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.

Phase problem

Note that experimentally, we can only measure the squared amplitude, I(q); we can never directly measure the true reciprocal-space, given by F(q). This is known as the phase-problem: our detector only measures the intensity of the scattered radiation, and not the phase of the scattered waves. As such, the scattering dataset is incomplete. If we had both the amplitude and phase of reciprocal-space, we could simply invert the data (using an inverse Fourier transform), and thereby recover the realspace structure of the sample. Lacking the phase information, we cannot do this. Instead, scattering data must be fit to candidate models, with concordant models being deemed 'more likely' to represent the sample. However, interpreting scattering data is a formally ill-prosed problem: there are multiple possible realspace models that will fit a particular experimental dataset. Abstractly, it is impossible to know which model is the right one. In practice, one can use auxilliary knowledge, such as other measurements (AFM, electron microscopy) or physical constraints, in order to exclude some candidate models, and thereby identify the correct sample structure.

There are many interesting research thrusts aimed at 'solving' the phase problem. For instance, one can conduct repeated measurements on the same (or highly similar) sample using different contrast conditions. For neutrons, this can be accomplished by isotopic substitution or by using polarized neutrons interacting with magnetic layers. For x-rays, one can vary the x-ray energy to modify the relative scattering contrasts. In any case, one then attempts to simultaneously fit the combined dataset in a self-consistent way; this narrows the range of possible solutions and thus recovers some phase information.

Alternatively, some methods exploit a high degree of coherence in the incident radiation in order to reconstruct the realspace structure (e.g. Coherent Diffractive Imaging).