Ronchigram defined by the smallest angle 2π

the aberration is the same, n is the order of the aberration, λ is the wavelength of the electron beam, α is the convergence angle, and φ is the azimuthal angle. Te popular Haider nota- tion [11] is also provided on To speed com- putation, we treat our specimen as a randomly generated noisy phase grating, which under the Eikonal approximation adds

m such that the phase shiſt of

random phases to the electron beam e iVnoise−⋅ πσ4

interaction parameter 2 me heπλ/ where me 2 a cutoff frequency at kmax

. Here, σ is the is the relativistic

() x

provides random white noise values between 0 and σ200 1

electron mass and e is the charge of an electron, and Vnoise(x) −

/2. Finally, the observed Ronchigram

is the electron probability density on the diffraction plane and is equivalent to the square of the modulus of the Fourier trans- form of the transmitted wavefunction g(k) = |FT{ψt

(x)}|2 .

Details Within the Ronchigram Key features in the texture of a Ronchigram’s fringes

Figure 2: Historical light Ronchigrams captured using Ronchi’s linear grat- ing technique initially developed in 1923 [1]. Interference fringes in (a) a per- fect optical system are parallel, and in (b) an astigmatic system are serpentine. Reprinted with permission from [2], The Optical Society.

associated with electron beams, while the light optics commu- nity more commonly refers to the “Ronchi test.”

A Portable Ronchigram Simulator To promote an intuitive understanding of electron Ronchi-

grams, and how they are affected by aberrations, we developed the web application (Figure 3). Tis app calculates the electron Ronchigram of a thin amorphous sample—the same conditions used during a typical alignment. Trough an intuitive graphical user inter- face, electron wavelength, objective aperture size, and up to 5th order aberrations can be specified or ran- domly generated. Concisely stated, the Ronchigram is the dif-

fraction pattern of a convergent beam focused on an amorphous specimen. We calculate the Ronchigram from the probe wavefunction described by the Fou- rier transform of the aberration function transmit- ted through (that is, multiplied with) an amorphous

where the lens aberration function, χ(k), is defined as: n+1

specimen potential: ψ =⋅t() {}FTee x

χ αφ π λ

(, ) = 2 ∑ nm, −iχ k

Cm n

Te terms Cn,m and φn,m () nmαφ φnm ,,

cos( ()) + 1

− describe a geometric

aberration in Krivanek notation [10], m is the degree of the aberration, which for cylindrically symmetrical aberrations is zero and for asymmetric aberrations is

2019 May •

Figure 3: The web application allows exploration of Ronchigrams on any laptop computer or mobile phone. The blue circle visible in the Ronchigram on the phone indicates the maximum aperture size for the highest resolution achievable.


−⋅ 4

iV x π σ noise() ,

include magnification and symmetry. For an in-focus beam, the center of a Ronchigram has high local magnification that represents a coherent, nearly aberration-free portion of the beam (blue circle in Figure 3). Moving further from the optic axis (that is, the center of the pattern), aberrations reduce the local magnification. Asymmetric aberrations (m > 0) break rotational symmetry. When aligning a microscope, the opera- tor tunes the currents through the electron lenses to minimize these aberrations. Te presence of lower-order aberrations affects the struc-

) unidirectionally stretches the region of high magnifica- tion and produces distinctive streaking, axial coma (C21 the center of the Ronchigram, and three-fold astigmatism (C23

ture and symmetry of the Ronchigram. Two-fold astigmatism (C12

) shiſts )

produces three-fold symmetric lobes (Figure 4). Experimen- tally, adjusting defocus can enhance the visibility of aberration symmetry. In a well-aligned aberration-corrected microscope, these lower-order aberrations can be nearly completely cor- rected, resulting in a large featureless region of high local magnification oſten with a characteristic six-fold structure attributable to residual higher-order aberrations [12,13]. While

keV with

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