Möbius strips are three-dimensional geometrical structures, fascinating for their peculiar property of being surfaces with only one "side"—or, more technically, being "nonorientable" surfaces. Despite being easily realized artificially, the spontaneous emergence of these structures in nature is exceedingly rare. Here, we generate Möbius strips of optical polarization by tightly focusing the light beam emerging from a q-plate, a liquid crystal device that modifies the polarization of light in a space-variant manner. Using a recently developed method for the three-dimensional nanotomography of optical vector fields, we fully reconstruct the light polarization structure in the focal region, confirming the appearance of Möbius polarization structures. The preparation of such structured light modes may be important for complex light beam engineering and optical micro- and nanofabrication.

Highly confined vectorial electromagnetic field distributions represent an excellent tool for detailed studies in nano-optics and high resolution microscopy, such as nonlinear microscopy 1 , advanced fluorescence imaging 2-4 or nanoplasmonics 5,6 . Such field distributions can be generated, for instance, by tight focussing of polarized light beams [7][8][9][10] . To guarantee high quality and resolution in the investigation of objects with sub-wavelength dimensions, the precise knowledge of the spatial distribution of the exciting vectorial field is of utmost importance. Full-field reconstruction methods presented so far involved, for instance, complex near-field techniques 11,12 . Here, we demonstrate a simple and straight-forward to implement measurement scheme and reconstruction algorithm based on the scattering signal of a single spherical nanoparticle as a field-probe. We are able to reconstruct the amplitudes of the individual focal field components as well as their relative phase distributions with sub-wavelength resolution from a single scan measurement without the need for polarization analysis of the scattered light. This scheme can help to improve modern microscopy and nanoscopy techniques.In the optical analysis of sub-wavelength objects such as cellular structures 13 , plasmonic particles 5,6,14 or single spins 3 , nano-optical tools including highly resolving microscopy techniques are used. Because such methods utilise complex and highly confined vector fields, the exact knowledge of the corresponding spatial field distributions is crucial. In the last decades, several techniques have been proposed to map these focal fields, such as using metal knife edges 15,16 to probe the total electric energy density distribution, fluorescence molecules 17 , tapered fibres 18,19 or tip-based methods 12 to image specific field orientations, or near-field scanning optical microscope (NSOM) techniques to extract amplitude and even phase information 11 . These NSOM-based methods require complex measurement and detection schemes and calibration procedures to allow for amplitude and phase a) Electronic mail: thomas.bauer@mpl.mpg.de mapping of individual field components. As an alternative approach for measuring phase information also a single particle scattering scheme was proposed recently 22 , where the authors show that Mie-scattering can distinguish the topological charge of vortex-beams.We now demonstrate a precise and easily implementable field reconstruction technique for highly confined field distributions created by arbitrary focusing systems, based on, what we call, Mie-scattering nanointerferometry. The basic concept of this reconstruction method for highly confined focal field distributions can be understood as follows. We use a metallic nanosphere on a glass substrate as local field sensor and scan it stepwise through the focal field distribution under investigation. If the particle size is chosen small enough, it will be excited by the local electric field E(r 0 ) only. Its response can be described,...

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