(Title stolen shamelessly from my postdoctoral advisor, who I assume will forgive me.)

As I’ve noted numerous times in previous posts, one of the fundamental properties that characterizes wave behavior (i.e. that makes a wave a wave) is wave interference.  When two or more waves combine, they produce local regions of higher brightness (constructive interference) and lower brightness (destructive interference), the latter involving a partial or complete “cancellation” of the wave amplitude.

Researchers have long noted that the regions of complete destructive interference of wavefields, where the brightness goes exactly to zero, have a somewhat regular geometric structure, and that the wavefield itself has unusual behavior in the neighborhood of these zeros.   In the 1970s this structure and behavior was rigorously described mathematically, and further research on this and related phenomena has become its own subfield of optics known as singular optics.  Singular optics has introduced a minor “paradigm shift” of sorts to theoretical optics, in which researchers have learned that the most interesting parts of a light wave are often those places where there is the least amount of light!

In this post we’ll discuss the basic ideas of singular optics; to begin, however, we must point out that most people have the wrong idea of what a “typical” interference pattern looks like!

The canonical demonstration of interference is Young’s double slit experiment, in which a spatially coherent light beam is incident upon a pair of holes (slits or small “pinholes”) in an opaque screen.  The light emanating from the two holes interferes as it propagates beyond, and the interference pattern can be observed on a second screen:

As can be seen from the second image above, the interference pattern observed on the screen is a set of bright vertical bands (constructive interference) separated by dark vertical bands (destructive interference).  It can be shown that there are vertical lines in the center of the dark vertical bands on which the light intensity is exactly zero: these lines are regions of complete destructive interference.  They are highlighted in red in the following figure:

If we move the measurement screen, we find that the interference pattern remains essentially unchanged: in three-dimensional space, therefore, the regions of complete destructive interference are vertically-oriented surfaces.

This experiment is is so familiar that we tend to think of “bright and dark bands” whenever the word “interference” is used; but does every interference experiment produce a similar result?

Let us make the simplest possible change to Young’s interferometer that we can imagine: we add a single extra hole, making it a “Young’s three-pinhole interferometer”!  The system and a typical interference pattern is illustrated below:

The intensity on the screen is a hexagonal pattern of bright spots interlaced with a hexagonal pattern of dark spots!  It can be shown that the centers of the darkest spots in the pattern are points of complete destructive interference; these zeros are illustrated as red dots for clarity below:

If we move the measurement screen, we find that the basic form of the interference pattern doesn’t change: in three-dimensional space, therefore, the regions of complete destructive interference are horizontally-oriented lines, and the pattern is fundamentally different than that of Young’s two pinhole interferometer.

What happens if we add even more pinholes?  The overall structure of the interference pattern will change, but the regions of complete destructive interference are still lines.  It seems that a “typical” interference pattern is one in which lines of zero intensity are present.  In technical terms, it is said that zero lines are a generic feature of wavefields; the zero surfaces of Young’s interference experiment are non-generic.

Under what circumstances does one see a “generic” interference pattern?  A very familiar example is the speckle pattern produced when a laser beam reflects off of a rough surface (picture from Wikipedia, brightness adjusted):

The countless dark spots are points of complete destructive interference, which are lines of zeros in three-dimensions.  The rough surface scatters light in all directions with different phases, acting in essence as a Young interferometer with a large number of randomly-placed “holes”.  Natural light does not produce such a speckle pattern because of its low spatial and temporal coherence.

The really intriguing aspect of the lines of complete destructive interference, however, is the behavior of the phase of the light wave in the neighborhood of these lines.  To describe this, we first briefly review what we mean by the phase of a wave, restricting ourselves to single-frequency (monochromatic) waves.

Let us start by considering monochromatic waves on a string, as might be generated by waving the end of the string up and down in a regular manner.  A snapshot of the string at an instant of time would look something like this, where the arrow indicates the overall direction of motion:

Because the “waving” of the string repeats itself regularly, we can characterize its state by an angle between 0 and 360°, with 0 representing the situation when the wave is at its peak:

This angle is what we refer to as the phase of the wave.  At the wave is at its highest; at the wave is at its lowest.  At the wave has no height but is moving upward; at the wave also has no height but is moving downward.  For future reference, we refer to those regions for which as the wavefronts of the wave.

We can also define a phase for an optical wave in three-dimensional space.  The simplest is a wave where the wavefronts are equally spaced planes in 3-D space; such a wave is known as a plane wave (here shown moving to the left):

A more complicated wave, such as light that has been distorted by passing through a piece of curved glass, will in general have curved wavefronts.  For instance, a top-down view of the behavior of the wavefronts on passing through a lens is illustrated crudely below:

Light is slowed down inside the glass of the lens, and the part of the wave that travels the farthest in the glass is delayed the longest, resulting in a net curvature of the wavefronts.

So what do the wavefronts look like around a generic zero line?  There are loosely two possibilities, depending on the orientation of the zero line with respect to the overall direction of the wave.

Let us first consider the case where the zero line lies in the same direction the wave is traveling.  If one looks at the wavefronts near the zero line, they actually form a helix around the central line:

If we were to make a movie of the motion of the wavefronts, we would find that they twist upwards in a right-handed sense, like a drill bit or the tip of a screw as it is screwed in.  This structure is referred to as a screw dislocation.

We can also consider the case where the zero line is horizontal to the direction of motion.  The wavefronts near the zero line have the following (rough) appearance:

If we look at a cross-section of this wave, we find that the zero line marks the beginning of a new wavefront which is wedged between two other wavefronts:

The blue dot indicates the location of the zero line.  This sort of structure is referred to as an edge dislocation.

There also can exist mixed dislocations, which are partly like a screw dislocation and partly like an edge dislocation; such structures occur when the zero line is intermediate between parallel and perpendicular to the direction of wave propagation.

The term dislocation is taken from the physics of crystal structures; the edge and screw dislocations of light are analogous to the types of defects that can occur in crystals.  An edge dislocation in a crystal is the case when an extra row of atoms is “wedged” into an otherwise regular crystal structure (source):

A screw dislocation in a crystal appears when the crystal is “torn” and the rows are shifted along the line of the tear.  The picture below shows an edge dislocation in the top row and a screw dislocation in the bottom row (source):

This analogy between dislocations in crystals and the structure of the phase near zero lines of light intensity was first noticed by Nye and Berry in their seminal 1974 paper¹ on the subject, “Dislocations in wave trains.”

In addition to studying the structure of the wavefronts, we may look at the behavior of the phase of the field near the zero lines.  Let us plot the phase in a plane perpendicular to the zero line; it turns out the image is the same for an edge or a screw dislocation:

The phase increases continuously as one traces a counterclockwise path around the zero point in the center.  All possible values of the phase come together at that central point, however, implying that there is no unique value of the phase along the zero line.  For this reason, the zero line is referred to as a phase singularity, and the general study of such objects is referred to as singular optics.

So the zero line can be referred to as a dislocation line, or a phase singularity: let’s give it one more name!  Referring again to the screw dislocation, we noted that, as time progresses, the wavefronts circulate or swirl around the zero line: in a very real sense, the light wave is “going in circles” around that central axis, very much like water swirling around a drain.  For this reason, these structures are also referred to as optical vortices.

These optical vortices are the generic, or typical, features of a wave interference pattern.  For instance, the following figure shows the intensity and phase of a speckle pattern produced by the interference of four randomly-oriented plane waves:

We can see numerous points of zero intensity, some of which are circled for clarity.  We can see the corresponding circles on the phase plot encircle structures that match the picture of the single vortex whose phase structure was depicted earlier.

Note the difference between the vortices in the solid circles and the vortices in the dashed circles.  If one follows a counterclockwise path along the dashed circle, the phase increases by 360°, but if one follows a counterclockwise path along a solid circle, the phase decreases by 360°.  Evidently there are two types of vortices; we dub the former right-handed vortices, and the latter left-handed vortices.  It is also possible to create vortices of higher order, where the phase increases or decreases by an integer multiple of 360° as one traces a counterclockwise path; these vortices are non-generic, and do not appear “naturally”.

These vortices have a number of remarkable properties.  First, it is to be noted that the generic vortices are stable under perturbations of the wavefield; that is, small modifications of wavefield do not make vortices “disappear”, or change their handedness.  Such modifications might involve transmission or reflection of the wavefield through a slightly distorted piece of glass, or the increase/decrease of the aperture through which a light wave is transmitted².  The location of the vortex in the wavefield and its orientation relative to the direction of motion might change, but the vortex itself will be fundamentally unchanged.

Second, we may associate a discrete “charge”, referred to as the topological charge, with any particular vortex.  This topological charge is simply the number of times the phase increases by 360° as one circles the vortex line; it can be a positive or negative value.

Third, related to the stability of vortices, this topological charge is conserved: the only way that vortices can disappear from a wavefield is if two vortices of opposite charge come together and annihilate.  Similarly, the only way new vortices can appear in a wavefield is in a “creation event” involving oppositely charged pairs.

This pair creation/annihilation is strikingly analogous to the creation/annihilation of fundamental particles in high-energy physics.  Like vortices, fundamental particles can only be annihilated if they are brought together with their corresponding antiparticle, e.g. an electron is annihilated when brought together with a positron.  No modern scientists have looked at vortex behavior as the basis of a unified theory of particle physics, but the idea has some history associated with it³.

For experimental studies and practical applications, to be discussed momentarily, it is relatively straightforward to “imprint” a laser beam with a single optical vortex at its center.  We have already noted above how a lens bends wavefronts by selectively slowing different parts of the wavefront; a vortex beam can be created by passing a laser beam through a spiral phase plate, an example of which is shown below (source):

This particular plate is designed to create a vortex in electromagnetic waves with a millimeter-scale wavelength; for optical waves, the plate must be much smaller (and more transparent), and typically can only be fabricated crudely with a “spiral staircase” shape.  If designed with the appropriate thickness, the wave passing through the thickest part of the plate ends up a full 360° out of phase with the wave that passes through the thinnest part.  The result is a so-called “donut” laser mode, which has a head-on intensity and phase pattern as follows:

The phase should look familiar; it is simply the phase pattern of a single vortex, as depicted earlier.

So why are we interested in such vortex laser beams, and optical vortices in general?  One of the most fascinating observations is that vortex beams with a net nonzero topological charge possess orbital angular momentum.

In a recent and very nice post, Chad at Uncertain Principles discussed an early experiment demonstrating that circularly polarized light possesses “spin” angular momentum.  This angular momentum can actually be used to spin objects — the light imparts a “twist” to the object.  As we have seen, an optical vortex also has a “twist” associated with its phase, and it can be shown that it possesses orbital angular momentum.  The difference between “spin” and “orbital” angular momentum can be understood by considering the motion of the Earth around the sun: the Earth’s rotation on its axis is its “spin” angular momentum and its orbit around the Sun is its “orbital” angular momentum.

This orbital angular momentum can also be used to rotate objects, in particular microscopic particles.  In a supplemental technique to optical tweezing, in which a focused light beam is used to trap and move particles, a vortex beam can be used as an optical spanner to rotate the particles.  The most clever application of this effect that I’ve seen to date was presented in a 2004 paper in Optics Express entitled, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays”⁴.  (The paper should be open access, and includes experimental videos.)  The paper describes a microscopic liquid pump driven by six vortices.  The vortex beams trap microscopic beads and force them to move in a circular path; the motion of the beads, in turn, forces liquid through the channel between the vortices:

I haven’t seen any follow-ups to this work in recent years, but the possibility of making microscopic light driven machines is clearly an intriguing one.

An understanding of vortices is also important for so-called “phase retrieval” problems.  The phase of a light field contains much important information for imaging and sensing applications, but visible light oscillates too rapidly for this phase to be directly measured.  Numerous computational techniques have been devised to estimate the phase from measurements of the intensity of the field, but these techniques are often thwarted by the presence of optical vortices (the phase structure of a field with many vortices ends up looking like a very confusing multilevel parking garage).

The discreteness of topological charge and the stability of optical vortices under perturbation has made a number of researchers suggest that vortices might be used as digital “bits” in a free-space optical communications system.  In principle, the use of vortices of different order would allow one to send multiple bits of information simultaneously in a communications channel, and the stability of the vortices could be used to transmit information without significant loss through atmospheric turbulence.  (I’ve done some work on optical vortices through turbulence with this application in mind⁵.)

Finally, we note that the study of singular optics provides a unifying framework for understanding many aspects of wave phenomena beyond the optical phase.  For instance, it has been known since the 1950s that vortices may also appear in the power flow of electromagnetic waves.  An example of this is from another paper of mine⁶ which studies the power flow of light in the immediate neighborhood of a slit in a metal plate:

The arrows indicate the direction of power flow, and the color indicates the amount of power flow.  At the points a, b, c and d it can clearly be seen that the power flow has a vortex behavior, and the center is a zero of power flow.

In optics, phase singularities can also be found in correlation functions and even in the description of the polarization of a light wave.   Singularities can also be found in other types of waves, such as acoustic waves, quantum mechanical wave functions, and even the periodic oscillations of the tides!  An early example of this was a paper by Proudman and Doodson, “The principal constituent of the tides of the North Sea,” Phil. Trans. Roy. Soc. A 224 (1924), 185-219, which shows clear vortices in the daily tides of the North Sea!  A modern illustration of these tides, showing a single vortex in the upper right corner where the tidal (phase) lines converge, is below (source):

Overall, singular optics gives researchers another way to look at and think about fields, understanding and characterizing them by the points where there is the least amount of light!

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¹J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336 (1974), 165-190.

² A classic paper demonstrating the creation/annihilation of vortices examined the structure of a focused wavefield as the aperture of the lens was changed; see G. P. Karman, M. W. Beijersbergen, A. van Duijl, and J. P. Woerdman, “Creation and annihilation of phase singularities in a focal field,” Opt. Lett. 22 (1997), 1503-1505.

³ In the mid-1800s, before there was any understanding of atomic structure, Lord Kelvin suggested that atoms might in fact be vortices in the light-carrying aether!  He was inspired by observations of vortices of smoke rings, and he would experimentally demonstrate their remarkable stability when he talked on the subject.  It is another fascinating “what-if” story to wonder what the history of atomic physics might have looked like if antiparticles had been discovered while the “vortex atom” model was still seriously considered.

⁴ Kosta Ladavac and David Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Exp. 12 (2004), 1144-1149.

⁵ G. Gbur and R.K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25 (2008), 225.

⁶ H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, “Creation and annihilation of phase singularities near a sub-wavelength slit”, Optics Express 11 (2003), 371.