Optical control, selection and analysis of population dynamics in ultrafast protein X-ray crystallography

Abstract

Ultrafast pump-probe X-ray crystallography has now been established at X-ray free electron lasers that operate at hard X-ray energies. We discuss the performance and development of current applications in terms of the available data quality and sensitivity to detect and analyse structural dynamics. A discussion of technical capabilities expected at future high repetition rate applications as well as future non-collinear multi-pulse schemes focuses on the possibility to advance the technique to the practical application of the X-ray crystallographic equivalent of an impulse time-domain Raman measurement of vibrational coherence. Furthermore, we present calculations of the magnitude of population differences and distributions prepared with ultrafast optical pumping of single crystals in the typical serial femtosecond crystallography geometry, which are developed for the general uniaxial and biaxial cases. The results present opportunities for polarization resolved anisotropic X-ray diffraction analysis of photochemical populations for the ultrafast time domain.

This article is part of the theme issue ‘Measurement of ultrafast electronic and structural dynamics with X-rays’.

1. Introduction to time-resolved protein crystallography: a separation of time scales

The advent of X-ray free electron laser (XFEL) facilities operating at keV photon energy suitable for molecular crystallography is revolutionizing the field of time-resolved structural biology. Slow, incoherent dynamics with 100 ps resolution are generally more readily accessible with current XFEL facilities than synchrotron facilities, particularly in pump-probe mode. A key technical reason for this is that the typical synchrotron geometry places limits on the level of photoconversion in the interaction region of the molecular crystals. Sufficient levels of photolysis are necessary in order to allow detection of light-induced differences, but there are a number of other considerations particularly the damage that is accumulated by repeated pump-probe cycles by both the optical laser and the X-ray exposures, in order to fill the dynamic range of the area detector. Time-resolved X-ray crystallography in pump-probe mode was pioneered and developed in the 1990s primarily by Keith Moffat and co-workers, who applied the Laue method to substitute for the inability to rotate the crystals on nanosecond time scales. These breakthrough results have been described and reviewed many times throughout the times of development, optimization and specific application [1–7]. These very important developments by Moffat, Ren and colleagues have delivered methods for processing and analysing of synchrotron Laue crystallography using pink beams to provide high-quality measurements of structure factor amplitudes [8–11]. While this was very successful, the Laue method intrinsically does affect the limiting resolution although accuracy can in favourable cases be comparable to corresponding monochromatic data. In the XFEL geometry using the now commonly used serial femtosecond crystallography (SFX) method, nanosecond laser excitation typically converts a much higher percentage of the micro-crystals. This was first seen with nanosecond flash excitation of crystals of the photoactive yellow protein (PYP) in comparison with corresponding synchrotron measurements [12–15]. It was seen that difference electron density (DED) maps exhibited more pronounced light-induced differences in the case of TR-SFX compared to the Laue method. Furthermore, analysis of the reaction initiation fractions indicated photoactive populations as large as 40% for TR-SFX, three to four times larger than that achieved with Laue synchrotron measurements; this was attributed to the increased size of the crystals required for Laue possessing such large optical density than optical pulses cannot penetrate the entire crystal let alone lead to uniform illumination. Another very important consideration is that the SFX method at XFELs is a single-shot technique, and that each crystal is fresh and not previously exposed. This removes the accumulating background differences as well as fall-off of intensity and resolution that would be caused by repeated laser excitation and X-ray exposure of the same crystal volume. Interestingly, the increased sensitivity to light-induced differences using the SFX method was in fact not readily predicted before the first convincing demonstrations for a number of reasons. Firstly, pump-induced structure factor amplitude differences ΔF are typically smaller than noise particularly with ultrafast excitation, such that the inverse Fourier transform after appropriate phasing of the data accumulates electron density differences to statistically significant levels in real space in well-correlated cases. The level of correlation between repeated measurements is increased by the lack of background from damage, but may be increased by inhomogeneity of crystal size and the level of isomorphicity. Another consideration is the strong fluctuation of the X-ray spectrum of the XFEL in SASE mode, its typical 40 eV bandwidth, as well as the small divergence relative to typical values for mosaic spread of protein crystals, considering typical focusing angles of 0.12–0.25 mrad used for SFX. These characteristics were previously investigated from single-crystal datasets with an application of a quasi-rotation method at the LCLS in the beginning period of exploring the XFEL-based protein crystallography results [16]. In addition, the absence of significant convergence in this study [16] together with pointing jitter adds noise to the structure factor amplitude determination in crystals with a very small mosaic spread which has as a consequence that all spots are partial and extraction of accurate structure amplitudes is hindered. A very workable approach in the processing of such data is to record the mean energy determined by a bent crystal spectrometer on a shot-to-shot basis which is routinely used in SFX crystallographic data processing [17–24]. Both Monte Carlo as well as post-refinement use the recorded mean energy as a single parameter needed to address the fluctuating X-ray spectrum, which has a very fine structure [22,25]. It has been surprising therefore that when running the XFEL in a self-seeded mode a strong improvement in the SFX data processing statistics was not seen [26] as would be expected. Reliable quantification of all physical contributions to the uncertainty of the measurement is in fact not yet obtained and future investigations would be of great interest to the SFX community. Instead, the field is relying on averaging as a principle technique when executing and processing multi-crystal SFX datasets of still images with high partiality, and predictions of the sensitivity of difference measurements cannot generally be made with sufficient statistical certainty.

General protocols have now been developed for the successful execution of flash-induced time resolved structural biology [27–29]. Provided that appropriate optical measurements are done on actual crystals, the technique is in principle only limited by the value of the primary quantum yield, if all other requirements are met for good-quality crystallography of micro-crystals. Of course, the amplitude of the real-space displacement, the scattering cross sections, X-ray wavelength, space group symmetry and dimensions and Debye–Waller factors are further considerations that determine the magnitude of the electron density differences. This is demonstrated from relatively recent results from both SACLA [29] and LCLS [30] that measured S-state differences in the core complexes of photosystem II crystals. These experiments are facilitated by the very high quantum yield of transformation of the oxygen evolving complex under nanosecond flash illumination.

Here, a fundamental separation is made between slow nanosecond and fast femto- and picoseconds time scales, in the application and analysis of pump-probe time-resolved crystallography. The differences are so significant that fast experiments represent a completely different area of research that has little correspondence with slow experiments in terms of the molecular dynamics that are prepared and detected. We emphasize that the ability to execute and analyse such experiments is only recently developed, and many physical aspects remain unexplored and uncharacterized. The unique characteristics of the new and emerging application of femtosecond time-resolved pump-probe SFX are the focus of this contribution, particularly a consideration of the distributions of populations on early time scales.

A fundamental difference between the time dependence of slow thermal processes in comparison with ultrafast reactions is that in the former case the population differences are governed solely by rate kinetics and linearity can be assumed for the subsequent analysis. Since the time progression simply modifies the concentrations of reactants, intermediates and products, the transient structures associated with the transition process are typically missed. Ultrafast time scales allow the possibility of accessing true molecular motion and that is one criterion for the separation of time scales. The actual time resolution needed will depend strongly on the material and reduced mass of the oscillators involved; Franck Condon motion of biological chromophores consisting of light atoms is often found to be faster than instrument response function that are typically on the order of 25–100 fs for spectroscopy instruments [31–33]. An observation of this type is yet to be made using pump-probe X-ray crystallography methods at XFELs.

For pump-probe delays that are within the vibrational dephasing times of systems under study, a key consideration is that the origin of structural motion is completely different from those obtained on slower incoherent time scale. A previous discussion considered the theoretical analysis of nuclear coherence under such conditions [34]. In the static picture, a straight-forward hole-burning analysis provides the connection between optical parameters and the ground state coherence and excited state coherence that is created with a brief optical pulse. Appropriate theory, which is taken from the Raman literature, may be applied to estimate the coherence magnitudes, phase and frequency dependence based on line-shape theory [35,36] which is known as the ‘Wigner phase space' representation [37,38]. Briefly, the calculation of ground state coherence is performed by taking the first moment of a cumulant expansion of the density matrix problem, in the first instance for a two-level system. The resulting response function is a linear response theory calculation that can explicitly include the laser carrier frequency, bandwidth and power as well as a model for the absorption band shape and lifetime parameters. While this method provides an estimate under appropriate experimental conditions, application for time-resolved X-ray crystallography adds at least two further assumptions and modifications. The linear response theory that takes a first moment of an expansion series is most accurate for weak interaction. In the XFEL geometry, however, strong excitation is invariably used in order to generate detectable populations. This in principle invalidates the response theory, as the photolysed population appears in the coherence calculation expressions as a linear term [35,36]. A time integration can replace the population calculation for the strong interaction, which we have shown previously [34]. Furthermore, the density matrix is written for a two-level system, such that population inversion is not possible and Rabi cycling would occur, if increasingly higher orders are included from the cumulant expansion. In practice, the magnitude of Franck Condon motion relative to optical bandwidths results in an effective four-level system for most systems under study, such that laser rate equations can be used to calculate the pumped system populations. Nevertheless, the results show that with appropriate high field correction, such calculations still provide a useful and intuitive estimate of the ground state coherence. The ground state coherence is impulsive and in the direction of the nuclear binding force, which determines the phase for a given carrier frequency and natural frequency. By contrast, the excited state coherence is considered to be non-impulse and is purely displacement driven. A typical situation considers the on-resonance excitation with a femtosecond pulse and laser spectrum that is narrower than the absorption line width. In this case, the total coherence, comprising of position and impulse, could, in fact, be minimized. With off-resonance excitation, the resulting ground state coherence could be multiple times the minimized value. This demonstrates that position, impulse and phase can be controlled by the carrier frequency, laser pulse power and electronic lifetime and dephasing properties [34].

An experiment could be proposed to manipulate the ground state nuclear coherence on ultrafast time scale in a controlled manner, based on the analysis and calculation using line-shape theory. Firstly, a pulse intensity, duration and laser spectrum is selected that is appropriate to generate low-frequency vibrational coherence within the dephasing time as well as a significant hole-burning amplitude. Using the high-intensity approximation equations [34], calculation of the spectral hole that is generated with pulses that are either below or above resonance conditions (figure 1) illustrate the result of using carrier frequency as a control parameter. The linear response theory predicts that while the resulting ground state total complex coherence amplitude $|A{1}_g|=\sqrt{(Q_g^2+P_g^2)}$ would be unmodified, the sign of the instantaneous position Q_g is inverted, and causing a phase ϕ = −tan⁻¹[P_g0/Q_g0] difference in excess of π. The calculation thus demonstrates that for ground state nuclear coherence, the initial motion will occur in opposite direction, with a change of sign of detuning frequency. This type of control could be applied in future experiments that aim to validate and quantify the respective ground and excited state contributions to atomic displacement. Such experimental conditions will require very precise timing information, as the coincidence time must be known with high precision in order to extract the phase. This is a considerable experimental challenge, as the level of jitter between the XFEL and laser is typically on the few-hundred femtosecond level [39,40]. Recent analysis of the accuracy with which coincidence time is determined using cross-correlation methods have placed limits on the precision [41]. Essentially, even when systematic errors such as, dispersion mismatch, X-ray timing jitter and X-ray intensity fluctuations are taken into account determination of absolute coincidence time still requires the application of a physical model to transient reflectivity. This modelling has a significant error, at least approximately 50 fs [42], in contrast to the accuracy of the relative arrival time measurement can be as good as approximately 10 fs [43]. This indicates that it is very likely the absolute phase will have a considerably larger uncertainty, while the relative phase could be determined with a precision to allow phase differences of the magnitude expected for conditions such as proposed here.

Figure 1.

Calculation demonstrating the effect of detuning on spectral ‘hole-burning' (a) when pumping a transition (peak cross section 1.68 × 10⁻¹⁶ cm²) with a 100 fs, 1.5 GW mm⁻² optical pulse. The unperturbed absorption line (black), the on resonance (red) and two detuned cases (blue) are shown [34]. Also shown the absolute ground state displacement (b) and phase (c). (Online version in colour.)

Another approach to minimize ground state coherence is to apply extremely short and broadband pulses, such that no specific frequency selection creates appreciable position or momentum. Attempts to achieve such conditions have been reported, using few femtosecond optical pulses that were analysed in terms of surviving excited state coherence exclusively [44]. This method is very challenging because it requires generation of dispersion-free pulses with flat spectral and temporal phase in addition to a very broad and smooth frequency distribution.

A commonly used approach to suppress ground state coherence relative to excited state coherence manipulates the photolysed population by using very weak excitation. Weak excitation reduces the population and thus reduces |A1_g| while the excited state coherence is displacement driven such that the pump-on minus pump-off differences contain excited state vibrational coherence selectively [35,36]. This method is very successfully applied to impulsive Raman spectroscopy (Vibrational Coherence Spectroscopy), but is however not directly suitable for conditions used for femtosecond time-resolved SFX unless orders of magnitude improvement of the photo-induced difference measurement is developed. Potentially, this regime may become accessible with MHz repetition rate experiments in the future. It is noted that LCLS-II is being developed to deliver such high repetition rate experiments in CW mode [45]. Detector frame rate limitations and sample delivery methods may dictate the effective data repetition rate for increased sensitivity of photo-induced difference measurements by X-ray diffraction.

Finally, an incoherent optical control process may additionally allow specific removal of excited state vibrational coherence. Such conditions may be achieved by the combination of a short femtosecond excitation followed by an intense picoseconds pulse resonant with stimulated emission. The resulting pump-dump-probe pulse sequence would selectively remove the excited state population through optical dumping, while the picoseconds pulse duration avoids the creation of secondary vibrational coherence. Application of such pulse sequences have been successfully shown for impulse Raman measurements [46], and in principle could be applied in the same manner to femtosecond time-resolved TR-SFX.

The discussion above has briefly summarized the physical origin of structural motion on ultrafast time scale, with the aim to discuss and assess experimental conditions for protein crystallography. The combination of motion of light elements, the real-space displacement [47] and large unit cells, that determine the magnitude of photo-induced structure factor differences, places specific limitations to our ability to extract nuclear coherence motions from crystallographic methods. It is only very recently that femtosecond time-resolved experiments with protein crystals have been demonstrated [12,48,49].

2. Multi-photon events in TR-SFX: calculation, detection and control

It is currently unavoidable to drive multi-photon processes under conditions that successfully resolve photo-induced electron density differences with femtosecond pulse duration. Essentially, the requirement to create at least approximately 10% photolysed populations in micrometre-sized crystals dictate the use of intense visible light pulses. Currently, one example exists in the literature, where careful and systematic optical spectroscopy experiments are available that are specifically analysed for the TR-SFX experimental conditions, which concerns the ultrafast crystallography of the photoactive yellow protein (PYP) [50]. A series of experiments were conducted and analysed that assessed the separation of the one-photon and diverse multi-photon processes. In the case of PYP, transient absorption experiments have measured the yield and associated cross sections of excited state absorption resulting in electron ejection and formation of a chromophore radical, in addition to stimulated emission processes. It was shown that both nonlinear transformations are particularly sensitive to the carrier frequency but also to second- and third-order dispersion. A systematic investigation of varying the optical properties including peak power, pulse duration, second- and third-order dispersion provided a series of transient absorption data of PYP that could be analysed for nonlinear transformation yields [50]. A simultaneous fit to a set of master equations of all the datasets for different power densities under defined and FROG characterized optical conditions, yielded the effective cross sections and yields of the linear and nonlinear transformations. These studies identified on-resonance conditions in the presence of positive second- and third-order dispersion as most optimal for suppressing the parasitic multi-photon transformations and maximizing the transient population of desired photointermediate, productive in the photoisomerization pathway. These early studies were conducted with solution samples of PYP, whereas measurements on crystalline PYP needed a different experimental configuration. Having selected the on-resonance carrier frequency as most promising, a femtosecond flash-photolysis measurement evaluated the eventual yield of reversible and non-reversible photoproducts using ms–s transient absorption using continuous wave probes on slurries of crystals that were adjusted to optical densities representative for typical TR-SFX conditions [51]. The analysis of the power density dependence of the linear and nonlinear yields was done analogous to the master-equation methods previously applied to transient absorption data [50]. It is important to note that the experimental evidence confirmed the initial rise of product formation with increasing power, and subsequently declined after rolling over with threshold intensities, as expected for the power dependence of linear and nonlinear optical processes. These experiments were analysed with equations essentially equivalent to open-aperture Z-scan analysis. In this approach, rate equations are fitted to power density dependence and separate single photon and multi-photon processes to obtain their corresponding effective cross sections under defined optical parameters. A representative result obtained for crystalline PYP found that for 130 fs pulses on resonance (λ = 450 nm, ground state cross section 1.68 × 10⁻¹⁶ cm² [50]) and power densities between 1 and 3 GW mm⁻² saw transient nonlinear cross sections between 4 and 8 × 10⁻¹⁷ cm⁻² and sequential double excitation cross sections approximately 12 × 10⁻¹⁹ cm². This resulted in S₁ populations between 25 and 35% and photoproduct yields between 5 and 7% due to the internal conversion quantum yield of 0.2 at this wavelength [50]. For longer pulses, the yields were increased (6–10% for 500 fs pulse of the same energy) due to the reduced the effect of double excitation from lower peak power.

3. Orientation analysis of linear and nonlinear phototransformations resulting from birefringence

In this contribution, we further evaluate the consequences of birefringence on the orientational photoselection under SFX conditions. Typically, the SFX method uses a stream of micrometre-sized crystals that are injected into the interaction region either with a liquid jet [52] or alternatively a droplet injector [53]. Depending on the crystal morphology there may be distributions of preferred orientations, for instance ‘needles’ tend to align in the direction of the jet, which can be detected from asymmetric radial modulations in the powder diffraction patterns. However, most orientations are present in SFX datasets, which aim to maximize completeness and statistics from many observations using either a Monte Carlo averaging approach or a specific post-refinement method. In order to prepare crystallographic analysis of pump-probe differences, the illuminated datasets are generally prepared by selecting a set of diffraction images that are found within a selected range of pump-probe delay according to time-stamping data [48]. As a result, all different orientations of optically pumped crystals are used in the eventual averaging for generation of the structure factor amplitudes that represent the time delay measurements. Such averaging already disregards differences in crystal size that will result in differences in the percentage population that is prepared due to attenuation of pump light with propagation through the crystal. Methods of crystallization as well as size selection techniques could minimize the differences associated with this inhomogeneity. Here, we consider that each different orientation will result in different photochemical populations. The basic consideration is that with linear polarization, for all crystal symmetries except cubic crystals, almost all crystal orientations will result in decomposition of the electric pump field into two orthogonal directions of polarization, which are the general solutions of wavevectors to Maxwell's polarized wave equations of anisotropic media. Intrinsic birefringence results in a phase velocity difference such that the resulting wave is circularly polarized. The analysis of field–dipole interaction of the transition dipoles present in the molecular structure, are however separated for the two orthogonal modes [54]. Triclinic, monoclinic and orthorhombic crystal symmetry results in a biaxial system which is trichroic. The modes are the solutions Maxwell's equations for linear polarization in an anisotropic media and with dictate the allowed polarizations ${\hat{\mathit{e}}}_1$ and ${\hat{\mathit{e}}}_2$.The total field can be decomposed as the following:

$$\begin{array}{r}E(z)={\hat{\mathit{e}}}_1{E}_1{e}^{-i{k}_{1}z}+{\hat{\mathit{e}}}_2{E}_2{e}^{-i{k}_{2}z}\end{array}$$ 3.1a

$$\begin{array}{r}\begin{array}{l}{k}_1=\omega \sqrt{{\mu }_o{ϵ}_1}=k_0{n}_{e_1}\\ k_2=\omega \sqrt{{\mu }_o{ϵ}_2}=k_0{n}_{e_2},\end{array}\}\end{array}$$ 3.1b

where E₁ and E₂ are the magnitudes of the field of in ${\hat{\mathit{e}}}_1$ and ${\hat{\mathit{e}}}_2$ whose initial magnitudes will depend the product overlap of the initial polarization with the allow directions. The absorption (A₁) along a particular optical axis ${\hat{\mathit{e}}}_1$ as a function of orientation is given as [54]

$$A_1=3\sum _{j\alpha }{ϵ}_{j\alpha }(\hat{\nu })c_{jd}⟨{({\hat{e}}_1\cdot {\hat{\mu }}_{j\alpha })}^2⟩,$$ 3.2

where $⟨{({\hat{e}}_1\cdot {\hat{\mu }}_{j\alpha })}^2⟩$ is the squared product of the ${\hat{e}}_1$ axis and dipole operator ($\hat{\mu }$) summed over all symmetry operations (${\tilde{O}}_k$) of a particular crystal form and absorbing species (α) and contributing electronic transition

$$⟨{({\hat{e}}_1\cdot \hat{\mu })}^2⟩=\frac{1}{n}\sum _{k=1}^n{({\tilde{O}}_k\cdot \hat{\mu }\cdot {\hat{e}}_1)}^2.$$ 3.3

The birefringence dependence on orientation and value of $⟨{({\hat{e}}_1\cdot \hat{\mu })}^2⟩$ will vary depending on the crystal type. In the absence of crystallographic symmetry in the triclinic case, two difficulties prevent structure-based analysis of field–dipole interaction. Firstly, the absence of symmetry in all cases, except those arrived at by pure chance, dictates that the optical directions which are the principal axes of susceptibility do not correspond to the crystallographic directions. Second, the absence of symmetry causes a wavelength dependence of the optical directions particularly near electronic resonance because of the anomalous dispersion region of the real part of the refractive index. Therefore, in order to make a structure-based analysis of birefringence response, an optical ellipsometry must be conducted for all frequencies of interest which further requires X-ray crystallographic face indexing as birefringence-based orientation is ambiguous in biaxial systems lacking twofold symmetry. As for $⟨{({\hat{e}}_1.\hat{\mu })}^2⟩$ the lack of symmetric means that there is no alignment of the crystallographic and optical directions

$$⟨{({\hat{e}}_1.\hat{\mu })}^2⟩={({\hat{e}}_1.\hat{\mu })}^2={(\sin {\delta }_{{\alpha }_j}\cdot \cos {\psi }_{{\alpha }_j})}^2.$$ 3.4

(a). Biaxial symmetry (triclinic, monoclinic or orthorhombic)

For monoclinic systems, the presence of a single twofold axis, by convention takes as the crystallographic b-axis, coincides at least one optical direction with a crystallographic direction. Furthermore, the twofold symmetry operator results in a wavelength independence as the wavevector surfaces must survive the symmetry operation for all frequencies. The orthorhombic symmetry is most straightforward, as twofold symmetry in each direction forces both coincidence of optical and crystallographic directions, as well as force the wavelength independence

$$\begin{array}{rl}⟨{({\hat{e}}_1.\hat{\mu })}^2⟩& =\frac{1}{n}{\sin }^2\hspace{0.17em}\delta \sum _{k=1}^{n=2}\cos \left(\varphi +\frac{2\pi k}{n}\right)\\ & =\frac{1}{2n}{\sin }^2\hspace{0.17em}\delta \left[n+\cos 2\varphi \sum _{k=1}^{n=2}\cos \left(\frac{4\pi k}{n}\right)-\sin 2\varphi \sum _{k=1}^{n=2}\sin \frac{4\pi k}{n}\right]\\ & ={\sin }^2\hspace{0.17em}\delta {\cos }^2\hspace{0.17em}\varphi ,\end{array}$$ 3.5

where δ and ϕ are the polar angles of the system aligned along the principle symmetry axis (figure 2). One method to find the directions of orthogonal modes involves a graphical construction of the wavefront section of the indicatrix normal to the wavevector. The normal directions of the wavefront section then correspond to the polarization directions of two modes. The optical indicatrix formula has the form of a three-dimensional ellipsoid

$$\frac{x^2}{n_1^2}+\frac{y^2}{n_2^2}+\frac{z^2}{n_3^2}=1,$$ 3.6

where n₁, n₂ and n₃ are the principal refractive indices of the crystal. For an incident wave with k-vector $\hat{\mathit{k}}=x_k,y_k,z_k$ the intersection a plane normal to $\hat{\mathit{k}}$ and the indicatrix will produce an ellipse (figure 3), where the semi major axes are the modified refractive indices of the two allowed polarizations of the decomposed wave. From the intersection of an ellipsoid and a plane defined by a normal unit vector that contains the origin of that ellipsoid will produce an ellipse with formula [55,56]

$$\frac{t^2}{A^2}+\frac{u^2}{B^2}=1;A=\frac{1}{\sqrt{{\beta }_1}};B=\frac{1}{\sqrt{{\beta }_2}},$$ 3.7

where (t, u) are the rotated axis in the plane of the ellipse, A and B are the semi major axes and β₁ and β₂ are the two solutions of the quadratic equation obtained by Lagrange's method of undetermined multipliers [55]

$$\begin{array}{rl}& {\beta }^2(x_k^2+y_k^2+z_k^2)-\beta \left[x_k^2\left(\frac{1}{n_2^2}+\frac{1}{n_3^2}\right)+y_k^2\left(\frac{1}{n_1^2}+\frac{1}{n_3^2}\right)+z_k^2\left(\frac{1}{n_1^2}+\frac{1}{n_2^2}\right)\right]\\ & +\frac{x_k^2}{n_2^2{n}_3^2}+\frac{y_k^2}{n_1^2{n}_3^2}+\frac{z_k^2}{n_1^2{n}_2^2}=0.\end{array}$$ 3.8

Figure 2.

Projections of the incident wavevector on the orthogonal optical axes of a lattice in the (a) biaxial $({\hat{e}}_1\ne {\hat{e}}_2\ne {\hat{e}}_3)$ and (b) uniaxial $({\hat{e}}_1={\hat{e}}_2\ne {\hat{e}}_3)$ and cases.

Figure 3.

The optical indicatrix for a biaxial (a) and uniaxial (b) systems. Important to note that in the uniaxial system the value of one the semi-major axis of the constructed ellipse is always equal n_o due to symmetry. Hence, the value of the other axis $(n_{\mathrm{e}}^{\mathrm{\prime }})$ only depends on the angle (θ) with the extraordinary axis (n_e).

A and B are the magnitudes of modified refractive indices, to determine the vectors of the two allowed polarization can be found numerically by rotating the plane of the ellipse so that values of A and B are solutions of (3.6) or analytically by use of Maxwell's equations. From [57], a plane wave travelling in a medium with orthogonal components must satisfy

$$\mathit{k}\times (\mathit{k}\times \mathit{E})+{\omega }^2{\mu }_0ϵ\mathit{E}=0.$$ 3.9

This translates into three linear homogeneous equations for the three components of E along the principle axes, which can be written in the following matrix form [57]:

$$\left[\begin{array}{ccc}{n}_1^2{k}_0^2-k_2^2-k_3^2& k_1{k}_2& k_1{k}_3\\ k_2{k}_1& n_2^2{k}_0^2-k_1^2-k_3^2& k_2{k}^3\\ k_3{k}_1& k_3{k}_2& n_3^2{k}_0^2-k_1^2-k_2^2\end{array}\right]\left[\begin{array}{c}{E}_1\\ E_2\\ E_3\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right].$$ 3.10

To determine the directions of allowed polarization we must determine the components (k_x, k_y, k_z) = (ku_x, ku_y, ku_z) and the elements of the matrix in (10) for the allowed wavenumbers k_A and k_B that have already been determined from (8) where k_A = A/k₀ and k_B = B/k₀. By solving two of the three equations in (10) we can determine the ratios E₁/E₃ and E₂/E₃ from which the direction of the corresponding electric field E can be obtained [57].

(b). Uniaxial symmetry (trigonal, tetragonal and hexagonal)

Uniaxial systems are those that include symmetry operators that are larger than twofold operators, except for the cubic case. Trigonal, tetragonal and hexagonal symmetry are all uniaxial, where the definitions of ordinary and extraordinary waves are applied. In each case, the axis corresponding to the high-order symmetry operation correspond to the optic axis. Depending on the intrinsic birefringence, a uniaxial system can either be positive or negative. In the uniaxial case, the indicatrix simplifies and due to symmetry of two principle refractive indices forces one axis of the ellipse projection to be invariant with $\hat{\mathit{k}}$, the so-called ordinary optical axis n_o. The complimentary axis ($(n_e^{\mathrm{\prime }})$) depends only on the angle of $\hat{\mathit{k}}$ with the principle ‘extraordinary' optical axis, n_e (figure 2).

$$n_0^{\mathrm{\prime }}=n_0$$ 3.11a

and

$$n_e^{\mathrm{\prime }}=n_e\cos \theta .$$ 3.11b

In the special case of θ = 0, 2π (i.e. $\hat{\mathit{k}}\times {\hat{\mathit{n}}}_{\mathit{e}}=1$), the ellipse projection of the indicatrix will become a circle and the system becomes isotropic. For all other values of θ, there will be a decomposition of the electric field along the two allowed polarization axis. The ${\widehat{{\mathit{n}}^{\mathrm{\prime }}}}_{\mathit{e}}$ vector is also only dependent on θ, with ${\widehat{{\mathit{n}}^{\mathrm{\prime }}}}_{\mathit{o}}=\hat{\mathit{k}}\times {\hat{\mathit{n}}}_{\mathit{e}}^{\mathrm{\prime }}$, the amplitudes of the decomposed electric field components depend only the directional cosines of θ and the initial polarization.

The values of $⟨{({\hat{e}}_n.\hat{\mu })}^2⟩$ also becomes simpler in uniaxial crystals, the two axis symmetry of the system means the dipole contributions in those axes cannot be separated and are averaged over all of ϕ, hence for a uniaxial crystal (trigonal, tetragonal or hexagonal).

$$\begin{array}{rl}⟨{({\hat{e}}_3.\hat{\mu })}^2⟩& ={\cos }^2\hspace{0.17em}{\theta }_d,\end{array}$$ 3.12

$$\begin{array}{rl}⟨{({\hat{e}}_2.\hat{\mu })}^2⟩& =⟨{({\hat{e}}_1.\hat{\mu })}^2⟩=\frac{1}{2}{\sin }^2\hspace{0.17em}{\theta }_d\end{array}$$ 3.13

$$\begin{array}{rl}& =\frac{1}{2}(1-{({\hat{e}}_3\cdot \hat{\mu })}^2),\end{array}$$ 3.14

where ${\hat{e}}_3$ is the by convention the high symmetry axis. Therefore, in the uniaxial case both the polarization normal modes and the values of $⟨{({\hat{e}}_n.\hat{\mu })}^2⟩$ depend only on their angle with respect to the high symmetry axis.

(c). Cubic symmetry

Cubic crystal symmetry is a separate class that does not show birefringence. With regard to photoselection in cubic crystals, these are the same as for isotropic samples up to second-order interaction. Therefore, analysis for isotropic conditions are generally applicable [55,56]. Thus, for each direction except precise coincidence of polarization with an optical axis, both uniaxial and biaxial symmetries cause a decomposition into two orthogonal modes while cubic crystals do not.

(d). Simulation of population dynamics in birefringent crystals

By analysing each mode separately, it is clear that a reduced portion of the incident beam reduces the populations accordingly. With unbalanced intensities, a power titration of the incident beam would result in an imbalance of one-photon and multi-photon processes. To demonstrate the significance of this (in the uniaxial case) a model was constructed similar to that found in [51] based on z-scan analysis. The system begins in its ground state (S₀) and is excited on resonance by a 100 fs optical pulse. As the pulse duration is significantly shorter than the excited state life times, multiple re-excitation events cannot occur. The magnitude of a photoproduct population is product of the internal quantum yield and population of a singularly excited S₁ state while double excitation to an S₂ state is effectively a loss mechanism. In order to maximize the photoproduct yield it is important to carefully select the energy density that maximizes the population of S₁. The populations of the states can be numerically calculated using

$$\begin{array}{rl}{N}_{S_0}(t)& =N_{S_0}(-\mathrm{\infty })\exp \left[\frac{{\sigma }_{S_0}F(t)}{\hslash \omega }\right],\end{array}$$ 3.15

$$\begin{array}{rl}{N}_{S_1}(t+\text{d}{t}^{\mathrm{\prime }})& =N_{S_0}(-\mathrm{\infty })\left(1-\exp \left[\frac{{\sigma }_{S_0}F(t)}{\hslash \omega }\right]\right)-N_{S_2}(t)\end{array}$$ 3.16

$$\begin{array}{rl}\text{and}{N}_{S_2}(t)& =N_{S_1}(t)\left(1-\exp \left(\frac{{\sigma }_{S_1}F(t)}{\hslash \omega }\right)\right),\end{array}$$ 3.17

where ${\sigma }_{S_n}$ is the isotropic cross section of state S_n, F(t) is the integrated flux of the excitation pulse with central frequency ω and $N_{S_0}(-\mathrm{\infty })$ is the population of the ground state before the pulse arrives. This model is only valid for excitation conditions where double excitation does not dominate, i.e. the population of the S₂ remains low. The birefringent case the two electric field components are combined with their respective cross-sections incorporating $⟨{({\hat{e}}_n.\hat{\mu })}^2⟩$, therefore

$${\sigma }_{S_0}F(t)=[⟨{({\hat{e}}_1.\hat{\mu })}^2⟩\cdot 3{\sigma }_{S_0}]F_{e_1}(t)+[⟨{({\hat{e}}_2.\hat{\mu })}^2⟩\cdot 3{\sigma }_{S_0}]F_{e_2}(t).$$

The factors of 3 in front is the ${\sigma }_{S_0}$ is to convert the orientationally averaged isotropic cross section previously obtained from solution and crystalline slurry measurements to the aligned value.

Figure 4 shows the populations of the three states as a function of pump pulse energy density in the situations where the polarization aligned vertically (θ = 0°) or horizontally (θ = 90°) with the crystal frame for different crystal orientations. In each case, the dipole is aligned very close to the high symmetry axis (${\theta }_d=\sim 0^{\circ }$). It can be seen that when the polarization and high symmetry axis align (figure 4,h) there is a strong pumping of S₁ at even low energy densities. While at higher energies double excitation begins to become relevant and S₂ is also populated at the cost of S₁ population. The opposite is true when there is perpendicular alignment (figure 4b,g), where pumping of S₁ and S₂ is heavily supressed requiring an order of magnitude increase in energy density to achieve the same level of pumping. In the intermediate cases, it is split between the two polarizations with one more dominant that the other.

Figure 4.

Calculated populations of S₀ (black), S₁ (red) and S₂ (blue) state for the e₁ (dashed), e₂ (dot-dashed) and the total (solid) as a function of energy density for vertical and horizontal polarizations in the crystal frame and different crystal orientation angles (θ). (Online version in colour.)

By taking a fixed pulse energy (1 mJ mm⁻²) and rotating the polarization and θ we can build a two-dimensional map of which combinations will lead to enhancement or suppression of pumping. Figure 5 shows that for this system (θ_d = 0) choice of polarization angle can greatly affect the variance that is seen in pumping over different orientations. This can be expanded further for systems with different values of θ_d. Figure 6 show it can be seen that this has a large effect of the distributions of the pumped regions. The special case where the dipole is perpendicular to the high symmetry axis (θ_d = 90°), results an almost uniform pumping of S₁ over all orientations and a complete suppression of S₂ for a pulse energy that would result in high variance for other values of θ_d.

Figure 5.

Calculation of the populations of S₀, S₁ and S₂ when excited by an optical pulse on resonance with the total energy density of 1 mJ mm⁻². The total (top), E_o (middle) and E_e (bottom) contributions are shown. The left column shows the normalized ratio of pulse energy in the respective components.

Figure 6.

Total populations of S₀, S₁ and S₂ for different values of θ_d, varied between 0 (top), π/2 (middle) and π/4 (bottom) for the same conditions as figure 5. This demonstrates the large effect the dipole orientation has on the distribution of state pumping.

We should note that these calculations do not take into account propagation inside the crystal or refraction at the crystal surface. We justify this as the protein crystals in a typical TR-SFX experiment are small (less than 50 µm path length) and have low absorption (less than 0.2 OD). A practical application of photoselection theory must additionally include modelling of propagation under conditions of liquid jet delivery and specific crystal morphology and its heterogeneity. An assessment of the phase difference resulting from typical values of intrinsic birefringence and optical path length in the TR-SFX geometry shows that this can generally be neglected [34]. In addition crystals are usually delivered into the X-ray interaction region inside a jet or droplet of their crystallization media (mother liquor) which has refractive index very similar (1.36 for PEG3350) to that of the protein crystal (1.3–1.4 [58]). We do consider that geometry of the droplet or jet may have an effect on localized pump intensity due to the combination of an air–liquid surface and the typical curvature of a droplet/jet acting like a lens could result in localized variance in pump intensity. This effect will be minimized for very thin jets (few micrometre diameter) where the diffraction limit and M² of the optical pulse will limit the maximum variation of optical intensity inside the drop. In this calculation we have only dealt with the uniaxial case however the same treatment could be applied to the biaxial case. The expected behaviour and distribution of populations is expected be more complex due to the addition degrees of freedom on the polarization modes and dipole orientation, depending on the presence or absence of twofold symmetry.

These calculations have demonstrated that in the case of uniaxial crystal there can be a large variation in the degree of excitation depending on the value of θ_d, due to the decomposition of the electric field. This could be an important consideration when selecting or modifying a crystal targets for TR-SFX studies, where it is favourable to have uniform controlled excitation for both boosting signal to noise and assignment.

4. Photoselection in reciprocal space: an opportunity for crystallographic analysis

Our second contribution considers the implications of inhomogeneity of photoselection (§3) under typical conditions of an SFX experiment. Specifically, each diffraction pattern that is indexed and integrated contains a number of diffraction spots with known Miller index that are stimulated where the orientation matrix is given for conditions that satisfy Bragg's Law. Of course, this is a selective portion of inverse space containing only those reflections that cut Ewald's sphere in the classical geometric construction of diffraction. Summing all diffraction patterns that are normally combined in SFX datasets into powder patterns, each reflection is measured multiple times and is found in Debye–Scherrer rings. Thus, different observations of a reflection having the same Miller index, found at different positions in the Debye–Scherrer rings, corresponds to a different orientation matrix. As a consequence, different measurements of the same reflection on the Debye–Scherrer rings correspond to very different photochemical populations. Here, we propose an explicit analysis where the orientation matrix from indexing is used in order to assign photochemical parameters that could potentially be used in order to extract the orientational dependence of photo-induced differences. In principle, this shows that both one- and multi-photon populations could be separated in this manner by enforcing an orientation matrix determined amplitude in the overall scaling process. The general procedure for determining the photochemical amplitudes for a given stationary diffraction pattern, from serial crystallography uses the orientation matrix given for the indexing result

$${\mathit{A}}^{\ast }=\left(\begin{array}{ccc}{a}_x^{\ast }& b_x^{\ast }& c_x^{\ast }\\ a_y^{\ast }& b_y^{\ast }& c_y^{\ast }\\ a_z^{\ast }& b_z^{\ast }& c_z^{\ast }\end{array}\right),$$ 4.1

where the subscripts x, y and z refer to the projections onto the laboratory axis of the reciprocal space lattice [a*, b*c*].

The laboratory space direction A = [a, b, c]^T is obtained by inverting the reciprocal-space orientation matrix.

$$\mathit{A}={\mathit{A}}^{-1}=\left(\begin{array}{ccc}{a}_x& a_y& a_z\\ b_x& b_y& b_z\\ c_x& c_y& c_z\end{array}\right).$$ 4.2

Next, it is necessary to define the optical k-vector direction, which is often (near-)collinear with the X-ray direction, in order to find the directions of the allowed polarized radiation modes and compute the rate equations as shown above. The general solution to this problem will show that the circumference of a Debye–Scherrer ring for a single reflection with Miller index h, k, l will trace out the resulting S₀, S₁ and S₂ amplitudes for a 2π trajectory of θ as shown in figures 5 and 6 for given geometries. Essentially, the detector polar coordinate will be directly proportional to these amplitudes that will thus be equal for all resolutions (figure 7). The rotation matrix R which must be applied to the laboratory orientation matrix A to place the indicatrix will depend on symmetry. In the case of uniaxial symmetry it is sufficient to know the direction of the three-, four- or sixfold axis however also needs a quantification of the magnitude of the intrinsic birefringence. Biaxial crystals present difficulty except for the orthorhombic case. For a triclinic crystal, this rotation matrix R must be determined experimentally by ellipsometry on all crystal directions and for each optical frequency, whereas the monoclinic case only provides an unambiguous solution to Maxwell's polarized wave equations in the absence of such measurements of the k-vector is orthogonal to the crystallographic b-axis which by convention carries the twofold symmetry. Only for orthorhombic crystals is a direct orientation of the indicatrix possible based on the reciprocal-space orientation matrix A*, because the directions if the principle susceptibilities align with the crystallographic directions. However, the magnitude of the intrinsic birefringence is also required to solve the normal modes in (equation (3.10)). The general procedure that is outlined thus provides an opportunity to analyse serial crystallography data by calculation of photochemical amplitudes. Practically, binning could be done in order to assemble stream files that contain diffraction images that are associated with defined ranges of photochemical amplitudes. The resulting streams of binned diffraction patters would, therefore, have the same signal-to-noise as the conventional SFX result, but allow a separation of time-resolved measurements on the basis of theoretical considerations as outlined in §§2, 3 and 4. This proposal, therefore, extends the existing analysis of time-resolved SFX that uses a Monte Carlo averaging. Here, we have shown that explicit methods exist in order to retrieve differences directly from X-ray crystallographic indexing.

Figure 7.

Pictographic representation of the photoselection scheme of a uniaxial crystal in an X-ray beam, demonstrating how a rotation of the crystal orientation angle θ will result in the same rotation in both reciprocal space and optical frame of the crystal. Therefore, a given Bragg reflection (pink/green) measured at different radial angles on its Debye–Scherrer cone will correspond in optical geometries in the crystal that result in weak (solid) or strong (solid) pumping for a fixed pulse energy. In this example, the radial distributions of S₁, S₂ and S₃ are taken from figure 6 where θ_d = 45° and pol = 0.

Data accessibility

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Authors' contributions

C.D.M.H. proposed calculation of photoproduct yields resulting from different crystal orientations. J.J.v.T. provided initial draft the article. C.D.M.H. performed the calculations, created the figures and the majority the word-processing. J.J.v.T. and C.D.M.H. worked equally on the scientific content and interpretation of calculations.

Competing interests

We declare we have no competing interests.

Funding

J.J.v.T. acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) [EP/M000192/1] and the Biotechnology and Biological Sciences Research Council (BBSRC) [BB/P00752X/1].