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e e p EELS PINEM (EEGS) 20 10 0 -10 -20 -30 -40 20 10 electron energy (eV) 0 -10 -20 -30 -40 electron energy (eV) Figure 1. Comparison of interactions in EELS and PINEM. a nanoparticle with external light and probing it with the energy gain process of a swift electron. Barwick et. al. demonstrated this technique for nanostructures with a femtosecond laser in ultrafast electron microscopy (UEM), and termed it photon-induced near field electron microscopy (PINEM), emphasizing the capability to spatially map the photon-electron interaction. 27, 28 Theoretical accounts have been given by various groups. 29–31 Here, we present the imaging aspect of PINEM, in particular, the capability to visualize the optical near field by spatially mapping the photon-electron interaction, and compare it to that by EELS. In EELS, swift electrons induce a transient electric field with a continuous spectrum of frequency and excite plasmons in nearby particles. In PINEM, a single-frequency optical pulse drives plasmons in particles, and electrons interact with confined plasmon fields (scattering). Therefore, PINEM maps the field of a single plasmon excited by light, whereas EELS maps the excitation efficiency of plasmons. We demonstrate this for a dielectric sphere, a silver nanorod, and a silver triangle, and discuss the plasmon field distribution with respect to PINEM images. 2.1 <strong>Electron</strong>-plasmon coupling 2. THEORETICAL In EELS (see Fig. 1), a moving electron imposes a spectrum of the electric field of Ẽ e (r, ω) = 2eω [ ɛveγ 2 exp i ω ] [ i z ɛ v e γ ɛ K 0 ( ω v e ) ( b ω ˆv e − K 1 γ ɛ v e ) b ˆb γ ɛ ] , (1) √ where b is the impact parameter, γ ɛ = 1 − ɛ v2 e c is the Lorentz contraction factor for an electron moving at 2 the velocity, v e , in dielectric medium, ɛ, and K n is the modified bessel function of the second kind. 22 This transient electric field excites plasmons in a nanoparticle nearby, and the electron loses an energy quanta of ¯hω. The plasmon field is determined by the geometry and material of the particle, and the transient field of a swift electron. Classically, the efficiency of the energy loss process is calculated by the rate of mechanical work performed on the electron by the electric field, Γ EELS (ω) = e π¯hω ∫ [ dtRe v e · {Ẽ (re , ω) e −iωt}] , (2) where r e = v e t and Ẽ is the plasmon field of the particle. In EELS, an electron both excites and probes plasmons, producing an efficiency map. A strong spatial dependence in Eqn. (1) results in a significant excitation of plasmon only when the electron trajectory is very close to the plasmon mode. The angular dependence of the radial component allows excitations of the plasmon modes of an antisymmetric field when the electron impinges inside the particle, unlike light which, in the Rayleigh regime, can only excite a dipole mode. 24 Proc. of SPIE Vol. 8845 884506-2
2.2 <strong>Photon</strong>-electron coupling Contrastingly, in PINEM (see Fig. 1), plasmons are excited by the linearly-polarized planar wave incident light of [ ] Ẽ p (⃗r, t) = Ẽ0ˆx (z − c (t + τ))2 exp [i (k p z − ω p t)] exp − 4c 2 σt 2 , (3) p where σt p is the photon duration, and τ is the delay between electron and photon pulses, and electrons interact with the plasmon fields and gain and/or lose multiple light quanta, n¯hω p . A detailed derivation of the transition probability of PINEM in a non-relativistic formulation was given in Ref. 30 and summarized in Ref. 32, where the time-dependent 1D Schrödinger equation of a free electron under a scattered electromagnetic wave was analytically solved to first order. The relativistic formulation was also given in Ref. 32, and it was shown that the PINEM probability in a non-relativistic formulation is, to first order, equivalent to that in a relativistic formulation as long as the relativistic velocity of an electron is used. The temporal filter aspect of PINEM was given in Ref. 33 and utilized to characterize the temporal spread and the energy-time correlation of an electron pulse in Ref. 34. When the total density of a propagating electron packet is given by P e (z ′ ) on a moving frame, z ′ ≡ z − v e t, the electron density in the nth order state becomes and the nth order transition probability, Q n , is given by Q n (z ′ ; τ) = |ξ n | 2 ⎛ ⎞ ξ n (z ′ ; τ) = ⎝ ˜F 0 ∣ ˜F ∣⎠ ∣∣ 0 P n (z ′ ) = P e (z ′ ) Q n (z ′ ; τ) (4) n ⎧ ⎨ e J n ⎩ − ∣ ˜F ∣ ∣∣ [ 0 exp − (z′ − v e τ) 2 ¯hω p 4veσt 2 2 p ] ⎫ ⎬ ⎭ , (5) where −e is the electron charge, and J n is Bessel function of the first kind. The exponential function describes the overlapping of electrons with the incident light pulse at the position of particle, z = 0, in a slowly-varying envelope approximation. 30 For a weak interaction, we obtain Q +1 ∝ ∣ ˜F ∣ 2 ( because J n (Ω) ≈ 1 Ω ) n n! 2 for a small argument. The field integral, ˜F , is defined as ∫ +∞ ˜F (x, y) ≡ v e dt ′′ Ẽ z (x, y, v e t ′′ , t ′′ ) (6) −∞ which is mechanical work (per charge) performed by an electromagnetic wave (of ω p ) on a traveling electron at z ′′ ≡ v e t ′′ , and therefore equivalent to Eqn. (2). By factoring out the time dependence, e −iωpt , Eqn. (6) is reduced to ∫ +∞ [ ( ) ] ˜F 0 (x, y) = dz ′′ Ẽ z (x, y, z ′′ ωp , 0) exp −i z ′′ (7) v e −∞ which becomes the Fourier transform of the z component of the scattered electric field at a spatial frequency ∆k ≡ ωp v 0 . The field integral is determined by the scattered light and the electron velocity, and it describes the magnitude of photon-electron interaction. Therefore, the task of PINEM image simulation reduces to calculating the light scattering of a nanoparticle and then evaluating Eqn. (7) as a function of position, (x,y). Note that the scattered electric field is a three-component vector quantity in 3D (x, y, z) space, while the field integral is a scalar quantity of 1D (z) projectile trajectories, in 2D (x, y) subspace. Similarly, an object is a 3D distribution of density, ρ (x, y, z), while a TEM image, I (x, y), is a set of projections (integrations of electron scattering density along z) in the x-y plane. Proc. of SPIE Vol. 8845 884506-3
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