1. Introduction

Coherent x-ray diffraction flash imaging using hard x-ray free electron lasers (XFEL’s) may enable imaging of single biological molecules [1]. The coherence properties of the light source determine the imaging parameters, such as the resolution length. XFEL’s exhibit very good transversal coherence, but the longitudinal coherence depend on their mode of operation [2, 3, 4]. The Linac Coherent Light Source (LCLS) is anticipated to be the first functional XFEL delivering photon pulses at the lengthscale of interatomic distances of about 1 Å [5]. LCLS will be operated in the self-amplified spontaneous emission (SASE) mode in which the initial random field of spontaneous undulator radiation is amplified to intense, quasicoherent radiation [2]. The temporal intensity profile of the LCLS is made up of about 250 SASE intensity spikes. Whereas each SASE spike is transversally and longitudinally quasicoherent, the phase relation between them is random, so that the longitudinal coherence of an LCLS pulse is not perfect. It can be shown that the coherence time associated with the first-order time correlation function is t_c=√π/σ_ω [6, 7], where σ_ω is the SASE bandwidth. For LCLS operated at a wavelength of 1.5 Å near saturation, t_c ≈ 300 as [8]. This is much shorter than the SASE pulse duration that is on the order of the electron pulse duration of tens to hundreds fs [5].

Coherent diffraction imaging has been demonstrated with light of much better temporal coherence than will be available at the LCLS. For example, third generation synchrotron light sources [9] and longer-wavelength FEL’s [10, 11] have been used successfully. Estimates for pulse requirements and resolution for imaging using SASE XFEL’s have not taken the characteristic temporal pulse structure of XFEL’s into account [12]. The effect of the limited temporal coherence on the image resolution can be estimated by comparing the coherence time t_c with the maximum time delay τ_max of two beams that are scattered into the highest resolution part of the diffraction pattern [13]. For a particle diameter d_max, τ_max=d_maxq_max/_ω, where ω is the angular frequency of the light, and q_max=2π/δ is the maximum diffraction scattering vector. The resolution δ can then be calculated to be

where λ is the wavelength and c is the speed of light. In this paper we will improve on this estimate and analyze the effect of the reduced temporal coherence on biomolecular diffraction imaging using SASE XFEL’s in more detail. We first develop the theoretical framework for the diffraction of short light pulses by molecules. We then use this model to calculate the inherent resolution limitations of diffractive imaging at SASE XFEL’s as a function of the particle size.

2. Diffraction model

The SASE XFEL diffraction pattern can be calculated as the sum of the intensities of the diffraction patterns produced by each SASE spike since their phase relation is random. We take the spike width as the coherence length t_c of each spike. It can be shown that the distribution of spike widths normalized to its average is given by

where ξ=Δτ/<Δτ>, Δτ is the rms spike width, <Δτ> is its average, a≈0.8685, and η≈9.510 [14]. We further assume that the sample does not change during the pulse, and that each SASE spike has a Gaussian temporal intensity distribution with a full-width at half maximum given by its coherence length t_c. We assume that the sample is a non-magnetic and linear elastic scatterer, and that the assumptions of scalar diffraction theory hold. The incoming electric field of a single SASE spike can be described as

where r⃗ is a position vector, t the time, and k⃗ the wave vector. σ is the temporal width of the SASE spike, and E ₀ a proportionality constant. The contribution of a volume element dV at position r⃗ to the total electric field E_rad at time t at a large distance R_r is

Here r ₀ is the Thomson scattering length, ρ the electron number density, k=2π/λ, λ the wavelength, and cos(ψ_r) the polarization factor. R_r=R ₀-r⃗ · k̂′ is defined in Fig. 1, where k̂ and k̂′ are the unit vectors in the direction of the incoming and outgoing light, respectively. The time t_ret=t - R_r/c=t ₀+r⃗ · k̂′/c determines when the volume element located at r⃗ emitted the light. Here, t ₀=t-R ₀/c.

 

Fig. 1. Sketch of the scattering geometry.

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If the detector is far from the object (R_r≫r), then R_r in the denominator of Eq. (4) can be approximated by R ₀, and ψ_r≈ψ ₀. The total scattered field at the detector can then be written as

with

q⃗≡k⃗-k⃗′, and V is the volume that contains the scatterer. The time-dependent second exponential term distinguishes this expression from the usual definition of the structure factor. The integral detector probability <n> is the average total number of photons at the detector and is proportional to ∫^∞ _-∞ I(t)dt ∝ ∫^∞ _-∞ |F(q⃗,t)|² dt. I(t)=Z ₀ k|E_rad|² is the intensity of the scattered light at time t, and Z ₀ the impedance of vacuum.

3. Effect of coherence on diffraction

We have quantified the effect of the limited temporal coherence of XFEL pulses on diffraction imaging by comparing the diffraction pattern of a sample irradiated by a sequence of short SASE spikes with the diffraction pattern of a sample irradiated by a fully-coherent pulse. The discrepancy of the two patterns is measured by the resolution-dependent R factor, defined as

where F ₀(u⃗) and F ₁(u⃗) are the modulus of the molecular structure factors in the scattering direction u⃗ of the molecule irradiated with an infinitely-long pulse and with a pulse of finite duration, respectively. We introduced a resolution dependence of the R factor by calculating the partial sums over all independent pixels up to a fixed resolution u_m. The R factor tends to increase at finer resolutions. In our calculations we assume that the sample is made up of N point-like atomic scatterers that are randomly distributed inside of a sphere. The electron number density is given by

where q_i is the number of electrons on each scatterer and r⃗_i their positions. Then Eq. (7) simplifies to

For computational reasons we calculate F(q⃗,t ₀) only in one dimension in reciprocal space for different times t ₀. This is sufficient since we are comparing diffraction patterns; if we would perform reconstructions, the diffraction patterns would have to be calculated in two or three dimensions, and the curvature of the Ewald sphere would have to be taken into account. We further assume that all atomic scatterer are identical. We then calculate F ₁(q) as the square root of the sum of |F(q⃗,t ₀)|² taken over all timesteps t ₀. Finally,

Figure 2 shows the R factor as a function of the image resolution length for different particle diameters. In these calculations we assumed a wavelength of 1.5 Å and an average coherence time t_c=300 as. The spike widths are assumed to follow Eq. (2). The value of R generally increases with decreasing resolution length because the contribution of the time-dependent second exponential term in Eq. (7) increases with q, and q is inversely proportional to the resolution length. The contribution of this exponential term also increases with increasing distance r, so that the R factor strongly increases with increasing particle diameter, as can also be seen in Fig. 2.

Typical R values for x-ray crystallographic data in the protein database [15] are less than 20%. Taking this value a an upper limit that determines the image resolution length [1], we see that for 3000, 1000, and 300 nm particles the resolution lengths are 9.7, 3.6, and 1.2 Å, respectively. The limited longitudinal coherence of SASE XFEL’s does not have a strong effect on the quality of the diffraction pattern of small particles less than ≈500 nm. For larger particles, however, the effect is significantly more pronounced, especially near atomic resolution. In an actual experiment the quality of the diffraction pattern will be further degraded by radiation damage to the sample, errors in the angular alignment of the molecule, and noise. Radiation damage and the effect of alignment errors are discussed in References [12] and [16], respectively. The noise in the diffraction pattern is anticipated to be reduced by taking the average over a large number of diffraction patterns.

 

Fig. 2. R factor as a function of the image resolution for different particle diameters. The different curves are labeled with the diameter in nm.

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We performed similar calculations at a wavelength of 15 Å and found that the resolution scales roughly with the wavelength according to Eq. (1), which means that at a ten times longer wavelength, ten times smaller particles are required to achieve similar resolution, or, equivalently, for a given particle size a ten time better resolution can be achieved by using a ten times shorter wavelength.

4. Conclusions

In summary,we have analyzed the effect of the limited longitudinal coherence of SASE XFEL’s on coherent diffraction imaging of single particles. We found that the temporal width of individual SASE spikes is the relevant parameter, and that its effect can be accounted for by a simple corrective term to the molecular scattering factor, as given by Eq. (7). We analyzed the effect of this modification on diffractive imaging of biological particles of relevant sizes at the LCLS and found that the resolution is strongly compromised for larger particles, and that atomic resolution imaging below a resolution length of 2 Å will not be possible for particles with a diameter larger than 500 nm. For smaller particles, the longitudinal coherence is sufficient to achieve atomic resolution lengths around 1 Å.