and, as a consequence, its phase becomes
Such points in space are referred to as
phase singularities. Even more interesting than the phase singularities themselves is what occurs in their vicinity:
The phase of the wavefield winds continuously around the singular point, taking
all possible values. This is analogous to
the direction that the local wind velocity
vector takes around the core of a cyclone.
This characteristic phase circulation is
the hallmark of a vortex and the source
of its unique physical properties.
An optical wavefield can be thought
of as the sum of many ideal plane waves
that propagate in various directions and
possess different relative phases. The
occurrence of vortices is then the outcome of the simultaneous superposition
of all these component waves at singular
points. Vortices form when the relative
phases of the constituent plane waves of a
wavefield interfere destructively, resulting
in points of zero field amplitude.
Vortices are thus points of darkness
embedded within the spatial domain of
the wavefield. When looked at in three
dimensions, however, they are revealed as
continuous threads or filaments that follow elaborate trajectories. Vortex threads
can exist as closed loops in space, formed
when two counter-rotating vortices are
first produced, then move apart from
each other and eventually annihilate
themselves by coming close together
again. Loops can also self-intersect into
convoluted structures resembling knots
as they evolve and even link up with
other vortex filaments.
Vortices are far more than a mathematical curiosity. In fact, the importance
of electromagnetic phase singularities
has been highlighted in many scientific
and technological contexts. As early as
the 1930s, P.A.M. Dirac explained their
origin and stated one of their physical
roles in his famous paper relating magnetic monopoles and the quantization
of electrical charge.
In medicine, for example, ventricular
fibrillation—a typically fatal cardiac
arrhythmia—is most often sustained
by a polarization vortex line within the
The phase circulation around a vortex
resembles the helical staircases at the
Sagrada Familia church in Barcelona. The
phase singularity is represented here by the
hole at the center of the stairwell.
heart’s tissue. In other words, a collection of phase singularities, which are
organized as a vortex filament in the
electromagnetic field that drives the
cardiac muscles to act synchronously, is
the mechanism behind suddenly chaotic
heart contractions, which in turn lead to
rapid cardiac arrest.
Under less drastic circumstances,
vortices are also well known in signal
processing for their central role in the
extraction of meaningful images out
of data from synthetic aperture radar,
electron microscopy, adaptive optics
and a number of other advanced imaging techniques that make use of phase
unwrapping, phase retrieval or numerical
Fourier transform algorithms.
The incidence of vortices in all these
cases is detrimental to the information
contained in the optical measurements.
This is because, when vortices are present, the measured phase cannot be determined in a unique way. Vortices result
in poor image reconstruction, since the
(Left) Vortices are represented by the dark holes in the region of lower field amplitude. A
line of vortices extends infinitely and exhibits alternating rotating directions. (Right, top)
In a Mathieu beam of order 3, the same number of vortices forms a line in the region
surrounded by the beam’s elliptical rings. (Right, bottom) However, in a Bessel beam of
the same order, a single vortex of charge 3 is located at the optical axis.