## Abstract

We study the nonlinear refraction of X-rays in highly ionized condensed matter by using a classical model of a cold electron plasma in a lattice of still ions coupled with Maxwell equations. We discuss the existence and stability of nonlinear waves. As a real-world example, we consider *beam self-defocusing* in crystalline materials (B, C, Li, Na). We predict that nonlinear processes become comparable to the linear ones for focused beams with powers of the order of *mc*
^{3}/*r _{o}* (≈10 GW), the

*classical electron power*. As a consequence, nonlinear phenomena are expected in currently exploited X-ray Free-Electron Lasers and in their future developments.

©2008 Optical Society of America

## 1. Introduction

Understanding nonlinear processes at the smallest accessible spatio-temporal scale is a frontier of modern research. In this respect, the new generation of X-ray Free-Electron Lasers (X-ray FEL) opens unprecedented possibilities: an example is nonlinear optics in the X-ray region (see e.g. [1, 2]). The new X-ray FEL are expected to deliver femtosecond pulses in the wavelength range 0.05 nm - 1 nm with peak power greater than 100 GW (see e.g. [3]). This corresponds to intensities up to 10^{23} W cm^{-2} for focused beams with 10nm spot-size. Even not taking into account relativistic effects and particle production (expected at intensities of ≈10^{26} W cm^{-2} [4, 5]), the underlying fundamental physical processes are in many respects unknown. Photons at the atomic-scale wavelength (*λ*≈0.1 nm) have macroscopic propagation lengths in condensed materials (even considering photoelectric absorption) and are sensitive to the granular structure of matter [6, 7, 8, 9, 10, 11]. Therefore, the physics of intense X-ray beams have to take into account nonlinear effects accumulating along large propagation distances in inhomogeneous environments. Within a general perspective, high-fluence X-ray photon bursts rapidly ionize the material, and the basic nonlinear processes are due to the hydrodynamics of the generated electron plasma (see e.g. [1, 4, 12, 13, 14, 15] and references therein). At hard X-ray wavelengths, the linear refractive index (due to plasma) differs from unity by an amount *δ* of the order of 10^{-5}. In this paper we investigate the nonlinear contribution to *δ*, which is an issue of high relevance to prepare the future science of X-ray FEL’s. In the early days of nonlinear optics, Bloembergen and others [16, 13] reported on similar analyses; here we extend those works in a two-fold way: (i) we largely generalize previous results beyond the lowest order approximation and by considering realistic aggregates of atomic systems, and (ii) we particularize to real world experiments with reference to the expected performances of the modern X-ray FEL in terms of power, losses and beam divergence. Within a purely classical formulation, we show that the nonlinear contribution to the index of refraction, expressed as *n*
_{2}
*I* for low intensity *I* (with *n*
_{2} being an effective Kerr coefficient [17]) becomes of the order of *δ* for diffraction-limited focused beams with powers of 10 GW (a basically wavelength-independent result, even including photo-absorption). With reference to the current experiments at FLASH [18], this effect can be observed for tightly focused beams. We evaluate the third order nonlinear susceptibility of the X-ray induced plasma and demonstrate the existence of stable nonlinear waves. Finally, we suggest a realistic experiment by quantifying the self-defocusing of an X-ray beam in real crystals (B,C,Li,Na).

## 2. Model and theoretical approach

X-ray propagation is described by Maxwell equations coupled to the hydrodynamic equations for a plasma of free electrons [12, 1]:

$$\nabla \times \mathbf{e}={-\mu}_{0}{\partial}_{t}\mathbf{h},$$

$${\partial}_{t}n+\nabla \xb7\left(n\mathbf{v}\right)=0,$$

$$({\partial}_{t}+\mathbf{v}\xb7\nabla )\mathbf{v}+\frac{q}{m}\left(\mathbf{e}+{\mu}_{0}\mathbf{v}\times \mathbf{h}\right)=0,$$

being **e**, **h** the time-dependent electromagnetic fields, **v** the electron mean velocity and **j**=-*qn*
**v** the current generated by the charge density *q*(*n*-*n*
_{0}), with *n* and *n*
_{0} the particle density of electrons and ions, respectively. We make the assumption that high energy photons instantaneously (i.e. at the attosecond scale) ionize the material, so that the electromagnetic field propagates in a periodically distributed *cold* plasma [14, 4]. The ion cores with density *n*
_{0}, in other terms, are assumed to be “frozen”; they only form the crystal lattice. In our picture, the XFEL electric field amplitude is sufficiently high to heavily perturb the nuclear electric field and induce multiple ionization. We reduce (1) to an integrable ordinary differential equation (we defer extensive mathematical details to future publications); our procedure begins by writing the current as **j**=**j**
_{L}+**j**
_{Δ}, being **j**
* _{L}* a linear contribution and

**j**

_{Δ}an arbitrary nonlinear term. In the linear regime

**j**=

**j**

_{L}and the Fourier domain (∂

*→-*

_{t}*iω*), Eqs. (1) are written in the

*canonical*form:

with *ω*
^{2}
* _{α}*=

*q*

^{2}

*n*

_{0}/

*mε*

_{0}. As detailed below, under typical experimental conditions (

*λ*≈0.1 nm and

*n*

_{0}≈10

^{30}m

^{-3}), the term 1-

*δ*, with

*δ*=

*ω*

^{2}

_{α}/

*ω*

^{2}, behaves as an effective periodic dielectric constant

*ε*(

_{r}**r**), whose period is settled by the crystal lattice constant

*a*. The spectrum of such a periodic system is then found by applying the Floquet-Bloch theorem [19]. The eigenmodes of (2) are Bloch modes

**Ẽ**,

_{k}**H̃**satisfying the Hermitian eigenvalue problem:

_{k}being
${\mathcal{L}}_{H}=\nabla \times \frac{1}{{\epsilon}_{0}{\epsilon}_{r}}\nabla \times $
and *ω*(**k**) the medium dispersion relation. Actually, only magnetic modes are to be calculated since **Ẽ _{k}** depends on

**H̃**via

_{k}*iωε*

_{0}

*ε*

_{r}**Ẽ**=∇×

_{k}**H̃**. Owing to the smallness of the dielectric perturbation

_{k}*ω*

^{2}

_{α}/

*ω*

^{2}, the spectrum 𝓢={

**H̃**,

_{k}**Ẽ**,

_{k}*ω*(

**k**)} of the canonical structure is completely determined the representation theory of space groups [19, 20]. Figure 1(a) shows the first Brillouin zone for the Face Centered Cubic (FCC) lattice, where the Brillouin zone is a truncated octahedron of height 4

*π*/

*a*. The dispersion relation is constructed from the free-space relation

*ω*(

**k**)=

*c*|

**k**|, folded over itself according to the dispersion periodicity

*ω*(

**k**)=

*ω*(

**k**+

**G**) [20], being

**G**an arbitrary integer translation of the reciprocal lattice unit vectors (Fig. 1). No gap opens in the wave spectrum and Bloch modes exist for all frequencies

*ω*. Each

**k**point defines a set of degenerate plane waves, which possess the same eigenfrequency

*ω*(

**k**) in the absence of periodicity. If properly combined, these waves form a basis for Bloch modes, whose expression is provided by the theory of projection operators [20]. Table 1 displays

*z*-propagating magnetic Bloch modes for

*ωa*/2

*πc*<1 (e.g., for

*λ*>

*a*) for the FCC and the simple cubic (SC) lattice. Since Bloch modes are constructed from eigenwaves of the free-space, they are always doubly degenerate (i.e., they are either

*x*or

*y*polarized).

Once the spectrum 𝓢 of the canonical structure is known, the reduction of (1) is performed by exploiting the following identity (we let **e**=2ℜ[**E**exp(-*iωt*)] and **h**=2ℜ[**H**exp(-*iωt*)]:

Equation (4) is a form of the reciprocity theorem of Maxwell equations [21] and does not contain any approximation, as it can be verified by direct substitution. In the absence of nonlinear effects (**j**
_{Δ}=0) Eq. (4) is identically zero due to the linear relation between **j**
_{L} and the electric field; conversely when **j**
_{Δ}≠0 Eq. (4) allows to derive an exact evolution equation for the field. This requires the relationship between the current shift **j**
_{Δ} and **E** and **H**. To this aim we solve the plasma fluid equations by an iterative expansion in the field components: the overall current is written as **j**=**j**
^{(1)}+**j**
^{(3)}, where **j**
^{(1)}=**j**
_{L}=(*iq*
^{2}
*n*
_{0}/*ωm*)**E** and **j**
^{(3)}=**j**
_{Δ} accounts for the third order nonlinearity. The second order terms can be neglected as far as the input direction does not satisfies phase matching conditions[14]; correspondingly the phase-effect in the forward scattering (detailed below) is negligible. By letting

$$\mathbf{v}=2\Re \left[{\mathbf{V}}^{\left(1\right)}{e}^{-i\omega t}\right]+2\Re \left[{\mathbf{V}}^{\left(2\right)}{e}^{-2i\omega t}\right],$$

we have (**E**
^{2}≡**E**·**E**):

The nonlinear current is given by **j**
^{(3)}=-*qn*
^{(2)}[**V**
^{(1)}]*-*q*[*n*
^{(1)}]***V**
^{(2)}. In the final expression of **j** there are terms including ∇*n*
_{0}, which account for out-of-axis nonlinear Bragg scattering.

## 3. Nonlinear waves and the *classical* electron power

In the following we will focus on the direct beam by retaining only terms that induce self-phase modulation in the input propagation direction. It is worthwhile remarking that the validity of our results is not limited to periodical samples: the features of the crystal lattice mainly influence diffracted orders, while the nonlinear effects on the forward propagating beam are expected to be observed for samples with comparable spatially-averaged charge density 〈*n*
_{0}〉, i.e., independently on the specific lattice structure. Correspondingly, we only retain 〈*n*
_{0}〉 terms in Eq. (6) and obtain

We consider a *z*-propagating beam with frequency below the first turning point of the band structure (see Fig. 1 for *a*/λ<1); in this case the treatment is simplified by the fact that only one Bloch function is involved. By exploiting Table 1, the field can be written as:

$$\mathbf{H}=-\sqrt{\frac{2}{{\eta}_{0}}}A\left(z\right)\hat{\mathbf{x}}{e}^{\mathrm{ikz}},$$

where
${\eta}_{0}=\sqrt{\frac{{\mu}_{0}}{{\epsilon}_{0}}}$
is the vacuum impedance. Bloch functions are normalized such that |*A*(*z*)|^{2} yields the beam intensity; correspondingly, Eq. (4) becomes

We remark that Eq. (11) differs from the standard SPM equation (see e.g. [16, 17, 22]) and holds true for rapidly varying amplitudes *A*(*z*). Its solution is found as *A*(*z*)=*A*
_{0}exp(*ik _{nl}z*), where

*k*is the nonlinear contribution to the wave-vector and satisfies

_{nl}Eq. (12) has always two real-valued solutions, as shown in Fig. 2(a) with *γ*≡*η*
^{2}
_{0}
*q*
^{4}〈*n*
_{0}〉/8*m*
^{3}
*ω*
^{5}. At variance with standard SPM [17, 22], the largest solution [continuous line in Fig. 2(a)] does not linearly grow with the power *A*
^{2}
_{0} and asymptotically tends to -*k* from above. This wave is forward propagating. Conversely, the other solution provides *k*+*k _{nl}*<0 [dot-dashed line in Fig.2(a)], representing a backward propagating wave. These two solutions at large powers have the same wave-vector; this can involve a nonlinear coupling between them, which strongly relies on the material lattice and will be considered in future work.

The stability of the two nonlinear waves is found by letting *A*(*z*)=[*A*
_{0}+*a*
_{1}(*z*)]exp(*ik _{nl}z*), with

*a*

_{1}(

*z*) a complex valued function. At the lowest order in

*a*

_{1}(

*z*) we obtain

$$\left[{k}_{\mathrm{nl}}+8{A}_{0}^{2}{\left(k+{k}_{\mathrm{nl}}\right)}^{2}\gamma \right]{a}_{1}+4\gamma {\left(k+{k}_{\mathrm{nl}}\right)}^{2}{a}_{1}^{*}=0.$$

While looking for exponentially amplified disturbances *a*
_{1}∝exp(*αz*), straightforward algebra leads to the conclusion that *α* is zero or imaginary for any value of *A*
_{0}. The found solutions are therefore always stable. The existence of stable nonlinear plane waves allows to identify an effective refractive index *n*, which depends on the beam intensity. The latter can be written as *n*=1-*δ*+*iβ*+*k _{nl}*/

*k*, being

*β*the absorption coefficient. The term

*k*/

_{nl}*k*conversely accounts for the nonlinear wave-vector; at the lowest order in the intensity for the forward propagating solution it reads

*k*/

_{nl}*k*=-|

*n*

_{2}|

*I*=-4

*γkI*, with

*n*

_{2}=-4

*kγ*<0 the effective Kerr coefficient. To determine the values of

*I*for which nonlinear effects become relevant, a noticeable figure of merit is the ratio between

*n*

_{2}

*I*and

*δ*that, after simple algebra, is written as

with *P*
_{0}=*mc*
^{3}/*r*
_{0}≅8.8 GW being the *classical electron power* and *r*
_{0}=*q*
^{2}/4*πε*
_{0}
*mc*
^{2} the electron radius. Equation (14) states that if a beam of power *P*
_{0} is focalized down to the diffraction limit (spot area of the order *πλ*
^{2}) then 𝒩≅0.5, i.e. the plasma nonlinear contribution is of the same magnitude of the linear one. This implies that nonlinear effects could be observed in the current FLASH experiments at DESY [6, 18] (where *P*≅10 GW) for tightly focused beams (waist≈100 nm), and certainly they will be observable with the next generation of X-ray FEL.

## 4. Self-defocusing and losses

We estimate the magnitude of nonlinear effects in the direct beam with reference to the simple case of self-defocusing. We consider a linearly polarized forward *z*-propagating beam with spatial profile *E*∝exp(-*r*
^{2}
_{⊥}/*w*
^{2}
_{0}) focused on a sample (here *r*
^{2}
_{⊥}=*x*
^{2}+*y*
^{2} and *w*
_{0} is the beam waist). The field is expected to display a power-dependent divergence in the far field. For an input intensity *A*
^{2}
_{0}(*r*
_{⊥})=*I*
_{0}
*exp*(-2*r*
^{2}
_{⊥}/*w*
^{2}
_{0}), the output beam acquires an additional transverse nonlinear phase profile given by Φ_{nl}(*r*
_{⊥})=*β*[*A*
_{0}(*r*
_{⊥})]*L*, being *L* the sample length. To the lowest order in the beam peak intensity *I*
_{0}, the forward propagating solution of Eq. (12) gives *β*≅-4*k*
^{2}
*γA*
^{2}
_{0}(*r*
_{⊥}). Since the most pronounced nonlinear effects are in proximity of the beam axis, one has

which implies that the sample acts as an effective thin lens with negative focal length

Correspondingly, the acquired divergence angle *θ* is given by

At any order in *I*
_{0}, we have

being *ξ*≡*kγI*
_{0}. Therefore, the divergence tan(*θ*) can be expressed as:

For large *ξ*, tan(*θ*) saturates at about 0.6*L*/*w*
_{0}, which defines the largest attainable divergence [Fig. 2(b)]. In the future generation of X-ray FEL, peak intensities of the order of 100 GW with source size of 60 *µ*m and divergence of 3 *µ*rad will be at disposal [3]. A beam with wavelength in the range 0.1-10 nm and intensity of ≈10^{22} W cm^{-2} (spot sizes ≈100 nm), displays a linear divergence of the order of 2 mrad; in order to have an observable nonlinear defocusing, we determine the intensity needed to induce a comparable beam divergence. For small divergence angles, owing to the linear relationship *θw*
_{0}=16*kγI*
_{0}
*L* between *I*
_{0} and the sample length *L*, the intensity needed to acquire a nonlinear divergence greater than the linear one is independent on the beam waist; this figure is therefore a good indicator to quantify nonlinear effects. In addition, in real materials the available effective length for the self-defocusing is compelled by photoelectric absorption and/or other material losses, whose strength strongly depends on the atomic number *Z*. In order to account for these effects, we consider the intensity needed to obtain a divergence comparable to the linear one on the sample characteristic loss length, the latter determined by two leading mechanisms: absorption [Fig. 3(a)] and electron-ion collisions [Fig. 3(b)] (electron-electron collisions are known to be negligible) [4]. Figure 3(a) displays the intensity needed to induce a nonlinear divergence of 2 mrad for waist *w*
_{0}=100 nm (or equivalently 20 mrad and for *w*
_{0}=10 nm) by including the absorption as quantified by means of tabulated data [23]. We estimated the average density of electrons as 〈*n*
_{0}〉=*Z*𝒩_{a}DM^{-1}
_{m}, where 𝒩* _{a}* is the Avogadro number,

*D*is the material density and

*M*the molar mass, thus obtaining 〈

_{m}*n*

_{0}〉≅10

^{30}m

^{-3}for all of the considered materials. Figure 3(b) deals with electron-ion collisions described by the conductivity ${\sigma}_{\mathrm{coll}}=\frac{3{\left(\mathrm{KT}\right)}^{\frac{3}{2}}}{2Z\sqrt{\frac{m}{3}}\pi {q}^{2}}$ , where

*K*is the Boltzmann constant and

*T*the plasma temperature. By comparing the two figures, one finds that electronic losses are dominant for wavelengths below 1 nm.

## 5. Conclusion

In conclusion, we predict that X-rays self-phase modulation is observable within the reported expected performances of future X-ray FEL sources for wavelengths of 1–10 nm, and in the current FLASH experiments at DESY. If tightly focused beams are used for retrieving structural information (e.g. in the foreseen single molecule diffraction experiments), we find that as far as the involved powers are of the order of *P*
_{0}=*mc*
^{3}/*r*
_{0}≅10 GW, distortion due to nonlinear refraction could be observable and influence the interpretation of the scattering pattern. This result is expected to be independent on the specific structure of the molecule.

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