Fourier Transform in Microscopy

In electron microscopy a lens can be placed behind a sample. Incident electron waves hit the sample and it creates scattering. Constructive interference and wave fronts are produced all over the sample (the lines on the picture). The lens gets parallel illumination and focuses it at a particular spot behind the lens. Other rays also hit the lens and each are focused at a particular spot behind the lens (the dot where intersection of lines can be seen). The spots on the back focal plane is the Fourier transform of the sample density (diffraction pattern). This will spread out again and it will interfere with other waves. Each ray represents a particular sine wave of a particular frequency; scattering at a different angle represents another. It will carry through and it will produce an image on the image plane. After it does Fourier transform, it will perform inverse Fourier transform (Fourier synthesis) wherein it will take each sine waves and add it up to reproduce the replica of the density sample. This copy can can be larger than the original sample.

fft_electron microscope