the vector vortex beam, we get the third singularity region. The shape of the third singularity region is a 2D area between the two dissimilar lines connecting the north and south pole of the sphere. Therefore, third angular momentum is a signularity that counts number of similar strips in a 2pi area. Therefore the concept of counting topological charge remains the same for all three angular momenta, always ℓ, phi and omega could acquires astronomically large values, online atomic spin which could have only two values.
Three types of singularity regions interact and they reshape singularity regions like coupled mathematical structure. h cut omega is for electromagnetic energy and the vortex energy is ℓxh cut. Sum of these two entergies tell that two different forms of electromagnetic waves could add directly.
The most beautiful thing about biological cavities is that they are not just dielectric resonator where the material resonates or the cavity resonator that traps, rather they are a new kind of resonator that shapes the transmitted and refracted energy beam along with the reflected one, the reflected ones or the optical vortices are only one modes of expression, there are transmitted vortex that are mechanical vortices and magnetic vortices are loops of fields they are invisible only seen with a magnetic paper. So, one of the greatest discoveries of biological materials is that it simultaneously produces three kinds of vortices, electrical in reflection mode, magnetic in reflectance mode and mechanical in transmittance mode.
When an ultrasonic pulse, containing, say, ten quasi-sinusoidal oscillations, is reflected in air from a rough surface, it is observed experimentally that the scattered wave train contains dislocations, which are closely analogous to those found in imperfect crystals (1).
(1) Nye, John Frederick and Michael V. Berry. “Dislocations in wave trains.” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 336 (1974): 165 – 190.
(2) Dutreix, C., Bellec, M., Delplace, P., & Mortessagne, F. (2021). Wavefront dislocations reveal the topology of quasi-1D photonic insulators. Nature communications, 12(1), 3571. https://doi.org/10.1038/s41467-021-23790-w