Digital Darkfield Displacements

Below is a (half kibipixel)^2 image portraying a simple square lattice that has been locally expanded by the presence of four spherically-symmetric coherent precipitates, each surrounded by a 1/r^2 displacement relaxation.

Below this is a power spectrum of the above image, showing Fourier transform amplitudes as a function of spatial frequency in the original image. The DC peak (electrical engineering parlance) or "unscattered beam" (diffraction lingo) is in the center of this pixel array.

Below this is a corresponding 16x16 array of logarithmic complex-color darkfield images. The array takes the same storage space as the image and its transform alone. Each image in the array is a reduced-resolution version of the original image, formed from Fourier coefficients corresponding to the position of the image in the array (relative to the pale red central brightfield, DC-peak, or "unscattered beam" image).

Each darkfield image is thus generated by a band pass filter (or back focal plane aperture) centered on a given frequency and direction, i.e. is associated with one reciprocal lattice vector. Within the darkfield images, brightness denotes the amplitude of that reciprocal lattice periodicity as a function of position in the image, while the color or "hue" (e.g. red, magenta, blue, cyan, green, yellow, red) denotes the phase deviation (relative to fixed) of that periodicity as a function of position in the image. It is the gradient of this latter (the local phase error or lattice displacement) that may be used to map lattice strain quantitatively in larger versions of these darkfield images.

If you look closely, within each darkfield image that corresponds to a diffraction spot in the power spectrum you will find reduced images of each precipitate. These images have a butterfly appearance, with an axis (or line of no contrast) orthogonal to the reciprocal lattice vector (i.e. to a line drawn between the brightfield image in the array's center, and the darkfield image of interest). This line of no contrast, also manifest in the gradient field associated with lattice strain, is a prominent feature of diffraction contrast from lattice defects in the transmission electron microscope, at least under "two-beam" conditions (i.e. when one set of lattice planes is diffracting much more strongly than any other).