62 J. ˇ Saroun and J. Kulda is represented by a rectangular, circular, or elliptical area at the interface between moderator and neutron channel with an associated neutron flux dis- tribution. In RESTRAX, the neutron flux is described either analytically as a Maxwellian distribution, or more accurately by a lookup table. In the latter case, a one-dimensional table with wavelength distribution is combined with two-dimensional tables describing correlations between angular and spatial coordinates. Such a table can be easily created by postprocessing of modera- tor simulation data, which results in a much more realistic model compared to the analytical description and allows for simulations of neutron fluxes on absolute scale. 4.3.2 Diffractive Optics Simulation of neutron transport through crystals in RESTRAX is based on a random-walk algorithm, which solves intensity-transfer Darwin equations [8] numerically, in principle for any shape of the crystal block. Details of the algo- rithm are described in [14]. It is based on the assumption of dominant effect of the mosaic structure on the rocking curve width, where mosaic blocks are treated as perfect crystal domains. However, the random walk is not followed through individual mosaic blocks, which would be an extremely slow process in some cases. Instead, the crystal is characterized by the scattering cross sec- tion per unit volume, σ(ε), which depends on the misorientation angle, ε of a mosaic block as σ(ε)=Qη −1 g(ε/η), (4.1) where η is the width of the misorientation probability distribution, g(x)andQ stands for the kinematical reflectivity. The diffraction vector depends on the misorientation angle and, in the case of gradient crystals, also on the position in the crystal, which can be expressed as G(r)=G 0 + ∇G · r + G(ε + γ), (4.2) where the second term describes a uniform deformation gradient and the third one the angular misorientation of a mosaic block parallel (ε) and perpendicular (γ) to the scattering plane defined by G 0 and incident beam directions. For a neutron with given phase-space coordinates, r, k, we can write the Bragg condition in vector form as [k + G 0 + ∇G · (r + kτ)+G(ε + γ)] 2 − k 2 =0, (4.3) where kτ is the neutron flight-path from a starting point at r. For the random- walk simulation, we need to find an appropriate generator of the random time-of-flight, τ . By neglecting second-order terms in (4.3), we obtain a linear relation between ε and the time-of-flight parameter, τ , ε = ε 0 + βkτ , (4.4) 4 Raytrace of Neutron Optical Systems with RESTRAX 63 where ε 0 = (k + G 0 + ∇G · r + Gγ) 2 − k 2 2Gk cos θ B and β = (k + G 0 ) ·∇G · k Gk 2 cos θ B . (4.5) Substitution for ε in (4.1) then leads to a position-dependent scattering cross-section, which, by integration along the flight path, yields the proba- bility, P (τ) that a neutron will be reflected somewhere on its flight path kτ. With the symbol Φ(ε/η) denoting the cumulative probability function corre- sponding to the mosaic distribution g(ε/η), we can express this probability as [15] P (τ)=1− exp  − Q β  Φ  ε 0 + βkτ η  − Φ  ε 0 η  . (4.6) Provided that we know the inverse function to Φ(x), we can generate τ by transformation from uniformly distributed random numbers, ξ.Letτ o be the time-of-flight to the crystal exit. Then the next node (scattering point) of the random walk would be τ = η kβ Φ −1  Φ  ε 0 η  − β Q ln (1 − ξP(τ 0 ))  − ε 0 kβ , (4.7) while the neutron history has to be weighted by the probability P (τ 0 ). In subsequent steps, the random walk continues in the directions k + G(τ)and k until the neutron escapes from the crystal (or an array of crystals) or the weight of the history decreases below a threshold value. Absorption is taken into account by multiplying the event weight by the appropriate transmis- sion coefficient calculated for a given neutron wavelength and material [16]. In Fig. 4.3, such a random walk is illustrated by showing points of second −10 0 10 0 1 2 3 −15 −10 −50 5 −4 −2 0 2 4 x [mm] y [mm] k+G k grad(Δd) [m −1 ] 0.0 0.1 0.2 0.3 Intensity / rel. units x [mm] Fig. 4.3. A map of simulated points of second and further reflections inside a Ge crystal, reflection 511, mosaicity η =6  , and deformation corresponding to a temperature gradient along y-axis, |∇G|/G =0.1m −1 . On the right hand, simulated spatial profiles of reflected neutron beam are plotted for different magnitudes of the deformation gradient 64 J. ˇ Saroun and J. Kulda and further reflections in a deformed mosaic Ge crystal and the resulting topography of the reflected beam. There are two important aspects of this procedure. First is the efficiency, because for usual mosaic crystals, only few steps are made in each history resulting in a very fast procedure. Second, both mosaic and bent perfect crys- tals can be simulated by the same algorithm. Indeed, in the limit η → 0, we obtain τ = −ε 0 (kβ) −1 and the neutron transport is deterministic, as expected for elastically bent crystals in the quasiclassical approximation [17]. In addi- tion, the weight factor in this case, P (∞)=1− exp(−Q|β| −1 ), is identical to the quantum-mechanical solution for the peak reflectivity of bent perfect crystals [18]. On the other hand, this model fails in the limit of perfect crys- tals (very small mosaicity and deformation), which would require another approach using dynamical diffraction theory. The crystal component is flexible enough for modeling most of the contem- porary neutron monochromators and analyzers as far as they can be described as a regular array of crystal segments with a linear positional dependence of tilt angles. More sophisticated multianalyzers (e.g., the RITA spectrome- ter [19]) featuring independent movements of individual segments can only be simulated in a step-by-step manner with the final result being obtained by a superposition of the partials. 4.3.3 Reflective Optics Raytrace of neutrons through various types of reflecting optics elements is a straightforward task, provided that we can treat the problem in the framework of the geometrical optics approximation and that we know the reflectivity function of the reflecting surfaces. With neutrons, the geometrical approx- imation is fully adequate for the simulation of transport through elements such as neutron guides or benders and the reflectivity of real Ni and super- mirror coatings can be determined experimentally. Mirror reflectivity can be thus stored in lookup tables and the problem is reduced to a geometrical description of the device, apart from computing issues related to numerical precision and convergence problems. Using this approach, RESTRAX can sim- ulate various neutron optics elements, such as curved neutron guides, benders, elliptic or parabolic multichannel guides and most recently also supermirror transmission polarizers. As an example, we present the simulation of multichannel supermirror guides aimed to focus neutrons onto small samples after passing through a doubly focusing monochromator [20]. Although RESTRAX can simulate two-dimensional grids of reflecting lamellae, for practical reasons we have con- sidered a multichannel device as a sequence of one-dimensional horizontally and vertically focusing sections (Fig. 4.4). Equidistant 0.5 mm thick blades were assumed to be curved either elliptically or parabolically, having reflect- ing surfaces on the concave sides with the reflectivity of an m = 3 supermirror. 4 Raytrace of Neutron Optical Systems with RESTRAX 65 0.5 m 0.3 m 0.5 m 80 mm 125 mm Fig. 4.4. The multichannel supermirror device with the dimensions indicated For elliptic guides, the number of blades was 20 and 30 for horizontal and ver- tical focusing, respectively. For the parabolic guide, the respective numbers were 14 and 22. Gaps between the blades and focal distances were defined by entrance and exit widths (or heights) of the guides. We have assumed that the entrance dimensions are equal to the ellipse minor axis in the case of elliptic profile. The simulations involved the entire beam path including a cold source with a tabulated flux distribution, straight 58 Ni neutron guide with cross section 6 × 12 cm 2 and a doubly focusing PG002 monochromator with 7 ×9 segments at the nominal wavelength 0.405 nm. A lookup table with the measured reflectivity of a real m = 3 supermirror was used to achieve a realistic description of the guide properties. Except for the multichannel guide and the horizontally focusing monochromator, the instrument layout corresponded to the IN14 spectrometer at the Institut Laue-Langevin in Grenoble. It is quite difficult to optimize the parameters of such a device analyti- cally, because it is not obvious how the focusing by the monochromator and the multichannel guide would link to each other and also what the penalty in terms of neutron transmission through the guide and what the effect of the relaxed instrument resolution would be. Some of the relevant parameters (crystal curvatures, guide focal lengths, and spacing between the lamellae) were optimized using the raytrace code and Levenberg–Marquardt techniques implemented in RESTRAX [20]. The results for an optimized parabolically shaped multichannel guide are shown in Fig. 4.5. In contrast to an experiment, Monte Carlo simulation permits one to investigate the beam structure in dif- ferent phase-space projections quite readily. For example, a projection in the plane of divergence angle and wave-vector magnitude can clearly resolve the directly transmitted and reflected neutrons due to their different dispersion relation, resulting from prior reflection on the monochromator. This effect is entirely hidden in other projections, as illustrated in Fig. 4.5. 66 J. ˇ Saroun and J. Kulda −20 −10 0 10 20 −20 −10 0 10 20 x [mm] y [mm] −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 k x [nm] −1 k x [nm] −1 k y [nm] −1 k z [nm] −1 −1.0 −0.5 0.0 0.5 1.0 15.2 15.4 15.6 15.8 Fig. 4.5. Simulated beam profiles at the sample in different real and momentum space projections for the optimized parabolic guide. The right-hand image permits one to easily distinguish directly transmitted neutrons in the central part from the reflected ones, due to their inverted dispersion relation 4.4 Simulations of Entire Instruments Ultimately, the matter of concern is in simulations of the entire neutron scat- tering instrument, which provide data relevant for instrument design and data analysis, such as neutron flux, beam structure in phase-space or resolution functions. Examples of RESTRAX applications in instrument development can be found in the literature [21–26]. In the following section, we give a brief summary of the raytrace method used to simulate TAS resolution functions. 4.4.1 Resolution Functions The intensity of a neutron beam scattered by the sample with a probability W (k i , k f ) and registered by the detector in a TAS configuration with the nominal settings of initial and final wave-vectors, k i0 , k f0 ,isgivenby I(k i0 , k f0 )=  W (k i , k f )Φ I (r, k i )P F (r, k f )drdk i dk f . (4.8) The function Φ I (r, k i ) represents the flux distribution of incident neutrons at a point r inside the sample while P F (r, k f ) is the distribution of probability that the neutron with phase-space coordinates (r, k f ) is detected by the ana- lyzer part of the instrument. Evaluation of this integral by the MC method is advantageous for two reasons: the high dimensionality of the integral and the fact that the latter two distributions in the integrand can be sampled directly by the raytrace technique. For this purpose, we set the scattering probabil- ity of the sample W (k i , k f ) = 1. The instrument response function is then obtained as an ensemble of (k i,e , k f,e ) vectors and their weights, p e ,which describe all possible scattering events detected by the instrument. They have the distribution given by the integral R(k i , k f )=  Φ I (r, k i )P F (r, k f )dr. (4.9) 4 Raytrace of Neutron Optical Systems with RESTRAX 67 -0.10 -0.05 0.00 0.05 0.10 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 (ξ 0 0) (ξ 0 0) ΔE [meV] -0.10 -0.05 0.00 0.05 0.10 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 ΔE [meV] Fig. 4.6. Resolution functions of the whole TAS instrument without (left)andwith (right) the multichannel guide. The center of the resolution function corresponds to elastic scattering at Q =(0, 0, 10) nm −1 Convolution with a scattering function, S(Q,ω), is carried out in analogy to the integral in (4.8) as a sum of the scattering function values over all events, I(Q 0 ,ω 0 )=  e k f,e k i,e S(Q e ,ω e )p e , (4.10) where Q = k f − k i and ω = ¯h 2m  k i 2 − k f 2  . Since the events can bear memory of initial and final spin states, this method makes it possible to distinguish resolution functions for the four combinations of initial and final neutron spin states. In Fig. 4.6, we show the resolution functions simulated for the TAS IN14 at the ILL, Grenoble, equipped with the multichannel guide described in the previous section. Inflation of the resolution volume as a result of beam com- pression by the multichannel guide is proportional to the gain in neutron flux at the sample. However, the resolution in energy transfer is not affected because the guide can be tuned to the monochromator curvature so that monochromatic focusing condition is fulfilled. References 1. M.W. Johnson, C. Stephanou, MCLIB: a library of Monte Carlo subroutines for neutron scattering problems, RAL Technical Reports, RL-78-090 (1978) 2. P.A. Seeger, L.L. Daemen, Proc. SPIE 5536, 109 (2004) 3.W.T.Lee,X.L.Wang,J.L.Robertson,F.Klose,C.Rehm,Appl.Phys.A 74(Suppl.), s1502 (2002) 4. P. Willendrup, E. Farhi, K. Lefmann, Physica B 350, e735 (2004) 5. G. Zsigmond, K. Lieutenant, S. Manoshin, H.N. Bordallo, J.D.M. Champion, J.Peters,J.M.Carpenter,F.Mezei,Nucl.Instr.Meth.A529, 218 (2004) 6. J. ˇ Saroun, J. Kulda, Physica B 234–236, 1102 (1997) 68 J. ˇ Saroun and J. Kulda 7. V. Sears, Neutron Optics (Oxford University Press, New York, Oxford, 1989) p. 259 8. V. Sears, Acta Cyst. A 56, 35 (1997) 9. J. ˇ Saroun, J. Kulda, J. Neutron Res. 6, 125 (1997) 10. J. ˇ Saroun, J. Kulda, Proc. SPIE 5536, 124 (2004) 11. P.M. Bentley, C. Pappas, K. Habicht, E. Lelievre-Berna, Physica B 385–386, 1349 (2006) 12. J.F. Breismeister, MCNPF: A general Monte Carlo n-particle transport code, Report LA-12625-M (LANL, Los Alamos, NM, 1997) 13. J.C. Nimal, T. Vergnaud, in Advanced Monte Carlo for Radiation Physics, Par- ticle Transport Simulation and Applications, ed. by A. Kling, F. Bar˜ao, M. Nakagawa, L. T´avora, P. Vaz (Springer, Berlin Heidelberg New York, 2001), p. 651 14. J. ˇ Saroun, Nucl. Instrum. Methods A 529, 162 (2004) 15. H.C. Hu, J. Appl. Cryst. 26, 251 (1993) 16. A. Freund, Nucl. Instrum. Methods. 213, 495 (1983) 17. A.D. Stoica, M. Popovici, J. Appl. Cryst. 22, 448 (1989) 18. J. Kulda, Acta Cryst. A 40, 120 (1984) 19. K. Lefmann, D.F. McMorrow, H.M. Rønnov, K. Nielsen, K.N. Clausen, B. Lake, G. Aeppli, Physica B 283, 343 (2000) 20. J. ˇ Saroun, J. Kulda, Physica B 385–386, 1250 (2006) 21. A. Hiess, R. Currat, J. ˇ Saroun, F.J. Bermejo, Physica B 276–278, 91 (2000) 22. J. ˇ Saroun, J. Kulda, A. Wildes, A. Hiess, Physica B 276–278, 148 (2000) 23. R. Gilles, B. Krimmer, J. ˇ Saroun, H. Boysen, H. Fuess, Mater. Sci. Forum 378– 381, 282 (2001) 24. J. Kulda, P. Courtois, J. ˇ Saroun, M. Thomas, M. Enderle, P. Flores P, SPIE 4509, 13 (2001) 25. J. Kulda, J. ˇ Saroun, P. Courtois, M. Enderle, M. Thomas, P. Flores, Appl. Phys. A 74(Suppl.), s246 (2002) 26. J. ˇ Saroun, T. Pirling, R.B. Rogge, Appl. Phys. A 74(Suppl.), s1489 (2002) 5 Wavefront Propagation M. Bowler, J. Bahrdt, and O. Chubar Abstract. The modelling of photon optical systems for third generation syn- chrotrons and free electron lasers, where the radiation has a high degree of coherence, requires the complex electric field of the radiation to be computed accurately, taking into account the detailed properties of the source, and then propagated across the optical elements – so called wavefront propagation. This chapter gives overviews of two different numerical approaches, used in the wavefront propagation codes SRW and PHASE. Comparisons of the results from these codes for some simple test cases are presented, along with details of the numerical parameters used in the tests. 5.1 Introduction In recent years, there has been an upsurge in the provision of new powerful sources of transversely coherent radiation based on electron accelerators. Free electron lasers (FELs) are providing coherent radiation from THz wavelengths to the ultraviolet, and there are projects in place to build FELs providing X- rays with the XFEL at HASYLAB in Hamburg, the Linear Coherent Light Source LCLS at Stanford and the Spring8 Compact SASE Source SCSS in Japan. Coherent synchrotron radiation (CSR) at wavelengths similar to or longer than the electron bunch is also produced by accelerating electrons. For CSR, the intensity is proportional to the square of the number of electrons in the bunch, hence very intense THz radiation is produced at bending magnets when the bunch length is of the order of a hundred microns, such as is required for FEL operation. Finally, the radiation from undulators, which provide the main sources of radiation in the new storage ring synchrotron radiation (SR) sources from UV to hard X-rays, has a high degree of coherence. Traditionally, ray tracing, based on geometric optics, has been used to model the beamlines that transport the SR radiation from the source to the experiment. This has provided a sufficiently accurate model for most situa- tions, although at the longer wavelength end of the spectrum some allowances for increased divergence of radiation due to diffraction at slits must be made. 70 M. Bowler et al. For the coherent sources, interference effects are important as well as diffrac- tion, and one needs to know the phase of the radiation field as well as the amplitude. Hence wavefront propagation, which models the evolution of the electric field through the optical system, is required. The full solution of the Fresnel Kirchoff equation for propagating the field is possible, but it is computationally intensive and approximate solutions are sought. One approximation applicable to paraxial systems is to use the method of Fourier Optics. The code SRW (synchrotron radiation workshop) generates the source radiation field and also allows for its propagation across “thin” optics. This code is described in Sect. 5.2. Beamlines at UV and shorter wavelengths require highly grazing incidence optics, and in this case the thin optic assumption may not be appropriate. The Stationary Phase method is applicable in this regime and is used to approximate the propagation in the code PHASE, described in Sect. 5.3. To cross-check both approximations, a Gaussian beam has been propa- gated across toroidal mirrors of different grazing angles and demagnifications, using both codes, and the size of the focal spots compared. These results are presented in Sect. 5.4 along with a study of the ability of both codes to handle astigmatic focusing. SRW and PHASE have both been used to model the beamline for trans- porting THz radiation from the Energy Recovery Linac Prototype (ERLP) at Daresbury Laboratory. This is described in Sect. 5.5. Finally Sect. 5.6 summarizes the results and looks at future needs for wavefront propagation simulations. The contribution of the COST P7 action has been in making two of these codes, PHASE and SRW, more widely known to the optics community, in running the test cases and in providing documentation to aid the new user. Two of the authors of these codes have joined with the COST P7 participants to write this chapter. 5.2 Overview of SRW The SRW software project was started at the European Synchrotron Radia- tion Facility in 1997 [1]. The purpose of this project was to provide users with a collection of computational tools for various simulations involving the pro- cesses of emission and propagation of synchrotron radiation. The SRW code is composed of two main parts, SRWE and SRWP, enabling the following: • Computation of various types of synchrotron radiation emitted by an elec- tron beam in magnetic fields of arbitrary configuration, being considered in the near-field region (SRWE) • CPU-efficient simulation of wavefront propagation through optical ele- ments and drift spaces, using the principles of wave optics (SRWP). 5 Wavefront Propagation 71 Thanks to the accurate and general computation method implemented in SRWE, a large variety of types of spontaneous synchrotron emission by relativistic electrons can be simulated, e.g., radiation from central parts and edges of bending magnets, short magnets, chicanes, various planar and ellip- tical undulators and wigglers. Either computed or measured magnetic fields can be used in these simulations. Simple Gaussian beams can also be easily simulated. The extension of this part of the code to self-amplified spontaneous emission (SASE) and high-gain harmonic generation (HGHG) is currently in progress. An SRWE calculation typically provides an initial radiation wave- front, i.e., a distribution of the frequency-domain electric field of radiation in a transverse plane at a given finite distance from the source (e.g., at the position of the first optical element of a beamline), in a form appropriate for further manipulation. After the initial wavefront has been computed in SRWE, it can be used by SRWP, without leaving the same application front-end. SRWP applies mainly the methods of Fourier optics, with the propagation of a (fully-coherent) wave- front in free space being described by the Fresnel integral, and the “thin” approximation being used to simulate individual optical elements – apertures, obstacles (opaque, semi-transparent or phase-shifting), zone plates, refractive lenses. If necessary, the calculation of the initial electric field and its further propagation can be programed to be repeated many times (with necessary pre- and post-processing), using the scripting facility of the hosting front-end application. 5.2.1 Accurate Computation of the Frequency-Domain Electric Field of Spontaneous Emission by Relativistic Electrons The electric field emitted by a relativistic electron moving in free space is known to be described by the retarded scalar and vector potentials, which represent the exact solution of the Maxwell equations for this case [2]:  A = e  +∞ −∞  β R δ(τ −t + R/c)dτ, ϕ = e  +∞ −∞ 1 R δ(τ −t + R/c)dτ, (5.1) where e is the charge of electron, c is the speed of light,  β =  β(τ) is the electron relative velocity, R is the distance between the observation point r and the instantaneous electron position r e (τ),R= |  R(τ)|,  R(τ)=r −r e (τ),tis the time in laboratory frame, τ is the integration variable having the dimension of time, and δ(x) is the delta-function. The Gaussian system of units is used in (5.1) and subsequently. [...]... at and beyond Table 5.4 Widths (RMS values) of Gaussian fits of horizontal and vertical cuts through the middle of the 620 μm beam at different locations in the beamline Position in beamline Horizontal width (mm) Vertical width (mm) SRW At nominal “focus” of first mirror At position of collimating mirror 2 m beyond collimating mirror PHASE SRW PHASE 10.9 38 .1 30 .9 11.1 37 .0 30 .05 12.0 43. 9 36 .3 11 .3 36.9... coupling modes: (a) resonant beam coupling through the cover layer (b) Front coupling directly into the guiding layer (c) Front coupling with prereflection and generation of a standing wave pattern 6 Theoretical Analysis of X-Ray Waveguides 93 In Sect 6.2 we will consider RBC coupling, analyzing in some detail the efficiency of WGs as a function of their geometrical and physical parameters (guiding layer... option, with an oversampling of 4, giving an input mesh for the propagation of 32 × 32 points over a 2 mm square As a starting point for the propagation, automatic radiation sampling was used with the accuracy 5 Wavefront Propagation PHASE 83 SRW Fig 5 .3 Focal spot for the 87.5◦ incidence mirror, 10:1 demagnification Intensities are in arbitrary units parameter set to 4, and 100 × 100 points were used to define... refers to the guiding (i.e., the median) layer, and 1 and 3, respectively, the upper and lower cladding layers In the X-ray range the refractive index is usually written as n = 1 − δ − iβ, 94 S Lagomarsino et al where δ and β are small compared to unity We can easily obtain them from the internet [27,28], in which they are calculated in the appropriate way from the atomic scattering factors as tabulated... the wavefield coupled into the waveguide and the two internally reflected fields produces standing waves in the direction of the surface normal; as a consequence, the intensity can by far exceed the intensity of the incident wavefield [31 ] For the resonance condition to be met, the angle of grazing incidence onto the waveguide surface, Φ0 , and the grazing angle onto the internal interfaces, Φ2 , must... Dmin is essentially independent of the wavelength and depends only on the cover material chosen The latter will be a heavier material or metal In this latter group Dmin varies very little and is about Dmin = 10 nm This number and (6.6) are also derived by Bergemann et al [11] in the rigorous treatment of the minimum spot size obtainable in a tapered double plate X-ray waveguide [33 ] The number Dmin... generated using automatic radiation sampling with an oversampling factor of 4 This generated 40 × 40 points over the initial grid of 45 mm by 45 mm Propagation through the 37 mm diameter aperture and along the beamline to beyond the collimating mirror was done using the automatic field sampling option with the accuracy parameter set to 4 For input into PHASE, a fixed grid of 51 by 51 points was used in SRW... 45 mm area and the output converted into PHASE field format The aperture is included in PHASE by setting the radius of a pinhole in the input source plane, defined in the input parameter file (fg34.par) An angular mesh of 201 points over ±50 mrads was used for the propagation In PHASE, if the mirrors do not aperture the beam, the transformation across the whole beamline can be carried out in one step... with respect to the optical element coordinates pi and qj of the elements i and j They are zero if |p − q| > 1 Each coefficient is a fourth order power series of the initial coordinates and angles and using these expansions the square root of the inverse of the determinant can be expanded with respect to the same variables In principle m can be determined within an explicit derivation of all eigenvalues... coupling a detailed study of the internal field structure is reported for a range of structured incident wave-fields 6.1 Introduction X-ray waveguides (WG) were first introduced in 1974 by Spiller and Segm¨ller u [1] who demonstrated the propagation of the X-rays in a waveguide composed of a BN film sandwiched between two layers of Al2 O3 In 1992, exploiting previous studies made by Bedzyk [2] on the X-ray . field and also allows for its propagation across “thin” optics. This code is described in Sect. 5.2. Beamlines at UV and shorter wavelengths require highly grazing incidence optics, and in this. k f 2  . Since the events can bear memory of initial and final spin states, this method makes it possible to distinguish resolution functions for the four combinations of initial and final neutron spin. featuring independent movements of individual segments can only be simulated in a step-by-step manner with the final result being obtained by a superposition of the partials. 4 .3. 3 Reflective Optics Raytrace