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Focusing characteristics of a 4 parabolic mirror light-matter interface
Lucas Alber,1, 2, Martin Fischer,1, 2, Marianne Bader,1, 2
Klaus Mantel,1 Markus Sondermann,1, 2 and Gerd Leuchs1, 2, 3
1Max Planck Institute for the Science of Light,Guenther-Scharowsky-Str. 1/ building 24, 91058 Erlangen, Germany
2Friedrich-Alexander-Universitat Erlangen-Nurnberg (FAU),Department of Physics, Staudtstr. 7/B2, 91058 Erlangen, Germany
3Department of Physics, University of Ottawa, Ottawa, Ont. K1N 6N5, Canada
Focusing with a 4 parabolic mirror allows for concentrating light from nearly the complete solidangle, whereas focusing with a single microscope objective limits the angle cone used for focusingto half solid angle at maximum. Increasing the solid angle by using deep parabolic mirrors comesat the cost of adding more complexity to the mirrors fabrication process and might introduceerrors that reduce the focusing quality. To determine these errors, we experimentally examine thefocusing properties of a 4 parabolic mirror that was produced by single-point diamond turning. Theproperties are characterized with a single 174Yb+ ion as a mobile point scatterer. The ion is trappedin a vacuum environment with a movable high optical access Paul trap. We demonstrate an effectivefocal spot size of 209 nm in lateral and 551 nm in axial direction. Such tight focusing allows us tobuild an efficient light-matter interface. Our findings agree with numerical simulations incorporatinga finite ion temperature and interferometrically measured wavefront aberrations induced by theparabolic mirror. We point at further technological improvements and discuss the general scope ofapplications of a 4 parabolic mirror.
Free space interaction between light and matter isincorporated as a key technology in many fields inmodern science. The efficiency of interaction influ-ences measurements and applications ranging fromvarious kinds of fundamental research to industrialapplications. New innovations and new types of highprecision measurements can be triggered by improv-ing the tools needed for a light-matter interface. Toachieve high interaction probability with a focusedlight field in free space, an experimental scheme us-ing parabolic mirrors for focusing onto single atomshas been developed in recent years . This schemerelies on mode matching of the focused radiation toan electric dipole mode (cf. Ref.  and citationstherein).
Focusing in free-space experiments is usually donewith state-of-the-art lens based imaging systems .Single lenses, however, suffer from inherent drawbackslike dispersion induced chromatic aberrations, opticalaberrations, and auto-fluorescence, respectively. Mostof these limitations can be corrected to a high de-gree by precisely assembling several coated lenses in alens-system, e.g. in a high numerical aperture (NA)objective. Although solving some problems, multi-lens-systems induce new problems such as short work-ing distances, low transmission for parts of the opti-cal spectrum, the need for immersion fluids, and highcosts, respectively. Therefore, multi-lens systems areoften application specific providing best performanceonly for the demands that are most important for theapplication.
These authors contributed equally to this work. [email protected]
Mirror based objectives are an alternative to lens-based systems and can overcome some of these prob-lems. The improvement is based on a mirrors in-herent property of being free from chromatic aberra-tions. The nearly wavelength independent behavioralso leads to a homogeneous reflectivity for a largespectral window. Comparing the reflectivity of mir-rors to the transmission of lens based objectives, mir-rors can sometimes also surpass lens-based systems.But surprisingly, they are rarely used when high in-teraction efficiency is required. This lack in applica-tion may be due to the fact that reflecting imagingsystems, like the Cassegrain reflector, cannot providea high NA. A high NA is however needed for matchingthe emission pattern of a dipole, which spans over theentire solid angle. The limitation in NA consequentlyconstitutes a limitation in the maximum achievablelight-matter coupling efficiency.
High NA parabolic mirrors (NA = 0.999) havemeanwhile been successfully applied as objectives inconfocal microscopy [9, 10], demonstrating the poten-tial for imaging applications. The parabolic mirror(PM) is a single optical element that, in theory, cancover nearly the complete 4 solid angle for tight fo-cusing . In this article we report on the detailedcharacterization of such a 4 parabolic mirror (4-PM), in which we sample the focal intensity distribu-tion with a single 174Yb+ ion, trapped in a stylus likemovable Paul trap .
In contrast to our previous studies , we measurethe response of the ion at a wavelength different tothe one used for excitation. This approach is standardin fluorescence microscopy and has also been used inexperiments with trapped ions . It renders unnec-essary a spatial separation of focused light and lightscattered by the ion, thus lifting the limitation of fo-cusing only from half solid angle as in Ref. . How-ever, we will find below that by using the solid angle
provided by our 4-PM the measured effective excita-tion point spread function (PSF ) is worse when usingthe full mirror as compared to focusing from only halfsolid angle. As outlined below, this is not a generalrestriction but specific to the aberrations of the mirrorused in our experiments. It is a challenge to determinethe aberrations of such a deep parabolic mirror and we discovered the full extent of these aberrationsonly when scanning the 3D field distribution with thesingle ion, revealing an error in the earlier interfero-metric measurements. Here, we present a reasonableagreement of the experiments with results of simula-tions incorporating a finite ion temperature and newinterferometrically measured wavefront aberrations ofthe parabolic mirror itself.
Despite of these aberrations, the efficiency obtainedhere for coupling the focused light to the linear dipoletransition of the 174Yb+ ion is better than reportedpreviously , using the full mirror as well as focus-ing from half solid angle. As a further improvement incomparison to Ref.  we keep the excitation of theion well below saturation making sure that the sizeof the ions wave function stays approximately con-stant as much as possible. All in all, the ion consti-tutes a nanoscopic probe with well defined propertiesthroughout the measurement range.
In the concluding discussion of this paper, theparabolic mirror is compared to other high NA focus-ing tools, especially to lens-based 4 microscopes. Itspossible field of application is discussed and furtherimprovements to the existing set-up are proposed.
II. SETUP AND EXPERIMENT
Our main experimental intention is to focus light toa minimal spot size in all spatial directions simultane-ously. The highest electric energy density that can berealized with any focusing optics is created by an elec-tric dipole wave . We therefore choose this type ofspatial mode in our experiment. The electric dipolewave is created by first converting a linear polarizedGaussian beam into a radially polarized donut modevia a segmented half-wave plate (B- Halle) [17, 18].Second, the radially polarized donut mode is focusedwith a parabolic mirror onto the trapped ion. This,in theory, enables us to convert approximately 91 %of the donut mode into a linear dipole mode . Theconversion efficiency is limited since the donut mode isonly approximating the ideal spatial mode that is nec-essary to create a purely linear dipole mode [2, 20] bybeing focused with the parabolic mirror. The donutmode, however, yields the experimental advantage ofbeing propagation invariant and comparably easy togenerate.
Our focusing tool, the parabolic mirror, is made ofdiamond turned aluminum (Fraunhofer Institute forApplied Optics and Precision Engineering, Jena) witha reflectivity of 64 % for the incident mode at a wave-length of exc = 369.5 nm. Its geometry has a focallength of 2.1 mm and an outer aperture of 20 mm in
FIG. 1. Optical set-up of the experiment and relevantenergy levels of 174Yb+. The cooling laser (blue) and therepump laser (red) are focused onto the ion through a holeat the backside of the parabolic mirror. AOM - acoustooptical modulator, DM - dichroic mirror, F - clean up fil-ter, 4-PM - 4 parabolic mirror, PMT - photo multipliertube, SHWP - segmented half-wave plate.
diameter. In total, the geometry covers 81 % of thecomplete solid angle. This fraction corresponds to94 % of the solid angle that is relevant for couplingto a linear dipole oriented along the axis of symme-try [11, 19]. Furthermore, the mirror has three boresnear its vertex: two bores with a diameter of 0.5 mmfor dispensing neutral atoms and for illuminating theion with additional laser beams, respectively; and onebore with a diameter of 1.5 mm for the ion trap it-self. The ion trap is a Stylus-like Paul trap similarto  with high optical access. The trap is mountedon a movable xyz piezo translation stage (PIHera P-622K058, Physik Instrumente) that is used for mea-suring the effective excitation PSF. The effective ex-citation PSF is defined as the convolution of the focalintensity distribution with the spatial extent of theion.
We measure the effective excitation PSF by prob-ing the focal spot at different positions. In order todo so, we use the translation stage to scan the ionthrough the focal spot with an increment of 25 nm.At each position, the incoming dipole mode excitesthe S1/2 - P