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

Bushuev, V.A.

doi:10.1107/S0909049508019602

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 15| Part 5| September 2008| Pages 495-505

doi:10.1107/S0909049508019602

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

V. A. Bushuev ^a ^*

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

(Received 25 September 2007; accepted 3 July 2008; online 22 July 2008)

A solution of the problem of dynamical diffraction 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.

Keywords: free-electron lasers; ultrashort X-ray pulses; X-ray optics; dynamical diffraction.

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 dynamical diffraction 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 τ₀ ≃ 0.1 fs, separated by time intervals Δt ≃ 0.3–0.5 fs. A typical pulse has a transversal size r₀ ≃ 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., 2001a ,b ; Graeff, 2004 ) and for the Laue case (Shastri et al., 2001b; 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., 2001a,b; Graeff, 2002; Malgrange & Graeff, 2003) or by a Gaussian (Shastri et al., 2001a; 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 dynamical diffraction of X-ray pulses with an arbitrary spatial and temporal structure, described by a field 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(iK₀r − iω₀t), which is incident on a single-crystal plate of thickness d. The field on the entrance crystal surface z = 0 can be written as

$[E_{\rm{in}}(x,t)=A_{\rm{in}}(x,t)\exp(iK_{0x}x-i\omega_0t),\eqno(1)]$

where A_in(x, t) is a slowly varying complex amplitude (the envelope of a wave packet), K_0x = K₀sinθ₀, K₀ = ω₀/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 = K₀γ₀, where γ₀ = cos(K₀·n) = cosθ₀. The angle of incidence of the radiation to the normal n is θ₀ = ψ − θ_B − Δθ, where θ_B is the Bragg angle for the central (average) frequency ω₀, which is determined by the expression 2K₀sinθ_B = h, where h is the modulus of the reciprocal lattice vector h = (hcosψ, −hsinψ), Δθ is the angular deviation from the exact Bragg angle, which is determined by the expression K₀h = −K₀hsin(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 γ₀ > 0. Representation of a pulse by the form (1) is correct as long as the characteristic cross-section size of a pulse r₀ $[\gg]$ λ, and its duration τ₀ $[\gg]$ λ/c.

Figure 1
Geometry in real space of X-ray pulse diffraction in the Bragg and Laue cases. 1, incident pulse E_in(r, t); 2, transmitted pulse E₀(r, t); 3, reflected pulse E_h(r, t) in the Bragg case; 4, reflected pulse in the Laue case; 5, crystal; θ₀ and θ_h are the angle of incidence of the initial pulse and the angle of reflection of the diffracted pulse, respectively, relative to the axis z.

Let us now write the field E_in(x, t) (1) in the form of a two-dimensional Fourier integral,

$[E_{\rm{in}}(x,t)=\textstyle\int\!\!\int{E}_{\rm{in}}(k_{0x},\omega)\exp(ik_{0x}x-i\omega{t})\,{\rm{d}}k_{0x}\,{\rm{d}}\omega,\eqno(2)]$

where

$[E_{\rm{in}}(k_{0x},\omega)=(2\pi)^{-2}\textstyle\int\!\!\int{E}_{\rm{in}}(x,t)\exp(-ik_{0x}x+i\omega{t})\,{\rm{d}}x\,{\rm{d}}t.\eqno(3)]$

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

$[q=k_{0x}-K_{0x},\quad\quad\Omega=\omega-\omega_0,\eqno(4)]$

one obtains a set of Fourier amplitudes of the field, E_in(k_0x, ω) = A_in(q, Ω), with

$[A_{\rm{in}}(q,\Omega)=(2\pi)^{-2}\textstyle\int\!\!\int{A}_{\rm{in}}(x,t)\exp(-iqx+i\Omega{t})\,{\rm{d}}x\,{\rm{d}}t.\eqno(5)]$

Expression (2) describes a set of plain monochromatic waves with amplitudes A_in(q, Ω), wavevectors k₀ = (k_0x, k_0z) and frequencies ω, where k_0x = K_0x + q, k_0z = (k₀² − k_0x²)^1/2 and k₀ = (ω₀ + Ω)/c, which are incident on a crystal surface. In accordance with the known results of the plane-wave dynamical theory 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,

$[E_g({\bf{r}},t)=\textstyle\int\!\!\int{B_g}(q,\Omega)A_{\rm{in}}(q,\Omega)\exp(i{\bf{k}}_g{\bf{r}}-i\omega{t})\,{\rm{d}}q\,{\rm{d}}\Omega,\eqno(6)]$

where B₀ = 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,

$[k_{gx}=K_{gx}+q,\quad\quad k_{gz}=\sigma_g(k_0^2-k_{gx}^2)^{1/2},\eqno(7)]$

where K_gx = K_0x + g_x, g = 0, h; σ_0,h = 1 in the Laue case; σ₀ = 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/K₀ and Ω/ω₀, 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)],

$[E_g({\bf{r}},t)=A_g({\bf{r}},t)\exp(i{\bf{K}}_g{\bf{r}}-i\omega_0t),\eqno(8)]$

where K_gx = K_0x + g_x = K₀sinθ_g, K_gz = σ_g(K₀² − K_gx²)^1/2 = K₀γ_g, γ_g = cosθ_g. The angle of diffraction reflection with respect to the crystal normal is θ_h = ψ + θ_B − bΔθ, where b = γ₀/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

$[A_g(x,z,t)=\textstyle\int\!\!\int{B_g}(q,\Omega)A_{\rm{in}}(q,\Omega)\exp(iS_g+iD_g)\,{\rm{d}}q\,{\rm{d}}\Omega,\eqno(9)]$

where

$[S_g(q,\Omega)=q(x-\tan\theta_gz)-\Omega(t-z/c\gamma_g),\eqno(10)]$

$[D_g(q,\Omega)=-\left[q-(\Omega/c)\sin\theta_g\right]^2z/(2K_0\gamma_g^3).\eqno(11)]$

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 ∼q², Ω² and qΩ 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 transmission coefficient T(q, Ω) in (6) and in (9) for a crystal with any thickness d in the general form. In the Bragg case,

$[\eqalign{&R=(R_1-pR_2)/(1-p),\cr&T=[\exp(i\varphi_1)-p\exp(i\varphi_2)]/(1-p),}\eqno(12)]$

where

$[\eqalign{&R_{1,2}=(\alpha_1\pm Q)/2C\chi_{\bar h},\quad\quad p=(R_1/R_2)\exp[i(\varphi_1-\varphi_2)],\cr& \varphi_{1,2}=k_0\varepsilon_{1,2}d,\quad\quad \varepsilon_{1,2}=(2\chi_0 +\alpha_1\pm Q)/4\gamma_0,\cr&\alpha_1=\alpha{b}-\chi_0(1-b),\quad\quad Q=(\alpha_1^2 + 4bC^2 \chi_h\chi_{\bar h})^{1/2}.}]$

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 = [k₀² − (k₀ + h)²]/k₀², determining the deviation from the exact Bragg condition, has the form

$[\eqalignno{\alpha(q,\Omega)={}&2\sin2\theta_{\rm{B}}\left[\Delta\theta-q/K_0\gamma_0\right.\cr&\left.+\,(\Omega/\omega_0)\sin\psi/\gamma_0\cos\theta_{\rm{B}}\right],&(13)}]$

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

In the Laue case,

$[B_g=A_g^{(1)}\exp(i\varphi_1)+A_g^{(2)}\exp(i\varphi_2),\eqno(14)]$

where

$[A_0^{(1,2)}=(Q\mp\alpha_1)/2Q,\quad\quad A_h^{(1,2)}=\pm Cb\chi_h/Q.]$

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(γ₀|γ_h|)^1/2/πC|χ_h|. In the case of a thick crystal (d $[\gg]$ Λ) this width is equal to Δθ_B = C|χ_h|/|b|^1/2sin2θ_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|/πdsin2θ_B. The reflection coefficient in the Laue case is maximal if the crystal thickness satisfies the condition d = (π/2)Λ(1 + 2n), 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 = $[\textstyle\int\!\!\int]$ I_gdxdt, does not depend on the distance z and the time t, which means conservation of energy during the pulse propagation in a vacuum,

$[W_g=(2\pi)^2\textstyle\int\!\!\int|B_g(q,\Omega)|^2|A_{\rm{in}}(q,\Omega)|^2\,{\rm{d}}q\,{\rm{d}}\Omega.]$

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

$[\eqalignno{A_{\rm{in}}(x,t)={}&\exp\left[-(x\gamma_0/r_0)^2+i\varphi_0(x)\right.\cr&\left.-\,(t-x\sin\theta_0/c)^2/\tau_0^2\right],&(15)}]$

where r₀ and τ₀ are the transverse size and the pulse duration of the incident pulse, respectively, φ₀(x) = α₀(xγ₀/r₀)² is the phase, and the parameter α₀ is equal to the phase at |x| = r₀/γ₀ [see Appendix B, expression (45)].

Depending on the ratio between r₀ and τ₀ it is possible to introduce the concept of a long pulse, for which pulse duration τ₀ $[\gg]$ (r₀/c)tanθ₀, and a short pulse with wide front with r₀ $[\gg]$ cτ₀cotanθ₀. In the first case, only a limited area of the crystal surface with |x| ≤ x₀ = r₀/γ₀ is involved in the scattering, whereas in the second case the incident pulse with size Δx ≃ cτ₀/sinθ₀ $[\ll]$ x₀ propagates along the crystal surface with speed c/sinθ₀, 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 Δθ₀ ≃ λ/πr₀ and spectral width ΔΩ₀ ≃ 2/τ₀ are comparable or even exceed the angular width Δθ_B and spectral width ΔΩ_B = Δθ_Bω₀cotanθ_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(χ₀)/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 r₀ → ∞ and τ₀ → ∞. In a real situation the approximation of a plane monochromatic wave will be realised at Δθ₀ $[\ll]$ Δθ_B and ΔΩ₀ $[\ll]$ ΔΩ_B, i.e. in the case of a source of size r_s $[\gg]$ λ/(πΔθ_B) [see expression (44) at α_s = 0] and of pulse duration τ₀ $[\gg]$ 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: r₀ $[\gg]$ 4 µm, τ₀ $[\gg]$ 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. τ₀ → ∞ in equation (15). In this case, A_in(q, Ω) = A_in(q)δ(Ω), α = 2sin2θ_B(Δθ − q/K₀γ₀) and

$[A_g(x,z)=\textstyle\int{B_g}(q)A_{\rm{in}}(q)\exp(iS_g+iD_g)\,{\rm{d}}q,\eqno(16)]$

where

$[S_g=q(x-\tan\theta_gz),\quad\quad D_g=-q^2z/(2K_0\gamma_g^3).]$

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 l₀ = cτ₀ $[\ll]$ r₀. 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 − xsinθ₀/c) and, in agreement with (5), A_in(q, Ω) = A_in(Ω)δ(q − Ωsinθ₀/c). As a result, from (9) it is easy to show that

$[A_g(x,z,t)=\textstyle\int{B_g}(\Omega)A_{\rm{in}}(\Omega)\exp(iS_g+iD_g)\,{\rm{d}}\Omega,\eqno(17)]$

where

$[S_g=(\Omega/c)\left[\sin\theta_0x+(1-\sin\theta_0\sin\theta_g)z/\gamma_g\right]-\Omega{t},\eqno(18)]$

$[D_g=-\Omega^2F_gz,\quad\quad F_g=(\sin\theta_0-\sin\theta_g)^2/(2K_0c^2\gamma_g^3).\eqno(19)]$

The substitution of expression q = Ωsinθ₀/c into (13) results in a known expression for the value of α in the case of an incident non-monochromatic plane wave,

$[\alpha(\Omega)=2\sin2\theta_{\rm{B}}[\Delta\theta+(\Omega/\omega_0)\tan\theta_{\rm{B}}].\eqno(20)]$

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 $[_g^{\,\prime}]$ , z $[_g^{\,\prime}]$ ) with transition rules x = x $[_g^{\,\prime}]$ cosφ $[_g^{\,\prime}]$ + z $[_g^{\,\prime}]$ sinφ $[_g^{\,\prime}]$ , z = z $[_g^{\,\prime}]$ cosφ $[_g^{\,\prime}]$ − x $[_g^{\,\prime}]$ sinφ $[_g^{\,\prime}]$ , in which the axis z $[_g^{\,\prime}]$ makes an angle φ $[_g^{\,\prime}]$ with the crystal normal n. This angle φ $[_g^{\,\prime}]$ is selected in such a way that the phase S_g (18) becomes independent of the transversal coordinate x $[_g^{\,\prime}]$ . From (18) it is easy to obtain

$[\tan\varphi_g^\prime=\gamma_g\sin\theta_0/(1-\sin\theta_0\sin\theta_g).\eqno(21)]$

Note that the axes of coordinates (x $[_h^{\,\prime}]$ , z $[_h^{\,\prime}]$ ) 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 $[_h^{\,\prime}]$ = 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 $[_g^{\,\prime}]$ /V_g), where V_g is the speed of the pulse along the longitudinal axis z $[_g^{\,\prime}]$ ,

$[V_g=c|\gamma_g|/(1-2\sin\theta_0\sin\theta_g+\sin^2\theta_0)^{1/2}.\eqno(22)]$

From expressions (19), (21) and (22) it follows that, for transmitted pulses (g = 0) both in the Bragg case and in the Laue case, φ₀′ = θ₀, V₀ = c and D₀ = 0. In other words the incident pulse with a wide front is transmitted along its initial direction K₀ 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 A₀(t − z₀′/c), which differs in the general case from A_in(t − z₀′/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 θ₀ 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 reciprocal lattice 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

$[\Delta\theta_h(\Omega)=-2(\Omega/\omega_0)\sin\theta_{\rm{B}}\cos\psi/\gamma_h.]$

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, sinθ₀ = 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 $[_h^{\,\prime}]$ /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_E²|F_hγ_h|)⁻¹, where in the approximation of the Gaussian forms for R(Ω) and A_in(Ω) the effective spectral width ΔΩ_E = ΔΩ₀ΔΩ_B/(ΔΩ₀² + ΔΩ_B²)^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 pulse duration, when the incident spectrum width exceeds the spectral width of the Bragg reflection ΔΩ₀ $[\gg]$ ΔΩ_B and therefore ΔΩ_E ≃ ΔΩ_B. If, for example, τ₀ = 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 τ₀ ≥ τ_B and with small transversal size r₀ ≃ Λ. Fig. 2 shows the space distributions of the modulus of amplitudes of the incident pulse |A_in(x_s, z_s)| with duration τ₀ = 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 l₀ = τ₀c = 3 µm, as well as its transverse size r₀ = 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. 2a). 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 pulse duration τ₀ exceeds the characteristic time τ_B = 2/ΔΩ_B ≃ 5.8 fs, where for a thick crystal τ_B = 2(Λ/c)sin²θ_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)].

Figure 2
Two-dimensional distribution of the amplitude of a Gaussian incident pulse |A_in(x_s, z_s)| (a) and the reflected pulse |A_h(x_p, z_p)| (b) in the symmetric Bragg Si(220) reflection (b = −1). Transversal pulse size r₀ = 10 µm, pulse duration τ₀ = 10 fs (l₀ = cτ₀ = 3 µm), central wavelength λ = 0.154 nm, thickness of crystal d = 50 µm, phase parameter α₀ = 2. Angle between the wavevector K_h and the normal N to the reflected pulse φ_h = 0. Distance from the crystal to the reflected pulse R = 3 m.

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 r₀ = 1240 µm, α₀ = 36.9 and angular divergence Δθ_s = 1.46 µrad.

Figure 3
Longitudinal sections of the intensity of reflected pulses I_h(z_p) along the normal N at x_p = 0 in cases of symmetric (a) and asymmetric (b) Bragg reflections. Long (τ₀₁ = 10 fs) and short (τ₀₂ = 1 fs) Gaussian pulses with amplitudes A₁ = A₂ = 1 [curves 1 and 2 in (a)

] are incident on a crystal with time interval Δt₁₂ = 20 fs. Thickness of the crystal d = 5 µm. (a) Dashed curve 3 is the total reflected pulse, angle φ_h = 0. (b) Distance from the crystal R = 0 (curve 1), R = 2 m (curve 2) and R = 5 m (curve 3). The asymmetry coefficient of the reflection b = −2, critical distance R_D = 1.6 m, angle of inclination of the reflected pulse φ_h = 23.65°.

It is clear that, starting from some duration of the incident pulse τ₀ < τ_B, only a part ΔΩ_B < ΔΩ₀ of the incident frequency spectrum ΔΩ₀ 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(Δt₁₂ + τ_B) ≃ −10 µm in Fig. 3(a), where τ_B = 2(d/c)sin²θ_B/|γ_h|, and Δt₁₂ 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. 3b).

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)].

Figure 4
Two-dimensional distribution of amplitude of the reflected pulse |A_h(x_p, z_p)| in the symmetric Laue case (b = 1). Distance from the crystal to the pulse R = 2 cm (a), and R = 10 cm (b). Duration of incident pulse τ₀ = 1 fs, crystal thickness d = 23.19 µm, critical distance R_D = 8.2 cm, angle of pulse inclination φ_h = −41.2°. For the incident pulse of longitudinal size l₀ = 0.3 µm there appears to be a δ-function on the background of the spatial distribution of the reflected pulse and consequently it is not shown in these figures. Other parameters: size of source r_s = 75 µm, phase parameter α_s = 2, distance from source to crystal z_s = 800 m, transversal size of incident pulse r₀ = 1240 µm, phase parameter α₀ = 36.9.

Figure 5
Longitudinal section of intensity of the reflected pulse I_h(z_p) at x_p = 0 in the symmetric Laue case (b = 1). Distance from crystal to pulse R = 0 (curve 1), R = 5 cm (curve 2) and R = 20 cm (curve 3). Duration of incident pulse τ₀ = 0.1 fs (l₀ = 0.03 µm), crystal thickness d = 23.19 µm, distance R_D = 7.9 cm, angle φ_h = −41.2°. Other parameters are the same as in Fig. 4

The second peak in the Bragg case at z_p ≃ −V_hτ_B, where speed V_h is defined in expression (22) and τ_B = 2(d/c)sin²θ_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 I₀(x, z, t) meanwhile practically coincides in form and intensity with the incident pulse, as the transmission coefficient T(Ω) = exp(ik₀χ₀d/2γ₀) 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., 2001a).

Figure 6
Two-dimensional distribution of amplitude of the reflected pulse |A_h(x_p, z_p)| in the asymmetric Bragg case (b = −2). Distance from crystal to pulse R = 5 cm (a), and R = 100 cm (b). Duration of incident pulse τ₀ = 1 fs, crystal thickness d = 5 µm, critical distance R_D = 64.5 cm, angle of pulse inclination φ_h = 23.65°. Other parameters are the same as in Fig. 4

Figure 7
Longitudinal section of intensity of the reflected pulse I_h(z_p) at x_p = 0 in the asymmetric Bragg case (b = −0.5). Distance from crystal to pulse R = 0 (curve 1), R = 50 cm (curve 2) and R = 100 cm (curve 3). Duration of incident pulse τ₀ = 0.1 fs, crystal thickness d = 5 µm, distance R_D = 63.6 cm, angle φ_h = −12.35°. Other parameters are the same as in Fig. 4

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

$[\Gamma_{\rm{in}}(\rho,\tau)=P^{-1}\left|\left\langle{A_{\rm{in}}(x,t)}A_{\rm{in}}^*(x+\rho,t+\tau)\right\rangle\right|,\eqno(23)]$

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 cross section, 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,

$[\Gamma_{\rm{in}}(\rho,\tau)=I_{\rm{in}}^{-1}\big|\textstyle\int\!\!\int|A_{\rm{in}}(q,\Omega)|^2\exp[i(q\rho-\Omega\tau)]\,{\rm{d}}q\,{\rm{d}}\Omega\big|,\eqno(24)]$

where

$[I_{\rm{in}}=\textstyle\int\!\!\int|A_{\rm{in}}(q,\Omega)|^2\,{\rm{d}}q\,{\rm{d}}\Omega.]$

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

$[\eqalignno{\Gamma_g(\rho,\tau)={}&I_g^{-1}\big|\textstyle\int\!\!\int|B_g(q,\Omega)A_{\rm{in}}(q,\Omega)|^2\cr&\times\exp[i(q\rho-\Omega\tau)]\,{\rm{d}}q\,{\rm{d}}\Omega\big|.&(25)}]$

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 ΔΩ₀ $[\gg]$ 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 thermal conductivity equation with distributed thermal sources in a crystal subsurface layer, it follows that the time of temperature propagation over a distance Δx is Δt ≃ (Δx)²/4a², where a² = λ_T/c_Tρ, λ_T and c_T are coefficients of thermal conductivity 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 Δx ≃ Λ, one obtains Δt ≃ 13 ns, and this is much longer than the duration of an X-ray pulse τ₀ = 0.1–200 fs (for diamond Δt ≃ 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 ΔT = Δθ_Bcotanθ_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 ΔΩ₀ $[\gg]$ 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 k₀ and k_gx in the expression for k_gz in (7) in the following form: k₀ = K₀(1 + ξ₁), where ξ₁ = Ω/ω₀, and k_gx = K_gx(1 + ξ₂), where ξ₂ = q/K_gx. As a result for the square-root expression in (7) we find that

$[k_0^2-k_{gx}^2=K_{gz}^2\left[1+2(a\xi_1-b\xi_2)+(a\xi_1^2-b\xi_2^2)\right],\eqno(26)]$

where K_gz² = K₀² − K_gx²,

$[a=(K_0/K_{gz})^2=1/\gamma_g^2,\quad\quad b=(K_{gx}/K_{gz})^2=\tan^2\theta_g.\eqno(27)]$

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 $[\ll]$ 1 the following expansion takes place,

$[(1+\xi)^{1/2}\simeq1+(1/2)\xi-(1/8)\xi^2.\eqno(28)]$

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

$[\eqalignno{\sigma_g(k_0^2-k_{gx}^2)^{1/2}={}&K_{gz}(1+a\xi_1-b\xi_2) -(1/2)K_{gz}\left[a(a-1)\xi_1^2\right.\cr&\left.-\,2ab\xi_1\xi_2+b(b+1)\xi_2^2\right],&(29)}]$

where K_gz = K₀γ_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) = tan²θ_g/γ_g². As a result, from (29) we find finally that

$[\eqalignno{k_{gz}={}&K_{gz}-q\tan\theta_g+(\Omega/c\gamma_g)\cr&-[q-(\Omega/c)\sin\theta_g]^2/(2K_0\gamma_g^3).&(30)}]$

If now in addition to k_gz (30) we consider the x-components of wavevectors k_g and the item ωt, the phase in the exponential in the integral (6) will have the following form,

$[{\bf{k}}_g{\bf{r}}-\omega{t}=({\bf{K}}_g{\bf{r}}-\omega_0t)+S_g(q,\Omega)+D_g(q,\Omega),\eqno(31)]$

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

$[E_{\rm{s}}(x_{\rm{s}},t)=A_{\rm{s}}(x_{\rm{s}},t)\exp(-i\omega_0t),\eqno(32)]$

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 ω₀ 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,

$[E_{\rm{s}}(x_{\rm{s}},t)=\textstyle\int\!\!\int{E_{\rm{s}}}(q,\omega)\exp(iqx_{\rm{s}}-i\omega{t})\,{\rm{d}}q\,{\rm{d}}\omega,\eqno(33)]$

where

$[E_{\rm{s}}(q,\omega)=(2\pi)^{-2}\textstyle\int\!\!\int{E_{\rm{s}}}(x_{\rm{s}},t)\exp(-iqx_{\rm{s}}+i\omega{t})\,{\rm{d}}x_{\rm{s}}{\rm{d}}t.\eqno(34)]$

Propagation of the pulse field E(x_s, z_s, t) in the region z_s ≥ 0 is described by the wave equation

$[\Delta{E}-(1/c^2)\,\partial^2E/\partial{t^2}=0,\eqno(35)]$

where Δ = ∂²/∂x_s² + ∂²/∂z_s² 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

$[E(x_{\rm{s}},z_{\rm{s}},t)=\textstyle\int\!\!\int{E_{\rm{s}}}(q,\omega)\exp(iqx_{\rm{s}}+ik_zz_{\rm{s}}-i\omega{t})\,{\rm{d}}q\,{\rm{d}}\omega,\eqno(36)]$

where k_z = (k² − q²)^1/2, k = ω/c.

From (32) and (34) it follows that E_s(q, ω) = A_s(q, Ω), where Ω = ω − ω₀. If the characteristic size of the source r_s $[\gg]$ λ, and the pulse duration τ₀ $[\gg]$ T, where λ is the wavelength of radiation and T is the period, then q $[\ll]$ k and Ω $[\ll]$ ω₀. In this case, with use of the expansion (28) for values of k_z in (36), we have

$[k_z\simeq K_0+\Omega/c-q^2/2K_0,\eqno(37)]$

where K₀ = ω₀/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,

$[E(x_{\rm{s}},z_{\rm{s}},t)=A(x_{\rm{s}},z_{\rm{s}},t)\exp(iK_0z_{\rm{s}}-i\omega_0t),\eqno(38)]$

where A(x_s, z_s, t) is the slowly varying amplitude of the pulse,

$[\eqalignno{A(x_{\rm{s}},z_{\rm{s}},t)={}&\textstyle\int\!\!\int{A_{\rm{s}}}(q,\Omega)\exp\left[iqx_{\rm{s}}-iq^2z_{\rm{s}}/2K_0\right.\cr&\left.-\,i\Omega(t-z_{\rm{s}}/c)\right]\,{\rm{d}}q\,{\rm{d}}\Omega.&(39)}]$

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,

$[A_{\rm{s}}(x_{\rm{s}},t)=\exp\left[-(x_{\rm{s}}/r_{\rm{s}})^2+i\varphi_{\rm{s}}(x_{\rm{s}})-(t/\tau_0)^2\right],\eqno(40)]$

where r_s is the size of the source in the plane z_s = 0, τ₀ is the pulse duration 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,

$[\textstyle\int\exp(-i\beta{x}+i\gamma{x^2})\,{\rm{d}}x=(i\pi/\gamma)^{1/2}\exp(-i\beta^2/4\gamma),\eqno(41)]$

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,

$[\eqalignno{A(x_{\rm{s}},z_{\rm{s}},t)={}&A_{\rm{s}}\exp[-(x_{\rm{s}}/r_0)^2+i\varphi_0(x_{\rm{s}})\cr&-(t-z_{\rm{s}}/c)^2/\tau_0^2+i\Phi_0],&(42)}]$

where A_s = 1/M^1/2, M = [(1 + α_sW)² + W²]^1/2, W = λz_s/πr_s² is the wave parameter,

$[\eqalign{&r_0=r_{\rm{s}}M,\quad\quad\varphi_0(x_{\rm{s}})=\alpha_0(x_{\rm{s}}/r_0)^2,\cr&\alpha_0=\alpha_{\rm{s}}+(1+\alpha_{\rm{s}}^2)W,\cr&\Phi_0=-(1/2)\arctan[W/(1+\alpha_{\rm{s}}W)].}\eqno(43)]$

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 r₀(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 φ₀(x_s), also a square-law function, depends on the transversal coordinate. The parameter of this phase α₀(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 α₀ ≠ 0, even at the initial plane wavefront, i.e. at α_s = 0. The phase Φ₀(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, Φ₀ = 0.

The width of the angular spectrum of a pulse (42) Δθ_s = Δq_s/K₀, i.e. the width of the function |A(q, Ω, z_s)| ≃ exp[−(q/Δq_s)²], where Δq_s = 2(1 + α₀²)^1/2/r₀ = 2(1 + α_s²)^1/2/r_s, does not depend on the distance z_s and is determined by the expression

$[\Delta\theta_{\rm{s}}=(\lambda/\pi{r_{\rm{s}}})(1+\alpha_{\rm{s}}^2)^{1/2}.\eqno(44)]$

The angular width Δθ_s (44) in the general case exceeds the diffraction divergence Δθ_d = (λ/πr_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 r₀ ≃ 820 µm, and the phase parameter α₀ ≃ 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 = xcosθ₀ − zsinθ₀, z_s = xsinθ₀ + zcosθ₀ + z₁, where θ₀ is the incident angle of the pulse on the crystal with respect to the normal n to the crystal surface, and z₁ 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

$[\eqalignno{A_{\rm{in}}(x,t)={}&\exp\left[-(x\gamma_0/r_0)^2(1-i\alpha_0)\right.\cr&\left.-\,(t-x\sin\theta_0/c)^2/\tau_0^2\right],&(45)}]$

where γ₀ = cosθ₀, and the time t is counted from the moment t₁ = z₁/c of incidence of the pulse maximum at x = 0, z = 0 on the crystal.

Figure 8
Geometry in real space of an asymmetric Bragg case (a) and Laue case (b) diffraction. Here K₀ and K_h are the average wavevectors of the incident and reflected pulses, respectively; N is the normal to the long axis of the reflected pulse; (x_s, z_s) is a Cartesian system of coordinates on the source surface, (x, z) is a system of coordinates on the crystal surface, the system of coordinates (x_h, z_h) moves together with the reflected pulse, axis z_h is directed along the wavevector K_h; axes of the moving system of coordinates (x_p, z_p) are directed along the main axes of the pulse, φ_h is an angle between the normal N to the reflected pulse and the direction of pulse propagation K_h.

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),

$[\eqalignno{A_{\rm{in}}(q,\Omega)={}&A_0\exp\left[-(q-\Omega\sin\theta_0/c)^2(1+i\alpha_0)/{\Delta}q_0^2\right.\cr&\left.-\,(\Omega/\Delta\Omega_0)^2\right],&(46)}]$

where ΔΩ₀ = 2/τ₀ is the spectral width of the incident pulse, Δq₀ = 2γ₀(1 + α₀²)^1/2/r₀ is the width of the angular spectrum in q-space, and amplitude A₀ = (1 + iα₀)^1/2/(πΔq₀ΔΩ₀). It is easy to show that Δq₀ = γ₀Δq_s, where Δq_s = 2(1 + α_s²)^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)],

$[R(q,\Omega)=\exp\left\{-\left[q-(\Omega/c)\sin\psi/\cos\theta_{\rm{B}}\right]^2/\Delta{q_{\rm{B}}^2}\right\},\eqno(47)]$

where Δq_B = K₀γ₀Δθ_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

$[A_h(x,z,t)=A_R\exp\left[-\Phi_1^2(1-i\alpha_0)-\Phi_2^2(1-i\beta_0)\right],\eqno(48)]$

where

$[\Phi_1=(x-z\tan\theta_h)\gamma_0/r_0,\eqno(49)]$

$[\Phi_2=\left[t-z/c\gamma_h-(x-z\tan\theta_h)\sin\theta_0/c\right]/\tau_R.\eqno(50)]$

Here, τ_R is the duration of the reflected pulse, which is determined by the following formula,

$[\tau_R=\tau_E(1+\beta_0^2)^{1/2},\eqno(51)]$

where

$[\eqalign{&\tau_E=(\tau_0^2+\tau_{\rm{B}}^2)^{1/2},\quad\quad\tau_{\rm{B}}=2/\Delta\Omega_{\rm{B}},\cr&\Delta\Omega_{\rm{B}}=\omega_0\Delta\theta_{\rm{B}}\,{\rm{cotan}}\,\theta_{\rm{B}},}\eqno(51a)]$

$[\eqalign{&\beta_0=(\tau_Z/\tau_E)^2,\quad\quad\tau_Z=2(F_hz)^{1/2},\cr&F_h=(\sin\theta_0-\sin\theta_h)^2/(2K_0c^2\gamma_h^3),}\eqno(51b)]$

$[A_R=(\tau_0/\tau_R)(1-i\beta_0)^{1/2}.\eqno(51c)]$

In order to obtain expression (48) it is considered that the ratio Δq₀/Δq_B = Δθ_s/Δθ_B $[\ll]$ 1, where Δθ_s = (λ/πr_s)(1 + α_s²)^1/2 is the width of the angular spectrum of the source radiation, and distance |z| $[\ll]$ z_F, where z_F = πr₀²/[λb²(1 + α₀²)].

From equations (51) it follows that the pulse duration τ_R after reflection from the crystal increases in comparison with τ₀ 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 ΔΩ₀ > ΔΩ_B, and therefore τ_B > τ₀. The second reason is connected to diffusion broadening of the part of the pulse duration 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 β₀ = 1,

$[R_D=(\tau_0^2+\tau_{\rm{B}}^2)/4|F_h\gamma_h|.\eqno(52)]$

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 sinθ₀ = sinθ_h, F_h = 0 and R_D → ∞. The critical distance R_D (52) increases with increase in τ₀, if τ₀ > 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 τ₀ $[\ll]$ τ_B the distance R_D does not depend on the duration of the incident pulse τ₀. Unfortunately in practice the distance R_D does not exceed several tens of centimetres in the Laue case at τ₀ ≤ 1–10 fs (see Fig. 9).

Figure 9
Dependence of the critical distance R_D on the asymmetry coefficient of reflection b at various durations of the incident pulse τ₀: curve 1, 1 fs; curve 2, 10 fs; curve 3, 20 fs; σ-polarization.

From the form of arguments Φ₁ and Φ₂ 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 θ₀ 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),

$[\eqalign{&x=ct\sin\theta_h+x_h\cos\theta_h+z_h\sin\theta_h,\cr&z=ct\cos\theta_h+z_h\cos\theta_h-x_h\sin\theta_h.}\eqno(53)]$

Then the functions Φ_1,2 in (48) will have the following form,

$[\Phi_1=bx_h/r_0,\quad\quad\Phi_2=(a_{0h}x_h+z_h)/(c\tau_R),\eqno(54)]$

where a_0h = (sinθ₀ − sinθ_h)/γ_h.

From equations (54) it can be seen that only in the symmetric Bragg geometry when sinθ₀ = 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 r₀ is much larger than its longitudinal size l₀ = cτ₀, 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

$[\eqalign{&x_h=x_p\cos\varphi_h+z_p\sin\varphi_h,\cr&z_h=z_p\cos\varphi_h-x_p\sin\varphi_h.}\eqno(55)]$

The angle φ_h is obtained from the condition that the coefficient of product x_pz_p in the expression Φ₁² + Φ₂² in (48) equals zero. As a result we find that

$[\tan\varphi_h=a_{0h}(1+\delta),\eqno(56)]$

where δ = (bcτ_R/r₀)²/(1 + a_0h²) $[\ll]$ 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.

Figure 10
Dependence of the inclination angle of the reflected pulse φ_h (1), of the incident angle θ₀ (2), and of the reflected angle θ_h (3) on the asymmetry coefficient of the reflection b.

So, the modulus of the amplitude of the reflected pulse in a moving system of coordinates (x_p, z_p) is

$[|A_h(x_p,z_p)|=(\tau_0/\tau_R)\exp\left[-(x_p/r_{\rm{T}})^2-(z_p/r_{\rm{L}})^2\right],\eqno(57)]$

where r_T = r₀/(|b|cosφ_h) is the transversal size of the pulse and r_L = V_Lτ_R is the longitudinal size of the pulse. Here V_L = ccosφ_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/cγ_h. From equation (56) it is easy to see that

$[\cos\varphi_h\simeq|\gamma_h|/(1-2\sin\theta_0\sin\theta_h+\sin^2\theta_0)^{1/2}.\eqno(58)]$

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. r₀ → ∞). For this reason propagation of the reflected pulse with projection speed V_T = csinφ_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 pulse duration 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 r₀/|b| onto this plane of the transversal size of the pulse: τ_tot = (r₀/c)|sinφ_h/b|. If, for example, r₀ = 800 µm, b = 1, then the angle of inclination φ_h = −41.2° and the total duration of the reflected pulse τ_tot = 2 × 10³ 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

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Volume 15| Part 5| September 2008| Pages 495-505

doi:10.1107/S0909049508019602

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

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

1. Introduction

2. Theory

3. Results and discussion

4. Conclusions

APPENDIX A

Calculation of projection kgz

APPENDIX B

X-ray pulse propagation in free space

APPENDIX C

Reflection of a Gaussian pulse

Acknowledgements

References

research papers

Calculation of projection k_gz