Magnetic monopole field exposed by electrons

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1 Mgnetic monopole field exposed y electrons A. Béché, R. Vn Boxem, G. Vn Tendeloo, nd J. Vereeck EMAT, University of Antwerp, Groenenorgerln 171, 22 Antwerp, Belgium Opticl xis Opticl xis Needle Smple Needle Ojective perture Lorentz imging system Bck focl plne ROI virtul source Reference virtul source Biprism Hologrm Imge plne Off-xis hologrphy Gouy phse experiment Supplementry Figure 1: Sketches of experimentl setups. () Off-xis hologrphy consists of superimposing reference wve with wve pssing through the smple. The resulting fringe pttern, the hologrm, crries informtion out the phse of the wve. () The Gouy phse experiment consists of cquiring through focus series in the ck focl plne. When pssing through the focus point, the shdow imge flips. The insertion of shrp edge locking prt of the em, here n ojective perture, llows etter visulistion of this imge inversion. In the cse of n oject illuminted with vortex em, s sketched here, the imge is not simply flipped ut lso rottes when going through focus in direction given y the sign of the oritl ngulr momentum. NATURE PHYSICS 1

2 um um Supplementry Figure 2: Evolution of the phse round the nickel mgnetic needle over the full perture. This phse mp ws clculted from finite element simultions tking into ccount the experimentl dimensions of the needle (the scle figures phse chnge of 2π). The volume mgnetiztion vlue ws dpted in the simultion to mtch the experimentl chnge in phse round the tip (Fig.2 ). This results in volume mgnetiztion of 1.72e5 A.m 1 insted of 4.88e5 A.m 1 expected for ulk Ni. This discrepncy cn e ttriuted to gllium implnttion during the FIB preprtion of the needle, reducing the effective volume of Ni which contriutes to the mgnetism. 2 NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION c d Supplementry Figure 3: Experimentl hologrms nd phse mps round the needle tip. () Experimentl hologrm cquired t the tip of the needle in field free conditions (Lorentz mode), the inset shows n enlrged view of the hologrphic fringes ner the tip of the needle. Note the sence of forking fringes s further explined in Supplementry Figure 7. () Hologrm of the sme region tken fter chnging the direction of the mgnetic field together with mgnified view of the tip s inset, lso reveling the sence of forking pttern in the hologrphic fringes. To chnge the direction of the mgnetic field, the needle ws tilted 3 out of plne nd n out of plne mgnetic field of.15 T ws pplied y slightly incresing the strength of the ojective lens. The lens ws turned off nd the needle ws tilted ck in plne efore cquisition of the new hologrm. (c) Phse mp otined from experimentl hologrms () with the mgnetic field pointing into the tip of the needle. The scle figures phse chnge from to 4π. (d) Sme imge tken with the mgnetic field pointing out of the needle. The reversl of the mgnetiztion chnges the hndedness of the phse distriution s expected. Across the needle, lck color is used s the phse scle sturtes due to the interction with the men inner potentil of the mteril. The color scle is chosen this wy to emphsize the mgneticlly induced phse. The sum of oth figures (c) nd (d) ws used to remove the influence of the electrosttic potentil nd to crete Fig 2.. NATURE PHYSICS 3

4 c Experimentl profile Simulted profile Phse chnge (rd) nm 2 nm Angulr position (rd) Supplementry Figure 4: Comprison of experimentl nd simulted phse shifts. () Experimentl phse mps round the tip of the needle. () Simulted phse mps corresponding t the sme position s (). (c) Phse profiles long circulr pth round oth experimentl () nd simulted () phse mps. The rdius of the integrting loop is 195 nm nd the profiles strts from the rrow sketched in () nd (). Note the ner liner phse decrese in oth experimentl nd simulted profiles s well s the reconnection of the phse over the needle which is needed for divergence free field. This reconnection hs sutle effect on the hologrms ut the mjority of the electron wve sees n zimuthl phse leding to cler vortex stte even though the phse is not discontinuous..3 5 Frctionl mode weighting OAM Modes Supplementry Figure 5: OAM Eigenmodes decomposition of the simulted phse in the full perture. The simulted phse presented in Supp. Fig. 2 ws used s source for the decomposition. Note the cler shift towrds negtive OAM, indicting the ction of the monopole field. The width of the distriution is cused y the fct tht the needle is conicl s opposed to n idel cylindricl needle which would led to single OAM mode. 4 NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION nm nm - 37 nm nm nm nm nm - 37 nm nm nm c nm nm - 37 nm nm nm d Focus 335 nm Experimentl profile Simulted profile Non mgnetic simulted profile Focus +323 nm Experimentl profile Simulted profile Non mgnetic simulted profile Reltive intensity (u.).8.6 Reltive intensity (u.) Position (pixels) Focus 37 nm Experimentl profile Simulted profile Non mgnetic simulted profile Position (pixels) Reltive intensity (u.) Position (pixels) Supplementry Figure 6: Experimentl nd simulted through focus series. () Experimentl through focus series of the electron intensity ner the diffrction plne. () Wve opticl simultions ssuming the theoreticl phse profile from Fig.2c. The simultion is simply Fourier trnsform of the complex wve in the needle plne with Fresnel defocus term to simulte the effect of defocus. Note the strong similrity with the experiment. In order to rule out tht the lck centrl prt in () is cused y shdowing effect, n lterntive simultion ssuming flt phse (no mgnetic effects) profile with totlly locking needle is given in (c). Indeed shdow of the needle is fintly visile ut contrry to the experiment the shdow is not truly lck nd disppers completely when in focus. These simultions provide strong evidence tht true vortex stte is produced. (d) Intensity profiles re given to further show the essentil effect of the destructive interference ner the center of the electron vortex. NATURE PHYSICS 5

6 c e 2 nm 2 nm 2 nm d f Supplementry Figure 7: Simultion of the effect of phse discontinuity on the hologrphic fringes. () Phse mp ssuming perfect zimuthl phse profile with m=1. () This phse profile leds to typicl forked hologrphic pttern. (c) Phse mp presenting continuous pproximtion to phse displyed in () with steep ut finite reconnection ner the ottom left corner. (d) Resulting hologrm which still resemles () ut upon detiled inspection the fork hs disppered nd ll fringes re single continuous lines s emphsized y the dotted lines. (e) Theoreticl phse shown lso in Fig. 2c. (f) Resulting hologrm. Note the similrities to (d) without ifurctions of the fringe lines in greement with the experimentl hologrms (Supp. Fig. 3, insets). These simultions only tke into ccount the mgnetic field effect while the electrosttic potentil inside the needle is not simulted for clrity. 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION Mgnetic needle True monopole m=-1 m=1 m=1 Supplementry Figure 8: Sketch of the fundmentl difference etween the interction with n pproximte monopole field nd true hypotheticl mgnetic monopole. () The pproximte monopole field will imprint continuously vrying phse on the incoming electron wve with n zimuthl phse profile tht is reconnected over the width of the needle. In the immedite vicinity of the needle there re therefore no phse singulrities nd no forked hologrphic fringes would occur if the phse is mesured in this plne (this is consequence of the field eing divergence free). Nevertheless, propgting the electron wve ehind the needle (going from top to ottom), the wve will develop vortex-ntivortex pir indicted y the lck ent rrow. This mens tht two singulrities rise with opposite winding numer. The positive singulrity remins on the opticl xis nd is surrounded y the mjority of electron density in the wve, while the opposite singulrity will split off nd shift to high ngles (depending on the steepness of the phse reconnection on the needle) nd to region where the electron density is nerly zero. The fct tht the mjority of the electron density is occurring round the min singulrity lso mens tht rel vortex stte with n OAM of h is creted. The unwnted opposite singulrity cn e excluded y mking use of n perture if needed. () In cse of true monopole field, the sitution is different nd single phse singulrity is creted strting from the monopole. This leds to the formtion of n electron vortex wve with OAM of h. Both setups therefore hve nerly the sme effect on the electron with the exception of the totl winding numer in the oservtion plne which is zero for n pproximte monopole field nd equls one for the true monopole field. Putting n perture to cut out the region of the opposite singulrity rings oth cses even closer together. For ll prcticl mtters, monopole field therefore ehves nerly identicl to the field produced y true monopole s long s we don t oserve the ner field re of the scttering. NATURE PHYSICS 7

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