Knots of Light

Around age six, we start learning how to tie our shoelaces, making knots that look like ribbons or possibly more complex forms, if we are a little clumsy. We use knots every day, but the type of knots we generally use are associated with physical objects, things we can touch. Although it can be hard to image, light can also be shaped in ways that form knotted confi­gurations, whose shape depends on the orbital angular momentum of the light. This parameter is responsible for making the beam of light twist around its own axis, generating different knot shapes, and expanding to a new degree of freedom that can carry valuable infor­mation.

Models of multicolored light twists in new knotted ways. (Source: ICFO)

Learning and mastering how to generate twisted light – light with orbital angular momentum – has been a thriving field of study for the past 20 years. Unlike spin angular momentum, which is associated with the polari­zation of light, orbital angular momentum is associated with the spatial distri­bution of the electric field. These two types of angular momentum can also be coupled, which results in a variety of light fields of different shapes with polari­zations that change from point to point.

The behaviour of light also becomes richer when it passes from oscil­lating at one single frequency to vibrating at many different frequencies. This introduces a broad array of polari­zation states, each describing a shape that can be traced by the electric field of the light over time. Combining this wider space of possi­bilities with the spatial variations produced by the orbital angular momentum should produce even more room for interesting connec­tions, but until now this has been an uncharted frontier: while there is a large body of research on structured light, it has been essen­tially focused on single-color fields.

Now, joint colla­borations by ICFO researchers have broken theoretical and experimental ground in this new field, uncovering new types of knots for twisted light and a new type of angular momentum. ICFO researchers Emilio Pisanty, Gerard Jiménez Machado, Veronica Vicuña Hernández, Antonio Picón and Alessio Celi, led by Juan P. Torres, have designed a beam of light with a polari­zation state that forms three-lobed trefoils at each point, by combining light of different frequencies (w and 2w), and making the trefoils connect to each other in a way such that the light beam, as a whole, has the shape of a knot.

These beams also exhibit a new kind of angular momentum, associated with the unusual symmetry of the beams, which remain invariant under rotations – but only when the polari­zation is rotated by a specific fraction of the rotation of the spatial dependence. They named this new quantity the torus-knot angular momentum, because of the type of knot in the beams. The researchers also implemented these beams experi­mentally, using nonlinear crystals to generate the beams, and they designed a nonlinear polari­zation tomo­graphy scheme to measure the trefoil shapes traced by the electric field. Their measurements show the presence of a new type of optical singularity which is topo­logically protected and robust against pertur­bations, caused by the different orientation of the polari­zation trefoils at different points around a circu­larly-polarized center.

Emilio Pisanty and Antonio Picón, led by ICREA Professor at ICFO Maciej Lewenstein, in colla­boration with researchers from the Laser Appli­cations and Photonics group at the University of Salamanca and from CU Boulder, show that this new optical singu­larity can be applied to nonlinear optics, even at the high-intensity extremes and in non-pertur­bative situations. They show, via theoretical simu­lations, that the high-order harmonics produced by the torus-knot beams at ultra-high intensities preserve the coor­dinated symmetry of the driving laser, forming twisted spirals of ultra-short pulses of light, and that the torus-knot angular momentum is conserved in the interaction. This new symmetry is essential in under­standing the production of shaped light at very short wavelengths, which can be used for novel appli­cations in micro­scopy, litho­graphy and spectro­scopy.

The results provide new frameworks and results that advance the study of structured light and non-linear optics. On one hand, the researchers were able to find new conser­vation laws for non-linear optics which hold even in extreme situa­tions where tens or hundreds of photons get combined to form single high-frequency photons. On the other, they analyzed the driving fields that make this possible and showed that they contain a new optical singu­larity, with a new degree of freedom that could be used to store valuable information, opening the possi­bility of using these new topologies of light for future communi­cation applications, among others. (Source: ICFO)

Reference: E. Pisanty et al.: Knotting fractional-order knots with the polarization state of light, Nat. Phot., online 10 June 2019; DOI: 10.1038/s41566-019-0450-2

Link: Quantum Optics Theory, ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels, Spain

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