Supporting Information for Relativistic effects in Photon-Induced Near Field Electron Microscopy

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1 Suorting Information for Relativistic effects in Photon-Induced Near ield Electron Microscoy Sang Tae Park and Ahmed H. Zewail Physical Biology Center for Ultrafast Science and Technology, Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 9115 Dated: May 9, 1 I. DIRAC EQUATION The Dirac euation for an electron in an electromagnetic wave is given by i Ψ t cα A φ βmc Ψ S.1 where e is the electron charge, A is the vector otential, and φ is the scalar otential. α and β are unit constant matrices chosen to satisfy the relativistic energy-momentum relation. or a one dimensional euation, along the z-direction, α, β, and Ψ are given by two comonents. as 1 α S. 1 1 β S.3 1 ψ1 Ψ S.4 ψ and en. S.1 becomes i t ψ1 ψ φ mc c ˆ z A z ψ1 c ˆ z A z φ mc ψ S.5 II. OURIER TRANSORM O GAUSSIAN UNCTION ourier transformation for differentiation shows the following roerty: n d 1 i n k n ˆ k ˆ k dz The Gaussian rofiles in momentum and osition saces are defined as, ] 1 Ĝ k; σ k ex k πσk then with σ z 1 σ k, they satisfy G z; σ z 1 πσz ex Ĝ 1 k k ; σ k i 1 k k Ĝ k k ; σ k σk z σ z ] S.6 S.7 S.8 G z; σ z e ikz S.9 d G z; σ z e ikz dz S.1 zewail@caltech.edu

2 III. RELATIVISTIC PINEM SOLUTION As in the non-relativistic case, the vector otential of the scattered wave by linearly olarized incident wave is given by A z 1 Ãz Ãz 1 Ẽz Ẽz 1 ] Im Ẽz S.11 iω iω ω The temoral deendence of the scattered electric field can be searated as Ẽ z, t Ẽ z, ex iω t] for a continuous wave. or an incident otical ulse of duration much longer than the electron transit time, the scattered wave in the vicinity of the nanostructure can be aroximated as ] Ẽ z, t Ẽ z, ex iω t τ t] ex 4σ S.1 Then, we obtain g z, t g z, t ex i v ]] t dt Im Ẽ z z t, ex iω t ] ex t τ ω t 4σ S.13 By substituting z z t in the integration and using the fact that Ẽ z, is only significant around z, we derive g z, t g z, t ex z t i dz Im Ẽ z z z, ex iω z ] ]] ex z z τ ω z t 4v σ ex i Im ex i ω ] z t z dz Ẽ z z, ex i ω ] ]] z ex z τ ω z t 4v σ By substituting t and t, the final state can be exressed as g z, g z, ex i Im ex i ω ] z dz Ẽ z z, ex ω Using the definition of ω g z, g z, ex dz Ẽ z z, ex i Im ω i ω z ] ]] ex z τ 4v σ ω i z ], en. S.14 becomes ]] ex i ω ] z ex z τ 4v σ S.14 S.15 The electron art in en. S.15 is identical to en. A1 in the revious ublication 1], and no relativistic correction is reuired when the relativistic velocity is used for both formulations, and corresonding k c and ω c for the classical formulation. Similarly, we Taylor-exand the exonential functions in en. S.15, re-substitute Im function by subtraction of its comlex conjugate, and rearrange it using the definition of a Bessel function Jacobi-Anger relation to obtain g z, g z, ex in ω ] z n ] J n ex z τ ω 4v σ S.16 IV. INAL DIRAC WAVEUNCTION We define the sinor vector as û k û k u 1 k u k u 1 k u k S.17 S.18

3 n ] and define ξ n z J n ω ex z τ and g 4v σ n z g z, ξ n z, such that g z, ex in ω ] z g n z 3 S.19 The final state wavefunction is retrieved by substituting f z, t g z t, t and f z, t 1 c iω γ f z,t ex i ω ] ω t in en. 17 of the main text at t, where ω γ mc, such that which becomes Ψ z, t Ψ z, t ex g z, û k i g z, γ mc û k ] in ω ] z g n z û v k i ex in ω z g n z γ mc ex i k z ω t] û k ex i k z ω t] S. S.1 where the differential term becomes ] ex in ω z g n z En. S.1 becomes Ψ z, t g n z û k g n z n ω g n z û k nω in ω ex in ω ] z g n z ex in ω ] z g n z γ k γ k g n z g n z i γ k i γ k S. ex in ω ] z ex i k z ω v t] ex in ω ] z ex i k z ω v t] ex in ω ] z t ex i k z ω v t] g n z û k g n z i n γ k g n z û k g n z i n γ k ex i k n z ω n t ] g n z û k g n z i n γ mc û k nω g n z û k g n z i n γ k n ex i k n z ω n t ] g n z û k n i g n z û k ex i k n z ω n t ] k kkn mcû γ k ex i k n z ω n t ] where k n k n ω, and ω n ω nω. An aroximation of û k n û k was used in the small range of k limit. In the momentum sace, the wavefunction becomes Ψ k g n z t ex i k n z ω n t ] û k k k û k k g n z t ex i k n z ω n t ] ; k u 1 k u k g n z u ; k k 1 k n u k kkn ; k S.3 S.4 S.5 S.6

4 4 The robability of each wavelet becomes P n kn 1 ω k n 1 ω kn 1 ω k n 1 ω dk g n ; k k n dk g n ; k k n dk g n ; k k n S.7 S.8 S.9 when each wavelet does not overla with the other, as g n ; k k n g m; k k m. En. S.9 is euivalent to the classical counterart: Ψ k P n g n ; k k n dk g n ; k k n dz g n z S.3 S.31 V. RELATIVISTIC MASS Relativistic transverse and longitudinal masses are m T m L γm v γ 3 m a v S.3 S.33 VI. CLASSICAL EQUIVALENCE The Schrödinger formulation of electromagnetic interaction is relativistically valid when classical momentum and energy are used as k c c m ω c T c 1 mv γ γ 1 mc γ γ 1 γ T S.34 S.35 or when the momentum with the relativistic transverse mass, m T γ m, and corresonding energy are used as k c c m T γ m S.36 ω c T c 1 γ m T v 1 γ mc γ 1 T S.37 γ In both cases, en. 4 in the main text is identical to en. 4, and en. 4 does not reuire a relativistic correction, as long as the actual velocity,, is used. Correct usage of arameters in various formulations are summarized in Table S.1. γ 1] S. T. Park, M. Lin, and A. H. Zewail, New J. Phys. 1,

5 5 TABLE S.1: Comarison of formulations m v i k i i/ ω i E i/ k ω Exeriment a m e.695c k.967m ec/ ω 1.391m ec / ω Dirac formalism m e k ω Schrödinger formalism m e k c.695m ec/ b ω c.4m ec / c ω Effective mass icture d m T 1.391m e k ω T.336m ec / e ω a or the exeriment, γ 1 T γ m ec is evaluated, and one obtains c 1, γ k m ec γ 1, and E ω γ m ec. b k c m c ω c mev d m T γ m e e ω T γ m ev ω ω ω ω ω