Is there an explicit connection between rolling-shutter images of rotating propellers and interference patterns with optical vortices? The rolling shutter effect is a neat fact of the geometry of modern CCD cameras and how they interact with objects that move faster than the camera can handle, and it's been beautifully explained by a couple of youtube videos, one at SmarterEveryDay (with a cool behind-the-scenes video to back that up) and one at standupmaths. 
These videos provide what I think is a good deal of a breakthrough on how we think about what we can do with the rolling-shutter effect, and the techniques they pioneer let you take (or simulate) exceptionally clean pictures like this one:

This is a simulated rolling-shutter picture of a rotating four-propeller, but what I notice is that it is incredibly close to the interference pattern that you get if you superpose an optical vortex with a plane wave, which can look kind of like this:


So, given this uncanny resemblance: can this similarity be traced to some deeper analogy between the mathematical description of the two phenomena? If so, how?
 A: Yes indeed, there is a connection. And, as one can imagine, it is a geometrical one. 
To make the connection, one needs to define three-dimensional spaces for the two scenarios. For the optical vortices, the space is simply the normal three-dimensional space, represented by the $x$, $y$ and $z$ coordinates, and we'll assume that the beam propagates in the $z$-direction. The phase factor of the vortex beam, which defines the wavefronts (surfaces of constant phase), is given (in cylindrical coordinates) by
$$ \psi_{\rm vort} = \exp( i\ell\phi-i k z) , $$
where $\ell$ is the order of the vortex (azimuthal index). For the figures in the question above $\ell=4$.
For the rotating four-blade propeller, one replaces the $z$-coordinate with time. The number of blades takes over the role of the azimuthal index $\ell$. In the process, we assume that the thickness of the propeller in its $z$-direction is of such a nature that it does not play a significant role in the observed pattern.
We'll start by describing the situation for the optical vortex first. In three dimensions, the wavefront of the optical vortex beam describes a (higher order) helical surface - single helix for first order vortex; double helix for second order vortex; and so forth. This can be expressed by
$$ \phi-\frac{k z}{\ell}={\rm constant}. $$
To observe the interference pattern shown in the figure above, one needs to let the vortex beam interfere with another beam - a reference beam - typically a plane wave. The fringes will only appear as in the figure, if this plane wave is tilted with respect to the plane of observation, which is perpendicular to the propagation direction. (The tilt needs to be larger than the largest tilt in the helical surface.) Otherwise the fringes will form spirals, which we don't observe. So the plane wave would have the expression
$$ \psi_{\rm pw} = \exp[i (k_y y + k_z z)] . $$
So now the planar wavefronts of the plane wave slice through the helical wavefronts of the vortex beam. Every point where these two beams are in-phase produces constructive interference, leading to a high intensity. The image in the figure only shows the intensity pattern of this interference in a particular plane, at say $z=0$:
$$ {\rm intensity}_{z=0} = |\psi_{\rm vort}+\psi_{\rm pw}|^2 = \frac{1}{2} + \frac{1}{2} \cos( \ell\phi- k_y y) . $$
So the fringes are observed for 
$$ \ell\phi- k_y y = {\rm constant} . $$
Now for the propeller. Here the motion of the propeller also produces a helix in the three-dimensional space that we defined (where $z$ is replaced by time):
$$ \phi-\frac{t\omega}{\ell}={\rm constant}, $$ 
where $\omega$ is the rotation speed. Moreover, the rolling shutter defines planar surfaces in this three-dimensional space that are tilted with respect to a plane of constant time. 
$$ y + v t={\rm constant}, $$ 
where $v$ is the shutter speed. So this is exactly analogues to the plane wave. In the image, one would only see the red of the propeller if the shutter-opening coincided with the location of a blade of the propeller. This is analogues to the constructive interference between the helical wavefront and the planar wavefronts. Again we only see one frame of this movie. Hence, a slice of the three-dimensional space for a fixed value of the time, say at $t=0$ (taking care to match the dimensions):
$$ \ell\phi - \frac{\omega}{v} y = {\rm constant}. $$
As a result, the two scenarios have percisely the same geometrical construction, provided that we replace the spatial propagation direction ($z$-direction) for the optical vortex beam with the time-dimension in the case of the propeller.


EDIT (by Frobenius under  flippiefanus' permission)


An image of the rolling shutter effect for a rotating four-blade propeller was produced  (with GeoGebra software, using the tools ''Animation On'' and ''Trace On''). On this image (red color) they superimposed the fringes curves of the interference optical vortex-plane wave (blue color) according to the above equation $\:\ell\,\phi\!-\!k_{y}\,y=c\:$,  for $\:\ell=4=\text{number of blades}\:$, $\:k_{y}=-10.6\:$ and three values of the constant c $=0,-6.2,-12.8$. 


Note that the previous equation in cartesian $\:x,y−$coordinates is
$$
x=\dfrac{y}{\tan\left(\dfrac{k_{y}\,y\!+\!c}{\ell}\right)}
$$




