Unveiling the OAM and Acceleration of Electron Beams Roy Shiloh 1 , Yuval Tsur 1 , Roei Remez 1 , Yossi Lereah 1 , Boris A. Malomed 1 , Vladlen Shvedov 2 , Cyril Hnatovsky 2 , Wieslaw Krolikowski 2 , and Ady Arie 1 . 1 Department of Physical Electronics, Fleischman Faculty of Engineering, Tel Aviv University, Tel Aviv 6997801, Israel 2 Laser Physics Centre, The Australian National University, Canberra ACT 0200, Australia Electron beams, specifically in a transmission electron microscope (TEM), are mainly used to investigate biological samples and materials. It was not until recently that investigation of special kinds of beams, namely vortex beams, has begun [1,2]. These beams are especially interesting because they carry orbital angular momentum (OAM) which may be coupled to the atomic wave-function, thus enabling probing of magnetic dichroism [3], for example. In light-optics these beams have long been known, and research into other types of beams, such as accelerating beams, is flourishing. Here we study the well known Airy beam [4,5] in the electron microscope – a shape-invariant, multi-lobed, non- spreading beam whose nodal trajectory follows a parabolic dependence, which has already been exploited in light-optics to overcome the diffraction limit implementing a “super-resolution” technique [6]. Where for the case of vortex beams the OAM property is of utmost importance, in this work * we develop a tool for easy measurement of the Airy’s nodal trajectory coefficient, which is the defining property of the Airy beam, derive an elegant analytic model and verify it by fabrication of the relevant amplitude masks and consequent measurement and analysis. Our results agree completely with the proposed model, which is derived without approximations, and nicely relates light- to electron-optics via the geometric ray-tracing technique. In the optics literature, the familiar form of the Airy beam is given by  [ 0 −1 (− 2 /4  2  0 3 )], where  0 is the transverse length-scale,   the de-Broglie k-vector and (, ) the transverse and longitudinal coordinates, respectively.  is the Airy function. The quantity 1/(4  2  0 3 ) is the nodal trajectory parameter, sometimes referred to as the acceleration of the Airy beam, due to the coordinate’s parabolic dependence. The Airy beam is easily generated on the optical axis using, for example, the amplitude mask depicted in Fig.1f. This mask, a computer-generated hologram, recreates both the object and image of the encoded pattern in the Fourier (or diffraction) plane, which is why we observe two Airy-like patterns in Fig.1b,d. The nodal trajectory coefficient could be directly calculated by measuring the distance between two lobes and fitting it to the Airy function’s zeros; this is difficult, however, since the Airy (a diffraction pattern) must be in focus for an accurate measurement, thus endangering the camera CCD. The accuracy also strongly depends on the resolution. A second method would be to take a focus series and follow the trajectory of the main lobe [5]. Instead, our method involves a cylindrical transformation, easily achieved in the TEM by using the stigmator lenses. It is interesting to note that the same astigmatic transformation is also useful for determining the orbital angular momentum of vortex beams. Mathematically, the cubic phase imposed by the mask on the astigmatic beam is Fourier- transformed, thus yielding the “astigmatic Airy” (see Fig.1), the curve of which is dependent on the nodal trajectory coefficient according to the following formula: ( 2  0 3 ) −1  (∓1) = ∓(3 + ℎ) 32 ⁄ ℎ  2 ( 2  0 3 ) −1  (±2) = ±(−3+ ℎ) 32 ⁄ ℎ  2 * Accepted to Physical Review Letters, 2/2015. 21 doi:10.1017/S1431927615000902 © Microscopy Society of America 2015 Microsc. Microanal. 21 (Suppl 3), 2015 Paper No. 0011