## research papers

## Diffraction of X-ray free-electron laser femtosecond pulses on single crystals in the Bragg and Laue geometry

^{a}M. V. Lomonosov Moscow State University, 119992 GSP-2 Moscow, Russia^{*}Correspondence e-mail: vabushuev@yandex.ru

A solution of the problem of

for X-ray pulses with arbitrary dimensions in the Bragg and Laue cases in a crystal of any thickness and asymmetry coefficient of reflection is presented. Analysis of pulse form and duration transformation in the process of diffraction and propagation in a vacuum is conducted. It is shown that only the symmetrical Bragg case can be used to avoid smearing of reflected pulses.### 1. Introduction

In the near future, X-ray free-electron lasers (XFELs) with wavelength λ ≃ 0.1 nm will become available for a wide community of users. Therefore it is of interest to consider as a tool for controlling and tailoring parameters of femtosecond pulses and for developing methods of X-ray laser pulse diagnostics. Three XFEL projects are now actively developed: the European XFEL Facility in Germany (Altarelli *et al.*, 2006), the LCLS (Linac Coherent Light Source) in the USA (Arthur *et al.*, 2002) and the SCSS (SPring-8 Compact SASE Source) in Japan (Tanaka & Shintake, 2005). In these machines, X-ray bunches of duration ∼100–200 fs leave an undulator as a result of self-amplification of spontaneous radiation of 15 GeV electrons. Theoretical calculations show that these pulses will have an irregular multiple-peak internal structure and consist of several hundreds of supershort independent sub-pulses of duration τ_{0} ≃ 0.1 fs, separated by time intervals Δ*t* ≃ 0.3–0.5 fs. A typical pulse has a transversal size *r*_{0} ≃ 50 µm at the undulator exit, angular divergence ≃ 1 µrad, peak power ≃ 10 GW and average power ≃ 40 W (Saldin *et al.*, 2004).

The analysis of diffraction of XFEL radiation has been restricted so far to the approximation of a plane (unlimited) wavefront for the Bragg case (Chukhovskii & Förster, 1995; Shastri *et al.*, 2001*a*,*b*; Graeff, 2004) and for the Laue case (Shastri *et al.*, 2001*b*; Graeff, 2002; Malgrange & Graeff, 2003). The time structure of the incident pulse has been approximated either by a δ function (Chukhovskii & Förster, 1995; Shastri *et al.*, 2001*a*,*b*; Graeff, 2002; Malgrange & Graeff, 2003) or by a Gaussian (Shastri *et al.*, 2001*a*; Graeff, 2004). Although giving some insight into the physics of short-pulse diffraction, such an approach cannot in principle take into account the presence of transverse mode structure and, even more essential, a non-uniform distribution of the field phase inside a pulse. However, such a phase distribution will inevitably arise at large, of the order of 100 m (Saldin *et al.*, 2004), distances from the undulator to the sample or monochromator crystal. Besides, all analysis so far has been limited to the reflected pulse field on the exit surface of a crystal, whereas significant practical interest is for spatial (transversal) and temporal (longitudinal) smearing of pulses during their further propagation in vacuum.

In the present article a general theory of **E**_{in}(**r**, *t*), on crystals with arbitrary thickness and asymmetry coefficient in the Bragg and in the Laue cases is developed. Such an approach allows us to analyse the structure of fields **E**_{g}(**r**, *t*) of forward-diffracted (transmitted, *g* = 0) and diffracted (reflected, *g* = *h*) pulses at any distance from the crystal, and also the degree of space and time coherence of these pulses and their relation with the statistical properties of the XFEL radiation field.

### 2. Theory

We shall consider diffraction reflection and transmission of a pulse of X-ray radiation *E*_{in}(**r**, *t*) = *A*_{in}(**r**, *t*) exp(*i***K**_{0}**r** − *i*ω_{0}*t*), which is incident on a single-crystal plate of thickness *d*. The field on the entrance crystal surface *z* = 0 can be written as

where *A*_{in}(*x*, *t*) is a slowly varying complex amplitude (the envelope of a wave packet), *K*_{0x} = *K*_{0}sinθ_{0}, *K*_{0} = ω_{0}/*c* = 2π/λ and *c* is the light speed in a vacuum; the axis *x* is directed along the crystal surface and the axis *z* is directed inside the crystal along the normal **n** to the surface (Fig. 1). The projection of the incident wavevector on the axis *z* is *K*_{0z} = *K*_{0}γ_{0}, where γ_{0} = cos(**K**_{0}·**n**) = cosθ_{0}. The angle of incidence of the radiation to the normal **n** is θ_{0} = ψ − θ_{B} − Δθ, where θ_{B} is the for the central (average) frequency ω_{0}, which is determined by the expression 2*K*_{0}sinθ_{B} = *h*, where *h* is the modulus of the vector **h** = (*h*cosψ, −*h*sinψ), Δθ is the angular deviation from the exact which is determined by the expression **K**_{0}**h** = −*K*_{0}*h*sin(θ_{B} + Δθ), and ψ is the inclination angle of reflecting crystal planes to the normal **n**. The restriction |ψ − θ_{B}| < π/2 on angle ψ follows from the condition γ_{0} > 0. Representation of a pulse by the form (1) is correct as long as the characteristic size of a pulse *r*_{0} λ, and its duration τ_{0} λ/*c*.

Let us now write the field *E*_{in}(*x*, *t*) (1) in the form of a two-dimensional Fourier integral,

where

Here and further on, all integrations are carried out over the infinite limits from −∞ to +∞. Substituting the field *E*_{in}(*x*, *t*) (1) into (3) and introducing new variables

one obtains a set of Fourier amplitudes of the field, *E*_{in}(*k*_{0x}, ω) = *A*_{in}(*q*, Ω), with

Expression (2) describes a set of plain monochromatic waves with amplitudes *A*_{in}(*q*, Ω), wavevectors **k**_{0} = (*k*_{0x}, *k*_{0z}) and frequencies ω, where *k*_{0x} = *K*_{0x} + *q*, *k*_{0z} = (*k*_{0}^{2} − *k*_{0x}^{2})^{1/2} and *k*_{0} = (ω_{0} + Ω)/*c*, which are incident on a crystal surface. In accordance with the known results of the plane-wave of X-ray diffraction, each single component wave in (2) is transmitted and reflected with the amplitude coefficients of transmission *T*(*q*, Ω) and reflection *R*(*q*, Ω). As a result we shall obtain the distribution of fields *E*_{g}(*x, z, t*) for transmitted (*g* = 0) and reflected (*g* = *h*) pulses at any point of space (*x, z*) outside the crystal and at any moment of time *t*,

where *B*_{0} = *T, B*_{h} = *R*.

Here it is taken into account that owing to a condition of continuity of the tangential components of the wavevectors at the entrance and exit crystal surfaces, the values of projections of wavevectors **k**_{g} in a vacuum will take the following form,

where *K*_{gx} = *K*_{0x} + *g*_{x}, *g* = 0, *h*; σ_{0,h} = 1 in the Laue case; σ_{0} = 1 and σ_{h} = −1 in the Bragg case; *z* ≤ 0 for a reflected pulse in the Bragg case and *z* ≥ *d* in the Laue case and for a transmitted pulse in the Bragg case. Contrary to the usual notation of wavevectors, namely denoting wavevectors in a vacuum by **K**_{g} and in the crystal by **k**_{g}, **K**_{g} denotes the average wavevectors of the pulses and **k**_{g} takes into account the *q*- and ω-spectra of the incident, reflected and transmitted pulses. Throughout this paper all wavevectors are restricted to a vacuum.

Note that the Fourier-transform-based approach, used here, is more simple and productive in comparison with the time-dependent Takagi–Taupin differential equations used by Chukhovskii & Förster (1995), Wark & He (1994) and Wark & Lee (1999).

Representing the square root (7) in the form of a series over small parameters *q*/*K*_{0} and Ω/ω_{0}, which is truncated discarding terms of the third order and higher, and substituting this result into the two-dimensional integral (6), we obtain a general expression for the electric fields of X-ray pulses [see Appendix *A*, equations (30) and (31)],

where *K*_{gx} = *K*_{0x} + *g*_{x} = *K*_{0}sinθ_{g}, *K*_{gz} = σ_{g}(*K*_{0}^{2} − *K*_{gx}^{2})^{1/2} = *K*_{0}γ_{g}, γ_{g} = cosθ_{g}. The angle of diffraction reflection with respect to the crystal normal is θ_{h} = ψ + θ_{B} − *b*Δθ, where *b* = γ_{0}/γ_{h} is the asymmetry coefficient of the reflection. In the Bragg case, γ_{h} < 0, *b* < 0, and angle ψ must satisfy the condition |ψ − π/2| < θ_{B}. The slow-varying amplitudes are

where

The phase *S*_{g} (10) determines the displacement of pulse centres in *x* and *t* with distance *z* from the crystal. The phase *D*_{g} (11), which is quadratic in [*q* − (Ω/*c*)sinθ_{g}] and proportional to *z*, describes the curvature of the wavefront and the diffraction smearing of pulses during their propagation in a vacuum. It is necessary to take into account the terms of the order of ∼*q*^{2}, Ω^{2} and *q*Ω to obtain a correct solution of expression (7) and to analyse diffraction broadening of pulses in space and in time. In all previous articles this extremely important aspect was not taken into account. Expression (11) describes the effect of the curvature of the asymptotes of the dispersion surface far away from the reflecting crystal for pulses limited in time and in space. The influence of curved asymptotes has been considered theoretically earlier, for example, by Bauspiess *et al.* (1976) in the case of the incident spherical X-ray or neutron waves on the interferometer. Integral (9) is equivalent to the integral formula of Kirchhoff–Helmholtz, generalizing the Huygens–Fresnel principle, since in the quasi-optical approximation the spherical wavefront of point sources can be replaced by a parabolic wavefront, which is justified in the paraxial region.

Let us write the reflection coefficient *R*(*q*, Ω) and *T*(*q*, Ω) in (6) and in (9) for a crystal with any thickness *d* in the general form. In the Bragg case,

where

Here χ_{g} are the Fourier components of the dielectric susceptibility of the crystal; *C* = 1 and *C* = cos2θ_{B} for σ- and π-polarized radiation, respectively. Parameter α = [*k*_{0}^{2} − (**k**_{0} + **h**)^{2}]/*k*_{0}^{2}, determining the deviation from the exact Bragg condition, has the form

where Δθ is the departure of the incident pulse from the Bragg angle.

In the Laue case,

where

The characteristic angular width Δθ_{B} of the diffraction reflection coefficients *R* (12) and *R* = *B*_{h} (14) depends on the ratio between the thickness of the crystal *d* and the extinction length Λ = λ(γ_{0}|γ_{h}|)^{1/2}/π*C*|χ_{h}|. In the case of a thick crystal (*d* Λ) this width is equal to Δθ_{B} = *C*|χ_{h}|/|*b*|^{1/2}sin2θ_{B}, or, in the other designations, Δθ_{B} = λ|γ_{h}|/πΛsin2θ_{B}. In the case of a thin crystal, when *d* ≤ Λ, the angular width of the reflection is Δθ_{B} ≃ λ|γ_{h}|/π*d*sin2θ_{B}. The reflection coefficient in the Laue case is maximal if the crystal thickness satisfies the condition *d* = (π/2)Λ(1 + 2*n*), where *n* = 0, 1, 2,….

The intensities of the transmitted and reflected pulses are determined by the expression *I*_{g}(*x, z, t*) = |*A*_{g}|^{2}. It is easy to show that the total energy of a pulse, *W*_{g} = *I*_{g}d*x*d*t*, does not depend on the distance *z* and the time *t*, which means conservation of energy during the pulse propagation in a vacuum,

As an example, we shall further consider everywhere a Gaussian incident pulse,

where *r*_{0} and τ_{0} are the transverse size and the of the incident pulse, respectively, φ_{0}(*x*) = α_{0}(*x*γ_{0}/*r*_{0})^{2} is the phase, and the parameter α_{0} is equal to the phase at |*x*| = *r*_{0}/γ_{0} [see Appendix *B*, expression (45)].

Depending on the ratio between *r*_{0} and τ_{0} it is possible to introduce the concept of a long pulse, for which τ_{0} (*r*_{0}/*c*)tanθ_{0}, and a short pulse with wide front with *r*_{0} *c*τ_{0}cotanθ_{0}. In the first case, only a limited area of the crystal surface with |*x*| ≤ *x*_{0} = *r*_{0}/γ_{0} is involved in the scattering, whereas in the second case the incident pulse with size Δ*x* ≃ *c*τ_{0}/sinθ_{0} *x*_{0} propagates along the crystal surface with speed *c*/sinθ_{0}, higher than the speed of light in a vacuum. It is the latter situation that will be realised for femtosecond pulses.

For narrow and short pulses the angular divergence Δθ_{0} ≃ λ/π*r*_{0} and spectral width ΔΩ_{0} ≃ 2/τ_{0} are comparable or even exceed the angular width Δθ_{B} and spectral width ΔΩ_{B} = Δθ_{B}ω_{0}cotanθ_{B} of a Bragg reflection. This leads to a sharp change in the form and to reduction of intensity of a reflected pulse, but also to its smearing in time as well as in space. The degree of smearing in the general case increases with the distance *z* (see §3).

### 3. Results and discussion

Let us explore some examples of typical pulse parameters. We shall always consider the 220 reflection of σ-polarized radiation with λ = 0.154 nm from a silicon single crystal at a departure angle Δθ = (1 − *b*)Re(χ_{0})/2sin2θ_{B}, which corresponds to the maximal reflected intensity, where θ_{B} = 23.65°. In the symmetric Bragg case (*b* = −1), the Bragg width for a thick crystal Δθ_{B} = 12.4 µrad, and the extinction length Λ = 2.16 µm. In the symmetric Laue case (*b* = 1), Λ = 4.92 µm. From a general point of view it is clear that for the reduction of heat absorption the thickness of a crystal should best be chosen small (*d* ≤ 1–3Λ), but at the same time large enough to provide sufficiently high X-ray reflection coefficient values.

The expressions (8)–(14) give a general solution of the problem of transmission and reflection of X-ray pulses in the Bragg and Laue cases. Let us discuss some special cases. If the field amplitude *A*_{in} does not depend on *x* and *t*, then, in agreement with expressions (5) and (13), *A*_{in}(*q*, Ω) = δ(*q*)δ(Ω), α = 2Δθsin2θ_{B}, and we find the well known result for a plane monochromatic wave: *A*_{g} = *B*_{g}(Δθ). Formally this means that in (15) one should assume *r*_{0} → ∞ and τ_{0} → ∞. In a real situation the approximation of a plane monochromatic wave will be realised at Δθ_{0} Δθ_{B} and ΔΩ_{0} ΔΩ_{B}, *i.e.* in the case of a source of size *r*_{s} λ/(πΔθ_{B}) [see expression (44) at α_{s} = 0] and of τ_{0} 2/ΔΩ_{B}. For example, for λ = 0.154 nm in the case of symmetric Bragg reflection Si(220) this leads to the following requirements: *r*_{0} 4 µm, τ_{0} 6 fs.

For a less restrictive approximation of a monochromatic X-ray beam, the amplitude *A*_{in}(*x*) depends only on one coordinate, *i.e.* τ_{0} → ∞ in equation (15). In this case, *A*_{in}(*q*, Ω) = *A*_{in}(*q*)δ(Ω), α = 2sin2θ_{B}(Δθ − *q*/*K*_{0}γ_{0}) and

where

Expression (16), which is valid at any *z* and *d*, is more general in comparison with that obtained earlier using the Green function method for diffraction reflection of a limited X-ray beam from a semi-infinite crystal at *z* = 0 in the Bragg case (Afanas'ev & Kohn, 1971) and at *z* = *d* in the Laue case (Slobodetzkii & Chukhovskii, 1970).

It is of further interest to analyse diffraction of a short pulse with a wide wavefront when the longitudinal size of the incident pulse *l*_{0} = *c*τ_{0} *r*_{0}. In this case it is possible to neglect boundary effects, *i.e.* to exclude the field dependence on the incident pulse transverse coordinate. Then *A*_{in}(*x*, *t*) = *A*_{in}(*t* − *x*sinθ_{0}/*c*) and, in agreement with (5), *A*_{in}(*q*, Ω) = *A*_{in}(Ω)δ(*q* − Ωsinθ_{0}/*c*). As a result, from (9) it is easy to show that

where

The substitution of expression *q* = Ωsinθ_{0}/*c* into (13) results in a known expression for the value of α in the case of an incident non-monochromatic plane wave,

For convenience, the analysis of the space and time structure of amplitudes *A*_{g}(*x, z, t*) (17) can be carried out in a new Cartesian system of coordinates (*x*, *z*) with transition rules *x* = *x*cosφ + *z*sinφ, *z* = *z*cosφ − *x*sinφ, in which the axis *z* makes an angle φ with the crystal normal **n**. This angle φ is selected in such a way that the phase *S*_{g} (18) becomes independent of the transversal coordinate *x*. From (18) it is easy to obtain

Note that the axes of coordinates (*x*, *z*) and (*x*_{p}, *z*_{p}) in Appendix *C* are parallel to each other; however, the system (*x*_{p}, *z*_{p}) moves with the speed of light in a vacuum along the direction of the wavevector **K**_{h} [see Fig. 8 and the formulae (53), (55)]. It is easy to be convinced that the angle φ = θ_{h} + φ_{h}, where the angle φ_{h} is defined from equation (56).

In the new coordinate system the phase *S*_{g} = −Ω(*t* − *z*/*V*_{g}), where *V*_{g} is the speed of the pulse along the longitudinal axis *z*,

From expressions (19), (21) and (22) it follows that, for transmitted pulses (*g* = 0) both in the Bragg case and in the Laue case, φ_{0}′ = θ_{0}, *V*_{0} = *c* and *D*_{0} = 0. In other words the incident pulse with a wide front is transmitted along its initial direction **K**_{0} with the speed of light and is not deformed during the pulse transmission in a vacuum, *i.e.* remains a plane non-monochromatic wave with time dependence *A*_{0}(*t* − *z*_{0}′/*c*), which differs in the general case from *A*_{in}(*t* − *z*_{0}′/*c*).

Quite a different situation takes place for the reflected pulses. In the general case the directions of the wavevector **K**_{h} and the normal **N** to the pulse do not coincide (see Appendix *C* and Fig. 8). This is stipulated by the fact that at a fixed incidence angle θ_{0} various spectral components of a field *A*_{h}(Ω) are reflected under different angles θ_{h}(Ω) to the crystal normal. As long as the vector **h** in **k**_{h} (7) has a non-zero projection *h*_{x} ≠ 0 along the *x* axis, part of the longitudinal impulse of the wavevector **k**_{h} is transferred to the crystal and the angle of reflection θ_{h}(Ω) is different for various spectral components Ω: θ_{h}(Ω) = θ_{h} + Δθ_{h}(Ω), where

Superposition of these plane waves gives rise to non-trivial propagation of the reflected pulse in a vacuum. Earlier the speed *V*_{h} (22) was not quite correctly named as the `group velocity' (Malgrange & Graeff, 2003). For a more detailed discussion of this, see Appendix *C*.

The only exception is the symmetric Bragg case (*b* = −1), for which ψ = π/2, sinθ_{0} = sinθ_{h}, Δθ_{h} = 0, *V*_{h} = *c* and *D*_{h} = 0, *i.e.* smearing of pulses in a vacuum does not take place. If |*b*| ≠ 1 in the Bragg case, and in any Laue case, *V*_{h} < *c* and the form of the pulse *A*_{h}(*t* − *z*/*V*_{h}) varies during the propagation in a vacuum (Figs. 3–7). For the Laue case, expressions (21) and (22) were given earlier by Malgrange & Graeff (2003) without, however, taking into account the pulse smearing effects, caused by the phase *D*_{h} (19), which is quadratic in Ω.

The distance *R*_{D} from a crystal along the wavevector **K**_{h}, at which a substantial smearing of the reflected pulse begins, is determined from the condition |*D*_{h}| ≃ 1, from which *R*_{D} ≃ (ΔΩ_{E}^{2}|*F*_{h}γ_{h}|)^{−1}, where in the approximation of the Gaussian forms for *R*(Ω) and *A*_{in}(Ω) the effective spectral width ΔΩ_{E} = ΔΩ_{0}ΔΩ_{B}/(ΔΩ_{0}^{2} + ΔΩ_{B}^{2})^{1/2}. The expression given above for the distance *R*_{D} coincides with equation (52) in Appendix *C*. The effect of smearing and broadening is increased with the reduction of when the incident spectrum width exceeds the spectral width of the Bragg reflection ΔΩ_{0} ΔΩ_{B} and therefore ΔΩ_{E} ≃ ΔΩ_{B}. If, for example, τ_{0} = 0.1 fs, then in the Bragg-case Si(220) reflection with *b* = −2 and wavelength λ = 0.154 nm the distance *R*_{D} ≃ 64 cm, and in the symmetric Laue case *R*_{D} ≃ 8 cm (see also Fig. 9).

We shall consider first the reflection of a long incident pulse of duration τ_{0} ≥ τ_{B} and with small transversal size *r*_{0} ≃ Λ. Fig. 2 shows the space distributions of the modulus of amplitudes of the incident pulse |*A*_{in}(*x*_{s}, *z*_{s})| with duration τ_{0} = 10 fs (*a*) and the reflected pulse |*A*_{h}(*x*_{p}, *z*_{p})| in the symmetric Bragg case (*b*). In this case the longitudinal size of the pulse *l*_{0} = τ_{0}*c* = 3 µm, as well as its transverse size *r*_{0} = 10 µm, are comparable with the X-ray extinction length Λ. Hereinafter the system of coordinates (*x*_{p}, *z*_{p}) moves together with the reflected pulse [see Fig. 8 and expressions (53), (55)]. After reflection the pulse is strongly stretched in the transverse direction *z*_{p} and its maximum intensity decreases by more than four times (Fig. 2*a*). The degree of distortion of the form of the pulse increases with increase in distance *R* from the crystal to the reflected pulse. Meanwhile the size and duration of the reflected pulse in the longitudinal direction remains almost unchanged. This is explained by that fact that the τ_{0} exceeds the characteristic time τ_{B} = 2/ΔΩ_{B} ≃ 5.8 fs, where for a thick crystal τ_{B} = 2(Λ/*c*)sin^{2}θ_{B}/|γ_{h}|. The duration τ_{B} is defined by a time delay of the waves reflected from a surface of the crystal and from an effective layer of the crystal of depth *z* ≃ Λ [see also expression (6) of Graeff (2004)].

All calculations whose results are shown in Figs. 2–7 are made on the basis of the general formula (9). For clarity, we shall further consider (see Figs. 3–7) the size of the source *r*_{s} = 75 µm, the parameter of the square-law phase of radiation of a source α_{s} = 2, and the distance from the source to the crystal *z*_{s} = 800 m. Then for an incident pulse we find that *r*_{0} = 1240 µm, α_{0} = 36.9 and angular divergence Δθ_{s} = 1.46 µrad.

It is clear that, starting from some duration of the incident pulse τ_{0} < τ_{B}, only a part ΔΩ_{B} < ΔΩ_{0} of the incident frequency spectrum ΔΩ_{0} will satisfy the diffraction conditions. This results in a sharp reduction of intensity in the case of short incident pulses in comparison with longer pulses.

As an illustration, Fig. 3 shows the intensity of reflection *I*_{h}(0, *z*_{p}) of the long and short pulses in the cases of symmetric and asymmetric Bragg reflections. In both cases the intensity of a short pulse after reflection considerably decreases, whereas the long pulse is more weakly deformed. The small peak in the region *z*_{p} = −*c*(Δ*t*_{12} + τ_{B}) ≃ −10 µm in Fig. 3(*a*), where τ_{B} = 2(*d*/*c*)sin^{2}θ_{B}/|γ_{h}|, and Δ*t*_{12} is the time interval between pulses, is connected by reflection of the short pulse from the bottom surface of the crystal (see also Malgrange & Graeff, 2003). In the symmetric Bragg case the intensity of the reflected pulse practically does not change for an increase in the distance *R* from the crystal to the pulse. At the same time, in the asymmetric Bragg case the maximal intensity of the reflected pulse decreases with an increase in the distance *R*: its width increases and the contribution of the short pulse to the total intensity becomes extremely small at *R* ≃ 0.5 m (see Fig. 3*b*).

Moreover, unfortunately for practical applications, for very short femtosecond pulses the duration and shape of a reflected pulse become almost independent of the incident pulse characteristics (Figs. 4–7) (see also Graeff, 2002, 2004; Malgrange & Graeff, 2003). Formally this can be seen from equation (17), as the smooth function *A*_{in}(Ω) can be taken outside of the integration. Therefore the form of a pulse on the crystal surface is determined by inverse Fourier transformation of the reflection coefficient *R*(Ω) [as can also be seen in the Green function in Chukhovskii & Förster (1995)].

The second peak in the Bragg case at *z*_{p} ≃ −*V*_{h}τ_{B}, where speed *V*_{h} is defined in expression (22) and τ_{B} = 2(*d*/*c*)sin^{2}θ_{B}/|γ_{h}|, arises owing to reflection from the rear surface of the crystal [see Fig. 6(*a*) at *z*_{p} ≃ −5.6 µm, and Fig. 7 (curve 1) at *z*_{p} ≃ −3 µm]. It is easy to see that any asymmetric Bragg case reflection and any Laue case reflection are not quite acceptable for diffraction tailoring of pulses, because already at distances *R* as short as 10–30 cm from the crystal the pulses become considerably diffused (Figs. 4–7). The transmitted pulse *I*_{0}(*x, z, t*) meanwhile practically coincides in form and intensity with the incident pulse, as the *T*(Ω) = exp(*ik*_{0}χ_{0}*d*/2γ_{0}) stays constant everywhere except the very narrow spectral slot |Ω| ≤ ΔΩ_{B} [see equations (12) and (14)]. It is obvious that the group of super-short statistically unconnected pulses with total duration τ_{p} < τ_{B} become merged into one wide asymmetric pulse of duration of the order of τ_{B} after a reflection (see also Shastri *et al.*, 2001*a*).

It is of interest to consider the space and time coherency of XFEL radiation, and the radiation of reflected pulses. The coherence function of an incident pulse is given by

where *P* = [*I*_{in}(*x*, *t*)*I*_{in}(*x* + ρ, *t* + τ)]^{1/2}, *I*_{in}(*x*, *t*) = 〈|*A*_{in}(*x*, *t*)|^{2}〉 (angular brackets mean the average over a sufficiently large time interval); Γ_{in}(0, τ) and Γ_{in}(ρ, 0) are the functions of time and space coherence, respectively, with Γ_{in}(0, 0) = 1. According to the calculations of Saldin *et al.* (2004), XFEL pulses are completely coherent over the whole and the coherence time, which is obtained after substitution of calculated pulses with amplitude and phase modulation in (23), has the value τ_{c} = 0.14 fs. More convenient for the analysis is the spectral representation,

where

It can be shown that the coherence functions of reflected and transmitted pulses are determined by the expression

If in the region of significant variation of the spectrum *A*_{in}(*q*, Ω) the coefficients *B*_{g} ≃ a constant, then the degree of coherence of the reflected pulse Γ_{g} ≃ Γ_{in}, *i.e.* the coherence remains preserved. For a short pulse, for which the spectral width ΔΩ_{0} ΔΩ_{B}, the time coherence, as follows from (25), is increased, *i.e.* a partial monochromatization takes place; however, at the same time the pulse intensity decreases.

One of the most serious problems in diffraction of the powerful XFEL pulses will be the very high thermal load on diffracting crystals. So far there is no exact solution for X-ray diffraction taking into account thermal heating, but it is possible to make some estimations. From analysis of the Green function of the Δ*x* is Δ*t* ≃ (Δ*x*)^{2}/4*a*^{2}, where *a*^{2} = λ_{T}/*c*_{T}ρ, λ_{T} and *c*_{T} are coefficients of and thermal capacity, respectively, and ρ is the crystal density. For silicon at temperature *T* = 300 K, λ_{T} ≃ 150 W m^{−1} K^{−1}, *c*_{T} ≃ 700 J kg^{−1} K^{−1}, ρ = 2.3 g cm^{−3} (Grigor'ev & Meilikhova, 1991). For Δ*x* ≃ Λ, one obtains Δ*t* ≃ 13 ns, and this is much longer than the duration of an X-ray pulse τ_{0} = 0.1–200 fs (for diamond Δ*t* ≃ 20 ns). Thus it is quite possible that the X-ray laser pulses are simply too short to influence their own diffraction scattering through heating. An increase in temperature of a crystal by Δ*T* = Δθ_{B}cotanθ_{B}/α_{T} ≃ 10 K, where α_{T} is the coefficient of linear expansion (for silicon α_{T} = 2.54 × 10^{−6} K^{−1}), results in displacement of the Bragg peak by an angle Δθ_{B}, which is not essential for short pulses with ΔΩ_{0} ΔΩ_{B}.

### 4. Conclusions

In conclusion, this paper presents a most general approach to the consideration of the diffraction of arbitrary X-ray pulses in crystals and their subsequent dispersion in space. The main attention is devoted to analysis of how space and time change the form and duration of short pulses depending on the distance from the crystal. It is shown that the unique opportunity to avoid distortion of the form and duration of a reflected femtosecond pulse is achieved by use of symmetric reflections in the Bragg diffraction geometry.

### APPENDIX A

### Calculation of projection *k*_{gz}

Let us present the values of the wavevectors *k*_{0} and *k*_{gx} in the expression for *k*_{gz} in (7) in the following form: *k*_{0} = *K*_{0}(1 + ξ_{1}), where ξ_{1} = Ω/ω_{0}, and *k*_{gx} = *K*_{gx}(1 + ξ_{2}), where ξ_{2} = *q*/*K*_{gx}. As a result for the square-root expression in (7) we find that

where *K*_{gz}^{ 2} = *K*_{0}^{ 2} − *K*_{gx}^{ 2},

Here γ_{g} = cos(**K**_{g}*·***n**) = cosθ_{g}, **K**_{h} = (*K*_{hx}, σ_{h}|*K*_{hz}|). In the Laue case, γ_{h} > 0, and in the Bragg case, γ_{h} < 0.

We shall consider now that at ξ 1 the following expansion takes place,

Then with use of expression (28), from equation (26) we find that

where *K*_{gz} = *K*_{0}γ_{g}. In view of an obvious form of *a* and *b* (27) it is easy to see that in (29) factors *a*(*a* − 1) = *ab* = *b*(*b* + 1) = tan^{2}θ_{g}/γ_{g}^{2}. As a result, from (29) we find finally that

If now in addition to *k*_{gz} (30) we consider the *x*-components of wavevectors **k**_{g} and the item ω*t*, the phase in the exponential in the integral (6) will have the following form,

where the linear (*S*_{g}) and square-law (*D*_{g}) phases are set by expressions (10) and (11), respectively.

### APPENDIX B

### X-ray pulse propagation in free space

The aim of Appendix *B* is to prove the validity of the form of the Gaussian incident pulse given in (15). In the plane of a source *z*_{s} = 0 (the exit window of the free-electron laser) is set a field

where *A*_{s}(*x*_{s}, *t*) is the complex slowly varying amplitude of the field, *x*_{s} is the transversal coordinate in the plane of the source and ω_{0} is the average frequency of radiation. It is required to find the field *E*(*x*_{s}, *z*_{s}, *t*) in any point in space (*x*_{s}, *z*_{s}) at the moment in time *t*.

We shall present the field *E*_{s}(*x*_{s}, *t*) (32) in the form of an expansion of the Fourier integral over the plane waves,

where

Propagation of the pulse field *E*(*x*_{s}, *z*_{s}, *t*) in the region *z*_{s} ≥ 0 is described by the wave equation

where Δ = ∂^{2}/∂*x*_{s}^{2} + ∂^{2}/∂*z*_{s}^{2} is the Laplace operator. In view of the boundary condition for the field *E*(*x*_{s}, 0, *t*) = *E*_{s}(*x*_{s}, *t*) in the plane *z*_{s} = 0 from (33) and (35) it is easy to see that

where *k*_{z} = (*k*^{2} − *q*^{2})^{1/2}, *k* = ω/*c*.

From (32) and (34) it follows that *E*_{s}(*q*, ω) = *A*_{s}(*q*, Ω), where Ω = ω − ω_{0}. If the characteristic size of the source *r*_{s} λ, and the τ_{0} *T*, where λ is the wavelength of radiation and *T* is the period, then *q* *k* and Ω ω_{0}. In this case, with use of the expansion (28) for values of *k*_{z} in (36), we have

where *K*_{0} = ω_{0}/*c* = 2π/λ is an average wavevector of the pulse. Substituting (37) into (36) we find the following expression for the pulse field in a plane *z*_{s},

where *A*(*x*_{s}, *z*_{s}, *t*) is the slowly varying amplitude of the pulse,

We shall consider now propagation in space of a Gaussian pulse for which it is possible to find simple analytical expressions. We shall present the amplitude (32) of the field on the source surface *z*_{s} = 0 in the following form,

where *r*_{s} is the size of the source in the plane *z*_{s} = 0, τ_{0} is the and φ_{s}(*x*_{s}) is the phase of the complex amplitude (40). Furthermore we shall consider that this phase is a square-law function of the coordinate *x*_{s}, *i.e.* φ_{s}(*x*_{s}) = α_{s}(*x*_{s}/*r*_{s})^{2}, where parameter α_{s} is equal to the phase at |*x*_{s}| = *r*_{s}.

For calculation of the Fourier amplitudes *A*_{s}(*q*, Ω) in (39) and for calculations of other integrals the known so-called main optical integral (Gradshteyn & Ryzhik, 1980) is used,

where β and γ are arbitrary complex values.

Substituting (40) into (34) and (39) leads to the following expression for the pulse amplitude at any plane *z*_{s},

where *A*_{s} = 1/*M*^{1/2}, *M* = [(1 + α_{s}*W*)^{2} + *W*^{2}]^{1/2}, *W* = λ*z*_{s}/π*r*_{s}^{2} is the wave parameter,

From expression (42) we can see that the initial Gaussian pulse keeps its form and duration in the process of propagation in space; however, the transversal size of the pulse *r*_{0}(*z*_{s}) increases *M* times in comparison with *r*_{s} with increase in distance *z*_{s} and with increase in phase parameter α_{s}. This phase parameter describes an initial curvature of the wavefront. The phase of the pulse φ_{0}(*x*_{s}), also a square-law function, depends on the transversal coordinate. The parameter of this phase α_{0}(*z*_{s}) increases with increase in distance *z*_{s} and with increase in parameter α_{s}, and also increases with reduction of the source size *r*_{s}. The phase parameter α_{0} ≠ 0, even at the initial plane wavefront, *i.e.* at α_{s} = 0. The phase Φ_{0}(*z*_{s}) does not depend on the transversal coordinate *x*_{s} and does not play an essential role during propagation and diffraction of the pulse. Later we shall consider for simplicity that in (42) *A*_{s} = 1, Φ_{0} = 0.

The width of the angular spectrum of a pulse (42) Δθ_{s} = Δ*q*_{s}/*K*_{0}, *i.e.* the width of the function |*A*(*q*, Ω, *z*_{s})| ≃ exp[−(*q*/Δ*q*_{s})^{2}], where Δ*q*_{s} = 2(1 + α_{0}^{2})^{1/2}/*r*_{0} = 2(1 + α_{s}^{2})^{1/2}/*r*_{s}, does not depend on the distance *z*_{s} and is determined by the expression

The angular width Δθ_{s} (44) in the general case exceeds the diffraction divergence Δθ_{d} = (λ/π*r*_{s}), related only to the size of the source *r*_{s}.

Theoretical calculations show (Saldin *et al.*, 2004) that on exit from undulator SASE1 (λ ≃ 0.1 nm) the full width of the pulse at half-height is equal to 90 µm, and the angular divergence of the beam is equal to 1.1 µrad. In our notation this means that *r*_{s} ≃ 76.4 µm and Δθ_{s} ≃ 0.93 µrad. From here and (44) it follows that the phase parameter α_{s} ≃ 2. Then with use of formulae (43) it is easy to show that at the distance *z*_{s} = 800 m the transversal pulse size *r*_{0} ≃ 820 µm, and the phase parameter α_{0} ≃ 24.

The relation between coordinates (*x*_{s}, *z*_{s}) on the source of an X-ray pulse and coordinates (*x, z*) on the crystal surface is determined by means of the following expressions: *x*_{s} = *x*cosθ_{0} − *z*sinθ_{0}, *z*_{s} = *x*sinθ_{0} + *z*cosθ_{0} + *z*_{1}, where θ_{0} is the incident angle of the pulse on the crystal with respect to the normal **n** to the crystal surface, and *z*_{1} is the distance from the source to the crystal (see Fig. 8). Then the amplitude of the field (42) on the crystal surface *z* = 0 will be

where γ_{0} = cosθ_{0}, and the time *t* is counted from the moment *t*_{1} = *z*_{1}/*c* of incidence of the pulse maximum at *x* = 0, *z* = 0 on the crystal.

### APPENDIX C

### Reflection of a Gaussian pulse

We shall consider diffraction reflection of the incident pulse *A*_{in}(*x*, *t*) (45) from the crystal. From the expression (45) in view of (41) we find the following expression for Fourier amplitudes of the incident pulse in (9),

where ΔΩ_{0} = 2/τ_{0} is the spectral width of the incident pulse, Δ*q*_{0} = 2γ_{0}(1 + α_{0}^{2})^{1/2}/*r*_{0} is the width of the angular spectrum in *q*-space, and amplitude *A*_{0} = (1 + *i*α_{0})^{1/2}/(πΔ*q*_{0}ΔΩ_{0}). It is easy to show that Δ*q*_{0} = γ_{0}Δ*q*_{s}, where Δ*q*_{s} = 2(1 + α_{s}^{2})^{1/2}/*r*_{s} is the width of the angular spectrum of the pulse in the plane *z*_{s} = 0 of the source.

For simplicity of the analysis of the form and the orientation of the reflected pulse we shall present the amplitude reflection coefficient *R*(*q*, Ω) in the integral (9) in the form of a Gaussian function [see argument α(*q*, Ω) (13) in (12) and (14)],

where Δ*q*_{B} = *K*_{0}γ_{0}Δθ_{B} is the width of the diffraction reflection curve.

We shall now substitute *A*_{in}(*q*, Ω) (46) and *R*(*q*, Ω) (47) into the general integral equation (9) for the amplitude of the reflected pulse, where *g* = *h*. As a result, using (41) we find that

where

Here, τ_{R} is the duration of the reflected pulse, which is determined by the following formula,

where

In order to obtain expression (48) it is considered that the ratio Δ*q*_{0}/Δ*q*_{B} = Δθ_{s}/Δθ_{B} 1, where Δθ_{s} = (λ/π*r*_{s})(1 + α_{s}^{2})^{1/2} is the width of the angular spectrum of the source radiation, and distance |*z*| *z*_{F}, where *z*_{F} = π*r*_{0}^{2}/[λ*b*^{2}(1 + α_{0}^{2})].

From equations (51) it follows that the τ_{R} after reflection from the crystal increases in comparison with τ_{0} for two reasons. The first reason is related to the finite quantity of the spectral width ΔΩ_{B} of the Bragg reflections. For sufficiently short pulses the spectral width ΔΩ_{0} > ΔΩ_{B}, and therefore τ_{B} > τ_{0}. The second reason is connected to diffusion broadening of the part of the τ_{Z} on increasing the distance from the crystal to the reflected pulse along the wavevector **K**_{h}. From the characteristic distance *R*_{D}, at which the intensity of the pulse |*A*_{R}|^{2} will decrease twice, it is possible to estimate from the equation β_{0} = 1,

From equation (52) it follows that diffusion broadening of the pulse is absent only in the case of symmetric Bragg reflection (*b* = −1), at which sinθ_{0} = sinθ_{h}, *F*_{h} = 0 and *R*_{D} → ∞. The critical distance *R*_{D} (52) increases with increase in τ_{0}, if τ_{0} > τ_{B} (see Fig. 9). The dependence of the value τ_{B} from the asymmetry coefficient of reflection *b* is shown in the insert of Fig. 9. For short pulses with τ_{0} τ_{B} the distance *R*_{D} does not depend on the duration of the incident pulse τ_{0}. Unfortunately in practice the distance *R*_{D} does not exceed several tens of centimetres in the Laue case at τ_{0} ≤ 1–10 fs (see Fig. 9).

From the form of arguments Φ_{1} and Φ_{2} in (48) we can see that the reflected pulse *A*_{h}(*x, z, t*) propagates with the speed of light *c* in a vacuum along the direction of the wavevector **K**_{h}. The orientation of the pulse in space is determined mainly by the angles θ_{0} and θ_{h}, *i.e.* by the asymmetry coefficient of the reflection *b*.

For analysis of this, we shall pass to the system of coordinates (*x*_{h}, *z*_{h}), which moves together with the pulse, and the axis *z*_{h} is directed along the wavevector **K**_{h}, *i.e.* this coordinate system is turned in relation to the laboratory system of coordinates (*x, z*) by angle θ_{h} (see Fig. 8),

Then the functions Φ_{1,2} in (48) will have the following form,

where *a*_{0h} = (sinθ_{0} − sinθ_{h})/γ_{h}.

From equations (54) it can be seen that only in the symmetric Bragg geometry when sinθ_{0} = sinθ_{h} and *a*_{0h} = 0 do the axes of the reflected pulse coincide with axes *x*_{h} and *z*_{h}. In all other cases the pulse propagates so that its axes are inclined by some angle φ_{h} relative to the direction of propagation **K**_{h} (see Fig. 8).

We shall consider the most interesting case of a wide and short pulse, whose transversal size *r*_{0} is much larger than its longitudinal size *l*_{0} = *c*τ_{0}, incident on a crystal. In order to find the angle φ_{h} between the normal **N** to the long axis of the reflected pulse and the direction of distribution **K**_{h} we shall pass to a new system of coordinates (*x*_{p}, *z*_{p}) by means of equations

The angle φ_{h} is obtained from the condition that the coefficient of product *x*_{p}*z*_{p} in the expression Φ_{1}^{2} + Φ_{2}^{2} in (48) equals zero. As a result we find that

where δ = (*bc*τ_{R}/*r*_{0})^{2}/(1 + *a*_{0h}^{2}) 1. The dependence of the angle of inclination φ_{h} of the reflected pulse from the asymmetry coefficient of the reflection *b* is shown in Fig. 10.

So, the modulus of the amplitude of the reflected pulse in a moving system of coordinates (*x*_{p}, *z*_{p}) is

where *r*_{T} = *r*_{0}/(|*b*|cosφ_{h}) is the transversal size of the pulse and *r*_{L} = *V*_{L}τ_{R} is the longitudinal size of the pulse. Here *V*_{L} = *c*cosφ_{h} is a projection of the pulse speed **c** on the axis *z*_{p}. Expression (57) represents, figuratively speaking, an instant photo-picture of the reflected pulse in the moment of time *t* = *z*/*c*γ_{h}. From equation (56) it is easy to see that

This expression can be found also from the condition of equality of the optical paths *L*_{ABC} = *L*_{DEF} (see Fig. 8).

Earlier (Malgrange & Graeff, 2003) the speed *V*_{L} was not quite correctly referred to as the group velocity. Such a discrepancy has arisen because in this work the incident and reflected pulses were considered infinite in the transversal direction along the axis *x*_{p} (*i.e.* *r*_{0} → ∞). For this reason propagation of the reflected pulse with projection speed *V*_{T} = *c*sinφ_{h} along this transversal direction could not be considered in principle.

The most important, and rather unfavourable for practical applications, feature of short X-ray pulse diffraction is the inclination of the reflected pulse relative to the propagation direction (see Figs. 8 and 10). The real τ_{tot}, *i.e.* the time of its passage through a plane, perpendicular to wavevector **K**_{h}, will be determined both by angle of inclination φ_{h} and by projection *r*_{0}/|*b*| onto this plane of the transversal size of the pulse: τ_{tot} = (*r*_{0}/*c*)|sinφ_{h}/*b*|. If, for example, *r*_{0} = 800 µm, *b* = 1, then the angle of inclination φ_{h} = −41.2° and the total duration of the reflected pulse τ_{tot} = 2 × 10^{3} fs, which exceeds the duration of the incident femtosecond pulse by some orders.

### Acknowledgements

The author is very grateful to D. Novikov for helpful discussions. The work was supported by the Russian Foundation Base Research (Grants No. 06-02-17249, No. 07-02-00324) and the International Science and Technology Center (Project No. 3124).

### References

Afanas'ev, A. M. & Kohn, V. G. (1971). *Acta Cryst.* A**27**, 421–430. CrossRef CAS IUCr Journals Web of Science

Altarelli, M. *et al.* (2006). Editors. Report DESY 2006–097. DESY, Hamburg, Germany. (http://xfel.desy.de/tdr/index_eng.html .)

Arthur, J. *et al.* (2002). LCLS Conceptual Design Report. LCLS, USA. (http://www-ssrl.slac.stanford.edu/lcls/cdr/ .)

Bauspiess, W., Bonse, U. & Graeff, W. (1976). *J. Appl. Cryst.* **9**, 68–80. CrossRef IUCr Journals Web of Science

Chukhovskii, F. N. & Förster, E. (1995). *Acta Cryst.* A**51**, 668–672. CrossRef CAS Web of Science IUCr Journals

Gradshteyn, I. S. & Ryzhik, I. M. (1980). *Table of Integrals, Series and Products.* New York: Academic Press.

Graeff, W. (2002). *J. Synchrotron Rad.* **9**, 82–85. Web of Science CrossRef CAS IUCr Journals

Graeff, W. (2004). *J. Synchrotron Rad.* **11**, 261–265. Web of Science CrossRef CAS IUCr Journals

Grigor'ev, I. S. & Meilikhova, E. Z. (1991). Editors. *Physical Values. Handbook.* Moscow: Energoatomizdat. (In Russian.)

Malgrange, C. & Graeff, W. (2003). *J. Synchrotron Rad.* **10**, 248–254. Web of Science CrossRef CAS IUCr Journals

Saldin, E. L., Schneidmiller, E. A. & Yurkov, M. V. (2004). Report TESLA-FEL 2004–02. DESY, Hamburg, Germany.

Shastri, S. D., Zambianchi, P. & Mills, D. M. (2001*a*). *J. Synchrotron Rad.* **8**, 1131–1135. Web of Science CrossRef CAS IUCr Journals

Shastri, S. D., Zambianchi, P. & Mills, D. M. (2001*b*). *Proc. SPIE*, **4143**, 69–77. CrossRef CAS

Slobodetzkii, I. Sh. & Chukhovskii, F. N. (1970). *Kristallografiya*, **15**, 1101–1107. (In Russian.)

Tanaka, T. & Shintake, T. (2005). *SCSS X-FEL Conceptual Design Report*, edited by Takashi Tanaka and Tsumoru Shintake. SCSS XFEL, R&D Group, RIKEN Harima Institute/SPring-8, Japan. (http://www-xfel.spring8.or.jp/SCSSCDR.pdf .)

Wark, J. S. & He, H. (1994). *Laser Part. Beams*, **12**, 507–513. CrossRef CAS

Wark, J. S. & Lee, R. W. (1999). *J. Appl. Cryst.* **32**, 692–703. Web of Science CrossRef CAS IUCr Journals

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