Electron Holography Axel Lubk Converting phase shifts to contrasts: - PowerPoint PPT Presentation

Electron Holography Axel Lubk

Converting phase shifts to contrasts: Fresnel imaging - area of increased area of reduced intensity intensity −𝜀𝑔 ⊙ ⊗ ⊗ +𝜀𝑔 area of reduced area of increased intensity intensity 2

Fresnel imaging: Pros & Cons Pro: Con:  simple  (partially) non-linear contrast  fast  defocus → unsharp images  sensitivity adjustable  quantification difficult (but possible)  sensitiv to dynamical scattering Can be overcome by Holography! (now) Recommended reading: 1. Völkl , Edgar, Allard, Lawrence F., Joy, David C. (Eds.) , Introduction to Electron Holography, Springer (1999).

1. Fundamentals of electron scattering a. Axial scattering b. Magnetic and electric Ehrenberg – Siday – Aharonov – Bohm effect 2. Fundamentals of Electron Holography and Tomography a. Holographic Principle (interference, reconstruction) b. Holographic Setups (inline, off-axis) and instrumental requirements c. Separation of electrostatic and magnetic contributions d. Tomographic reconstruction of 3D electric potential and magnetic induction vector field from tilt series of projections

How do fields act on electrons waves? ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ t ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 𝛾 𝛾 𝛾 = − 𝑓𝑢 −𝐶 𝒛 𝛾 = − 𝑓𝑢 𝐹 𝒚 deflectio semiclassics 𝐶 𝒚 𝐹 𝒛 𝑛𝑤 0 2 𝑛𝑤 0 n angle initial 𝛾 = 1 𝑙 𝛼𝜒 velocity 𝜒 = 𝑓𝑢 𝜒 = 𝑓𝑢 phase 𝛸 ℏ 𝐵 𝑨 ℏ𝑤 0 shift 𝑞 = 𝑛𝑤 0 = ℏ𝑙 electrostatic magnetic vector potential potential momentum wave number

Ƹ How do fields act on electrons waves? * kinetic momentum 𝑞 2 ො reduced Klein-Gordon equation operator 𝐹Ψ = 2𝑛 − 𝑓𝑊 Ψ (high-energy approximation) 𝑞 = −𝑗ℏ𝛼 + 𝑓𝑩 ො Ψ = 𝑓 𝑗𝑙 𝑨 𝑨 𝜔 2 − 2𝑛𝑓𝑊 𝜔 paraxial approximation −2𝑙 𝑨 ℏ Ƹ 𝑞 𝑨 𝜔 = 𝑞 ⊥ 2 • 𝜖 𝑨 𝜔 ≈ 𝑗 − Ƹ 𝑞 ⊥ + 𝜏𝑊 − 𝑓 small-angle scattering ℏ 𝐵 𝑨 𝜔 2ℏ 2 𝑙 𝑨 • no backscattering  2D time-dependent Schrödinger equation 𝜖 𝑨 𝜔 ≈ 𝑗 𝜏𝑊 − 𝑓 axial approximation ℏ 𝐵 𝑨 𝜔 (wavelength << object details) • very small angle scattering ℏ𝑤 𝑊 − 𝑓 𝑓 𝜔 = 𝑓 𝑗𝜒 𝜔 0 → 𝜒 = න ℏ 𝐵 𝑨 𝑒𝑨 object * It is a good exercise to do derivation by yourself.

Phase shift by electric potential s 1 s 2 𝜒 𝑜𝑒𝑡 = 𝑓 𝜒 = 𝑙 න න V𝑒𝑨 ℏ𝑤 𝑡 2 −𝑡 1 object refractive index

Detectable phase shift * source − 2𝜌 𝑓 Ԧ Δ𝜒 = 𝜏 න 𝑊 𝑒𝑡 ර 𝐵(Ԧ 𝑠)𝑒Ԧ 𝑡 ℎ V 1 V 2 𝑡 2 −𝑡 1 𝑡 2 +𝑡 1 electric magnetic s 1 s 2 2𝜌 𝑓 Δ𝜒 = 𝜏 𝑊 p,1 − 𝑊 − ℎ Φ p,2 detector phase difference  * Why can we only detect phase differences?

Detectable phase shift source 𝑓 Δ𝜒 = 𝜏 න 𝑊 𝑒𝑡 − ර 𝑩𝑒𝒕 ℏ V 1 V 2 𝑡 2 −𝑡 1 𝑡 2 +𝑡 1  electric magnetic s 1 s 2 𝑓 Δ𝜒 = 𝜏 𝑊 p,1 − 𝑊 − ℏ Φ p,2 detector phase difference 

Detectable phase shift source 𝑓 Δ𝜒 = 𝜏 න 𝑊 𝑒𝑡 − ර 𝑩𝑒𝒕 ℏ V 1 V 2 𝑡 2 −𝑡 1 𝑡 2 +𝑡 1  electric magnetic s 1 s 2 𝑓 Δ𝜒 = 𝜏 𝑊 p,1 − 𝑊 − ℏ Φ p,2 detector For the magnetic phase shift a Lorentz force phase difference  is not required at the electron trajectories !

Ehrenberg - Siday – Aharonov - Bohm Effect Proposal: Ehrenberg & Siday 1949 Aharonov & Bohm 1958 Experiment: Möllenstedt & Bayh 1962 Time Increasing Magnetic Flux

Magnetic phase shift  i t Φ(𝑦) x  f ref   z 𝜒(𝑦) = 𝑓 𝜒(𝑦) ℏ Φ(𝑦) Φ = ℏ = 2𝜌 magnetic flux quantum for 𝑓

Summary: object exit wave Phase object  Amplitude object a a exp[ i  ]

Summary: object exit wave phase modulation  (𝑦, 𝑧) : micro- /nanofields • electric • magnetic amplitude modulation 𝑏(𝑦, 𝑧) : • scattering into large angles • interference effects • inelastic scattering

1. Fundamentals of electron scattering a. Axial scattering b. Magnetic and electric Ehrenberg – Siday – Aharonov – Bohm effect 2. Fundamentals of Electron Holography and Tomography a. Holographic Principle (interference, reconstruction) b. Holographic Setups (inline, off-axis) and instrumental requirements c. Separation of electrostatic and magnetic contributions d. Tomographic reconstruction of 3D electric potential and magnetic induction vector field from tilt series of projections

Dennis Gabor Easter 1947, on the tennis court: ... and all of sudden it came to me, without any effort on my side. Interference and diffraction are mutually inverse 1902-1979 Nobel Prize 1971 Electron Holography measures phases

Dennis Gabor Holography Object wave hologram Image wave interference diffraction

Common Forms of Electron Holography focal series inline off-axis transport of intensity J.M. Cowley, 20 forms of holography , Ultramicroscopy 41 (1992), 335-348

Holography - Dennis Gabor ´ s idea object object exit wave propagation prop wave

Holography - Dennis Gabor ´ s idea object object exit wave propagation prop wave h o l o g r a m

Holography - reconstruction of wave object PROBLEM  INVERSE object exit wave back-propagation prop wave h o l o g r a m

Holography: basic scheme 𝑠 𝜔

Holography: recording hologram 𝑠 ℎ𝑝𝑚 = (𝜔 + 𝑠)(𝜔 + 𝑠) ∗ = 𝜔𝜔 ∗ + 𝑠𝑠 ∗ + 𝜔𝑠 ∗ + 𝜔 ∗ 𝑠 𝜔

Holography: reconstruction of wave 𝜔 𝜔 𝜔 ⋅ ℎ𝑝𝑚 = (𝜔𝜔 ∗ )𝜔 + (𝑠𝑠 ∗ )𝜔 + (𝜔𝑠 ∗ )𝜔 + (𝜔 ∗ 𝑠)𝜔 = 𝜔(𝜔𝜔 ∗ + 𝑠𝑠 ∗ ) + 𝑠 ∗ (𝜔𝜔) + 𝑠(𝜔𝜔 ∗ ) 𝑠 ∗ ∗ -wave 𝜔 -wave 𝑠 𝑠 -wave 𝑠 modulated in modulated in modulated in amp 𝜔; 𝑠 amp/phase 𝜔 amp 𝜔

Holography: reconstruction of wave 𝜔 𝜔 ∗ 𝑠 ⋅ ℎ𝑝𝑚 = (𝜔𝜔 ∗ )𝑠 + (𝑠𝑠 ∗ )𝑠 + (𝜔𝑠 ∗ )𝑠 + (𝜔 ∗ 𝑠)𝑠 = 𝑠(𝜔𝜔 ∗ + 𝑠𝑠 ∗ ) + 𝜔(𝑠 ∗ 𝑠) + 𝜔 ∗ (𝑠𝑠) 𝑠 𝑠 ∗ -wave 𝑠 -wave 𝜔 -wave 𝜔 modulated in modulated in modulated in amp 𝜔; 𝑠 amp 𝑠 amp/phase 𝑠

Plane reference wave r 𝜔 𝜔 ∗ 𝑠 ⋅ ℎ𝑝𝑚 = (𝜔𝜔 ∗ )𝑠 + (𝑠𝑠 ∗ )𝑠 + (𝜔𝑠 ∗ )𝑠 + (𝜔 ∗ 𝑠)𝑠 = 𝑠(𝜔𝜔 ∗ + 𝑠𝑠 ∗ ) + 𝜔(𝑠 ∗ 𝑠) + 𝜔 ∗ (𝑠𝑠) 𝑠 𝑠 𝜔 ∗ 𝑠 -wave -wave 𝜔 -wave modulated in modulated in amp 𝜔 phase 𝑠

Where to take the hologram ? Object plane ? Fresnel region Fraunhofer region Fourier plane In principle : „ where “ is not essential, but with electrons we are „ coherency- limited“ .....

Where to take the hologram ? Inline Holography Scattering Regimes Reconstruction Schemes    Illumination k 2 / Differential Defocus / Transport of Intensity Reconstruction Defocus Series Fresnel (near field) Reconstruction Fraunhofer (far field) Fraunhofer Holography Figure from Lee, Optics Express Vol. 15, Issue 26, pp. 18275-18282 (2007)

1. Fundamentals of electron scattering a. Axial scattering b. Magnetic and electric Ehrenberg – Siday – Aharonov – Bohm effect 2. Fundamentals of Electron Holography and Tomography a. Holographic Principle (interference, reconstruction) b. Holographic Setups (inline, off-axis) and instrumental requirements c. Separation of electrostatic and magnetic contributions d. Tomographic reconstruction of 3D electric potential and magnetic induction vector field from tilt series of projections

Transport of Intensity Reconstruction Paraxial Eq.  ( r , ) z i     ( r , ) z    z 2 k Continuity Eq. / Transport of Intensity Eq.   ( r , ) z 1      j ( r , ) z     z k density / intensity phase 1             ( r , ) z ( r , ) z    2 arg     k experimental data from 2 slightly defocussed images                  z z z z z    2 O z   z 2 z           z z , z z              z z z z       2 z O z 2

Transport of Intensity Reconstruction simpliefied TIE reconstruction minimal model   ( r , ) z 1      ( r , ) j z     z k 1          ( r , ) z ( r , ) z     k phase object   ( r , ) z const.     r ( , ) z      ( r , ) z    z k Poisson problem (e.g., solve with periodic boundary conditions)