The cool thing about science is that, even in the areas that you think you are pretty knowledgeable, surprises abound. This is what keeps me turning up to work (occasionally) and (even more occasionally) committing the crime of science writing. In this case, I get to combine a work interest (using light to measure stuff) with one of last century's passing fads (light with orbital angular momentum).

I'm being a little unfair to the community of researchers who play with twisted light. In the '90s, twisted light was a big deal. Then after the echoes from the gasps of amazement died down, it proved to be difficult to sustain broad interest. Nevertheless, there is a core of researchers who are still investigating and finding uses for these strange light fields. In a recent paper, some of those researchers have shown that imparting orbital angular momentum to a light field allows one to make certain types of measurements with an accuracy that would otherwise be impractical, if not entirely impossible.

In many types of optical measurements, we rely on the accurate alignment of two coordinate systems. Imagine a light beam that gets split in two. One light beam goes to some reference, where its polarization is measured. The second light beam has something done to it, and then its polarization is measured by some other instrument.

Polarization is a measurement that tells us about the spatial orientation of the electromagnetic field of the light and how it evolves in time. For instance, a linearly polarized light field implies that the electric field oscillates in a single plane—at one instant in time, the field is at a maximum in one direction, while at another there is no electric field at all; later, it is maximum in the opposite direction. Circularly polarized light, on the other hand, has an electric field amplitude that stays constant, but it rotates in a circle as the light propagates.



When we measure polarization, however, we use apparatuses that measure the intensity of light after it has been filtered at a specific orientation. Clearly, any statement we make about the polarization of a light field depends on the orientation of the filters—in other words, on the orientation of our coordinate system. In our example above, if one instrument is tipped at an angle to the other, then they will never agree on the polarization state unless we have a purely circularly polarized light field.

For a very precise measurement, we will want the two filters aligned as perfectly as possible. How accurately can this be done? The answer with normal light is that the accuracy goes as the square root of the number of photons in the light beam. The longer you measure, the more photons you get, and the more accurate you get. But your accuracy only increases slowly with increasing photon numbers—more precisely, it increases with the square root of the photon number. This is called the shot-noise limit, and it is the limiting factor in all classical experiments.

To achieve a more accurate measurement, you need to use specially prepared entangled states of light. Their accuracy is a bit more complicated, but if you could entangle N photons and only make one measurement, then the accuracy will be proportional to N (as opposed to the square root of N in the classical case). That is, the more photons you entangle, the more accurately you can measure. One can also make multiple measurements, so, if you make nine measurements with N, you get an accuracy that is three times better. That is, we get a sort of mix of the classical situation and the quantum situation. The accuracy scales linearly with the number of photons used in a single measurement and as the square root of the number of measurements.



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This sounds promising then. If you want more accuracy, entangle more photons. Unfortunately, that is easier said than done. So, an alternative would be nice. And this is exactly what this multinational team of researchers has achieved.

Light that is... twisted

Now that we understand the limitations of the measurements, we need to take a break and understand what makes twisted light different. From a quantum mechanical perspective, this is very easy. Every particle is defined by a set of quantum numbers that define its energy, momentum, spin momentum, and orbital angular momentum. Light typically has an energy defined by its frequency and a momentum defined by its wavelength.

The polarization is defined by the spin momentum. Spin momentum can be +1 or -1, corresponding to the two different directions the electric field vector rotates for circularly polarized light—other polarizations are admixtures of these two components. But what about orbital angular momentum? Normally, that is zero. However, if you can make it non-zero, then you have twisted light. Furthermore, unlike spin momentum which takes on just two values, orbital angular momentum can take on integer values as large or small as you like.

What does such a light beam look like? The name gives it away in a sense. As the picture illustrates, at any particular plane across the light field, the electric field is always pointing slightly inward. So, unlike an ordinary light beam where the phase is nearly constant across the light field—that is, the wave has the same starting point in an oscillation everywhere across the beam—here it varies in multiples of 180 degrees across the beam. The electric field spirals around like a corkscrew; hence, twisted light. The quantum number tells how sharp the spiral is, while the sign tells us the direction of the spiral.

If you think carefully about the spiral nature, you will see that at the very center of the beam, the electric field has to be in one of two possible phases. The only way it can do that is with a field of zero amplitude, so the beam has a dark spot in the center. If the quantum number is larger, the spiral is tighter and there are additional locations where the phase must take on two values, so the beam breaks up into a ring of spots surrounding a central dark region.

Using the twist to amplify a measurement

How does this help measurements? It comes down to the symmetry of the beam. Let's take an example, one where we add three units of orbital angular momentum to the light beam. We start off with light that is linearly polarized (so the electric field oscillates in a plane) parallel to the table and pass it through an optical element to add the orbital angular momentum. The beam that emerges has four spots that form a dark cross. If we do the same thing with a light beam that is linearly polarized vertically, then the beam with orbital angular momentum still has four spots, but they form a dark X.

The spots have rotated by 45 degrees—or have they? They could also have rotated by 135 degrees, or 225 degrees, or 315 degrees. In fact, the rotation is 135 degrees (the polarization of the incoming light beam has been rotated by 90 degrees, so it is 45+90 = 135). Our measuring equipment can't tell in that case, but if the polarization of the incoming light beam is rotated just 30 degrees from horizontal, our measuring equipment will report a rotation three times as large (90 degrees).

What this means is that small rotations are amplified in the measurement process by the amount of orbital angular momentum imparted to the light field. In the example above, our measurement of small polarization rotations is three times more sensitive than that which can be achieved using an ordinary light field.

We can use this to make measurements that involve tiny changes in polarization orientation or to ensure that a measurement apparatus has a consistent alignment. The results will be much more accurate than thought possible, all by using a relatively trivial measurement procedure. Indeed, with large amounts of orbital angular momentum, the sensitivity increases further. Even better, the sensitivity is functionally the same as you would get using entangled photons—that is, we can achieve increases in accuracy that scale linearly with the imparted orbital angular momentum and as the square root of the number of measurements.

For our example above, we gained a factor of three in sensitivity using a beam with four units of angular momentum (the four points of the X). To achieve the same with an entangled state, we would need to entangle three photons, which is doable. On the other hand, it is trivial to impart 100 units of angular momentum to a light beam, but it's nearly impossible to entangle 99 photons with each other. That is a big practical difference.

There is also a cost to this approach, however. In the example with four units of angular momentum, the rotation is three times larger, which means that at 30 degrees, we measure a 90 degree rotation. Here, the symmetry comes back into play. If the rotation is marginally larger than 30 degrees, say 40 degrees, we will not be able to tell if this is a 20 degree rotation or a 40 degree rotation. That means that you are increasing accuracy at the expense of dynamic range. Effectively, you need to know beforehand that the angle you want to measure is small, which is often the case. Of course, you can imagine a more complicated scheme, where you first use a beam without angular momentum to determine the approximate rotation and then use the appropriate orbital angular momentum charge to make a more accurate measurement.

To give you an idea of how exciting this idea is, at the first opportunity, I convinced a student that we should see if this could be applied to some of our surface science measurements. She is going to do some calculations to see if our measurements are compatible and, hopefully, conduct some test experiments. I have hopes that we can achieve sensitivities much better than current commercial instruments.

Nature Communications, 2013, DOI: 10.1038/ncomms3432