Non-integer optical modes in a Möbius-ring resonator

Transcription

1 Non-integer optical modes in a Möbius-ring resonator S. L. Li 1, L. B. Ma, 1, * V. M. Fomin 1, S. Böttner 1, M. R. Jorgensen 1, O. G. Schmidt 1,2,3, * 1 Institute for Integrative Nanosciences, IFW Dresden, Helmholtzstr. 20, Dresden, Germany 2 Material Systems for Nanoelectronics, Chemnitz University of Technology, Reichenhainer Str. 70, Chemnitz, Germany 3 Center for Advancing Electronics Dresden, TU Dresden, Georg-Schumann-Str. 11, Dresden, Germany *Corresponding authors: L.B.M. (l.ma@ifw-dresden.de), and O.G.S. (o.schmidt@ifw-dresden.de). Abstract: In-plane polarized light experiences a non-trivial topological evolution as it propagates resonantly in a Möbius ring resonator. The resultant geometric phase varies continuously when changing the light ellipticity, which leads to constructive interference for a non-integer number of wavelengths, and therefore to the occurrence of an arbitrary fractional number of optical modes. The geometric phase in Möbius-ring resonators is topologically robust and implies excellent intrinsic fault-tolerance. PACS numbers: Hz, Ex, Da When propagating light experiences non-trivial topological evolution in parameter space, it acquires geometric phase [1, 2]. This geometric phase has been explored in open light paths by recording polarization rotations in helical waveguides [3-5] and by measuring the interference fringe shift in a nonplanar Mach-Zehnder interferometer[6]. A closed light path, such as that in whispering-gallery-mode (WGM) microcavities [7], was assumed to preclude the existence of geometric phase due to the absence of spin-orbit coupling. Here, we demonstrate the generation of topologically protected geometric phase for light confined in a special type of optical microcavity the Möbius-ring resonator. The geometric phase causes light to constructively interfere for a non-integer number of wavelengths manifesting itself through the appearance of non-integer optical modes. The Möbius-ring [8] is a fascinating loop structure which is well-known for its onesided topology. While the behaviour of electronic waves in Möbius-ring structures has been 1

2 theoretically studied for Möbius aromaticity [9] and twisted semiconductor strips [10], the topological behaviour of light circulating in Möbius-rings has remained unexplored. In this Letter we investigate the topological impact on constructive optical interference in a Möbiusring made of a twisted dielectric strip. The strip thickness d is assumed smaller than the wavelength λ of the considered light, i.e. d < λ/n, so that the electric field is stringently confined within the strip during propagation. The strip width is taken larger than the wavelength and the strip length is chosen to be in the micrometer range to support optical modes in the visible spectral range. It is well known that constructive interference of two waves requires a phase difference corresponding to an integer number of their wavelengths. When light propagates around a closed trajectory (with a perimeter of L), optical resonances form by selfinterference after a full-round trip when a multiple number m of wavelengths matches L (mλ = L), where the integer number m is the mode number. These kind of resonances are known from many types of optical structures, in particular from WGM microcavities such as microspheres, micro-disks, and micro-rings[7]. The integer m implies that an even number (N = 2m) of antinodes is accommodated in the microcavity because each wavelength possesses two antinodes. We start our discussion with linearly polarized light for WGM resonances in ring-type resonators. Calculations for a cylindrical ring cavity confirm the expected optical resonant modes with wavelength λ = L/m and an even number N = 2m of antinodes. In contrast to this, optical resonant modes supported by the Möbius-ring accommodate an odd number (2m-1) of antinodes, which is equivalent to a half-integer number (l = m-1/2) of wavelengths interfering in the ring resonator, and which we correspondingly index by a half-integer mode number (m- 1/2). Figure 1 shows examples of antinode patterns for optical resonances in a cylindrical ring 2

3 and a Möbius-ring with antinode numbers N=16 (m=8) and N=15 (l=7.5), respectively. Both ring structures are formed from strips equal in size and with the same refractive index n. FIG. 1 (color online). Constructive self-interference of light with integer and non-integer number of wavelengths for a cylindrical ring and a Möbius-ring (with air as surrounding medium). (a) Crosssectional resonant antinode amplitude profiles calculated in a cylindrical ring (left) and Möbius-ring (right) resonator with numbers of antinodes N=16 and N=15, respectively. The odd numbers of antinodes are a result of constructive self-interference of light with a half-integer number of wavelengths. (b) Calculated resonant spectra including two resonant modes for the cylindrical ring and Möbius-ring. The modes labelled with half-integer azimuthal mode numbers correspond to the Möbius ring. Both ring structures are formed from strips equal in size (320 nm wide, 2510 nm long, 80 nm thick) with a refractive index of n = 3.5. The occurrence of non-integer optical modes in the Möbius-ring contradicts the conventional picture of constructive interference. Even though the length of the optical path in the cylindrical ring and the Möbius-ring is the same, the mode energies are different [see Fig. 1(b)]. This unusual phenomenon is explained below by the presence of a geometric phase. The phase factor in a physical wave system can be divided into two parts: the dynamical phase determined by the system energy, which reflects the system s evolution in time, and the geometric phase, which memorizes the evolution path in a parameter space. The geometric phase does not occur in trivial topologies, while the dynamical phase is always 3

4 present and evolves over time. Light in conventional WGM optical cavities [7] experiences trivial topological evolution, and therefore geometric phase is not generated. However, when a linearly polarized light enters the Möbius-ring structure considered here, the optical electric field is forced to remain parallel to the plane of the twisted strip, and consequently the polarization orientation continuously varies along the twisted strip during propagation [see Fig. 2(a)]. This behaviour represents an adiabatic parallel transport of linearly polarized light in a smoothly curved Möbius-ring [11]. FIG. 2 (color online). Constructive interference of light with a half-integer number of wavelengths caused by the presence of a geometric phase of π after parallel transport of polarization along a Möbius ring. (a) Sketch of parallel transport of linear polarization along a Möbius-ring in real space. (b) In polarization vector space the linear polarization state evolves along the equator of a Poincaré sphere. (c) Τhe generation of a geometric phase of π in addition to the dynamical phase of (2m-1)π leads to constructive interference with a half-integer number of wavelengths. The solid circles represent optical resonant antinodes. Solid blue curve represents the light wave carrying the dynamical phase. Dashed red curve represents the wave-flip caused by the geometric phase. The green arrow points at the starting point while the black arrows indicate the propagation direction in the Möbiusring. For the adiabatic transport of a degenerate physical system, the geometric phase can be quantified by the corresponding solid angle in parameter space determined by either the wave vector k, which forms a sphere in the momentum space, or the wave polarization state, which 4

5 spans a Poincarésphere. The geometric phase is equal to the solid angle subtended by the trace of the wave vector at the origin of the momentum space [3-5] and a half of the solid angle subtended by the loop of the polarization vector at the origin of a Poincarésphere [2, 12]. For the Möbius ring, the k-direction changes during light propagation along the circular trajectory and k evolves along a great circle on a sphere in the momentum space resulting in a solid angle Ω = 2π. However, the angle of 2π does not provide any observable phase changes, so the evolution of k does not contribute to the geometric phase in WGM microcavities. For the parallel transport of in-plane polarized light [described by ), where and are right and left circular bases] in a Möbius-ring, the continuous variation of the polarization orientation can be visualized by a closed loop along the equator of the Poincarésphere [see Fig. 2(b)]. One full round trip generates a geometric phase equal to half of the solid angle Ω/2 = π for the right and left circular polarization bases as ). Figure 2(c) shows the optical self-interference for half-integer mode numbers in a Möbius-ring. Apparently, the dynamical phase, which changes by 2π for one wavelength, cannot accomplish constructive interference in the Möbius-ring because it is exactly off-phase in the case of half-integer numbers of wavelengths. However, the presence of the geometric phase π leads to an effective wave-flip, which precisely compensates for the unmatched dynamical phase. It is therefore the geometric phase π that allows for constructive interference in the Möbius-ring in spite of the presence of non-integer number of wavelengths. In this sense, the occurrence of the geometric phase changes the conventional requirement that a difference equal to an integer number of wavelengths is needed for constructive interference. Remarkably, the geometric phase is wavelength independent, since it is only related to the intrinsic topological property of the physical evolution. In the model shown in Fig. 2, a single light wave self-interfering after one round-trip is assumed. An alternative way to describe the resonances in a ring cavity considers two light 5

6 waves starting from the same point but propagating in opposite directions. In this model the two beams meet after a half round-trip and interfere with each other [13]. Both considerations result in an equivalent loop on the Poincarésphere [12], and therefore lead to the same result. A geometric phase represents the global topological feature of a physical evolution, which is therefore invariant to local distortions. In particular, the geometric phase in a Möbius-ring is topologically robust, since all topologically-equivalent paths will project to the same loop on the Poincarésphere. These topologically-equivalent paths exist for instance, when the Möbius-ring is deformed by stretching and twisting the ring structure, or transformed by twisting the strip with an odd number of half-twists. As shown in Fig. 3, in the case of linearly polarized light, the geometric phase of π is invariant in both cases and causes waves to interfere with a non-integer number of wavelengths. FIG. 3 (color online). Topological protection of the geometric phase in transformed Möbius-ring structures. From left to right: Optical resonant profiles in a three-half-twisted Möbius-ring, a Möbiusring, a stretched and twisted Möbius-ring. A number of antinodes N=15, that corresponds to 7.5λ over the perimeter, is shown as an example. The topological robustness of the geometric phase is a result of the gauge invariance in the adiabatic evolution [14]. The gauge invariance, in turn, indicates that the geometric phase is an important property of a physical system. It has been shown that geometric phases can act as quantum logic gates in quantum computation [15]. A topologically-protected geometric phase therefore implies an intrinsic fault-tolerance towards environmental perturbance, which is of profound importance for practical applications. 6

7 Following the analysis above, consider the introduction of light with different ellipticities e ( ) propagating in the Möbius ring. In this case, the major axis orientation of the ellipse is confined locally to stay in-plane in the twisted strip, performing a cyclic evolution. The propagation of differently polarized light in the Möbius ring projects a series of loops on a Poincarésphere, varying from the equator (linear polarization, e = 1) to the two poles (circular polarization, e = 0), as shown in Fig. 4(a). The left (right) handed chirality is defined as that of the Möbius strip along the propagation direction, which is found the southern (northern) hemisphere of the Poincarésphere. The solid angle (Ω) subtended by the corresponding loop varies from 0 to 2π (in the northern hemisphere) and 2π to 4π (in the southern hemisphere), and consequently the geometric phase (γ=ω/2) varies from 0 to π and π to 2 π, respectively[2, 12]. The optical mode number can now be defined as, where M i is an integer number. By correlating the solid angle Ω, and therefore the geometric phase γ, with the polarization ellipticity e spread over the Poincarésphere [16], the optical mode number behaves as shown in Fig. 4(b). When changing the ellipticity from linear (e=1) to circular (e=0) the optical mode symmetrically approaches the neighbouring integer numbers owing to opposite light chiralities. The mode number rapidly changes from a half integer towards a whole integer when e changes from 0.5 to 1. At e=0, the right and left handed circular polarizations are degenerated at integer mode numbers due to trivial topology. This trivial topological effect is due to the arbitrary rotational symmetry of the circle depicted by the electric field vector in circular polarization. In this way, the optical mode number is continuously tuned into an arbitrary fractional number other than only a half integer. Alternatively, an arbitrary fractional number can also be realized, for example, in a Möbiusring made of an anisotropic inhomogeneous medium where the geometric phase might be no longer restricted to π due to a non-abelian evolution [17]. 7

8 FIG. 4 (color online). Optical mode number continuously tuned into an arbitrary fractional number. (a) The propagation of differently polarized light in a Möbius ring projects to a series of loops on a Poincarésphere, which leads to a variation of the geometric phase from 0 to 2 π. (b) Optical mode number m (here: between 7 and 8) as a function of light ellipticity e. For elliptically polarized light (0<e<1), each ellipticity leads to two different fractional mode numbers due to opposite chiralities (left and right). The linearly polarized light (e=1) results in a half integer mode number, while circularly polarized light (e=0) results in integer mode number due to a trivial topological effect. The geometric phase naturally evolves from solving Maxwell s equations in the Möbius-ring structure. We found that certain structural features are required for the occurrence of a geometric phase in Möbius-rings. For example, the ultra-thin strip (of subwavelength thickness) is critical for delivering topology information to the propagating light. When the strip is too thick (greater than or equal to the wavelength), even numbers of antinodes emerge because the polarization is no longer strictly required to rotate along the twisted strip structure, and therefore does not undergo a pure non-trivial topological evolution. In summary, this work demonstrates the generation of geometric phase for light propagating in the closed light path of a Möbius-ring resonator. The presence of geometric phase breaks the paradigm that a light-path difference equal to an integer number of wavelengths is required for constructive interference. The Möbius-ring structures considered here are of micrometer size, which is much smaller than previously reported for helical waveguide structures [3, 5, 6]. Such small sized optical components seem well-suited for a new generation of on-chip photonic devices with excellent topological robustness. Our work introduces non-trivial topology to the field of optical microcavities, and may lead to many promising applications in nanophotonics and quantum information technologies. 8

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