Fourier transform

The Fourier transform is a mathematical transformation that appears in many branches of physics. One feature of this transform is that periodic signals in the input stream are converted into well-defined peaks in the output Fourier space.

The Fourier transform is central in scattering. In scattering, an incident wave travels through a sample. All of the entities in the sample act as scattering sources, giving rise to secondary waves that interfere with one another. This wave-interference phenomenon is essentially physically performing the Fourier transform operation. Thus, the observed scattering pattern (patterns of constructive and destructive interference) is the Fourier transform of the realspace density profile probed by the wave. The full 3D reciprocal-space is the Fourier transform of the sample's structure.

Mathematical form

The Fourier transform is typically given by:

${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx$

The transform inverts the units of the input variable. For instance, when the input stream represents time, the Fourier space will represent frequency (1/time). When the input stream represents space, the Fourier space will represent inverse-space (1/distance). The Fourier transform can also be inverted:

$f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{2\pi i\xi x}\,d\xi$

Scattering

The fundamental equation in scattering is:

${\begin{alignedat}{2}I(\mathbf {q} )&=\left\langle \left|\sum _{n=1}^{N}\rho _{n}e^{i\mathbf {q} \cdot \mathbf {r} _{n}}\right|^{2}\right\rangle \\\end{alignedat}}$

Where the observed scattering intensity (I) in the 3D reciprocal space (q) is given by an ensemble average over the intensities of all (N) scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the scattering contribution ($\rho _{n}$ denotes the scattering power) of the N entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous density of the scattering $\rho (\mathbf {r} )$), we can write an integral over all of real-space:

${\begin{alignedat}{2}I(\mathbf {q} )&=\left|\int \limits _{V}\rho (\mathbf {r} )e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} V\right|^{2}\\\end{alignedat}}$

The inner component can be thought of as the reciprocal-space:

${\begin{alignedat}{2}F(\mathbf {q} )&=\int \limits \rho (\mathbf {r} )e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} V\\\end{alignedat}}$

As as be seen, this is mathematically identical to the Fourier transform operation previously described.