Abstract
The Möbius strip, a fascinating loop structure with onesided topology, provides a rich playground for manipulating the nontrivial topological behaviour of spinning particles, such as electrons, polaritons and photons, in both real and parameter spaces. For photons resonating in a Möbiusstrip cavity, the occurrence of an extra phase—known as the Berry phase—with purely topological origin is expected due to its nontrivial evolution in parameter space. However, despite numerous theoretical investigations, characterizing the optical Berry phase in a Möbiusstrip cavity has remained elusive. Here we report the experimental observation of the Berry phase generated in optical Möbiusstrip microcavities. In contrast to theoretical predictions in optical, electronic and magnetic Möbiustopology systems where only Berry phase π occurs, we demonstrate that a variable Berry phase smaller than π can be acquired by generating elliptical polarization of resonating light. Möbiusstrip microcavities as integrable and Berryphaseprogrammable optical systems are of great interest in topological physics and emerging classical or quantum photonic applications.
Main
The Berry phase (also called the ‘geometric phase’), a nonintegrable phase factor originating from a nontrivial evolution of a physical system in parameter space^{1}, plays a fundamental role in various fields ranging from condensed matter physics^{2,3}, acoustics^{4}, highenergy physics^{5}, cosmology^{6}, quantum information^{7} to optics^{8}. In optics, the Berry phase can be acquired by the nontrivial evolution of either the polarization state or the wave vector in its corresponding parameter space, with purely topological origin^{8,9,10,11,12,13}. Investigation of the manipulation of polarization states was pioneered by S. Pancharatnam, and the generated geometric phase is now also called the Pancharatnam–Berry phase^{1,10}. Generation of a Berry phase based on wavevector evolution has been explored by recording polarization rotations in the open light paths of helical^{14,15,16,17} or outofplane curvilinear waveguides^{18}. Notably, in recent years, the generation of the topologically protected Berry phase has also been studied in specially designed threedimensional (3D) optical microcavities with light circulating along a closed path, for example in asymmetric whisperinggallerymode microcavities^{19}, outofplane microrings^{20} and microrings with embedded angular scatterers^{21}, which show great potential for onchip integrated topological photonic devices.
The Möbius strip^{22}—a fascinating loop structure wellknown for its onesided topology—also symbolizes the topological twist of the band structure in topological insulators^{23}. The Möbius topology plays important roles in multiple disciplines, ranging from the generation of symmetrybreaking Möbius soliton modes in a magnetic medium^{24} to the extraordinary behaviour of electronic waves in Möbius aromaticity^{25} and twisted semiconductor strips^{26}, as well as anomalous plasmon modes formed in metallic Möbius nanostructures^{27}. To impose the Möbius topology on photons, the optical field is twisted by liquidcrystal qplates as cavityfree systems^{28}. However, investigating the topological phenomena of light resonating in a real Möbiusstructured cavity remains highly desirable. Very recently, a Möbiusstrip cavity composed of a twisted dielectric strip was explored as a platform for the investigation of nonEuclidean optics^{29}. However, the optical Berry phase, the key topological phenomenon, has not been experimentally demonstrated, although the existence of a Berry phase in a Möbius cavity was theoretically predicted a long time ago^{30,31}. Theoretical studies have shown the occurrence of Berry phase π in an ideal Möbiusstrip cavity, resulting in constructive selfinterference of halfinteger number modes (that is, accommodating halfinteger numbers of wavelengths) in a closedpath trajectory^{30}. The halfinteger number of wavelengths, which has a purely topological origin, contradicts the wellknown resonant condition in conventional optical or plasmonic cavities.
The topological behaviour of circulating light in Möbiusstrip waveguiding systems has not yet been explored experimentally, preventing achieving new insights into observable optical phenomena for topologybased signal processing and communications. In this Article we report the experimental observation of optical Berry phases occurring in Möbiusstrip microcavities with tailored crosssectional geometry. In contrast to previous theoretical predictions, where only phase π occurs, here we observe and reveal programmable Berry phases ranging from π to 0 for resonant light waves with linear to elliptical polarization, in carefully designed Möbiusstrip resonators. As the quantum holonomy, the Berry phase generated in a compact optical Möbius system is particularly promising for geometric quantum mechanics and its applications, such as simulation, metrology, sensing and computation.
Results
Principles
The discussion starts with the optical polarization states in an ideal Möbius strip, in which the strip thickness T is much smaller than the strip width W (Fig. 1). Considering linearly polarized light resonating in the Möbiusstrip waveguide, the optical electric field is guided and forced to remain in the plane of the twisted strip. Such a twistedstrip waveguide functions similarly to some freespace optical components, such as a halfwave plate^{1} or a Dove prism^{32}, which affect the orientation of the polarization states. As a result, the polarization continuously reorients along the twisted strip during propagation, which we term the ‘inplane’ (IP) mode. The IP mode represents an adiabatic cyclic evolution of linearly polarized light in a smoothly curved waveguide. The acquired phase factor of the light wave can be divided into two parts, the dynamic phase and the Berry phase (alternatively called the geometric phase). The dynamic phase reflects the system’s evolution in time, which is determined by the optical path and the system’s curvature. In contrast, the Berry phase memorizes the evolution path in the parameter space, which is independent of the dynamic phase. Using a phenomenological model, the occurrence of Berry phase in a Möbius strip can be directly visualized by the parallel transport of a vector along the twisted strip. For reference, we investigate a comparable 3D ring cavity (‘curved strip’) with the same curvature as that of a Möbius strip but without the Möbius topology (Supplementary Fig. 1 and Supplementary Note 1). In contrast to the onesided Möbius strip with a single surface, the curved strip is topologically identical to conventional microring cavities. As shown in Fig. 1, although the vector orientations are kept in parallel with each other at each local position during transport, the vector flips by an angle π after a full trip around the Möbius strip, whereas there is no such a flip in the curved strip.
For the adiabatic cyclic evolution of a degenerate physical system there are usually two analytical ways to quantify the occurring Berry phase via the corresponding solid angle in parameter space. One way is determined by the wave vector k, which forms a sphere in momentum space. The Berry phase is equal to the solid angle enclosed by the trace of the wave vector at the origin of momentum space^{14,15,16}. The other way is related to the wave polarization state vector, which spans the Poincaré sphere. The Berry phase is equal to half of the solid angle enclosed by the loop of the polarization vector at the origin of the Poincaré sphere^{11,13}.
For linear polarization, the polarization state can be described by \({s\rangle }={\frac{1}{\sqrt{2}}(+\rangle +\rangle )}\), where +〉 and −〉 are right and left circular bases. The continuous variation of the polarization orientation can be visualized as a closed loop along the equator of the Poincaré sphere (Fig. 1, left), resulting in a solid angle, Ω = 2π. Hence, a Berry phase as large as half the solid angle, Ω/2 = π, is generated for the right and left circular polarization bases as \({s^{\prime} \rangle }={\frac{1}{\sqrt{2}}({\rm{e}}^{i{{\uppi }}}+\rangle {\rm{e}}^{i{{\uppi }}}\rangle )}\) (refs. ^{11,12,13}). For optical resonances in curvedstrip cavities, the trajectory of the polarization evolution on the Poincaré sphere is topologically trivial and thus does not generate any Berry phase, as illustrated in Fig. 1. The similar photon propagation trajectory and propagation constant render the curvedstrip cavity a perfect reference for our study of quantifying the Berry phase.
Light ellipticity in Möbius strips
Twophoton polymerizationbased direct laser writing was used to fabricate dielectric Möbius and curvedstrip microrings using the negative photoresist IPDip (Methods and Supplementary Fig. 2). The polymerized IPDip is transparent in the visible to nearinfrared spectral range, and is thus suitable for supporting optical resonances in this range. Figure 2a,b presents scanning electron microscopy (SEM) images of Möbius and curvedstrip cavities with identical design parameters, T/W ≈ 0.33 (T = 0.9 μm).
To understand the behaviour of light propagation and resonances in alldielectric Möbius microcavities, 3D numerical simulations based on the finiteelement method were carried out (Methods). Figure 2 (bottom panels) presents the calculated mode profiles in typical Möbius and curved strips with T/W = 0.33 (T = 0.9 μm), respectively. The electricfield orientation of the resonant light rotates along the twisted strips, as can be seen in the magnified images of the horizontal and vertical sites of the Möbius and curved strips. In the Möbiusstrip cavity, an odd number of antinodes (N = 115) is calculated, implying constructive interference with a halfinteger mode number (M = N/2 = 57.5) of wavelengths. The constructive interference is offphase in the case of halfinteger numbers of wavelengths, and the phase mismatch is precisely compensated by the presence of Berry phase π, which leads to a 180° waveflip. In the curvedstrip cavity, the even number (N = 116) of antinodes corresponds to the conventional constructive interference with an integer number (M = 58) of wavelengths.
As a key approximation in an ideal Möbius strip, strip thickness T is assumed to be much smaller than the wavelength of the considered light in the waveguiding medium (λ/n), so the optical field is strictly confined within the strip during propagation. In reality, the thickness of a Möbiusstrip cavity can only be shrunk to a finite value, because of the need for sufficient optical confinement and lowloss light propagation. When T is comparable to W, the twisted crosssection does not provide rigorous inplane confinement of the optical electric field. This leads to two competing effects. On the one hand, due to electromagnetic inertia, the electric field tends to maintain its orientation as it propagates along the strip. On the other hand, the tilted crosssection forces the electric field to rotate along the twisted strip. As a result, the main electricfield orientation tilts away from the strip plane, generating inplane and outofplane components (Fig. 3a) with a phase retardation due to the different effective refractive indices for the electricfield components in and perpendicular to the strip plane (Supplementary Fig. 3 and Supplementary Note 1). An elliptical polarization is thus formed in the crosssection of the Möbiusstrip cavity, as illustrated in Fig. 3a,b.
The resonant modes were characterized by measuring transmission spectra using an evanescently coupled tapered nanofibre—a widely adopted approach for nearfield delivery and probing of light waves (Methods, Supplementary Fig. 4 and Supplementary Note 2)^{33}. The polarization states of the resonant light in the Möbius and curvedstrip microcavities were examined by tuning the input optical polarization, step by step. For the Möbiusstrip cavity with T/W ≈ 0.67, the measurement shows an elliptical polarization state (Fig. 3c). This is in sharp contrast to the linear polarization state for the Möbiusstrip cavity with T/W ≈ 0.33. The ellipticity as a function of T/W was further studied by numerical simulations (Fig. 3d). The resonant light is almost linearly polarized when T/W < 0.5, and becomes elliptically polarized when T/W > 0.5. A further increase in T/W leads to a ‘toruslike’ring structure (T/W ≈ 1). In this case, the optical electric field is no longer forced to rotate along the structure with a squareshaped core (Supplementary Fig. 6 and Supplementary Note 4), so the light does not undergo any nontrivial topological evolution.
The evolution of elliptically polarized resonant light projects a loop onto the Poincaré sphere away from the equator, depending on its ellipticity (Fig. 3e). With solid angle Ω changing between 2π and 0, Berry phase φ_{B} is generated for the right and left circular polarization bases as \({s^{\prime} \rangle }={\frac{1}{\sqrt{2}}({\mathrm{e}}^{i{{{\varphi }}}_{{\mathrm{B}}}}+\rangle +{\mathrm{e}}^{i{{{\varphi }}}_{{\mathrm{B}}}}\rangle )}\). Accordingly, by engineering the polarization ellipticity, one can develop a strategy to generate a variable Berry phase affecting both the eigenvalues and eigenstates of Möbiusstrip cavities. For T/W ≈ 1, the unmodified and nonrotating polarization state leads to the absence of the Berry phase, indicating a negligible Möbiustopology effect on the resonant light. Berry phaseinvolved constructive interferences in a Möbius strip are described further in Supplementary Note 5.
Manipulating Berry phases
Controlling the crosssectional dimension of a Möbius waveguiding system provides great flexibility in supporting the optical resonances of elliptically polarized light and hence manipulating the acquired Berry phases. The acquired Berry phases are summarized in Fig. 4a by comparing the resonant modes measured in pairs of Möbius and curvedstrip cavities with various T/W values (Methods). In the case of T/W = 1, no wavelength offsets are observed between the Möbius and curvedstrip cavities, indicating the absence of the extra phase, that is, the Berry phase. As T/W decreases, clear increasing resonantmode offsets emerge in the resonant spectra measured for the Möbius and curvedstrip cavities, providing direct evidence of an increasing Berry phase acquired in the Möbiusstrip samples.
The simulation results in Fig. 4b reveal a consistent trend and confirm that the Berry phase can be freely tuned in the range from π to 0 by shaping the waveguiding crosssection of a Möbius strip. T/W ratios ranging between ~0.65 and ~0.85 are well suited to tuning the Berry phase, as the ellipticity of the resonant light is particularly sensitive to T/W in that range. The Berry phase φ_{B} extracted from the resonantmode spectra is slightly lower than that derived from simulations over the whole T/W ratio range, as the measured resonantmode offsets are not as large as those from theoretical calculations. In an ideal case, the resonantmode offset is solely determined by the presence of the Berry phase, and the Möbius and curvedstrip cavities generate an identical dynamic phase as a result of having the same size and curvature. In practical sample fabrication, a substantial curvature discrepancy exists between the Möbius and curvedstrips due to the structuralsymmetryinduced inherent strain, which reduces the mode offsets for the two types of cavity. As such, a deviation between experimental and simulated phase values constantly exists in the entire investigated T/W ratio range except for T/W = 1, showing the same evolution trend of Berry phase as a function of T/W.
Discussion
The above mainly focuses on circulating light with an optical electric field parallel to the strip plane (IP modes). However, resonant modes with an optical electric field perpendicular to the strip plane, termed ‘outofplane’ (OP) modes, can also be well supported for generation of the Berry phase at sufficiently large values of T. Similar to IP modes, OP modes with linear and elliptical polarizations can be formed for the generation of a variable Berry phase. Simulation and experimental results for OP modes and the associated Berry phases, presented in Supplementary Figs. 8 and 9 and Supplementary Notes 6 and 7, show similar behaviour to IP modes.
As a topological effect arising in cyclic adiabatic evolution, Berry phases for photons have been observed and systematically investigated using alldielectric optical Möbiusstrip cavities. The variable crosssection of the waveguiding strip produces different responses of light towards the twisted topology. Curvedstrip cavities with the same curvature were taken as a topologically trivial counterpart to extract the Berry phase. By controlling the crosssectional strip dimension, the role of the Berry phase in resonance modes has been revealed both experimentally and theoretically, offering a guide for manipulating the ellipticities of circulating light waves and the resultant Berry phase in the range of π to 0.
The topological robustness of the Berry phase is a result of gauge invariance in the adiabatic evolution^{34}. The Möbius waveguiding structures operate in the optical wavelength range and are of micrometre size, much smaller than those investigated in previous reports using helical waveguides and other openlightpath systems^{15,16,35}. Such a miniaturized optical component is suitable for a new generation of onchip integrable systems with excellent topological robustness for both fundamental physics and practical applications. The programming of optical ‘Möbiosity’ as a new route to lighttopology shaping has the potential to serve as a versatile knob for alloptical manipulation of both classical bits and qubits, and implies promising functionalities such as supporting optical framed knots as information carriers^{36}, and quantum logic gates^{37,38} in quantum computation and simulation.
Methods
Numerical simulation
The 3D numerical simulations (COMSOL Multiphysics Wave Optics module) were carried out for both Möbius and curved strips. The key parameters were designed to meet the requirements for practical fabrication procedures and operation at telecommunication wavelengths (radius R = 10 μm, width of 0.9–4 μm and thickness of 0.6–1.5 μm). Models were generated using MATLAB with the equations provided in Supplementary Note 1. Opticalmode fields were simulated in the wavelength range 1,500–1,560 nm. IP and OP excitations were provided by a local line current source at the waveguide core oscillating along the corresponding directions. The local polarization state was extracted by analysing the electricfield distributions at the field maxima point of the waveguide’s crosssection spanning from one mode antinode to node, and is presented using a projected polar plot.
Device fabrication
Pure quartz glass substrates were cleaned by immersion in acetone and isopropyl alcohol, ultrasonicated (Elmasonic S, Elma Schmidbauer), and dried with N_{2} gas. The microcavities were fabricated using 3D direct laser writing (Photonic Professional GT, Nanoscribe) on cleaned quartz substrates. The laser lithography photoresist was dropcast onto the substrates. The principle of the direct laser writing system is based on multiphoton lithography and 3D scanning with a focused laser beam within the sample volume. A highresolution galvanometer mirror system was used for laserbeam scanning in the x–y plane, which allowed each volumetric pixel to be positioned with an accuracy of 10 nm. The scanning was controlled by DeScribe software in Nanoscribe, which was also used to design the structures. During the fabrication process, the negative photoresist (IPDip, Nanoscribe) was polymerized by twophoton absorption (at 780 nm). After exposure, the structures were developed in mrDeveloper (mrDev 600, Micro Resist Technology) followed by rinsing in isopropanol, which resulted in fabricated structures bonded onto the substrate. The structures were then dried using a critical point drier (Autosamdri931, Tousimis Research Corporation) to remove the solvent. Each fabricated Möbius and curvedstrip microstructure was supported by two ~20µmhigh pylons to avoid optical leakage to the substrate.
Device characterization
For SEM imaging, the samples were sputtercoated with a thin (~8 nm) Cr layer to avoid charging and to improve contrast. The SEM images were acquired using an NVision40 system (Carl Zeiss) using a primarybeam energy of 5 keV. Optical transmission measurements were conducted using an evanescently coupled tapered fibre. The optical setup and details of preparing the tapered fibre are provided in Supplementary Note 2. A wavelengthtunable infrared laser (Tunics 100SHP, Yenista) was used as the light source, with a scanning range covering 1,500–1,600 nm. The transmission signal was measured using an InGaAs switchablegain photodetector (PDA20CSEC, Thorlabs). Resonances with a maximized extinction ratio of >10 dB were obtained by finely adjusting the coupling gap spacing. The polarization state of the incident laser light was controlled by a fibrebased threepaddle polarization controller. Launching of linearly polarized light with a tilted orientation was carefully examined by connecting a fibrecoupled polarimeter (PAX1000IR2, Thorlabs) at the output. Polarization mapping was carried out by adjusting the polarization angle of the input laser light in steps of 10°. The Berry phase was estimated by deriving the compensated phase value (that is, the resonant wavelength offset) for spectral matching of the resonance modes in Möbius and curvedstrip cavities with identical structural parameters.
Data availability
The main data supporting the findings of this study are available within the Supplementary Information. Additional data are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank H. Xu and R. Engelhard for technical support and S. Kiravittaya for helpful discussions. L.M. acknowledges financial support from the WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter–ct.qmat (EXC 2147, projectID 390858490) and the German Research Foundation (MA 7968/21). O.G.S. acknowledges financial support from the Leibniz Program of the German Research Foundation (SCHM 1298/261). J.W. acknowledges support from the National Natural Science Foundation of China (grant no. 62105080). V.M.F. acknowledges financial support from the German Research Foundation (project no. FO 956/61).
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L.M., O.G.S. and J.W. conceived the study, inspired by an initial investigation with S.L. and V.M.F. J.W., S.L., C.H.L., R.T. and V.M.F. performed the theoretical calculations and analysis. S.V., L.S. and M.M.S. fabricated the samples. S.V. and J.W. conducted the optical experiments. S.B. conducted SEM characterizations. J.W., L.M., Y.Y. and S.L. analysed the data. J.W. and L.M. wrote the manuscript. All authors discussed the results and contributed to the manuscript.
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Wang, J., Valligatla, S., Yin, Y. et al. Experimental observation of Berry phases in optical Möbiusstrip microcavities. Nat. Photon. 17, 120–125 (2023). https://doi.org/10.1038/s41566022011077
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DOI: https://doi.org/10.1038/s41566022011077
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