## Twisting light into a Möbius strip

Some months ago, I wrote a post introducing the subfield of optics known as singular optics.  Singular optics is concerned with the behavior of wavefields in the neighborhood of regions where the intensity of the wave is zero, and the “phase” of the wave is therefore singular.  The zeros typically take the form of lines in three-dimensional space, and surfaces of constant phase often form a spiral around this line, and circulate around it as time evolves:

This circulation of the phase has led to such structures being known as “optical vortices”.  It can be shown that such vortices are stable features of a wavefield; that is, they are resistant to distortions of the wave induced by focusing, propagation through atmospheric turbulence, etc.

One of the fascinating aspects of the development of singular optics is that it provides a different perspective on optical waves.  Instead of considering light as an extended field “flowing” through space, singular optics allows us to view it as a topological structure, and to characterize any field by its structure.  What that means, roughly, is that we can in a sense talk about the “shape” of a wavefield, and look at the possibility of creating wavefields of unusual shape.

This was done for optical vortices in 2001, when Dennis and Berry demonstrated1 theoretically that the zero lines of optical vortices can be produced in the form of knots or links in a wavefield. Not long after, Dennis showed that zero lines could also be braided in the form of a pigtail braid2.

More recently, other authors have started considering what other sorts of topological features might be achievable in wavefields.  Topology is a branch of mathematics that is concerned with what properties of an object are preserved under distortions that don’t include tearing or gluing of the object.  A typical way to highlight this is to note that, in topology, a sphere and a cube are in a sense equivalent shapes:

If we imagine the cube to be fashioned of clay, we can squish and shape the clay to make a cube without tearing or gluing the clay at any point.  Similarly, a coffee cup and a torus (donut) are equivalent:

Both the cup and the donut have a single hole in them, and one can be deformed into the other preserving the hole.  However, a sphere and a donut are not topologically equivalent objects, because one must tear a hole in a sphere to make the donut hole or one must glue shut a hole in a donut to make a solid sphere.  Some shapes are fundamentally different from one another, in that they must be ripped in order to be made to match.

The archetypical example of this fundamental difference is a one-sided surface, known as a Möbius strip (picture from Wikipedia):

Whereas “ordinary” surfaces have two sides, like a sheet of paper, a Möbius strip has only one side: if you start on one side of the strip and follow a path along it, you will eventually find yourself on the other side of the strip!  Recently, Möbius strips were in the news, as a nanoscale strip was constructed by researchers out of DNA.

With the advent of singular optics and its emphasis on the structural properties of wavefields, it was perhaps inevitable for someone to investigate whether it is possible to make Möbius strips in optics.  It turns that that it is possible, but one must take advantage of a different sort of optical singularity of a wavefield, known as a polarization singularity.

I have discussed the polarization of light in a “basics” post in some detail; to talk about Möbius strips of light, we’ll need to review some aspects of polarization first.

The term “polarization” refers to the transverse wave nature of light; that is, broadly speaking, the electric and magnetic fields of a light wave oscillate perpendicular to the direction of wave motion:

The arrows indicate the direction and strength of the electric (E) and magnetic (H) fields along the line of propagation.  As the electric field is generally the active ingredient in a light field, i.e. the one that actually is involved in chemical reactions, heating, etc., the behavior of the electric field vector is the referred to as the state of polarization.

If one were to look at the electric field depicted above head-on, the arrow of E would oscillate up-and-down, tracing out a straight line in time.  This is what is known as a state of linear polarization.  It is also possible, however, for an electric field to trace out a circular or elliptical path in time, in states known as circular polarization and elliptical polarization.  The head-on view of each of these cases are illustrated below:

The fourth case, unpolarized light, is the case where the light field has random fluctuations and has its state of polarization change in time; we will not consider that case further here.

For a monochromatic (single frequency) wavefield, it can be shown that the polarization at any point in space must be linear, circular or elliptical.  From this observation, it also follows that the interference of two monochromatic wavefields always produces linear, circular, or elliptical polarization; elliptical polarization is the most common outcome.

Though light is said in general to be transverse, it is to be noted that this doesn’t necessarily mean that the electric field is always transverse to the direction of motion.  A general beam of light consists of many planar transverse waves, and the sum of those electric field vectors may include a component that is along the direction of motion.  For instance, suppose we look at three plane waves superimposed on one another, as schematically illustrated below:

The polarization of beam 2 points out of the page.  The vertical parts of the polarization of beams 1 and 3 will cancel out, but the horizontal parts of the polarization point in the same direction.  If we consider the path of beam 2 to be the average direction of travel (average of directions of travel of 1-3), we see that there must be a small component of the electric field that points in the direction of travel.  In traditional optics theory, where the light considered is primarily a narrow, highly directional beam, this small longitudinal component of the polarization is neglected and the light beam is said to be paraxial.  For paraxial light, the electric field vector lies entirely in a plane transverse to the direction of motion.

Okay, that’s the review of polarization; now let’s make some additional observations!

In optics experiments, we tend to look at light that is uniformly polarized — that is, it has the same state of polarization over the entire cross-section of the light beam.  For a light beam of circular cross-section and uniform horizontal polarization, we have:

The brightness indicates the intensity of the light.  The red lines indicate the path of the electric field in their immediate neighborhood; at each point of the field shown the polarization is horizontal.

There is no reason, however, why we could not create a light beam with non-uniform polarization; that is, the state of polarization is different at different points in the beam’s cross-section.  An example of this is a radially polarized field, which may be depicted as follows:

Radially polarized fields have been shown to have interesting and useful focal properties in recent years, and nonuniformly polarized fields in general have properties that uniformly polarized fields do not.

One particular feature of note in the radially polarized field is the intensity minimum at the center of the beam.  At the center of the beam is a polarization singularity; all around the center, the polarization has different orientations of the polarization vector, and the only possible behavior of the field at the middle is to have zero intensity and therefore undefined polarization.  This is comparable to the case of optical vortices: the center of a vortex is of zero intensity, and all around the center of the vortex the phase takes on all possible values.  Generally, it may be said that the intensity zero of a phase singularity or polarization singularity “hides” the singular behavior of the wave.

There are two types of singularities3 related to polarization, however, for which the field intensity does not vanish.  We have noted that the most general state of polarization of a  monochromatic wavefield is elliptical polarization; an ellipse may be characterized by the length of its major axis $L$ and its minor axis $l$, as well as the orientation of said axes:

Consider the major axis alone: it defines an orientation in space, and this orientation is well-defined everywhere except when the polarization is circular.  For circular polarization, it may be said that there is no major axis or, equivalently, every diameter of the circle serves equally well as a major axis.  A point of circular polarization is a singularity in the orientation of the ellipse axis.  The typical manifestation of this singularity is a line in three-dimensional space, or a point in a transverse cross-section of an optical field, known as a C-point.  The polarization ellipses around a C-point take on certain standard forms, illustrated below4:

We will not discuss these singularities in detail here, except to note that the major axes of the polarization ellipses define lines, and these lines may be extended (out of the page) to form surfaces in three-dimensional space.

The idea of twisting these surfaces of major axes into Möbius strips was introduced early this year by the Israeli researcher Isaac Freund, in the main citation of this post.  Freund considered the superposition of two laser beams, an ordinary Gaussian beam with clockwise circular polarization and a first-order vortex beam with counter-clockwise circular polarization:

The superposition of these two beams produces a beam with generally elliptical polarization everywhere in the transverse plane, with the exception of the central axis: because the intensity of the vortex beam is zero on axis, only the Gaussian beam’s polarization contributes and it is purely circularly polarized on axis.  Therefore this combined beam has a C-line along its central axis.

If both beams are paraxial, then the major axes of the polarization ellipses all lie within the transverse plane and nothing interesting happens.  Freund, however, considered the case where the two beams have their axes inclined by a small angle with respect to one another.  Then the combined non-paraxial field has polarization ellipses which in general have their axes oriented in different directions in three-dimensional space.

Freund considered the change in orientation of the polarization ellipses as one travels along a closed circular path surrounding the central axes of the combined field.  He found that the polarization ellipses trace out a multi-twist Möbius strip, where the number of twists depends upon the radius of the circular path — the larger the circular path, the larger the number of twists in the strip.

An ordinary Möbius strip can be constructed by taking an ordinary loop of paper, cutting it, twisting one end 180°, and reattaching it:

If one twists the end an extra 360° before reattaching it, one gets a triple-twist Möbius strip:

This process can be continued to get multi-twist Möbius strips of higher order.  Freund demonstrated that the polarization ellipses followed along a centrally-located circular path around a non-paraxial polarization singularity produce multi-twist Möbius strips, with an order that depends upon the radius of the path.

Explaining why one gets Möbius strips in such a light field is a much more difficult proposition.  Without going into much detail, we note that the direction of the major axes of the polarization ellipses in Freund’s field depend on the relative phases between the two parts of the field, namely the Gaussian field and the vortex field.  The phase of the vortex field, however, is a function of position, and the twists in the Möbius strip arise from the behavior of the vortex field’s phase.

Freund suggests, and demonstrates elsewhere5, that these Möbius strips appear around any C-line in a non-paraxial optical field. Considering that polarization singularities are common features of an electromagnetic wave, appearing even in light coming from the sky on a sunny day (discussion in paper here), it seems that we’ve always been “surrounded” by optical Möbius strips — we just didn’t know it!

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1 M.R. Dennis and M.V. Berry, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. Lond. A 457 (2001), 2251-2263.

2 M.R. Dennis, “Braided nodal lines in wave superpositions,” New. J. Phys. 5 (2003), 134.

3 The second singular type of polarization singularity are surfaces on which the polarization is linear; this is a singularity of the ellipticity of the polarization ellipse. We will not discuss such singularities further here.

4 Figure adapted from British researcher J.F. Nye’s groundbreaking paper on polarization singularities, J.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. Lond. A 389 (1983), 279-290.

5 I. Freund, “Optical Möbius strips in three-dimensional ellipse fields: I. Lines of circular polarization,” Opt. Commun. 283 (2010), 1-15.

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Freund I (2010). Multitwist optical Möbius strips. Optics letters, 35 (2), 148-50 PMID: 20081950