X-ray focusing by bent crystals: focal positions as predicted by the crystal lens equation and the dynamical diffraction theory

The location of the beam focus when monochromatic x-ray radiation is diffracted by a thin bent crystal is predicted by"crystal lens equation". We derive this equation in a general form valid for Bragg and Laue geometries. This equation has little utility for diffraction in Laue geometry. The focusing effect in the Laue symmetrical case is discussed using concepts of dynamical theory and an extension of the lens equation is proposed. The existence of polychromatic focusing is considered and the feasibility of matching the polychromatic and monochromatic focal positions is discussed.

to correct errors found in (Chukhovskii & Krisch, 1992) for the Laue geometry. A new formula valid in Bragg and Laue geometry is obtained, using the same geometrical approach as in (Chukhovskii & Krisch, 1992).
The CLE has wide applicability in Bragg geometry. However, its use for Laue geometry is limited to very thin crystals, because it ignores a basic dynamical focusing effect also found in flat crystals, as described in section 3. The applicability of the lens equation in symmetrical Bragg geometry is discussed in appendix D. The CLE concerns the focusing of monochromatic radiation, and is in general different from the condition of polychromatic focusing. The particular cases where these two different focusing conditions coincide are discussed in section 4. A final summary is given in section 5.

The crystal lens equation revisited
The lens equation will be derived in Bragg or Laue geometry, with source S and focus F in real or virtual positions (see Fig. 1). Consider a monochromatic x-ray or neutron beam from a real or virtual point-source S. The origin of coordinates O is chosen at the point of the crystal surface such that the ray SO, of wavevector k 0 , is in geometrical Bragg incidence. It gives rise outside the crystal to a diffracted ray of wavevector k h = k 0 + h, where h is the reciprocal lattice vector in O, and | k h | = | k 0 | (see Fig. 2). This is valid in both transmission geometry (Laue) or reflection geometry (Bragg) for both plane and curved crystals 1 .
1. Schematic representation of the different diffraction setups with real or virtual source in Bragg or Laue cases: a) real source, real focus (red) in Laue case or virtual focus (blue) in Bragg case, b) real source, virtual focus (red) in Laue case or real focus (blue) in Bragg case, c) virtual source, real focus (red) in Laue case or virtual focus (blue) in Bragg case, d) virtual source, virtual focus (red) in Laue case or real focus (blue) in Bragg case. L 0 = SO is the distance source to crystal and L h = OF is the distance crystal to focus.
The inward normal to the crystal surface in O is n, and ϕ 0 = ( n, k 0 ) is the oriented angle from the vector n to the vector k 0 . Similarly, ϕ h = ( n, k h ). Without loss of generality ϕ 0 is positive; θ B is the Bragg angle (always positive). In the case of symmetric geometry(asymmetry angle α = 0) we find ϕ 0,h = ±θ B in Laue or ϕ 0,h = (π/2) ∓ θ B in Bragg. Otherwise, the asymmetry angle α is defined as the angle of rotation of the vector h from its direction in the symmetrical case. In Laue case ϕ 0,h = α ± θ B ; in Bragg case ϕ 0,h = α ∓ θ B + π/2, therefore 2θ B = |ϕ 0 − ϕ h | in both cases, 2α = ϕ 0 + ϕ h in Laue case and 2α = ϕ 0 + ϕ h − π in Bragg case.
When moving the point of incidence O to P over an arbitrary small distance s along the curved crystal surface (see Fig. 2), h and n are changed into h and n , respectively.
The incident wavevector k 0 has the direction of SP . It is diffracted into k h . The projections of the vectors k h and k 0 + h on the crystal surface are equal (conservation of the parallel components of wave-vectors). ϕ 0,h are changed into ϕ 0,h = ϕ 0,h +∆ϕ 0,h .
Furthermore, in the present case of cylindrical bending of very thin crystal, the surface projection of h is constant (the angle between h and n is constant). This implies that Schematic view of the relevant parameters in focusing by a bent crystal in Bragg geometry.
The source distance L 0 = SO is set as positive if the source is on the incidence side of the crystal (real source) or negative if the source is on the other side (virtual source) (see Fig. 1). The radius of curvature R c is set as positive if the beam is incident on the concave side of the bent crystal. The focus distance L h is set as positive if the (real or virtual) focus F is situated on the incidence side on the crystal. With these conventions, ( n, n ) = s/R c , 0 L 0 = s cos ϕ 0 , h L h = s| cos ϕ h |, where 0,h are the angles between k 0,h and k 0,h . Using the relationship we obtain and The crystal lens equation valid in both Bragg and Laue cases, is finally obtained by inserting these expressions in equation (1) | In the Laue symmetrical case (cos ϕ h = cos ϕ 0 ) it predicts L h = L 0 (for a real source, the focus is virtual at the same distance as the source) and, in the particular case of L 0 = +∞, a plane incident wave is diffracted into a plane wave.
The crystal lens equation (5) obtained here is different from the equation given in (Chukhovskii & Krisch, 1992) 2 . Both equations are equivalent for the Bragg case (cos ϕ h < 0), which is also considered by Snigirev & Kohn (1995). They are not equivalent in the Laue case.
Note that we used in this section the same notation as (Chukhovskii & Krisch, 1992), where R c is positive for a concave surface, used to focus in Bragg case. For the rest of the paper, we also use the notation: p ← L 0 , q ← −L h R ← −R c , θ 1 ← ϕ 0 and θ 2 ← ϕ h , which is more convenient for Laue crystals, because real focusing is obtained when the beam coming from a real source is incident on the convex side of the bent crystal (with positive R).
Equation (5) is obtained here using a geometrical ray optics approach. It can also be deduced from a wave-optics approach as shown in Appendix A.

Dynamical focusing in Laue geometry
The applicability of the CLE for the Laue case is limited to very thin crystals. The dynamical theory (see book (Authier, 2003)) predicts "new" focal conditions, even for flat Laue crystals. This is analyzed here in the framework of the Takagi-Taupin equations, hereafter TTE (Takagi, 1962;Takagi, 1969;Taupin, 1964;Taupin, 1967).
Section 3.1 deals with the derivation of the "influence functions" (Green functions) which represent the wavefield generated in the crystal by a point-source on the crystal entrance surface.
In section 3.2, the approach to dynamical focusing in the symmetric Laue case (Kushnir & Suvorov, 1982;Guigay et al., 2013) is extended to asymmetric geometry.
The effects of anomalous absorption (Borrmann effect) are obtained in parallel. The new concept of "numerically determined focal length" of a flat crystal, denoted as q dyn , is introduced.
In section 3.3, a lens equation for a bent Laue symmetrical crystal of finite thckness, expressed in terms of q dyn is established. Its predictions are shown to be in agreement with numerical calculations.
In section 3.4, we make the verification that the formulation for the Laue asymmetric case by (Guigay & Ferrero, 2016) is in agreement with the CLE (equation (5)) in the limit of vanishing crystal thickness.

Influence function derived from Takagi-Taupin equations
The x-ray wavefield inside the crystal is expressed as the sum of two modulated plane waves with slowly varying amplitudes D 0,h ( x). The spatial position x is expressed in oblique coordinates (s 0 , s h ) along the directions of the k 0 and k h = k 0 + h vectors, which are the in-vacuum wave-vectors of modulus k = 2π/λ, where λ is x-ray wavelength.
h is the Bragg diffraction vector of the undeformed crystal. In such conditions, the differential TTE are where χ 0 , χ h , and χh are the Fourier coefficients of order 0, h and − h of the undeformed crystal polarisability. The polarization factor c (c = 1 for σ-polarization and c = cos 2θ B for π-polarization) is omitted from now on. u( x) is the displacement field of the deformed crystal. In the case of cylindrical bending we have where A and the φ 1,2 functions are defined in Appendix C. This a "constant strain gradient" case (Authier, 2003) meaning that ∂ 2 ( h. u)/(∂s 0 ∂s h ) is constant. In terms of the functions G 0,h (s 0 , s h ) defined by the TTE have a simpler form An incident monochromatic wave of any form can be expressed as a modulated where the first exponential term stands for the effects of refraction and normal absorption, s 0,h = s 0,h − σ 0,h ; Ω = k 2 χ h χh/4 and the M -function is the Kummer function (a confluent hypergeometric function) defined by the convergent infinite series This type of TTE solution was already obtained by different methods (Petrashen', 1974;Katagawa & Kato, 1974;Litzman & Janáček, 1974;Chukhovskii & Petrashen', 1977). (11) is the phase shift acquired by scattering at the point of coordinates (s 0 , σ h ) along the incident ray. We can say that the kinematical (single-scattering) approximation of equation (11) is and the full multiple scattering is D h = D h,kin M .

Dynamical focusing and Borrmann effect in a flat, asymmetric, Laue crystal
Dynamical focusing by flat Laue crystals (without bending) was predicted by Afanas'ev & Kohn (1977) and verified experimentally by (Aristov et al., 1978;Aristov et al., 1980a;Aristov et al., 1980b) in the case of symmetrical geometry. The theory was extended to the asymmetric case by Kohn et al. (2000). The application of dynamical focusing to high-resolution spectrometry was proposed by Kohn et al. (2013).
The amplitude D h (ξ) is zero outside the interval −a < ξ < a, and is proportional to (Kato, 1961), In the case |Za| 1 the asymptotic approximation can be used in the central region |ξ| a where a 2 − ξ 2 ≈ a − ξ 2 2a . We thus obtain in this central region the approximation where the two exponential terms are related to the two sheets of the dispersion surface.
The function exp(−iZξ 2 /(2a)) represents a converging wave if Re(Z) > 0 (divergent if (Re Z < 0). A double, real and virtual, focusing effect is thus expected at opposite distances ±q 0 from the crystal, with This equation is present in (Kohn et al., 2000;Kohn et al., 2013)  is approximated by |χ hr | or |χ h |.
The moduli of the two terms in equation (16) are proportional to exp(∓aIm (Z)), respectively. This is the expression of anomalous absorption (Borrmann effect). Two focal positions will be observed for small absorption, but only one for strong absorption, as shown in Fig. 4.
The reflected amplitude at any distance q from the crystal can be calculated numerically, without the approximations used above, by the Fresnel diffraction integral The "axial intensity profile" |D h (0, q)| 2 shows in general two strong maxima at distances q 1,2 = ±q dyn < q 0 (Fig. 4). This difference is a cylindrical aberration effect related to the approximations used to obtain equation (17). The parameter q dyn , which depends on the crystal thickness, is the "dynamical focal length" obtained numerically, thus non-approximated (contrary to q 0 ). As an example, some numerical values are given in Table 1.
The focusing condition for a source at a finite distance p from the crystal can be obtained by considering that propagation in free-space and propagation in the flat crystal are space-invariant, therefore expressed as convolutions in direct space or simple multiplications in reciprocal space. Therefore, they can be commuted. This allows to merge the free-space propagation before and after the crystal. The focusing condition is therefore On the contrary, propagation through a bent crystal is not space-invariant because the IF is not only dependent on the variables (s 0 , s h ), but also on the variables (σ 0 , σ h ) because of the factor exp[−i h. u(s 0 , σ h )] in equation (11).

A new lens equation for a bent crystal of finite thickness in symmetrical Laue geometry
In symmetrical Laue geometry, the factor exp(iχ 0 (s 0 + s h )) in equation (11) is constant on the crystal exit surface and will be omitted. Equation (11) is (see Appendix B) Let us consider the incident amplitude D inc (τ ) = exp(ikτ 2 /(2p)), where τ is a coordinate along the axis Oτ normal to k 0 (see Fig. 3). On the exit surface, using s 0 = (ξ + a)/ sin 2θ B and σ h = −τ / sin 2θ B , and the notation R = R cos θ B we obtain from equations in appendix C, in the case α = 0 Using the integration variable η = ξ − τ , the amplitude along the ξ-axis is, with The wave amplitude at a distance q downstream from the crystal is obtained using a Fresnel diffraction integral similar to equation (18). We thus have a double integral over η and ξ . The ξ integration is performed analytically (Guigay et al., 2013) and it turns out that where L = p + q, p −1 e = p −1 + R −1 , q −1 e = q −1 − R −1 and L e = p e + q e . The focal positions are given by L e = ±q dyn . This can be written as Translating equation (24) in the notation of section 2 If q dyn is set to zero, we obtain L h = L 0 , the same result as the lens equation (5).
Equation (25) can be considered as a "modified lens equation" which takes dynamical diffraction effects into account in symmetric Laue geometry. We do not know an equation like equation (25) for the general case of asymmetrical Laue diffraction.
Examples of numerical calculations using equation (23)  Alternatively, provided that the parameter q dyn has been previously determined numerically by a plot similar to Fig. 4, the focal positions can be given directly by equation (25). The results are in very good agreement with the focal positions obtained obtained numerically in Fig. 5. An important advantage in using the new CLE is that the same value of q dyn can be used for any value of the radius of curvature and for any value of source distance.
We are often interested in real focusing (q > 0) of an incident beam from a very distant real source, for instance in dispersive EXAFS beamlines. Suppose 0 < R ≤ q dyn . When p increases from zero to infinity, q 1 decreases from q 1 = R q dyn /(q dyn + R ) to q 1 = R q dyn (q dyn −R ). Simultaneously, q 2 decreases from q 2 = R q dyn /(q dyn −R ) to q 2 = R q dyn (q dyn +R ). For very large p-values, we have the simple relation q 1 +q 2 ≈ 2R in good agreement with the numerical results in Fig. 5. It can be seen from equation (23) that the intensity function |D h (s 0 , s h )| 2 as a function of ξ is symmetric around ξ c = −aqL e /(2q e R ). This denotes a lateral shift of the intensity profile from its position for the unbent crystal (the axial intensity profiles of Fig.5a and 5b are actually plotted as a function of (ξ − ξ c ).

Semianalytical approach in asymmetric Laue geometry and its CLE limit
The generalization of equation (22) to asymmetric Laue geometry is (Guigay & Ferrero, 2016) Here, φ(ξ, η) is calculated from the term exp(−i h. u(s 0 , σ h )) in equation (11) with with parameters µ 1,2 , a 1,2 and g given in Appendix C. The reflected amplitude D h (ξ, q) at distance q downstream from the crystal is again obtained as in equation (18), therefore by double integration over η and ξ . The ξ -integration can be again performed analytically. The remaining η-integration involving the Kummer function is carried out numerically (Guigay & Ferrero, 2016). We consider this approach as semi-analytical, in contrast to the approach based on a numerical solution of the TTE (Nesterets & Wilkins, 2008).
It is interesting to study analytically the limit of this semi-analytical formulation in the case of vanishing crystal thickness (a → 0) because the comparison with lens equation (5) represents a validity test of the semi-analytical formulation. In the limit (a → 0), the Kummer function is equal to unity in equation (26), and the integral can be replaced by 2a times the integrand evaluated at η = a = 0, therefore This is the expression of the amplitude of a cylindrical wave focused at the distance q such that Using the identity which is derived in Appendix C, the focusing condition is or, which is the CLE (equation (5)) for the Laue case, with the correspondence p → L 0 , q → −L h , R → −R c , θ 1 → ϕ 0 and θ 2 → ϕ h
Using equations (3) and (4) we obtain Equation (33) is usually referred to as the "geometric focusing" condition for bent crystals. It is also applied in the case of flat crystals (Sanchez del Rio et al., 1994).
Like in equation (5), the crystal thickness does not appear in equation (33). The combination of equations (5) and (33) gives which is verified either in the symmetric Bragg case (cos ϕ h + cos ϕ 0 = 0), or if cos ϕ 0 /L 0 = | cos ϕ h |/L h = 1/R, which is the Rowland condition. The Rowland condition is therefore necessary for the coincidence of equations (5) and (33)  On synchrotron dispersive EXAFS beamlines, the use of a Bragg symmetric reflection by a bent polychromator at a large distance from the source guarantees the focusing of a broad bandwidth (up to 1 keV) on a small spot (Tolentino et al., 1988) at a distance close to L h = (R c sin θ B )/2.
Laue polychromators are also used in synchrotron beamlines. In symmetric Laue geometry, condition (5) should be replaced by equation (25), which is L h ≈ R c cos θ B + (R c cos θ B ) 2 /q dyn if the source distance is very large. Coincidence with (33) is then obtained if R c = −q dyn /(2 cos θ B ), which means real focusing at the distance |L h | = q dyn /4 with beam incidence in the crystal convex side (R c < 0). If |L h | is fixed, the required conditions are |R c | = 2|L h |/ cos θ B and q dyn = 4|L h |. The last condition should be fulfilled by choosing the crystal thickness, as in (Mocella et al., 2004;Mocella et al., 2008).
Another polychromatic condition for Laue geometry has been introduced more recently (Martinson et al., 2015;Qi et al., 2019;Qi et al., 2021). The energy components of a polychromatic ray traversing a bent Laue crystal with finite thickness meet the Bragg condition at different positions along the ray path. They are diffracted with different Bragg angles, therefore they exit in different directions, giving raise to a polychromatic focus from a single ray. The "magic condition", under which single ray focusing and geometric focusing (equation (33)) would coincide, is achieved by the adequate choice of the asymmetry. The magic condition is independent of the crystal thickness (Qi et al., 2021). We observe that the magic condition (equation (19) in (Qi et al., 2021)) and the modified lens equation (25) are both satisfied in the particular case of symmetric Laue geometry in Rowland configuration.

Conclusions and future perspectives
The crystal lens equation (CLE, equation (5)) based on the conservation of the parallel component of the wavevector in the diffraction process has been revisited. It includes all cases of symmetric and asymmetric Laue and Bragg geometries. It differs from the previous formulation (Chukhovskii & Krisch, 1992) in the Laue case. However, in Laue geometry, the lens equation can be only applied if the crystal is so thin that important effects resulting from the dynamical theory of diffraction, like the focusing of the Borrmann triangle, can be neglected. We derived the modified lens equation (25)  Under a deformation field u( r), the crystal polarizability is taken as χ( r − u( r)), where χ( r) is the polarizability of the non-deformed crystal. The Fourier components of the electric susceptibility χ h,h are multiplied by the phase factors exp(− h. u( r)) and exp(∓i h. u( r)), respectively, in the TTE. In the case of a very thin crystal, the ray reflected at position x on the bent crystal surface is simply affected by the phase factor which is obtained using u(x) = −(x 2 /(2R c )) n and n. h = n.( k h − k 0 ) = k(cos ϕ h − cos ϕ 0 ).
In the case of the undeformed crystal, the incident amplitude exp[ikτ 2 /(2L 0 )], along the axis Oτ is translated into along the axis O ξ (see Fig. 3). This is combined with equation (35) corresponding to a real or virtual focus if the phase of this function is negative or positive respectively.
Using the convention defined in section 2 (see Fig. 1 Comparing equations (37) and (38), we finally obtain which is equivalent to the lens equation (5).
Appendix B Derivation of the influence functions equation (11) Considering G 0,h (s 0 , s h ) as functions of s 0 and s h , the TTE (10) are We define the functions F 0,h (s 0 , s h ) such that The equations (41) are rewritten as The refracted amplitude is F ref = Eδ(s h ). Equations (43) can be written in the form of integral equations: We can combine equations (44) into a single integral equation for F h only where Ω = k 2 χ h χh/4 is used. By iteration starting from According to the definition of the Kummer function (equation (12)), the series in brackets is equal to M ( Ω iA + 1, 1, −iAs 0 s h ). Using the known relation M (a, b, z) = e z M (b − a, b, −z), together with equations (40) and (42) we obtain Using equation (9b) and which, including equation (8), is equation (11).
In the symmetric Laue case, for which A = 0, we obtain from equation (46) consequently, equation (11) becomes In the opposite case when A >> Ω, an approximated solution could be obtained by considering the simplified integral equation (13) of section 3 is obtained.
Appendix C Expression of the phase factor in Laue geomety and derivation of equation (30) .

Appendix D Relevance of the lens equation in the symmetric Bragg case
Let us consider the case of a flat non-absorbing crystal plate (without bending), in symmetrical Bragg geometry. The fact that experimental results and also numerical calculations (Honkanen et al., 2018) of Bragg diffraction with plane crystals do not show any focusing effect (contrary to Laue case), can be loosely explained by the following intuitive approach. Consider that any geometrical ray emitted from a real distant point-source produces a reflected ray at the point of incidence on the crystal surface, with a reflectivity coefficient r equal to the complex reflectivity of the incident plane wave having the same glancing angle of incidence θ = θ B +∆θ as the geometrical ray under consideration Note that |r(∆θ)| 2 is the usual diffraction profile. Taking the origin of coordinates at the point corresponding to θ = θ B , the reflected wave-amplitude along an axis Oξ situated in the diffraction plane and perpendicular to the reflected direction, at negligible distance from the crystal, may be approximated by setting ∆θ = ξ/p in equation (58): No focusing effect is expected from this amplitude distribution, because the phase function arcsin(ξ sin 2θ/(p|χ h |)) is an odd function of ξ, thus it does not have a secondorder term characteristic of a focusing effect. The first-order term produces a lateral shift of the image. There is no equivalent to the dynamical focusing length q dyn introduced in the Laue case. The reflected beam is indeed divergent, as in the case of a usual mirror.
The focusing properties of cylindrically bent crystals in symmetric Bragg geometry were simulated by (Sutter et al., 2010), using a finite-difference method, and by (Honkanen et al., 2017;Honkanen et al., 2018), using a finite-element method for numerical solution of the TTE. The obtained phase distribution of the reflected wavefront shows a parabolic shape, with concavity inversion as compared to the parabolic phase distribution of the incident wavefront. This is a clear indication of a single real focusing effect, which is indeed confirmed by simulating the reflected wave propagation. The obtained focusing distances are indeed in good agreement with the CLE which is L −1 0 + L −1 h = 2/(R c sin θ B ) in this case.

Synopsis
A crystal lens equation is deduced to address the location of the focus when monochromatic x-ray radiation encounters a bent crystal. It is extended using dynamical theory of diffraction for Laue symmetrical diffraction. Combination of polychromatic and monochromatic focusing is also discussed.