X-ray pulse stretching after diffraction

In this article, the effect of stretching of short X-ray pulses after symmetric or asymmetric diffraction on crystal systems is studied. This is used to determine the optimal experimental arrangement to minimize the pulse stretching during diffraction.

Abstract

The development of ultrashort X-ray pulse sources requires optics that keep the pulse length as short as possible. One source of pulse stretching is the penetration of the pulse into a crystal during diffraction. Another source is the inclination of the intensity front when the diffraction is asymmetric. The theory of short X-ray pulse diffraction has been well developed by many authors. As it is rather complicated, it is sometimes difficult to foresee the pulse behavior (mainly stretching) during diffraction in various crystal arrangements. In this article, a simple model is suggested that gives a qualitatively similar shape to the diffracted pulse which follows from exact theory. It allows proposal of what experimental arrangement is optimal to minimize the pulse stretching during diffraction. First, the effect of pulse stretching due to penetration into a crystal surface is studied. On the basis of this, the pulse profile change during diffraction by two crystals, either symmetric or asymmetric, is predicted.

1. Introduction  

The development of ultrashort X-ray pulse sources (Stepanov & Hauri, 2016 ▸) stimulated development of the diffraction theory of such pulses. An interesting effect in this context is the slight stretching of pulses after diffraction. The reason for this is that the impinging pulse generates diffracted radiation through a certain volume of the traveled path inside the crystal. Consequently, the diffracted pulse will be longer than the impinging pulse. So far, many authors have applied the dynamical theory of X-ray diffraction from a perfect crystal, assuming that the pulse length is substantially shorter than the extinction depth, or they simply considered the pulse shape as the δ function (Shvyd’ko & Lindberg, 2012 ▸; Chukhovskii & Förster, 1995 ▸; Shastri et al., 2001b ▸; Wark et al., 2000 ▸; Graeff, 2002 ▸; Malgrange & Graeff, 2003 ▸; Chapman & Nugent, 2001 ▸). This theory is rather complicated and it proves difficult to foresee what should be done to keep the diffracted pulse short, especially when using a more complex crystal arrangement.

This article is focused on possible stretching of short (∼fs or shorter) incident X-ray pulses after diffraction as a result of the penetration depth and intensity-front inclination. It is prevalently supposed that the pulse will be long enough to be described as a monochromatic wave. Shvyd’ko & Lindberg (2012 ▸) did not explicitly calculate the prolongation but it is possible to extract the lateral profile of the diffracted pulse from their article. An important outcome is that the profile is approximately a rhomboid (with rounded corners). One side of the rhomboid is perpendicular to the direction of the diffracted beam and the other side is perpendicular to the diffracting crystallographic planes. One can see several other satellite rhomboids, whose intensities are at least one order of magnitude lower, and their origin is connected with the assumption that the shape of the incident pulse is the δ function and the pulse is very narrow.

Shvyd’ko & Lindberg (2012 ▸) calculated the diffraction from one crystal. However, very often a system of two (or even more) crystals is used in experiments. A theory for a simple two-crystal system is, for example, presented in the work of Shastri et al. (2001a ▸). To investigate the effect of two or more crystal systems on the stretching of the diffracted pulse is obviously important. Fortunately, it is shown in this article that even a very simple geometrical model also gives a rhomboidal profile of the diffracted pulse, and thus it may give an approximate description of what occurs when the pulse penetrates into the crystal. Moreover, it allows one to easily extrapolate the model to more complex crystal systems.

2. Pulse penetration  

2.1. Symmetric diffraction  

In this article, pulses with lengths that are at least an order of magnitude shorter than the extinction depth but far from being approximated as the δ function will be studied. Each pulse may capture thousands of atomic layers. Furthermore, it is supposed that the shape and the direction of the incident pulse do not change as it travels inside the crystal.

Let us suppose that a narrow beam consisting of X-ray pulses is diffracted from a perfect crystal and that this diffraction is symmetrical (Fig. 1 ▸). The pulse penetrates into the crystal and it generates diffracted radiation at every point of its path. First, a narrow element of the incident pulse will be considered. Fig. 1 ▸ shows that the diffracted elements from all points of the path inside the crystal are distributed along the line perpendicular to the diffracting planes and move as a whole in the direction of the diffracted beam. Moreover, they have an equal phase. The space occupied by the diffracted photons generated by the incident-pulse element at any moment resembles a thin rod that is perpendicular to the diffraction planes. In this article, this will be called the ‘penetration rod’ (PR). It is inclined with respect to the normal to the direction of the diffracted beam (Fig. 1 ▸).

Figure 1.

An element of the impinging short pulse after diffraction, which has the shape of a thin rod perpendicular to the diffracting planes.

The length of the PR is 2e and the transverse dimension of the diffracted beam element (i.e. the projection of the PR onto the normal to the diffracted beam direction) is 2ecosθ, where e is the penetration depth and θ is the Bragg angle. Supposing that this penetration depth e is equal to the extinction depth Λ₀, then (Caciuffo et al., 1987 ▸; Authier, 2001 ▸)

where γ₀ and γ_H are the direction cosines of the incident and diffracted beam with respect to the crystal-surface normal, C is the polarization factor, K is the wavenumber and χ_H is the susceptibility. The formula is valid for non-absorbing crystals.

The pulse length Δ (the projection of the PR onto the diffracted beam direction) is

and is minimal in the case of small θ. It represents the pulse stretching. Here, the pulse length and the transverse dimension (width) are the space occupied by photons, i.e. the intensity distribution is neglected. Evidently, the maximum intensity is at the top of the PR.

Synchrotron monochromators usually consist of two parallel crystals adjusted to a nondispersive (parallel) setting. (The second crystal gives the diffracted beam its original direction.) As shown in Fig. 2 ▸, the profile of the pulse element diffracted from both crystals has again the shape of a PR perpendicular to the diffracting planes. Its length is extended twice and is the sum of the PR lengths of both crystals. This represents the stretching of the pulse by 4esinθ. Fig. 2 ▸ is a schematic drawing only and it again does not take the intensity into account. To get the real length of the diffracted element, the intensity distribution along the PR should be calculated.

Figure 2.

Because of pulse penetration into the crystal, each element of the narrow impinging short pulse is broadened and prolonged after diffraction. After diffraction from two symmetric crystals, a new PR (drawn in red) is generated, which is again perpendicular to the diffracting planes and whose length is the sum of the PR lengths of both crystals. The prolongation is due to the inclination of the new PR with respect to the normal to the beam direction.

The extinction depth in the hard X-ray region and for crystals used as monochromators is of the order of micrometres. This means that the effect of the extinction depth may increase the pulse length up to ∼30 fs (again, neglecting the intensity).

So far, a very narrow element of the impinging beam has been considered. In reality, a beam has a finite width (i.e. the transverse dimension), which is larger than or comparable to the penetration depth. The intensity front of the impinging beam is perpendicular to the beam axis (no inclination). As was shown above, the penetration of every pulse element into the crystal creates a PR perpendicular to the crystal surface after diffraction. After diffraction of the whole impinging pulse, the individual PRs superpose and create a rhomboid. One side of the rhomboid is perpendicular to the beam direction and its length is a measure of the beam width. The other side is perpendicular to the diffracting planes and its length is a measure of the penetration depth. The maximum intensity is at the front of the pulse. This corresponds qualitatively to the work of Shvyd’ko & Lindberg (2012 ▸). Fig. 3 ▸ shows how the real diffracted short pulse is broadened and prolonged as a result of the penetration depth. The effect of pulse stretching will be minimal if the intensity front perpendicular to the direction of the beam is as parallel as possible to the normal to the diffracting planes. In other words, the stretching is minimal if the rhomboid is squeezed and becomes slim. This condition will be obviously fulfilled for small Bragg angles θ for a given λ. (In general, the pulse length can also be shortened by using suitable crystals and diffracting planes to shorten the extinction depth.)

Figure 3.

A real impinging short pulse, having finite width, has a rhomboidal shape after diffraction. The prolongation is caused by the penetration of the incident beam into the crystal. The diffraction is symmetrical. IF stands for intensity front.

After the diffraction from two crystals, the PR is doubled, as shown in Fig. 2 ▸. However, this is only a schematic depiction. Let us suppose that the diffracted intensity decreases exponentially with depth. To get the intensity distribution after two crystals, it is necessary to make a convolution of two rhomboids. The resulting intensity distribution shows that the maximum intensity is not at the front of the pulse anymore and the pulse stretching is approximately doubled as compared with the single-crystal case (Shastri et al., 2001a ▸).

So far it has been supposed that the impinging pulse is long enough to be regarded as a monochromatic wave. However, very short pulses that are approximated by the δ function represent polychromatic beams. According to Shvyd’ko & Lindberg (2012 ▸), the incident wavefield with a bounded wavefront is a superposition of plane waves propagating at different angles of incidence. One may expect that the crystal selects the wavelengths and sends the corresponding beams in different directions. The satellite diffractions originated in this way might then be shielded by slits. However, the precise calculation (Shvyd’ko & Lindberg, 2012 ▸) does not support this simple idea. The authors show that there are weak satellite diffractions traveling in the same direction but their distances from the main reflection path are different. If this is correct then these satellite reflections may be at least partially shielded.

3. Pulse penetration and intensity-front inclination  

3.1. Asymmetric diffraction – one crystal  

As mentioned above, the effect of the penetration depth on the diffracted pulse length should be minimal if symmetrical Bragg diffraction is used and the Bragg angle is as small as possible (for a given λ). The rhomboid remains slim in this case. For example, in the case of Si(220) diffraction and Cu Kα radiation, the penetration depth is ∼1 µm. The pulse stretching, i.e. the projection of PR (2 µm) onto the diffracted beam direction, is ∼0.8 µm, which represents ∼2.6 fs. This may be reduced when using (111) diffraction. The use of smaller Bragg angles is impossible with Si crystals and the given λ.

If it is not possible to fulfill the condition of small θ then it may be possible to use strongly asymmetric diffraction. It is known that with strongly asymmetric diffraction the penetration depth decreases to zero (Authier, 2001 ▸). The dependence of the penetration on the angle α between the surface of the crystal and the diffracting crystallographic planes is determined by γ₀γ_H in equation (1). This is shown in Fig. 4 ▸. This figure also shows that the penetration is small in the case of small θ. So in the case of symmetric crystals and small θ, two mechanisms are present that decrease the pulse stretching: the small penetration according to Fig. 4 ▸ and the parallelism between the PR and the normal to the beam direction. Fig. 4 ▸ shows that for highly asymmetric diffraction the penetration is low. However, there is another problem: asymmetric diffraction inclines the intensity front (beam profile) (Chubar et al., 2016 ▸; Zholents et al., 1999 ▸).

Figure 4.

The expression γ₀γ_H from equation (1) determines the behavior of the penetration depth for a given λ. It is seen that for highly asymmetric diffraction the penetration depth approaches zero.

Here, inclination means the deviation from the normal to the beam direction. Let us suppose that the pulse with the intensity front perpendicular to the beam direction diffracts from the crystal with an asymmetry angle α. The situation is schematically shown in Fig. 5 ▸.

Figure 5.

Asymmetric Bragg diffraction inclines the intensity front. (Penetration is not shown.)

The intensity front of the impinging beam and its inclination after the diffraction are indicated in red. The penetration depth of the impinging radiation into the crystal is neglected for the moment. The intensity front of the impinging radiation is inclined by an angle β, where

The cross section of the exit beam is increased by a factor sin(θ + α)/sin(θ − α).

When the Bragg angle θ is close to α (the strongly asymmetric case) and thus β is close to θ, the inclined intensity front (beam profile) after diffraction is almost perpendicular to the diffracting crystallographic planes. Taking the penetration into account, the PR is again exactly perpendicular to the diffracting planes, as in the case of symmetric diffraction, and its length is 2e. Consequently, the two sides of the rhomboid are almost parallel and the rhomboid is narrow. Thus the pulse remains narrow but inclined (Fig. 6 ▸) by angle θ. The PR behaves as the prolongation of the intensity front.

Figure 6.

The inclined intensity front represents the prolongation of the pulse. This effect may be reduced if the sample surface is parallel to the intensity front of the pulse.

The pulse projection onto the beam direction represents the pulse prolongation, which may be substantial if the sample is placed perpendicular to the beam. However, if the sample is inclined in such a way that its surface is parallel with the profile of the pulse (Fig. 6 ▸), the sample surface may be irradiated simultaneously by all parts of the pulse. The effect of pulse stretching due to the inclination is then minimal. Unfortunately, this kind of asymmetry broadens the diffracted beam. Shvyd’ko & Lindberg (2012 ▸) proposed using the effect of the intensity-front inclination for time measurements of ultrafast processes.

Asymmetric diffraction, as shown in Fig. 5 ▸, was also studied by Shvyd’ko & Lindberg (2012 ▸). This figure shows that the beam profile is narrow and long and is inclined from the normal to the beam direction by ∼28°. This value exactly corresponds to equation (3) and indicates that our simple model is correct.

3.2. Asymmetric diffraction – two crystals in nondispersive (+, −) setting  

Probably the only simple way to cancel the inclination of the intensity front is to use another asymmetric crystal as the second crystal (or to use a channel-cut crystal). Fig. 7 ▸(a) represents a strongly asymmetric double-crystal monochromator. The first asymmetric diffraction inclines the intensity front, which is now almost perpendicular to the diffracting planes. It is obvious that the second diffraction reverses this effect, and the intensity front becomes again perpendicular to the exit-beam direction. In other words, using two strongly asymmetrically cut crystals in a nondispersive setting with parallel diffraction surfaces cancels the inclination effect. The penetration in the first crystal creates a PR with a length of 2e, which is small because of the strong asymmetry. This PR leaves the first crystal in such a way that it is still perpendicular to the diffracting planes and is almost parallel with the intensity front. It represents a prolongation of the intensity front after the first diffraction. Its length after diffraction on the second crystal may be neglected since it is reduced by sin(θ − α)/sin(θ + α). This means that, after diffraction from both crystals, the pulse stretching is in practice influenced only by the penetration in the second crystal, which is small because of the high asymmetry. However, as seen in Fig. 7 ▸(b), the rays which penetrate into the second crystal have to travel a long way to escape the crystal after diffraction. One may speculate whether all photons diffracted from the deeper layers under the surface leave the crystal. If not then the penetration effect on the second crystal may be further reduced. In any case, quantifying this effect would require detailed study.

Figure 7.

A double-crystal setting with strongly asymmetric diffraction cancels the intensity-front inclination (a) created by the first crystal. (b) The rays which penetrate into the crystal have to travel a long way, after diffraction, to escape the crystal.

Given the above, it can be concluded that in order to give the diffracted beam its original direction, and to minimize pulse stretching due to penetration at the same time, a strongly asymmetric (+, −) double-crystal monochromator with parallel diffraction surfaces seems to be the only choice. Moreover, the double-crystal monochromator with asymmetry shown in Fig. 7 ▸ has a high wavelength acceptance.

3.3. Asymmetric diffraction – two crystals in dispersive (+, +) setting  

It is well known that when a high degree of monochromaticity is needed a double-crystal monochromator with a (+, +) setting is often used. If the crystals diffract symmetrically (diffracting planes are parallel with the surface) and the Bragg angle is small, the pulse stretching is small as well, almost as in the case of the above-mentioned (+, −) setting. However, the direction of the diffracted beam is different from the direction of the impinging beam. To make these directions the same, a four-crystal arrangement (Bartels, 1983 ▸) is often used. Unfortunately, every crystal contributes to pulse stretching, even if this stretching is small. If the Bragg angle is high and the diffraction is symmetrical, the stretching will be maximal. This is because the angles between the intensity front and PR in both crystals are relatively high and thus the rhomboids are thick.

If it is not possible to use a small Bragg angle, then a double asymmetrically diffracted (+, +) crystal monochromator may also be used, but this is more complicated. As was discussed above, asymmetric diffraction inclines the intensity front and thus stretches the pulse. Two asymmetric crystals in a (+, +) setting may even enhance this effect. As seen in Fig. 8 ▸, after diffraction from the first crystal, the intensity front becomes inclined. When changing ψ the inclination after the second crystal also changes, but for certain ψ calculated according to equation (4) this inclination becomes zero and the intensity front is again perpendicular to the beam direction:

(Here γ = 2θ.) Fig. 8 ▸ shows this situation for θ = 40°, φ = 70° and ψ = 26.35°. However, this cancelation of the intensity-front inclination is at a cost of the increase of the transverse dimension of the beam. Such a pulse may be focused by a parabolic mirror, keeping the pulse length short.

Figure 8.

The (+, +) X-ray crystal monochromator designed according to equation (4) so that it does not incline the intensity front.

Such a monochromator transmits a narrow wavelength band and minimizes the pulse stretching but the direction of the exit beam is different from the direction of the impinging beam. The beam may also be directed in the opposite direction (from the right to the left). In this case, the exit beam is very narrow. This kind of monochromator seems to be the optimal solution to keep the pulse short. The use of the Bartels arrangement is also possible here. In this case, equation (4) does not have to be fulfilled.

4. Conclusions  

In the case of X-ray pulse diffraction by crystals, the penetration into the crystal may significantly influence the length of the diffracted pulse; it broadens and stretches the pulse. A similar effect may occur when asymmetric diffraction is used and the intensity front is inclined. In this article, an attempt was made on the basis of a simple geometrical model to propose how to minimize this beam prolongation. By comparing this model with exact calculations (Shvyd’ko & Lindberg, 2012 ▸), it is believed that the model can give reliable results concerning geometry (e.g. pulse profile). From the model, some suggestions follow for given wavelengths. Firstly, the use of symmetrical Bragg diffraction and a small Bragg angle should minimize pulse stretching due to the penetration depth. Two symmetrically cut crystals in a non-dispersive (or dispersive) setting may also be used. The nondispersive (+, −) setting gives the diffracted beam its original direction but it slightly increases the stretching.

If it is not possible to use small Bragg angles, strongly asymmetric diffraction may be used. In this case, the use of two asymmetrically cut crystals in a nondispersive (+, −) setting and with parallel diffraction surfaces is necessary to cancel the wavefront inclinations caused by every crystal. It is shown that even a high-resolution dispersive (+, +) setting may cancel the intensity-front inclination if relation (4) between angles is fulfilled. If it is not fulfilled, the crystal system may create a strong intensity-front inclination which prolongs the pulse.