An Atlas of Fourier Transforms

The relation between real and Fourier space is essential to the study of matter. This is especially true for electron microscopy which leverages real and reciprocal space for the measurement of materials, instrument operation, and data analysis. Thus, a deep intuition for Fourier transforms across dimensions is required for experts in the field. However, mastery of the frequency domain can take years, sometimes decades, to develop [1]. Here, we introduce a library of Fourier transforms as an advanced and introductory reference or education manual. The Atlas of Fourier Transforms is a curated compilation of 2D structures and their Fourier transforms. For beginners, the atlas is an advanced course in Fourier transforms—images graphically explain the underlying mathematics. As the atlas progresses, so does the complexity of each transform. Concepts in symmetry, rotation, translation, addition, multiplication, order, and disorder are introduced and combined. For the experienced, it provides an essential reference to a comprehensive dictionary of structures and their Fourier transforms. To illustrate, Figure 1 shows two representative pages taken from The Atlas of Fourier Transforms. Pages are presented as diptych pairs: on the left, the real-space structure, and on the right, the corresponding Fourier transform. While each real-space structure is made from real values, the structure in Fourier space contains complex values comprised of amplitude and phase. Here, the amplitude and phase are mapped to colors on the Munsell sphere: the amplitude is mapped to an intensity value between black and white; phase is represented by the color hue. In the quantum mechanics of electron scattering, only the amplitude is observable, however, the phase often plays an important role in the structure of measured amplitudes. This atlas provides a referenceable intuition into the arrangement of phase in Fourier space. Each row in Figure 1 teaches a concept about emergent structure in Fourier space. The first row of images highlights basic symmetry and linearity of Fourier transforms. The Fourier transform of a half circle is two-fold symmetric, like its real space structure. Reflecting a half circle reflects the Fourier transform. Adding two half circles results in a full circle, which is rotationally invariant. In reciprocal space, an Airy disk—with the same rotational symmetry—emerges as the complex phases of half-circles cancel. The second row introduces periodicities. Placing two circles vertically produces a vertical sinusoid, while placing circles horizontally forms a horizontal sinusoid in Fourier space. Note the wavelength is inversely proportional to the distance between circles. In column 3, we form a rectangular array that Fourier transforms into a lattice-like structure hinting at the origin of crystalline diffraction patterns. The third row of images shows the relationship between convolution in real space and multiplication in the frequency domain. The image shows a periodic triangular mask when applied to a crystal lattice structure. This is a multiplication in real space, which results in a convolution of the two transforms in k-space (i.e., frequency space). The last row contains a subset of aperiodic structures contained in the atlas. The first image shows regularly spaced circles along a spiral. The second image highlights polycrystallinity. The last image shows a topological defect with 2π phase winding. The atlas also includes other disorders and their transforms, such as stacking faults, interstitial disorder, and non-invertible crystal structures. Real space images alongside their 2D Fourier transforms. Real space images include simple shapes, periodic masking of crystalline structures, aperiodic structures, and disordered structures.