Lensless imagingXFEL coherence and coherent scattering Due to its nearly monomode operation, the SwissFEL will produce radiation with a high degree of transverse coherence (see Infobox). To a good approximation, the radiation propagates as a Gaussian beam, with a divergence angle given by θ ~ λ/4πr0 ~ 1 μrad, for a wavelength λ = 1 Å and an rms source (electron beam) radius r0 ≈ 20 μm. The longitudinal coherence is poor, however, in par ticular for the unfiltered SASE radiation; the corresponding longitudinal coherence length ξlong = λ2/2Δλ is of the order of 50 nm, corresponding to a bandwidth Δλ/λ = 10-3. The longitudinal coherence can be increased by seeding of the SwissFEL electron beam or by filtering the photon beam with a monochromator. As discussed in , for thin samples and reasonably small scattering angles, good transverse coherence is sufficient to produce speckle, and hence to permit lensless imaging.
A comparison of the scattering of incoherent and coherent radiation by a collection of small objects is shown in Figure III.2. The Debye-Scherrer rings of the incoherent scattering pattern contain little information – basically the average separation d of the scattering objects. In contrast, coherent radiation incident in a beam spot with diameter a yields a rich pattern of interference speckles, each with an angular extent which is inversely proportional to a. Only the intensity of the radiation in the speckles is measured in a detector; the phase of the scattered radiation is lost. In order to obtain from the scattering pattern the distribution of scatterers in real space (projected onto a plane perpendicular to the incoming beam direction), one needs to perform an inverse Fourier transform, but first one needs to recover the lost phase information. This is the famous phase problem in scattering and crystallography, and its solution provides the form of the object without the requirement of diffraction-limited X-ray lenses. Several approaches to phase retrieval have been developed. One is to allow the scattered radiation to inter fere with an unscattered reference beam, thus per forming holography, as described for magnetic scattering in Chapter I. Other methods involve per forming oversampling and the iterative application of constraints . The coherent scattering pattern from an infinite array of identical objects with period d consists of a series of isolated (Bragg) peaks (representing terms in a Fourier series describing the array) at scattering angles satisfying sin θ = qλ/4π, where the scattering vector q can take the values qn = 2πn/d (n = 0, ±1, ±2, ±3, ...). The scattering thus samples the array at the scattering vector interval Δq = 2π/d, and only a single measured quantity, the scattered peak intensity, is available for each Fourier component. A finer sampling of the scattering pattern of the periodic array is not possible, since the scattering between Bragg peaks is zero. A single object of spatial extent a will, on the other hand, produce a continuous coherent scattering (speckle) pattern, which can in principle be sampled at an arbitrarily large number of scattering vectors. This oversampling can yield the additional information requried to recover the scattering phase. The phase-retrieval process is generally per formed in an iterative manner, by making an initial assumption for the phases and repeatedly Fourier and inverse-Fourier transforming between the real and scattering (reciprocal) space representations. Constraints are applied in each iteration cycle. In reciprocal space, it is required that the calculated scattering intensity agree with the measurements. In addition, one or more real-space constraints are applied: such as that scattering from the sample is zero outside a delimiting boundary, that the sample’s electron density is non-negative, or that the scattering is concentrated in isolated “atoms”. A fur ther method of per forming phase-retrieval, the ptychographic method, is described later. The 1-dimensional case discussed above can easily be extended to a 2-dimensional projection, and if a series of scattering measurements can be recorded for different orientations of the sample, to a 3-d diffraction pattern.
Fig. III.2. Whereas the scattering of incoherent radiation, e.g. from a synchrotron, yields only the average spacing d of a collection of scatterers
(a), the scattering of coherent radiation (with beam diameter a) from the SwissFEL produces a rich speckle pattern (b), which can be
inver ted to obtain the exact distribution of scatterers (from ).
Resolution limits in bio-imaging
Of particular interest in nanoscale imaging are biological objects, down to the level of the atomic structure of macromolecules. The principal limit to the resolution achievable with X-ray or electron scattering by bio-materials is imposed by radiation damage. (The investigation of biological samples with electron microscopy and diffraction is treated in an Infobox.) Howells et al.  have investigated both theoretically and experimentally the resolution limits for imaging non-periodic (i.e., noncrystalline) bio-materials (see Fig. III.3). Making use of the classical electron radius
Fig. III.3. The required and maximum tolerable doses for imaging a
bio-material sample (H50C30N9O10S1) against a background of
water . The water background is accounted for by setting ρe =
ρe(sample)-ρe(water) and using μ(water) and ρm (water). The required
dose is calculated according to the Rose criterion, as described
in the text, for 1 (solid line) and 10 keV (dashed line) Xrays.
Measurements of the maximum tolerable dose have been
compiled from experiment for various X-ray and electron scattering
and the complex electron density
(related to the complex index of refraction of the scatterer, n = 1 – δ – iβ), the number of incident X-rays per unit area required to scatter P photons into a detector from a 3-dimensional voxel of the sample with dimensions (d × d × d) is given by
This corresponds to a dose, measured in Grays (with 1 Gy = 1 J/kg of deposited energy) of
where ρm is the mass density and μ is the inverse absorption length. The Rose criterion for the detection of features in the presence of background noise requires that the signal exceed the rms noise level by a factor of 5. Assuming Poisson shot noise, this criterion requires P = 25. The solid and dashed lines in the plot show the dose required to achieve a par ticular resolution d for a bio-material with composition H50C30N9O10S1 and density ρm = 1.35 g/cm3, against a water background, for X-ray energies hν of 1 and 10 keV, respectively. Note that the required dose depends on the resolution as d-4. Also shown in the plot are the empirical results of analyses of electron and X-ray scattering experiments, giving the maximum allowable dose at resolution d. When translated to integrated X-ray flux, these points lie close to the 1010 photons/μm2 damage limit often quoted for bio-materials studied at atomic resolution. (It should be noted that small samples, such as nano- and 2d-crystals or single molecules, may show a significantly higher damage limit, due to the immediate escape from the sample of primary photoelectrons.) Of particular interest is the point of intersection of required and tolerable dose, corresponding to 109 Gy and a resolution of d = 10 nm. This is the approximate limit for imaging a single non-periodic sample at a synchrotron. Better resolution requires exposing many equivalent copies of the sample or taking advantage of coherent scattering from the large number of identical unit cells in a crystalline sample. Consider the following cases: a) imaging a bio-material in a water background with 1 keV X-rays to a resolution of 2 nm, and b) imaging a pure bio-material sample with 10 keV X-rays to 0.2 nm. Simple application of the Rose criterion above predicts the required integrated X-ray flux into a 100 × 100 nm2 spot for a) and b) to be 1011 and 7 × 1015 photons, respectively. The former corresponds to a single SwissFEL shot, indicating the real possibility of single-shot (projection) imaging of sub-cellular organelles at nm-scale resolution. The latter integrated flux, corresponding to 70'000 shots, suggests that individual biomolecules could be lensless-imaged to 0.2 nm resolution within several minutes of measuring time. But this is under the assumption that radiation damage can somehow be avoided, and, for the individual molecules, that the scattering statistics can be accumulated (see below).
Electron microscopy and diffractionAlternatives to the SwissFEL for per forming structural determinations on individual biomolecules and 2d-bio-crystals are the techniques of electron tomography and electron diffraction. Electrons (100-500 keV) are scattered by matter more strongly than are hard X-rays, hence requiring the use of thin samples (< 1 μm). In a large sample, the amount of energy deposited by inelastic scattering events per useful elastic scattering event is 1000 times less for electrons than for X-rays . In sub-µm samples, this factor is less, due to the escape of photoelectrons. In order to preserve samples for long enough in the electron beam to collect useful statistics, it is necessary to flash cool the sample, generally by plunging it into liquid ethane. Electron tomography  uses electron microscope images of many identical, individually-measured molecules, in their natural environment, taken at various tilt angles, to reconstruct the 3d-structure. The problems involved in data analysis are similar to those that will be encountered with single-molecule lensless imaging at the SwissFEL. The 3d-resolution presently achievable with electron tomography is 1–2 nm, limited by radiation damage and the change of the beam focus with sample tilt for regions slightly displaced from the tilt axis. Electron diffraction  can be per formed on cryocooled crystalline samples which are sufficiently thin. An attractive application is to 2d-membrane crystals, for which structural determinations have been performed down to 0.2 nm resolution . Time-resolved electron diffraction measurements of bio-molecular structure will require improvements in transverse coherence and successfully dealing with space-charge effects . Besides the elimination of radiation damage by using shor t SwissFEL pulses, principal advantages of X-ray imaging over electron techniques are the relaxed demands on sample thickness and accurate sample tilting. A fur ther point is related to the much stronger scattering of electrons than X-rays: whereas a simple kinematical approach, which neglects multiple scattering, is justified in interpreting X-ray scattering, much more complicated dynamical theories must be used for the electron techniques.
Particle injection and laser orientation
Figure III.i4 shows a scenario for using polarized laser light to per form two-axis orientation of anisotropic protein molecules, such as the tobacco mosaic virus (TMV) .