Knowledge about optical resonances determined by the topology of the Möbius strip

In the current edition of nature photonics, Prof. Dr. Oliver G. Schmidt, Dr. Libo Ma and their partners present a strategy for observing and manipulating the Berry optical phase in Möbius ring microcavities. In their research paper, they discuss how an optical Berry phase can be generated and measured in dielectric Möbius rings. Furthermore, they present the first experimental proof of the existence of a variable Berry phase for linearly or elliptically polarized resonant light.

A Möbius strip is a fascinating object. You can easily create a Möbius strip by rotating the two ends of a strip of paper 180 degrees and connecting them. Upon closer inspection, you realize that this tape has only one surface that cannot be distinguished between inside and outside or below and above. Due to this special topological property, the Möbius strip has become the subject of countless mathematical discourses, artistic representations and practical applications, for example in paintings by MC Escher, as a wedding ringor as a drive belt to use both sides of the belt equally.

Optical ring resonators

Closed bands or rings also play an important role in optics and optoelectronics. However, up to now they have not consisted of Möbius strips and are not made of paper, but are instead made of optical materials, for example silicon and silicon dioxide or polymers. These “normal” rings are not measured in centimeters either, but in micrometers. If the light with a certain wavelength propagates in a microring, constructive interference causes optical resonances to occur. This principle can be exemplified by a guitar string, which produces different pitches at different lengths: the shorter the string, the shorter the wavelength, and the higher the pitch.

An optical resonance or constructive interference occurs exactly when the circumference of the ring is a multiple of the wavelength of the light. In these cases, the light resonates in the ring, and the ring is called an optical ring resonator. Conversely, the light is highly attenuated and destructive interference occurs when the circumference of the ring is an odd multiple of half the wavelength of the light. Therefore, an optical ring resonator enhances light of certain wavelengths and strongly attenuates light of other wavelengths that do not “fit” the ring. In technological terms, the ring resonator acts as an optical filter that, integrated into a photonic chip, can selectively “sort” and process light. Optical ring resonators are central elements of optical signal processing in today’s data communication networks.

How polarized light circulates in the Möbius strip

In addition to wavelength, polarization is an essential property of light. Light can be polarized in various ways, for example linearly or circularly. If light is propagated in an optical ring resonator, the polarization of the light does not change and remains the same at all points in the ring.

The situation fundamentally changes if the optical ring resonator is replaced by a Möbius strip or, better still, a Möbius ring. To better understand this case, it helps to consider the detail of the geometry of the Möbius ring. The cross section of a Möbius ring is usually a thin rectangle in which two edges are much longer than their two adjacent edges, such as a thin strip of paper.

Now suppose that linearly polarized light circulates in the Möbius ring. Because the polarization prefers to align in the direction of the long cross-sectional side of the Möbius ring, the polarization rotates continuously up to 180 degrees while passing completely around the Möbius ring. This is a big difference from a “normal” ring resonator, where the polarization of light is always maintained.

And that’s not all. The polarization twist causes a shift in the phase of the light wave, so that optical resonances no longer occur at full wavelength multiples that fit into the ring, but at odd multiples of half the wavelength. vibe. Part of the research group had already theoretically predicted this effect in 2013. This prediction, in turn, is based on the work of physicist Michael Berry, who introduced the eponymous “Berry phase” in 1983, describing the change in the phase of the light whose polarization changes as it travels.

First experimental evidence

In this article published in the journal nature photonics, the Berry phase of light circulating in a Möbius ring is revealed experimentally for the first time. For this purpose, two rings with the same diameter were made. The first is a “normal” ring and the second is a Möbius ring. And as predicted, the optical resonances in the Möbius ring appear at different wavelengths compared to the “normal” ring.

The experimental results, however, go far beyond previous predictions. For example, linear polarization not only rotates, but also becomes more and more elliptical. Resonances do not occur exactly at odd multiples of half the wavelength, but generally at non-integer multiples. To find out the reason for this deviation, Möbius rings with decreasing bandwidth were made. This study revealed that the degree of ellipticity in the Polarization and the deviation of the resonant wavelength compared to the “normal” ring became progressively weaker as the Möbius strip became narrower and narrower.

This can be easily understood because the special topological properties of the Möbius ring merge with the properties of a “normal” ring when the width of the band is reduced to that of its thickness. However, this also means that the Berry phase in Möbius rings can be easily controlled simply by changing the band design.

In addition to the fascinating new fundamental properties of Möbius optical rings, new technological applications are also opening up. The tunable optical Berry phase in Möbius rings could serve for all-optical data processing of classical bits and qubits and support quantum logic gates in quantum computation and simulation.