Ray tracing - ESRF

Ray tracing 

❑ A powerful tool which allows “optics experiment” with the 

computer.This technique allows the accurate simulation of the optical 

components of the BL in the design phase. 

❑ There are many commercial programs to perform ray tracing in 

the visible region, suitable for camera objective, microscopes etc. 

❑ For x-ray applications, specially with synchrotron radiation 

the choice is SHADOW. It can raytrace any x-ray optical system and 

predict image shapes, intensities and resolving powers with great 

accuracy 

Manuel Sánchez del Río 

SHADOW proved to be an essential computer tool for: 

❑ Simulate beamline optics (Mirror and Crystal Optics) 

❑ Understand properties of Optical Elements (i.e. Laue crystals) 

❑ Check if operation performances are as expected 

Computer simulations for X-ray optics (5 of 20) 

Luso, May1998

Ray tracing in Optics can be easily done with computers. 

source 


image 

mirror 

Two aspects: 


❑ Rays emerging from the source 

[source generation] 

❑ Rays through Optical Elements 

[Geometrical and Physical models] 

Luso, May1998

What CAN be calculated with SHADOW? 

❑ Beam Cross sections (focal spot, etc). Effects of: 

❑ Energy resolution 


❑ source characteristics (dimensions, depth, emittances...) 

❑ vignetting (finite optical elements, slits, etc.) 

❑ aberrations 

❑ thermal deformations 

❑ waviness and roughness errors of mirrors 

❑ Flux & Power (number of photons at sample, absorbed/transmitted power, etc.) 

❑ Other parameters (i.e polarization transfer) 


PSI, Feb 1998

What CANNOT be calculated with SHADOW? 

(Ideas for future developments...) 

1) Physical models: 

❑ Sources: ➟X-ray tubes. ➟Elliptical undulators. ➟Tapered undulators. ➟General wigglers. 

➟Wiggler exact calculation. 

❑ Optics: ➟Extended energy range (E> 100 keV). ➟Compton absorption cross section. 

➟Coherence (holography, phase contrast, speckle patterns...). 

❑ Gratings: ➟Efficiency model. 

❑ Crystals: ➟Any crystal. ➟External Rocking Curve. ➟Transmitted beam (phase shifters, 

diamond monochromators). ➟Improved mosaic model (transition Perfect -> 

Mosaic). ➟Bent crystal reflectivity. ➟Beam penetration in the crystal. 

❑ New devices: ➟Bragg Fresnel Lenses. ➟CRL. ➟Non-sequential systems (i.e. Multiple 

mirror arrays). 

2) Program structure: 

➟Migration Fortran -> C ?. ➟Dynamic arrays (any number of rays). ➟Test set. ➟➟User 

program macros. ➟Global optimization. ➟Improve user-friendly interface 



PSI, Feb 1998

1- Simplest case: The spherical source. 

Source Models in SHADOW 

❑ 1st Solution. Grid Generation of Rays: 

x = (0,0,0) 

v = (0, cos 2π(i/N), sin 2π(i/N) ) 

i=0,...N 

❑ 2nd Solution. Random Generation of Rays: 

x = (0,0,0) 

v = (0, cos (2πnr), sin (2πnr) ) 

n r = random number in [0,1] with flat probability 

distribution function given by the computer random 

generator. 

Manuel Sánchez del Río -9- 

Experiments Division 

Programming Group 

Grenoble, January 18, 1994 

EUROPEAN SYNCHROTRON RADIATION FACILITY

2- General case: The Gaussian [or any kind of distribution f ] generator. 

❑ 1st Solution. Deterministic generator: 

❑ 2nd Solution. Stochastic generator: 

x = (0,0,0) 

v = (0, cos θi , sin θi) I = f (θi ) 

θi = θmin + (i/N) (θmax - θmin) i = 0, ..., N 

x = (0,0,0) 

v = (0, cos θ i , sin θ i) 

θ i are numbers generated randomly but following 

the gaussian [or f] distribution. 

How to get them? 






The inversion method 

f(φ) 

cdf(φ) 

φ 

φ 

1 

0 

❑ Given the probability distribution, i.e. 

f ( φ) 

= 

1 

------------- exp i 

κ 2π 

φ2 

2κ2 ⎛– -------- ⎞ 

⎝ ⎠ 

❑ Calculate the Cumulative Distribution Function (cdf) 

cdf ( φ) 

= f ( φ)dφ 





φ 

∫ 

0 

❑ Calculate the inverse function 

Φ( n) 

cd f 1 – = ( n) 

If now n samples an interval [0,1] with flat 

probability distribution, Φ(n) samples an interval 

(-∞, +∞) with probability distribution f. 

❑ If n ∈[n1 ,n2 ] then the φ interval is [φ1 ,φ2 ] with 

n1=cdf(φ1) and n2=cdf(φ2) EUROPEAN SYNCHROTRON RADIATION FACILITY

The synchrotron sources 

e - 

ϑ 

Ψ 


Dipoles Wigglers Undulators 

hϖ 

Source Size 0.45mm(H)x0.30mm(V) 

Source Divergences 

hϖ 

e - 

Bending Magnet and Wiggler Radiation Undulator Radiation 


hϖ 

Undulator Divergences 

Luso, May1998

Tracing an Optical Element (i.e. MIRRORS) 

1) Geometrical Model 

source = { ray i , i=1...N } 

trajectory: x = x start + t v 

Surface equation 


f(x,y,z) = 0 

=> Intercept point 

source 

ray = ( x, ...) v, 


n = grad(f) 

Optical Element 

Ray tracing equation. Here (Mirrors) the specular reflectivity is applied: 

θin = θout ; vin , n and vout coplanar. In vectorial notation: vout = vin - 2 vin 2) Physical Model 

Reflectivity ~ probability of observation or intensity after reflection. 

We must include a new variable in the ray: Iout = Iin R(E,θ,...) 

In fact, R is polarization-dependent, so it is better to store the 

I = | Es exp(iφs ) + Ep exp(iφp ) | 2 

Electric field: 

ray i = ( x, v, i, flag, E s , E p , φ s , φ p , φ g , 2π/λ) 

v in 

v out 

image 

Luso, May1998

Focus is not perfect because of : 

❑ aberrations 

❑ source dimensions and depth [emittances] 

❑ waviness and roughness errors of mirrors 

All these effects can be studied by ray-tracing. Example: 

circular source (diameter: 10 μm) 

divergences: 100 μrad 

p=30m 

θ=89.8 deg 

q=10m 

Toroidal Mirror 

Ellipsoidal Mirror 




Ferrara, March 11 1994 


The model is not sufficient: The ray may not exist after the mirror [it is absorbed] 

We must extend the model 

E s out = E s in * R s 

E p out = E p in * R p 

θ inc=89.9deg 

Reflectivity ~ probability of observation 

or intensity after reflection. 

We must include a new variable in the ray: I 

I out = I in R(E,θ,...) 

In fact, R is polarization-dependent, so it is 

better to store the Electric field: 

I = | E s exp(iφ s ) + E p exp(iφ p ) | 2 

ray i = ( x, v, i, flag, E s , E p , φ s , φ p , φ g , 2π/λ) 

This permits also to work with polarized light, analyze the polarization degree 

and polarization properties through the beamline. 

|Es|=|Ep| φp = φs + π/2 

circular pol. light 

Rs, Rp and phase change are given by a physical model of reflectivity. For mirrors 

the Fresnel equations are coded in the program. 






Other O.E.s can be easily modeled: 

❑ Multilayers:Mirror + “New” Reflectivity curve. 

Geometry: Reflectivity: 

θ graz = 0.42 deg 

n pairs (100) 

Substrate (Si) 

❑ CRYSTALS: Symmetric Bragg Case =Mirror + “New” Reflectivity curve given by the DTD. 


v in 

θ 

v out 

v out = v in - 2 v in⊥ 

Light element Si (41.4 A) 

Heavy element W(10.3 A) 






Other O.E.s can be easily modeled: 

❑ LENSES: interface between two mediaof refractive indeces n 1 and n 2 

n 1 

n 2 

k in k out 

❑ CRYSTALS: Symmetric Bragg Case = Mirror + “New” Reflectivity curve given by the DTD. 

θ 

u n 

n 1(k in x u n) = n 2(k out x u n) 


v in 

v out 

v out = v in - 2 v in 

[Hetch, Optics] 






mλ = d (sin θ in + sin θ out ) 

or 

k out = k in + G || 

Gratings: Geometrical Model 

k in 

G || =m (2π/d) u || 


Computer simulations for X-ray optics (Ray-tracing and More) 

LSBarcelona, 13 Dec 1996 

θ in 

[k 1 ,k 3 ] k 1 

k 3 

k 2 

slit 

θ out k out 

m=2 

m=1 

m=0 

m=-1 

m=-2 


Gratings: Monochromators 

❑ They combine the dispersive effect of the grating + focusing effect of curvature 

❑ Can be analyzed by ray-tracing 

❑ Focusing equations: Generalized case of mirror’s focusing equations 

Efficiency [%] 

0.4 - 

0.0 - 

1 1 

-- + -- 

p q 

cosθ1 + cosθ2 

----------------------------------- 

Rs Manuel Sánchez del Río -12- 

Computer simulations for X-ray optics (Ray-tracing and More) 

LSBarcelona, 13 Dec 1996 

= 

( cosθ1 

) 2 

( cosθ2 

) 

-------------------p 

2 

+ --------------------q 

0 100 

Energy [eV] 

200 

= 

cosθ1 + cosθ2 

----------------------------------- 

Rt (Not considered in SHADOW) 


Mirror optics 

Surface errors and their implementation in SHADOW 

❑ Figure: (macroscopic ondulations of the surface, giving the surface shape (plane, spherical...), 

and deformations produced by the manufacturing errors, thermal loads and mechanical 

stress. 

In SHADOW we use a mapping of the surface (from FEM, etc.) or include directly the error 

parameters (surface radius,...) 

❑ Slope errors: generally considered as sinusoidal-like variations of the surface shape. It 

spreads the image in the focal plane due to the deflection of the incident rays at different 

angles. 

In RT tracing we use a reflection model with a mapping of the surface (d~mm>>λ, ie., geometrical 

optics regime) 

❑ Roughness: Random irregularities in the micro or sub-microscopic scale, which depends 

on the manufacture processes and materials. It increases the scattering of the photon beam, 

producing a blur of the image in the focal point. 

In RT, as it is impossible to characterize deterministically the surface, a statistical approach 

based on the diffraction theory is used. 



PSI, Feb 1998

Mirror optics: Thermal Analysis (figure error) 


Power density on first crystal 

Spot without thermal load 

Crystal deformation 

Spot with thermal load 


BL 2 SHADOW/ 

ANSYS calculation. 

Source: 

Wiggler 

λu = 125 mm 

N = 12 

E = 6.04 GeV 

σx = 69 μm 

σz = 47 μm 

εx = 610 -7 rad.cm 

εz = 610 -8 rad cm 

Optics: 

C,Be,Al attenuators 

Flat Mirror 

Double-xtal monochromator 

First crystal: cryogenically 

cooled 

θ B = 3.78 degrees 

(sagittally focusing 

M=1/3) 

Cylindrical mirror 

(M=1/4.3) 

Power on first crystal: 

2.6 kW 

PSI, Feb 1998

Mirror optics: waviness (slope error) analysis 

Focal spot simultions for BL1 

(Microfocus): 

Source 

Undulator placed in a low beta 

section 

λu = 46mm 

N = 33 

E = 6.04 GeV 

σx = 59 μm 

σz = 11 μm 

εx = 4 10 -7 rad.cm 

εz = 4 10-9 rad cm 

III harmonic @ 7144 eV 

Ellipsoidal mirror M=1/10 

Slope error 0.8” 


Slope 

error: 

Sinusoidal 

Toroidal Mirror Ellipsoidal Mirror 

Ellipsoidal Mirror + 0.8” slope error 


C. Riekel et al. 

Rev Sci Instrum 63 

974-981 (1992) 

Multi- frequency 

M. Sanchez del Rio et al. 

Nucl. Ins. Meth. A319 

170-177 (1992) 

PSI, Feb 1998

Mirror optics: 

Slope/Figure errors with real profiles I 

R. Signorato & M. Sanchez del Rio, SPIE Proceedings 3152 (1997) 

(ID12a) 


F1 

7.64 m 


.3 m 

M=0.2 

42.77 m 

❑ R1 = 4809 m (concave) 

❑ R2 = 154232 m (convex) 

❑ Expected focal position ~ 17m 

❑ Theoretical focal FWHM = 37 x 0.2 ~ 8 μm 

❑ Incuding Slope Error = 5μrad x 8m x 2 mirrors = 80 μm 

Results: 

❑ 22 μm FWHM 

❑ Satellite structures 

❑ Best focus fount at 7.64 m from M2 

source 

37 μm 

PSI, Feb 1998

Slope/Figure errors with real profiles II 

Mirror metrology 

❑ Slopes measured at the ESRF LTP 

❑ Standard data reduction (linear detrending, slope error rms, PSD) done with XOP. 

❑ Profiles used to create a surface mapping (transversal replication) to be included in 

SHADOW (by its PRESURFACE tool). 



PSI, Feb 1998

Slope/Figure errors with real profiles III 

Ray-tracing/Experiment comparison 



A = experimental 

B = calculated 

PSI, Feb 1998

Slope/Figure errors with real profiles IV 

Focal images at positions E (@7.62m from M2) and T (@17.14 m from M2) 



PSI, Feb 1998

Imaging properties of spherically bent crystals 

Backlighting scheme 

Mesh grid 

X-ray source 


Image of source 

Film (image of mesh) 

Ray tracing of curved crystal optics (16 of 30) 

Rowland circle geometry 

2R R=R~>100-180 mm 

θ B ~ 80-88 deg 

λ = 6.63 A 

Resolution better than 10 µm 

Ref: M. Sanchez del Rio, A. Ya Faenov, 

V. M. Dyakin, T. A. Pikuz, S. A. Pikuz, 

V. M. Romanova and T. A. Shelkovenko, 

Physica Scripta 55, 735-740 (1997) 

Weimar, Oct 4, 1999

Imaging properties of spherically bent crystals 

Ray tracing simulation 


θ B [deg] c [cm] R [cm] a [cm] b [cm] 

Conf A 86 9.98 10.00 6.25 25.00 

Conf B 86 18.56 18.60 11.63 46.5 

Conf C 80 9.85 10.00 6.25 25.0 


A central 

A lateral 

B central 

point 

source 

10 µm 100 µm 

2 2.1 2.1 2.9 

5 5.5 9.4 ~25 

3 3 4.2 ~18 

B lateral 4 4.3 7.7 ~25 

C central 

3.3 3.3 3.3 4 

Weimar, Oct 4, 1999 

1000 

µm 

C lateral 10 11 18 No resolution

Collimation of X-rays (from plasmas) 

Ray tracing simulation of spherically bent crystals 

We studied the effects of 

1) Surface shape and geometrical errors 

2) Crystal size (entrance pupil) 

3) Source depth abd width 

4) Incident angle 


Ref: M. Sanchez del Rio, M. Fraenkel, A. 

Ziegler, A. Ya. Faenov and T. A. Pikuz, 

Rev. Sci. Instrum. 70(3), 1614-1620, 1999 


0.6 

0.4 

0.2 

0.0 

-0.2 

-0.4 

80 

60 

40 

20 

0 

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 

-0.6 

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 

0.6 

0.4 

0.2 

0.0 

-0.2 

-0.4 

140 

120 

100 

50 µm source 

200 µm source 

-0.6 

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 

Weimar, Oct 4, 1999

Introduction V: 

Perfect crystals in asymmetric diffraction 

In Reflection (Bragg) mode 

S1 

Δθ 1 


Δθ 2 

S 2 

Ray tracing simulations for crystal optics (7 of 23) 

In Transmission (Laue) mode 

M. Sanchez del Rio & F. Cerrina M. Sanchez del Rio et al. 

Rev. Sci. Instrum 63 (1) 936 (1992) Nucl. Ins. Meth. A347 338 (1994)] 

S 1 

k 1 

d 

θ B 

k 2 

S 2 

SPIE, July 22, 1998

Theory I 

Geometrical model 

S 1 

k' 1 

k1 


θ 1 

k 1 

d 

θ B 

B H 

B H 

k' 2 

1 

= ⎛– -- ⎞uB ⎝ d⎠ 

α>0 

θ 2 

k 2 

k2 

S 2 


❑ Elastic scattering: k2 = k1 ❑Boundary conditions: k2 = k1 + BH ⇔ 

– sinθ2 

= sinθ1 

Gratings 

λ 

– -- sinα 

d 

mλ = d’ (sin θ in + sin θ out ) 

or 

k out = k in + G || 

d/sinα = d’ /m 

Tracing condition (Grating eq.) 

k in 

G || =m (2π/d’) u || 

θ in 

θ out k out 

m=-2 

m=-1 

m=0 

m=1 

m=2 

SPIE, July 22, 1998

Theory II 

Physical model: Dynamical Theory of Diffraction 

C 1 

C 2 

= 

= 

e iϕ – 1t o 

e iϕ – 2t o 


z 

= 

❑ BRAGG case 

(Zachariasen 1945) 

(Compact, general (thin/thick, Laue/Bragg), with absorption, general expression for τ, F H ) 

valid for perfect flat crystals 

R( θ) 

❑ LAUE case R( θ) 

ϕ1 2π koδ'o = ----------γ 

o 

ϕ2 2π koδ''o = ------------γ 

o 

1 – b b 

-----------Ψ 

2 o – --τ 

2 

q = bΨHΨ H 

γ o 

b ≈ -----γ 

H 

where 

δ' 

⎛ o⎞ 

= 

⎝δ''o⎠ τ 

1 

-- 

b 

IH = ----- = 

Io 1 

-- 

b 

IH = ----- = 

Io Ray tracing simulations for crystal optics (9 of 23) 

1 

-- 

b 

x1x2 c ( 1 – c2) --------------------------------c2x2 

– c1x1 1 

-- 

b 

x1x2 c ( 1 – c2) --------------------------------x2 

– x1 1 

-- ψ 

2 o z q z 2 

⎛ – ⎛ ± + ⎞⎞ 

⎝ ⎝ ⎠⎠ 

2 

2 

x 

⎛ 1⎞ 

= 

⎝x2⎠ 1 

---- B 

2 

ko 2 H 2 ke = ( + ( o ⋅ BH ) ) τ ≈ 2( θ– θB) sin2θ 

Ψ H 

= 

z q z 2 

– ⎛ ± + ⎞ 

⎝ ⎠ 

----------------------------------ψ 

H 

e2 mc2 --------- 1 

--- 

V 

λ2 

----- ( f 

π 

o + f ' + f '') 

e 

n 2πi H r ⎛ ( ⋅ n) 

⎞ 

∑ 

K 

⎝ ⎠ 

n ∈ unitcell 

SPIE, July 22, 1998

Perfect crystals IV 

Tracing a polychromatic divergent bem on a flat symmetrical xtal 


point source: image plane 

1: 3 Ι Energy lines 

2: continous spectrum 

θ Β 


PSI, Feb 1998

Benchmarks I 

Phase space 

Study the changes induced by a Si 111 crystal in the (z,z’) space, E= 8 keV, α = 5 deg 

z 

z z' 

z' 


Dependency in energy 


SPIE, July 22, 1998

Benchmarks II 

Du Mond diagrams (J.W.M. Du Mond, Phys. Rev. 52 (1937) 872) 

(+,+) Si 111, 12 keV 



SPIE, July 22, 1998

Benchmarks III 

Few equations 

Focusing equations for a monochromatic beam (like for gratings): 

1 

p 


+ 1 

q = sin 1 + j sin 2j 

Rs 

For Bragg symmetrical crystals ( 1 = 2 = 0): 

Spectral resolution: 

0 

= E 

E0 

= cot 0 

1 1 

+ 

p q = 2 sin 0 

; 

Rs 

; 

sin 2 1 

p + sin2 2 

q 

1 

p 

+ 1 

q = 

= sin 1 + j sin 2j 

Rt 


2 

Rt sin 0 

q 

! 2 D +( geom + ss) 2 s 

cot 0 = ! 2 D + 

p 

R sin 1 

, 1 src + s1 

p 

2 

(1) 

(2) 

cot 0 

(3) 

SPIE, July 22, 1998

High resolution I 

Collimating pre-mirror (Δ src ~ 0) 

Mirror (3 mrad) and (+,-) Si111. E=10 keV, source 85 μm 185 μrad 


Mirror E[eV ] I [ rad] 

No mirror 8.47 0.99 1008 38 158 24 

Parabolic 1.27 0.05 1004 20 3.9 0.3 

Elliptical 1.27 0.05 1004 20 3.9 0.3 

Cylindrical 1.28 0.06 1002 18 4.4 0.3 

Cylindrical+1" slope 1.44 0.15 1000 20 20.7 2.9 

Cylindrical+2" slope 2.33 0.12 1008 18 38.5 3.6 


SPIE, July 22, 1998

High resolution II 

Rowland bending (p/(R sin θ B )= 1) 


θ B +δ 

θ B 

θ B −δ 

(+,-) Si111, 10 keV, source 85 μm 185 μrad 

System I1 I2 E1[eV ] E2[eV ] 

at- at 413 14 307 12 8.92 0.55 8.71 0.69 

Rowland(1:1)- at 414 25 65 9 1.44 0.06 1.23 0.24 

Rowland(concave)-Rowland(convex) 399 12 296 10 1.44 0.07 1.32 0.06 

Out-Rowland(1:3)- at 408 18 36 3 8.2 0.9 2.1 0.5 


θ B 

θ B 

SPIE, July 22, 1998

High resolution III 

Normal incidence (Masciovecchio et al, NIM B111 (1996) 181) 


ΔE=0.28 meV (e -M =0.13) 

ΔE=0.45 meV (e -M =0.2) 

ΔE=0.5 meV [experimental] 

Si 13,13,13, ~ 25 keV, θ B =89.98 deg, R=250 cm 


SPIE, July 22, 1998

High resolution IV 

Multicrystal dispersive system 


E=14413 eV 

Si 10,6,4 α=20 deg 


Si 4,2,2 α=0 

Si 10,6,4 α=-20 deg 

ΔE=6.3 meV (6.7 from refs) 

Si 9,7,5 α=75.4 deg 

Si 9,7,5 α=-75.4 deg 

ΔE=1.65 meV (1.57 theoretically from ref) 

Nested monochromator 

Isikawa et al. RSI 63 (1992) 1015 

Chumakov et al. SPIE 3151(1997) 262 

Alp et al. APS Res. 1(1998) 9 

Highly asymmetric 

monochromator 

Chumakov et al. NIM A383(1996) 642 

SPIE, July 22, 1998

Focusing optics I 

Polychromatic focusing with plane Laue crystals I 


s 1 

λ,θ Β 

λ’,θ’ Β 

λ”,θ” Β 

Pseudo-focusing effect of a Laue crystal for a white incident divergent beam. The selective 

reflections on the Bragg planes convert the divergent beam into a convergent one. 

❑ magnification factor M 

s 

2 

Δθ 

1 

= ---- = --------- = 

s 

1 

Δθ 

2 

❑ best focus expected at a distance of s2 = Ms1 1 

-- 

b 

❑ Symmetrical Laue: M=1 ( i.e. a 1:1 focussing device) 

❑ Asymmetrical Laue: M=1/b => M(E) => Dispersive System=> Focus size is a function of ΔE 


s 2 

SPIE, July 22, 1998


Polychromatic focusing with plane Laue crystals II 


Undulator Source 

top view 

28 m 

Layout of the experimental station at the Troika 

beamline. 

The monochromator is composed by the Laue 

diamond and Bragg Ge crystals in the nondispersive 

arrangement. 

Inset: Incidence angles for the x-ray beam when 

the diamond crystal is set to diffract photons 

with energy of 9100 eV 

slit 

1 m 


Ge 

image planes 

112 cm 

transmitted beam 

 

125.6 

 

15.6 55 

Diamond Crystal 

diffracted beam 

SPIE, July 22, 1998


Polychromatic focusing with plane Laue crystals III 



M. Sanchez del Rio et al, Rev Sci Instrum, 66 (11) 

5148- 5152, Nov 1995 

0.4 cm 

0.3 cm 

0.2 cm 

0.1 cm 

0.05 cm 

0.8 cm 

SPIE, July 22, 1998

Focusing optics III 

Sagittal bending 

Two aspects that can be analyzed: 

1) The effects of surface shape: 

• ❑ Anticlastic effect 

• ❑ Alignment 

• ❑ Loss of cylindrical shape 

• ❑ Spurious errors left by ribs 

• ❑ Conical surface: Ice & Sparks, JOSA 

A11 (1994) 1265 


p q 

2) beam transmission as a function of 

the angular divergence [Sparks et al., 

NIM 172 (1980) 237]. The 

M=1:3=0.33 is optimal to focus divergent 

beams. 

M=q/p=1:3 


5 mrad 

2.5 mrad 

1 mrad 

SPIE, July 22, 1998

Diffraction & Coherence I 

Fraunhofer diffraction pattern from a slit 

Coherent & 

collimated beam (8 keV) 


10 cm 

Fraunhofer Diffraction by a slit 

Diffraction Pattern 


25 microns 

5 microns 

12.5 microns 

SPIE, July 22, 1998

Diffraction & Coherence II 

Fraunhofer diffraction after a symmetrical/asymmetrical crystal 

The conservation and propagation of transversal coherence can be studied from the visibility of the diffraction patterns 

Example: Degradation of the transversal coherence when using asymmetric crystal (which produce chromatic aberrations) 

[S. Brauer et al., J. Synchr. Rad. 2 (1995) 163] 

Si 111, α=0, α=0.15 deg 

8 keV, ΔE~ 1 eV 


α=0 

8000.0eV 

8000.5 eV 


α=0.15 deg 

SPIE, July 22, 1998

Introduction 

They are formed by a large number of small perfect crystallites of microscopical or submicroscopical 

size, which are oriented almost, but not exactly parallel to each other 

The most extensively used is Highly Oriented Pyrolithic Graphite (HOPG), produced by 

Advanced Ceramics (USA), Optigraph (Russia) and Panasonic (Japan). Typical mosaicity values 

go from 0.1 to 0.5 deg 

❑ Wider reflectivity curve than perfect crystals ~ ∆E/E ~ 10 -2 

❑ Lower peak reflectivity but higher integrated reflectivity 

❑ Well established theory Bacon & Lowde Acta Crystall. 1 (1948) 303 and Zachariasen’s book 

(1945) 

❑ The theory fails when using narrow beams due to possible macrostructures of about 40 x 400 

microns: Freund et al. SPIE 2856 (1996) 68 

❑ Macrostructures have been seen by x-ray topography: Ohler et al., NIM B129 (1997) 257 

❑ We studied the focusing properties [SPIE vol. 3448, 246-255, 1998]. We found that the existence 

or not of macrostructures depend on the particular sample [SPIE vol. 3773, 1999]. 



Weimar, Oct 4, 1999

Focusing properties (von Hamos geometry, 1932) 

TOP 

source 

S 

SIDE 

ω 

w 


ww 

ω' = ω + 2 τ sin θ Β 

SS 

image 


❑ Mosaic crystals present a defocusing effect in the 

plane perpendicular to the diffraction plane 

❑ They present a parafocusing effect in the 

diffraction plane: 

• ❑ focusing of monochromatic radiation in M=1:1 

• ❑ Chromatic aberrations in the diffraction plane 

• ❑Aberrations produced by the beam penetration in the crystal bulk 

Weimar, Oct 4, 1999

Mosaic Crystals: 

are formed by a large number of small perfect crystallites of microscopical or 

submicroscopical sizes, which are oriented almost, but not exactly parallel to each other. 

Assumptions: 

1-The relative displacements of the crystallites are assumed large compared with the 

x-ray coherence width: No relationship between the scattering from various cristallites. 

2- Scattering planes are defined by the direction of their normal, and the distribution of 

the normals of the different crystallites is in good approximation gaussian with standard 

deviation κ 

3- κ is large compared with the Darwin width or width of the diffraction profile crystallite. 

θ D 

θ D 

Reflection Mode: 

focussing 1:1 in 

diffraction plane. 

Transmission Mode 






0.4 deg fwhm 

Bragg case 

Mosaic Crystals diffraction profiles 

0.2 deg fwhm 

17 keV 

8 keV 

3 keV 

Diffraction profile in Bragg case: 





r( Δ) 

a 

= ------------------------------------------------------------------------------ 

1 + a + 1+ 2acoth( 

A 1 + 2a) 

Diffraction profile in Laue case: 

r( Δ) 

= sinh( 

Aa) 

exp( 

– A( 1 + a) 

) 

with: 

w( Δ)Qsp 

; 1 

a = ---------------------- = 

-- 

μ μ 

A 

η sp 

μt 

= ------------sin 

θ B 

; = w( θ– θD)Qsp ; 

1 

---------- e 

2π 

Δ2 2κ 2 

– ⁄ ( ) ( Fx ⁄ vo) 2 λ 3 

---------------------------sin2θ 

B 

Q p = Qs cos2θ 

B 


Ray tracing modeling with SHADOW 

k in , |k|=1 

y 

β 

θ D 

α 

φ 


z 

Ref: M. Sanchez del Rio et al. Rev. Sci. Instrum. 63 (1992) 932 

k out , |k|=1 

x 


The geometrical model: 

1) Force the ray to be reflected by a crystallite following 

the Bragg law. The crystallite normal lies in a 

cone. The crystallite normal is selected by Monte 

Carlo considering the Gaussian distribution of crystallites 

2) Include penetration (secondary extinction) by a 

MFP model. 

The Physical Model: 

Reflectivity by theory of Bacon&Lowde 

3 energy lines resolved spatially 

Aberrations due to secondary extinction 

Weimar, Oct 4, 1999

0.2 mrad 

1 mrad 

side view: 

mosaic 

perfect 

10m 10m 

top view: 

mosaic 

perfect 

mosaic 

perfect 





12 cm 

0.5 cm 


Beam evolution in the ⊥ plane using the unfocused beam 

Ref: M. Sanchez del Rio et al. SPIE proceedings 3448 246 (1998) 

-0 cm 13 cm 60 cm 100 cm 200 cm 

E= 8 keV Detector = photographic film 

❑ Expected divergence ∆ = 2 τ sin θ B = 2 (0.25 deg) sin (13.4 deg) ~ 2 mrad 

❑ From fits up to 30 cm ∆ = 2.6 mrad 

❑ From fits up to longer distances ∆ >> 2.6 mrad. Problems originated by the film? 



Simulation 200 cm 

∆ = 2.5 mrad 

Weimar, Oct 4, 1999

Parafocusing effect 

Simulation: 0.51 mm FWHM with τ=0.25 deg, t=0.5 mm 


BM F source 

X 

30 m 

20 cm 40 cm 60 cm 80 cm 

96 cm 100 cm 120 cm 140 cm 


A 

B 

Monochromator 

9 mY 

C 

Secondary 

Source 

Z 

Mosaic crystal 

D 

E 

K 

Weimar, Oct 4, 1999

Topographic studies on Highly Oriented Pyrolithic Graphite 

Ref: A. Tuffanelli et al. SPIE proceedings 3773 (1999) 


40 µm 

E= 18 keV x 


Weimar, Oct 4, 1999

string_1 

string_2 

chromoso 

me pool 

756. 

fitness - 

function 

Global optimisation of Optical Systems with SHADOW 

string_3 

string_4 

string_n 

Genetic Algorithms 

x[0]= 1234.5678 x[1]= 9876.5432 x[3]= 564738 ........ 

str[0] = 1110010111000011010100010101001 

1111111111100000 

1111111111100000 

ready 

1111111111111 

000000000000 

mating 

✄ 

Illustration of the Genetic Algorithm work cycle 

numbers are 

represented by 

bit-strings 

child-chromosome 

111111110000 

random 

number 

generator 

111111111110000000000 

mutation 

1. Select a new configuration - the higher the ’temperature’ the larger the range of the new parameters. 

2. Calculate the cost of the new config. (call the function to analyze with the current config.) 

3. Decide if you wanna accept this new config. by following rules: 

* if cost < last_accepted_cost => ACCEPT 

* if cost > last_accepted_cost => ACCEPT with a probability prob(tmp) 

5. Repeat procedure (either with the new or the old config.) until thermal equilibrium is reached 

6. If temp > 0 : Decrease temperature and restart loop 

The decreasing of the temperature is the most influential part of the system. It also is responsible for the speed and the 

accuracy of the system: 


Experience at ESRF 

SRI’95. Workshop 6, Oct.18 

cost(x1) 

Simulated Annealing 

x1 

cost(x1) 

accuracy ~ speed -1 

Principle of the Simulated Annealing method 

x1 


Theoretically calculated mirror profiles with cylindrical and elliptical shapes (top panel) and the intensity 

distribution at the image plane (bottom panel) for both elliptical (left) and cylindrical (right) cases. FM 

values are 9.91 106 for the elliptical and 8.33 105 for the cylindrical one. 

Some mirror profiles of table 4 after different gereations 

(top) and the intensity distribution of the best chromosome 

after 15000 generations(bottom). 


Experience at ESRF 

SRI’95. Workshop 6, Oct.18

Ray tracing - ESRF

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