josep nicolas claude ruget carles...

$: Josep Nicolas Claude Ruget Carles Colldelramaccelconf.web.cern.ch/AccelConf/medsi2016/talks/motc01_talk.pdf · A pinhole, lenses (refraction), mirrors (reflection), Fresnel lenses$
J.Nicolas MEDSI’16

Optics and mechanics of mirror benders

Josep NicolasClaude RugetCarles Colldelram


1. Basics of x-ray mirror optics2. Characteristics of elliptic mirrors3. Mirror bender calculations and metrology4. Mechanical design of mirror benders


1. Basics of x-ray mirror optics§ Focusingwithreflectiveoptics§ Surfaceerror,aberration


0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Photon Energy (keV)

Tran

smis

sion

0 5 10 15 20 250.999

0.9995

1

1.0005

1.001

Photon Energy (keV)

Refra

ctio

n in

dex

re(n

)

Transmission of1mmofSilicon

Refractive indexfor Silicon

Refractive index of x-rays


Total reflection

102 103 10410-8

10-6

10-4

10-2

100Au

Photon Energy (keV)

Refra

ctio

n in

dex

re(n

)

db

n=1-d-ib𝑛" sin 𝛼" = 𝑛( sin 𝛼(

𝜃* = 2𝛿�


Mirror 400mm(toroidal) Mirror 1200mm(plane elliptic)

Amirror inits vessel


Anopticalsystemimages(stigmatically)an objectpointA,ontoanimagepointA’ ifalltheraysthatdepartfromA,andreachtheopticalsystem,gatheratA’,independently oftheirdirection.

objectpointImagepoint

Opticalsystem

Examplesoffocusingsystemsare:Apinhole,lenses(refraction),mirrors(reflection),Fresnellenses(diffraction).In(almost)allthecasesthefocusingelementdeviatesthedirectionoftheincomingrays.

To focus (v.)


Rays vs Waves

Rays arethelinesthatindicatethedirectionofpropagationofenergy,directionofvariationofphase

X-Raysarewaves(andphotonsaretheirquanta),andtobepreciseoneshoulddowavefront propagation.

Geometricaloptics,isanapproximation,basedonpropagationofrays.

AWavefront isthesurfaceofpointsatthesameopticalpathdistancefromtheemissionpoint.(forus,atthesamedistance),i.e.phaseisconstantacrossasurface

“Raysareperpendiculartowavefronts”

Rays

Wavefront


hν=24eV– λ=50nm Raytracing

6m1m

Pointsource Toroidal mirror100x10mm2

100μm

Althoughraytracingcannotaccountfordiffractionorinterferenceeffects,itreproducesaccuratelytheimagesifenergyishighenough(shortwavelength)

hν=123eV– λ=10nm

Ray tracing vs wavefront propagation


6m1m

130x6μm2

gaussian sourceToroidal mirror100x10mm2

100μm

Thesourcesizeoftheelectronbeamreducesthecoherenceofthebeam,andreducesthecontrastofwaveeffects.

hν=24eV– λ=50nm Raytracinghν=123eV– λ=10nm

Effect of the source size


object

mirror

image

Ellipsoidal mirror

Focusing with mirrors

“Thesumofthedistancestothetwofociisconstantforallthepointsofanellipse”


Stigmatic imagingAnellipsoidalmirrorfocusespoint-to-pointstigmatically(oranastigmatically)

Ellipsoidal mirror

Localcurvatureisnotconstant

Sagittalcurvaturevariesfromupstreamtodownstream


The approximation ofthe toroidal mirror tothe ellipsoid isaccurate only inthe centerofthe mirror.

Rays far from it arenot properly focused.

Toroidal aberration

Aberrations

Toroidal


u"

v"

h"

p"

q"

α"

β"

M"

A"

B"

Aberration computing

ℎ 𝑢, 𝑣 =𝑢(

2𝑅 +𝑣(

2𝑟

𝑂𝑃𝐷 𝑢, 𝑣 = 𝐌 − 𝐀 + 𝐁 −𝐌

Approximatetorusequation

Opticalpathdifference


𝐀 −𝐌 = 𝐴= −𝑀=( + 𝐴? − 𝑀?

( + 𝐴? − 𝑀?(�

𝐹AB 𝑝, 𝑞, 𝛼, 𝛽, 𝑅F =1

𝑛!𝑚!𝜕AKB𝐹𝜕𝑢A𝜕𝑣BL M,M|O,P,Q,R,ST

𝐹UV 𝑢, 𝑣|𝑝, 𝑞, 𝛼, 𝛽, 𝑟F = W W 𝐹AB 𝑝, 𝑞, 𝛼, 𝛽, 𝑟F 𝑢A𝑣BX

BYM

X

AYM

Image conditionWe need to compute the optical distance between object and mirror, and between mirrorand image.

It can be expressed as a Taylor series


Main aberrations• F20 meridional defocusing*

• F02 sagittal defocusing

• F30 primary coma*

• F12 astigmatic coma

𝐹(M =cos( 𝛼2

1𝑝 +

1𝑞 −

2𝑅 cos 𝛼

𝐹M( =121𝑝 +

1𝑞 −

2 cos 𝛼𝑟

𝐹"( =sin𝛼2

1𝑝 −

1𝑞

1𝑝 +

1𝑞 −

cos 𝛼𝑟

𝐹\M =sin𝛼 cos( 𝛼

21𝑝 −

1𝑞

1𝑝 +

1𝑞 −

1𝑅 cos 𝛼

F20 F30 Will appear again on the tutorial


acos211

Rqp=+

Meridional focusing

a

Sagittal focusing

p

q

R

a

p

q

r

-1 -0.5 0 0.5 1-200

-100

0

100

200

300

incidence angle variation (mrad)

foca

l dis

tanc

e va

riatio

n (m

m)

SagittalMeridional

The basic equation for paraxial optics is the focus condition

rqpacos211

=+

Varying the incidence angle


dZ

dP

dzI

When one changes the pitch of a focusing mirror, one steers the beam, but also changesthe incidence angle (therefore the focusing).

Special cases. 1:1 magnifiction. Flat mirrors

Varying the incidence angle

For a fixed figure Two motors (pitch, z)control four optical parameters (focus,Beam angle, spot position, footprintcenter / coma)


Figure error: slope


Figure error for coherent sources (a case)

Position [ m]-5 0 5

Inte

nsity

[a.u

.]

0

0.2

0.4

0.6

0.8

1

1.2

Just consider the caseinwhich the mirror is perfectalmost everywhere except for astep inthe center.

-slope erroris very small,

- There is nolighton axis due todestructiveinterference between the two sides ofthewavefront

For diffraction limited imaging (FELs),oneneeds tocontrolthe wavefront distortion,which is given by the residualheight error.


Figure error: height error

Δ𝑤 = 2Δℎ𝑐𝑜𝑠𝛼

Interference effects inimage formation aregiven by the Strehl ratio(actualpeakintensity tothe errorfreepeak intensity)

Δ𝑤

Δℎ

𝑆 ≅ 1 −2𝜋𝜆 Δ𝑤 f

ghi

(

Δℎ fghi

≤ 𝜆1 − 𝑆�

4𝜋𝑐𝑜𝑠𝛼ExampleLCLSII – S=0.97- 2.1mrad -13keVè 0.4nmrms


designed for the resonant inelastic X-ray scattering (RIXS)experiments at MAX IV require an aperture length of 280 mmwith 100 nrad r.m.s. slope error. In addition, elliptical-cylinder-shaped focusing optics of 20 nrad r.m.s. residual slope devia-tion up to a length of 1000 mm are proposed to focus photonsat the Single Particles, Clusters and Biomolecules (SPB)experimental station at the European XFEL (Mancuso et al.,2013). In contrast to such long focusing mirrors, the length ofKirkpatrick–Baez (KB) mirror substrates at sources likePETRA III or MAX IV will be in the range 100–200 mm(Kalbfleisch et al., 2010; Johansson, 2014). Such opticalcomponents of elliptical and hyperbolic shape are alreadyavailable today (Matsuyama et al., 2012; Siewert et al., 2012)and allow diffraction-limited focusing of hard X-ray photonswithin nanometre focus size (Mimura et al., 2010). All theseoptical components used in tangential focusing have fairlylarge radii of curvature. On the other hand, sagittal focusing,which was proven to preserve brilliance very well, especially inthe soft X-ray regime, results in highly curved surfaces ofcylindrical, toroidal or ellipsoidal form setting new challengesfor manufacturing.

Dedicated metrology instrumentation of comparable accu-racy has been developed to characterize such optical elements.Second-generation slope-measuring profilers like the Nano-meter Optical component measuring Machine (NOM)(Siewert et al., 2004; Yashchuk et al., 2010; Nicolas & Martinez,2013; Assoufid et al., 2013) allow the inspection of reflectiveoptics up to a length of 1.5 m (Alcock et al., 2010) with anaccuracy better than 50 nrad r.m.s. It has taken the place of thewell known Long Trace Profiler-II (LTP) (Takacs et al., 1987)as a fundamental tool for the inspection of optics. It should benoted that metrology does not only provide characterizationof optical components, but such data can be directly used inrealistic beamline modelling (Samoylova et al., 2009).

2. On the precision of optical elements to guide andfocus X-rays

Synchrotron optical components are long shaped (up to 1.3 m)and used under the grazing-incidence condition (Wolter, 1952)

which makes them difficult to measure and thus to manu-facture. The impact of shape imperfections at optical compo-nents in the long spatial frequency error regime (from about1 mm to aperture length) on the imaging performance of anX-ray focusing system is related to the induced local phaseshifts in the reflected beam. The latter distort the wavefrontand cause the converging beam to have phase errors.Assuming a small source and a large distance between sourceand mirror, the acceptable p.v. mirror height variations !h arelimited to a few single nanometres only,

ð2!="Þ !hðxÞ!! !! sin # # 1; ð1Þ

with # being the grazing angle and " the wavelength of theX-ray beam (Samoylova et al., 2009). A further criterion for abeamline performance is the r.m.s. wavefront distortion whichshould be $r.m.s. < "/14 or better. This leads to a conditionknown as the Marechal criterion (Marechal, 1947), where theacceptable root-mean-square height error for a number ofoptical components over all spatial frequencies present withinthe residual surface errors is given by

!hrms $"

14ffiffiffiffiNp

2#; ð2Þ

where N is the number of reflecting surfaces in the system(Siewert et al., 2012). Clearly, the requirements on surfacequality become linearly more difficult to achieve withdecreasing X-ray wavelength, hence the difficult challenge ofmaking hard X-ray reflective optics of sufficient quality.Practically, mirrors of sub-nm r.m.s. figure quality for thelong-, mid- and high-spatial figure error need to be manu-factured and hence measured. These conditions have been themotivation to improve deterministic finishing technology aswell as metrology capabilities. To demonstrate the currentstate of metrology, we will discuss the inspection of a super-polished focusing mirror pair for beamline P06 at PETRA III.Finishing technology like ion beam figuring (IBF) (Schindleret al., 2003; Thiess et al., 2010) and elastic emission machining(EEM) (Yamauchi et al., 2002) allow the substrate topographyto be controlled on an atomic scale. Of course, this is onlyrealistic if precise topography data are available. Mirrorsfinished by use of EEM have recently shown <1 nm r.m.s.figure accuracy on a length of up to 350 mm (corresponding to50 nrad r.m.s. slope deviation) (Siewert et al., 2012) and haveallowed diffraction-limited focusing at the CXI experiment atLCLS (Boutet & Williams, 2010). However, upcoming opticslike the 1 m-long KB-focusing mirrors (Kirkpatrick & Baez,1948) at the European XFEL with a required residual slopeerror of 20 nrad r.m.s. (this corresponds to about 1 nm p.v.figure error!) will represent a challenge for both metrologyand finishing technology. To reach such a quality, the grav-itational sag and clamping forces need to be accounted for inthe mirror substrate so as to provide the required figure shapewhen mounted at the beamline. In addition, diffraction-limited sources translate into very high power densities onoptical components and additional care is required to preservethe mirror shape as well as possible, even by means of activeoptics. Whereas heat-load-induced deformations are very

new science opportunities

J. Synchrotron Rad. (2014). 21, 968–975 F. Siewert et al. % Characterization of ultra-precise X-ray optical components 969

Figure 1Improvement of mirror quality in terms of slope error during the last twodecades (based on data and measurements performed at the BESSY-IIOptical Metrology Laboratory).

Some reference values of figure error

Siewert etal.JSR21,968-975(2014)

Quickly evolved inthe recent past:

• Development ofmirror metrology

• Deterministic polishingtechniques:

Most (usual)companies candelivermirrors better than 0.5µradrms,onwhatever figureandlength.

For flats erroris sub-nanometer,andslope errorbelow 100nrad,cost anddelivery time.

Sagittal curvature always difficult


Causes of figure error• Fabrication error(figuring,

polishing metrology):wellbelow 0.5µradrms

• Gravity sag.

• Clamping mechanics.

• Stressinduced by coolingscheme.

• Thermal deformations.

• Bendererrors,ellipseapproximatino errors

• Coating stress!Bakeoutdeveloped stresses (?)

mirror

Cooling pads

Cooling tube supports

mirror

Cooling pads

Cooling tube supports

Mirroralone:R=331.039m /0.163μrad RMSInholder:R=338.303m /0.533μrad RMS(Firstiteration)


MisalignmentsNoerror

Error∆q=0.5% Pitcherror50µrad Yaw error50µrad

6m1m

PointsourceEllipsoidat3deg100x10mm2

FOVis 100µm


By decoupling verticalandhorizontalfocusing,one canobtain aberration freefocusing byusing high quality mirrors polished flat,with low slope error,andmechanically bent ontoplano-elliptic figure.

Focused beam

50μm

10μm

Kirkpatrick-Baez configuration


• Decouple horizontalandverticalplanes

• Astigmatic sources or images

• Nosagittal curvature

• Allow different stripes

• Moreaccurate metrology

• Relaxes alignment requirements (nosagittal curv.)

• If bender

• Start from flator spherical mirrors

• Adaptive focus andprimary comma

• Several foci

• Focus independent ofalignment

Some KB advantages


2. Characteristics of elliptic mirrors§ Geometryoftheellipse.§ Height,slope,curvature.§ Polynomialapproximation.


𝑎 =𝑝 + 𝑞2

Description of an elliptic mirror

Majorsemi-axis

𝑐 =12 𝑝( + 𝑞( − 2𝑝𝑞 cos2𝛼�

Halfdistancebetweenfoci(lineareccentricity)

𝑒 = 1 −2

𝑝 + 𝑞 𝑝𝑞 cos( 𝛼�

eccentricityMinorsemi-axis

𝑏 = 𝑝𝑞� cos𝛼

pq

A

A’

x

y

α

c

a

b ψ

tan𝜓 =𝑞 − 𝑝𝑞 + 𝑝 cot𝛼

Orientation


Equations of the ellipse

αp

q

A(-psinα,pcosα)

m

A’(qsinα,qcosα)

x

y

𝐫t − 𝐫B + 𝐫tu − 𝐫B = 𝑝 + 𝑞

𝑥 + 𝑝sinα ( + 𝑦 − 𝑝cosα (� + 𝑥 − 𝑞sinα ( + 𝑦 − 𝑞cosα (� = 𝑝 + 𝑞

Vectorequationoftheellipse

Implicitequationoftheellipse(asafunctionofcoordinates)


Explicit equations of the ellipse

𝑦 𝑥 = −𝑥 𝑝( − 𝑞( sin 2𝛼 − 4 𝑝 + 𝑞 cos𝛼 𝑝𝑞 − 𝑝𝑞� 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥(�

𝑝( + 6𝑝𝑞 + 𝑞( − 𝑝 − 𝑞 ( cos2𝛼

Concavebranchoftheexplicitsolutionoftheellipse

• Notethaty(0)=0• Thisisanexact solution!

𝑑𝑦𝑑𝑥 𝑥 = −

2 𝑝 + 𝑞 cos𝛼 𝑝 − 𝑞 sin 𝛼 −𝑝𝑞� 𝑝 − 𝑞 sin 𝛼 + 2𝑥𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥(�

𝑝( + 6𝑝𝑞 + 𝑞( − 𝑝 − 𝑞 ( cos2𝛼

Slopefunctionoftheellipse

• Notethaty’(0)=0• Thisisanexact solution!


Curvature and imaging conditionCurvatureoftheellipse

• Notethatitisnotconstant

1𝑅 𝑥 =

𝑑(𝑦𝑑𝑥( 𝑥 =

𝑝𝑞� (𝑝 + 𝑞) cos𝛼2 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥( \ (⁄

1𝑅M

=~𝑝 + 𝑞) cos𝛼

2𝑝𝑞

Curvatureatthecenteroftheellipse

2𝑅M cos𝛼

=1𝑝 +

1𝑞

• Itisthefocusingcondition(mer)


Taylor expansion of the ellipseDefinition

with

Inthecaseofanellipse,thecoefficientsarefunctionsofp,q andα

𝑦 𝑥 = W𝐸A𝑥AX

�YM

𝐸A = 1𝑛!𝑑A𝑦𝑑𝑥A�=YM

𝐸( =cos𝛼4

1𝑝 +

1𝑞 =

12𝑅M

𝐸\ =sin 2𝛼16

1𝑞( −

1𝑝(

𝐸� =E\(

4E(5 − csc( 𝛼 + E(\ sec( 𝛼 𝐸� =

E\\(7 − 3 csc( 𝛼)4E((

+ 3E((E\ sec( 𝛼

𝐸� =E\�(21 − 14csc( 𝛼 + csc� 𝛼)

8E(\+ E(E\((7 − csc( 𝛼) sec( 𝛼 + 2E(� sec� 𝛼


Aberrations, revisited

𝐸\ =sin 2𝛼16

1𝑞( −

1𝑝(

𝐹(M =cos( 𝛼2

1𝑝 +

1𝑞 −

2𝑅 cos 𝛼

𝐹\M =sin 𝛼 cos( 𝛼

21𝑝 −

1𝑞

1𝑝 +

1𝑞 −

1𝑅 cos𝛼

𝐸( =cos𝛼4

1𝑝 +

1𝑞

𝐹(M = 2cos𝛼cos𝛼4

1𝑝 +

1𝑞 −

12𝑅

𝐹\M = −2cos𝛼sin 2𝛼16

1𝑞( −

1𝑝(

Remember:

Defocusisgivenbythequadratictermoftheellipsepolynomial

IfF20=0Primarycomaisgivenbythecubictermoftheellipsepolynomial

𝐹(M = 2cos𝛼 Δ𝐸(

Remember: 𝐹\M = −2cos𝛼 Δ𝐸\


3. Mirror bender calculations and metrology

§ Euler-Bernoulliequation§ Rectangularfootprintcase§ Customfootprintcase


Elastic beam theoryEuler-Bernoulliequation

Secondmomentofinertia

• z(x)isthedeformationinducedbytheload.[m]

• M(x)isthedistributionofbendingmomentsalongthemirror

• EistheYoung’smodulusofthebody.[N/m2]

• I(x)isthesecondmomentofinertiaofthemirrorsubstratesection.[m4]

𝐼 𝑥 = � 𝑧(𝑑𝑥𝑑𝑧��F�A

𝐼?? =𝑏ℎ\

12

Forarectangularsection

𝐸 · 𝐼 𝑥𝑑(

𝑑𝑥( 𝑧 𝑥 = 𝑀 𝑥 𝑀 𝑥 = W 𝑥 − 𝑥F 𝐹F

�

F|=T�=

Bendingmoments

𝑏ℎ


Integration (rectangular mirror footprint)

𝑧 𝑥 = 𝑐� + 𝑐"𝑥 + � �𝑀 𝑥′𝐸𝐼 𝑥′ 𝑑𝑥′

="

�� (⁄𝑑𝑥"

=

�� (⁄ 𝑧 𝑥 = 𝑐� + 𝑐"𝑥 + 𝐸(𝑥( + 𝐸\𝑥\

GeneralSolution toEuler-Bernoulli’equation For constant section mirror the solutionis athird degree polynomial

𝑀 𝑥𝑥

𝐸( =3𝑎𝐸𝑏ℎ\ 𝐹� + 𝐹�

𝐸\ =2𝑎

𝐸𝑏ℎ\𝐿 𝐹� − 𝐹�

𝐹�𝐹�

𝑎 𝐿 − 𝑎 𝐿


𝐸( =3𝑎𝐸𝑏ℎ\ 𝐹� + 𝐹�

𝐸\ =2𝑎

𝐸𝑏ℎ\𝐿 𝐹� − 𝐹�

Rectangular mirror footprint• The sumofforces controls the main curvature (andthe

defocusing)

• The difference between the forces controls theeccentricity (andprimary coma)

120100

Eccentricity (E3) [m-2]

806050

10-6

-4-3

3210

-1-2

100

Force Downstream [N]

120

Force Upstream [N]

100

Curvature (E2) [km-1]

806050

0.0260.0240.022

0.020.0180.0160.014

100

Force Downstream [N] Force

Upstream [N]

𝐸(𝐸\

=𝐸(,��𝐸\,� �

+ 𝐸(� 𝐸(�𝐸\� 𝐸\�

𝐹�𝐹�

Measurement No1 2 3 4 5 6

Fit E

rror [

rad

rms]

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

FU=50.1 NFD=50.0 N FU=100.0 N

FD=49.9 N FU=99.9 NFD=100.0 N

FU=50.1 NFD=99.8 N

FU=120.3 NFD=99.7 N

FU=60.3 NFD=80.3 N

Accuracy of the linear fitE2E3

fitbetterthanabout50nrad rms(metrologylimited– tooquick)


High demagnification ellipsesTheerrorofthecubicpolynomialapproximationisgivenby:

𝑦¡¢¢£¢ 𝑥 =

−𝑥 𝑝( − 𝑞( sin 2𝛼 − 4 𝑝 + 𝑞 cos𝛼 𝑝𝑞 − 𝑝𝑞� 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥(�

𝑝( + 6𝑝𝑞 + 𝑞( − 𝑝 − 𝑞 ( cos2𝛼 −cos𝛼4

1𝑝 +

1𝑞 𝑥( −

sin 2𝛼16

1𝑞( −

1𝑝( 𝑥\

Position [mm]-200 -150 -100 -50 0 50 100 150

Hei

ght E

rror [

nm]

-100

0

100

200

300

400FU=7 N, FD=269 N,Error:6.267 μ rad

For ALBA’s LOREAbeamline HFM10:1,400mmlong,this erroris inthe order of6.3urad rms

𝑦¡¢¢£¢ 𝑥 = 𝑦¡¤U*¥ 𝑥 − 𝐸(𝑥( − 𝐸\𝑥\

Forhighdemagnificationellipses,higherorderaberrationsaresignificant


Varied width mirrors

Byinsertingtheexactcurvatureprofile,wehavethedefinitionoftherequiredmirrorwidth.

𝐸𝑏 𝑥 ℎ\

12𝑑(𝑦𝑑𝑥( 𝑥 = 𝑀M + Δ𝑀

2𝑥𝐿

𝐸𝑏 𝑥 ℎ\

12𝑝𝑞� (𝑝 + 𝑞) cos𝛼

2 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥( \ (⁄ = 𝑀M + Δ𝑀2𝑥𝐿

𝑏 𝑥 = 𝑏M1𝑏M𝑀M +

1𝑏MΔ𝑀

2𝑥𝐿

24 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥( \ (⁄

𝐸ℎ\ 𝑝𝑞� (𝑝 + 𝑞) cos𝛼

Deformationequationincludescurvature:

Thesolutiondependsontheforcesonecanapply.

Oftenb(x)isjustapproximatedbyaloworderpolynomialvariation

𝑀M =12 𝑎"𝐹" + 𝑎(𝐹(

Δ𝑀 =12 𝑎"𝐹" − 𝑎(𝐹(


Examples of varied profile mirrors

-200 0 200

-20

0

20

-200 0 200

-20

0

20

-200 0 200

-20

0

20

-200 0 200

-20

0

20

Mirrorwidthprofilefora10:1demagnificationmirror,fordifferentbendingmomentcombinations.


Model of deformation

𝐸𝑏 𝑥 ℎ\

12𝑑(𝑦𝑑𝑥( 𝑥 = 𝑀M + Δ𝑀

2𝑥𝐿

Deformationofthemirror,inthiscase,isNOTapolynomial.

𝐸𝑏 𝑥 ℎ\

12𝑑(𝑦𝑑𝑥( 𝑥 = 𝑎

𝑥𝐿 −

12 𝐹� − 𝑎

𝑥𝐿 +

12 𝐹�

𝐸𝑏 𝑥 ℎ\

12𝑎𝑑(𝑦�𝑑𝑥( 𝑥 =

𝑥𝐿 −

12

𝐸𝑏 𝑥 ℎ\

12𝑎𝑑(𝑦�𝑑𝑥( 𝑥 = −

𝑎𝑥𝐿 −

12

𝑦 𝑥 = 𝐹�𝑦� 𝑥 + 𝐹�𝑦� 𝑥

Position [mm]-150 -100 -50 0 50 100 150

Hei

ght E

rror [

nm]

-40

-30

-20

-10

0

10

20Base Functions

hU(x)hD(x)

…butitisstillalinearcombinationoftwofunctions.


Position [mm]-150 -100 -50 0 50 100 150

Res

idua

l Hei

ght [

nm]

-5

0

5

10

15

20

Residual Error#1: 0.330 urad#2: 0.385 urad#3: 0.401 urad#4: 0.378 urad#5: 0.377 urad#6: 0.341 urad

Metrology of varied width mirror benders

Measurement No1 2 3 4 5

Fit E

rror [

N rm

s]

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

FU=50.1 NFD=50.1 N

FU=50.4 NFD=249.6 N

FU=50.4 NFD=399.3 N

FU=199.8 NFD=400.7 N

FU=201.0 NFD=250.0 NAccuracy of the linear fit

FUFD

Afterfittingdatatotheproposedbaseoffunctions,residualfigure(polishingerror)isequalforallbendingpositions


4. Mechanical design of mirror benders§ Functionalitiesofbender§ Parasiticforces§ Cyl benders,U-type,4-point,scissor,

bimorph,…


A zoo of mechanical solutions

(b) cantilever spring bender

(a) “s” spring bender

Mechanical design approaches

M. R. Howells, et al., Theory and practice of elliptically bent x-ray mirrors, Opt. Eng.39(10), 2748-61 (2000)

Beam theory applied to a bent structure:

xL

CCCC

x

yxEI 2121

2

2

2)(

!!

+=

"

"

E is Young’s modulus of the mirror material

I(x) is the moment of inertia as a function of

position along the beam, or mirror

C1

and C2 are the end couples producing

bending moments

12)(

3

0

hbIxI ==

bird-like shapeanti-clastic bending

Slide: 9ACTOP08, Trieste 10 October 2008A. Rommeveaux

KB development at ESRF

0

2

4

68

10

12

14

2001 2002 2003 2004 2005 2006 2007 2008

Nu

mb

er o

f b

end

ers

Different configurations: ¾ Single bender (13)¾ KB 170-170 (8)¾ KB 170-96 (8)¾ KB 170-300 (3)¾ KB 300-300 (5)

Since 2001: 61 benders tested and pre-shaped at the metrology lab22 with multilayer coatings

ID23 5 x 7 µmID27 2 x 4 µmID13 0.3 x 0.4 µmID22 76 x 84 nmID19 45 nm in Vertical

Graded multilayer *96 mm 170 mm

300 mm

Spot size

* O.Hignette

et al., AIP Conf Proc.-

January 19, 2007 -

Volume 879, pp. 792-795

Spot size achieved is generally close to geometrical limit (i.e.

not limited by optics aberrations)


LTP characterization of Long mirror benders

1.

Mirror intrinsic slope error characterization (out of bender)2.

Mirror clamped on mechanics: •

Evaluation of deformation induced by mechanics and correction if

possible•

Adjustment, optimization of gravity compensators•

Verifying safety switches, range of curvature 3.

Hysteresis cycles4.

Long term stability

Anti-twist Correction

Twist Simulation

before twist correction

after twist correction

Anti-twist mechanisms

Anti-twist correction

should not stress the

mirror substrate

in other directions

Visualization and correction of

surface twist with ZYGO GPITM

with virtual pivot

A – stabilizing bracket

Cylindrical bender ESRF-Scissor bender

S-spring bender

Canti-lever bender

Four-point bender

SLAC HiRes


Functionalities of a bender

1. To introduce the forces that deform themirror to the required ellipse withintolerances.

2. To minimize parasitic forces thatintroduce undesired deformations

3. To introduce forces that deform themirror to compensate gravity sag orother errors

Mechanical design of benders







• Fixtheapplicationpointanddirection

• Demagnify motordisplacementstonanometerresolutiondeformation


Apply controlled forces



Flexurehingespointtothe“rotation”point.Mirrorisglued(monolithic)

Lever(+bending?)demagnifiesmotorlinearmotiontoanangleconstraintonthemirrorends.




Pivotincenterofcontactsurfaceensuresforceisnormaltothesurface,andfixesapplicationpoint

(Notvisible)Forceisexertedbyahellicoidal springincompression





Demagnificationofmotordisplacementbyelasticbendingofthecantilever


Twist Simulation






mirror substrate

in other directions



with virtual pivot






Demagnificationofmotordisplacementbyarigidlongleverarm


KB development at ESRF

0

2

4

68

10

12

14

2001 2002 2003 2004 2005 2006 2007 2008

Nu

mb

er o

f b

end

ers

Different configurations: ¾ Single bender (13)¾ KB 170-170 (8)¾ KB 170-96 (8)¾ KB 170-300 (3)¾ KB 300-300 (5)

Since 2001: 61 benders tested and pre-shaped at the metrology lab22 with multilayer coatings

ID23 5 x 7 µmID27 2 x 4 µmID13 0.3 x 0.4 µmID22 76 x 84 nmID19 45 nm in Vertical

Graded multilayer *96 mm 170 mm

300 mm

Spot size

* O.Hignette

et al., AIP Conf Proc.-

January 19, 2007 -

Volume 879, pp. 792-795

Spot size achieved is generally close to geometrical limit (i.e.

not limited by optics aberrations)








• Avoidtwist• Avoidforcesalonglongitudinal

axis


Control parasitic forces


axis


Twist Simulation






mirror substrate

in other directions



with virtual pivot


Virtualpivottomanuallyadjustthetwistofoneendofthemirrorduringmetrologytests


Torsion (twist) adjustment


Twist Simulation






mirror substrate

in other directions



with virtual pivot


Therearebenderswhicharetorsionfree,andotherswhichallowcorrectingit



axis

Rolledarticulation(ononesideonly)toallowthemirrorsettleitstwistfreely.

Bendingforceactuatorsallowpitchandrolltoavoidintroducingundesireddeformations

Control parasitic forces







• Gravitysagcompensation


Gravity sag compensation

• Gravitysagcompensation

Manuallyadjustablesprings,forgravitysagcompensation






Adaptive optics

• Gravitysagcompensation• Adaptiveoptics

Piezo– electriccorrctorsembeddedinthemirrorbulk

PiezoCeramics

Silicon

Silicon

HVelectrodes



Nanometerresolution–stiff-pushing/pullingmotors

Adaptive optics



Force-stabilized– motorizedcorrectors.

Adaptive optics


Position [mm]-200 -100 0 100 200

Heig

ht e

rror [

nm]

-5

-4

-3

-2

-1

0

1

2

3

4

5

AchievedOptimalDifference

Position [mm]-200 -100 0 100 200

Hei

ght e

rror [

nm]

-50

-40

-30

-20

-10

0

10

20

30

40

50

SlopeError: 0.115μradrmsSurfaceError:0.858nmrms

Using 4actuators,500mm

Initialslopeerror 0.87μradrmsInitialsurfaceError:23.2nmrms

Difference tomodel-baseoptimum of0.08nm

Adaptive opticsResults of the ALBA-SENER nanobender


Position [mm]-200 -100 0 100 200

Hei

ght e

rror [

nm]

-10

-8

-6

-4

-2

0

2

4

6

8

10R=2.94kmR=3.56kmR=3.98km

The correction is preserved within the nanometer with variations oftheradius ofcurvature of35%.

Adaptive optics


Acknowledgements

ALBA Claude RugetCarles CollderamIgors SicsPablo Pedreira

SLACDaniele CoccoDaniel MortonCorey HardinMay Ling Ng

Valeriy Yashchuck (ALS)Ray Barrett (ESRF)

SENERIRELEC


Backup slides – gravity sag correction§ Rectangularfootprintcase§ Customfootprintcase


Gravity sag calculation

F1 F2

-ρLbhg

z

ya1

a2

L

1. Equilibriumofforcesandtorques:todeterminealltheforces.2. Functionofmomenta:todeterminetheRHSoftheE-Bequation.3. Integration.4. Boundaryconditions:todeterminetheintegrationconstants.


Equilibrium of forces

¦𝐹" + 𝐹( − 𝑔𝜌𝐿𝑏ℎ = 0

𝑎"𝐹" + 𝑎(𝐹( −𝑔𝜌𝐿(𝑏ℎ

2 = 0

F1 F2

-ρLbhg

z

ya1

a2

L

𝐹" = 𝑔𝑚𝑎( − 𝐿 2⁄𝑎( − 𝑎"

𝐹( = 𝑔𝑚𝐿 2⁄ − 𝑎"𝑎( − 𝑎"

• Allforcesandtorquesmustaddtozero,otherwisethemirrorwouldbeacceleratinglinearlyorangularly.

• Torquesarecalculatedfromtheoriginofcoordinates(notthecenterofthemirror)


Function of moments

RegionI.(onlyweight)

𝐹ª 𝑥 = −𝑔𝜌𝑏ℎ𝑥

𝑎ª 𝑥 =𝑥2

𝑀� 𝑥 = 𝐹ª 𝑥 𝑥 − 𝑎ª 𝑥

= −12𝑔𝜌𝑏ℎ𝑥

(


Function of moments

RegionI.(onlyweight)

𝐹ª 𝑥 = −𝑔𝜌𝑏ℎ𝑥

𝑎ª 𝑥 =𝑥2


𝑀�� 𝑥


(

+ 𝐹" 𝑥 − 𝑎"+ 𝐹( 𝑥 − 𝑎(

Function of moments

RegionII.(weight,F1)

RegionIII.(weight,F1,F2)

𝑀�� 𝑥


(

+ 𝐹" 𝑥 − 𝑎"


Integration (1)• Eachregionhastobeintegratedseparately,becausethefunctionisnot

continuous.• Inallthecasesthefunctionofmomentaisapolynomialoforder2.Whichwhen

integratedtwicebecomesapolynomialoforderfour.

𝑧« 𝑥 = 𝑧M + 𝑧"𝑥 + 𝑧(𝑥( + 𝑧\𝑥\ + 𝑧�𝑥�

𝑑𝑧«𝑑𝑥 𝑥 = 𝑧" + 2𝑧(𝑥 + 3𝑧\𝑥( + 4𝑧�𝑥\

𝑑(𝑧«𝑑𝑥( 𝑥 = 2𝑧( + 6𝑧\𝑥 + 12𝑧�𝑥(

𝑘 =𝐸𝑏ℎ\

12𝐸𝑏ℎ\

12𝑑(𝑧«𝑑𝑥( 𝑥 = 2𝑘𝑧( + 6𝑘𝑧\𝑥 + 12𝑘𝑧�𝑥(

Bydefining: Wehave:


𝐴( = 0𝐴\ = 0

𝐴� = −𝑔𝜌𝑏ℎ24𝑘

𝑀�(𝑥) = −12𝑔𝜌𝑏ℎ𝑥

(

𝐴M, 𝐴"

RegionI.

Implies

2𝑘𝑧( + 6𝑘𝑧\𝑥 + 12𝑘𝑧�𝑥( = 𝑀 𝑥

Integration (2)

Unknown

𝑀��(𝑥)= −𝑎"𝐹" + 𝐹"𝑥

−12𝑏𝑔ℎ𝑥

(𝜌

𝐵( = −𝑎"𝐹"2𝑘

𝐵\ =𝐹"6𝑘

𝐵� = −𝑏𝑔ℎ𝜌24𝑘

RegionII.

𝐵M, 𝐵"

𝑧® → 𝐴F, 𝐵F, 𝐶F

RegionIII.

𝑀�� 𝑥= −𝑎"𝐹" − 𝑎(𝐹( + (𝐹"+ 𝐹()𝑥 −

12𝑏𝑔ℎ𝑥

(𝜌

𝐶( = −𝑎"𝐹" + 𝑎(𝐹(

2𝑘

𝐶\ =𝐹" + 𝐹(6𝑘

𝐶� = −𝑏𝑔ℎ𝜌24𝑘

𝐶M, 𝐶"

Implies Implies

Unknown Unknown

Notationchange!!!


Boundary conditions1.Continuitybetweenregions+ attbese pointsdeformationiszero

𝑧� 𝑎" = 0𝑧�� 𝑎" = 0𝑧�� 𝑎( = 0𝑧�� 𝑎( = 0

𝑧� 𝑎" = 𝑧��(𝑎")𝑧�� 𝑎( = 𝑧��(𝑎()

2.Derivability(continuityof1stderivative)betweenregions

ThesystemislinearonA0,A1,B0,B1,C0,C1,andcanbesolvedwithuniquesolution(usingMathematica).

è SeeMathematicaNotebook:GravitySagCalculation.nb


−500 0 500−300

−200

−100

0

100

200

300

Position [mm]

Surfa

ce h

eigh

t [nm

]

BesselEnds

Example of gravity sag correctionAmirrormeasuredwithsupportsattheBesselpointsandattheends.Gravitysagisremovedinbothcases.


Gravity sag of arbitrary profile mirror

𝐸𝑏 𝑥 ℎ\

12𝑑(𝑧«𝑑𝑥( 𝑥 = 𝑀« 𝑥 +𝑀" 𝑥 +𝑀( 𝑥

Thewidthofthemirrorisnotconstant

MG:gravity,M1:support1,M2:support2

CalculatingMG

• Massattheleftofx:

• Masscenter:

M1,M2

𝑚� 𝑥 = 𝜌ℎ � 𝑏 𝑥u 𝑑𝑥u=

�±/(

𝑥� 𝑥 = � 𝑥′𝑏 𝑥u 𝑑𝑥u=

�� (⁄� 𝑏 𝑥u 𝑑𝑥u=

�� (⁄³

𝑀F 𝑥 = 𝑥 − 𝑥F 𝐹F𝜃 𝑥 − 𝑥F

𝜃 𝑥 Heavyside Stepfunction

𝐼M 𝑥

𝐼" 𝑥𝐼M 𝑥


Gravity sag of arbitrary profile mirror

𝑑(𝑧«𝑑𝑥( 𝑥 = 12

𝑀« 𝑥 +𝑀" 𝑥 +𝑀( 𝑥 𝐸𝑏 𝑥 ℎ\

• Thewidthofthemirrorisnotconstant.

• Theequationisno-longertheintegralofapolynomial.

• Itcanbeintegratednumerically.

• Allthefunctionsofxbecomevectors(lineararrays)

MG:gravity,M1:support1,M2:support2

𝑓 𝑥 → 𝑓 𝑥A → 𝑓 𝑛Δ𝑥 → 𝑓A


CalculatingMG(x)

• Massattheleftofx:

• Masscenter:

𝑚� 𝑥 = 𝜌ℎ � 𝑏 𝑥u 𝑑𝑥u=

�±/(

𝑥� 𝑥 = � 𝑥′𝑏 𝑥u 𝑑𝑥u=

�� (⁄� 𝑏 𝑥u 𝑑𝑥u=

�� (⁄³

ThesebecomenumericalintegralsIfyoulikeyoucanapplymoresophisticatedintegrationmethods,Simpson’srule,Spline,…

Moment of inertia of the weight

𝑀« 𝑥 = −𝑔𝑚� 𝑥 𝑥 − 𝑥� 𝑥 𝑀« A = −𝑔 𝑚� A 𝑥A − 𝑥� A

𝑚� A = 𝜌ℎΔ𝑥 W 𝑏B

A

BY"𝑥� A = W 𝑥B𝑏B

A

BY"

W 𝑏B

A

BY"

µ

AndthefinalexpressionofMG(x)isalsoavector


Moments of inertia of the supportsCalculatingM1(x),M2(x)

• NeedtoknowtheactualvaluesofF1 andF2 (notjustitsexpressionasafunctionofmirrordimensions).

• TheycanbewrittenincloseexpressionusingtheHeavisidefunctionstepfunction

𝐹" = 𝑔𝑚M𝑥( − 𝑥M𝑥( − 𝑥"

𝐹( = 𝑔𝑚M−𝑥" + 𝑥M𝑥( − 𝑥"

m0:totalmassx0:masscenterofthemirror

𝑀F 𝑥 = 𝑥 − 𝑥F 𝐹F𝜃 𝑥 − 𝑥F

𝑀F A = 𝑥A − 𝑥F 𝐹F𝜃 𝑥A − 𝑥F


Integration• Onceallthefunctionsofxareevaluated,theycanbenumericallyintegrated

• Notethat,sinceweusecloseexpressions,therearenoboundaryconditionsbetweenzones.

• Whichbecomes

• Averageheightandaveragepitchareundetermined.Often,thesearesimplysettoaveragezeroastheydonotcarryinformationaboutthemirrorfigure,butaboutitsposition.

𝑧 𝑥 = 𝑐� + 𝑐"𝑥 + � � 12𝑀« 𝑥′ + 𝑀" 𝑥′ + 𝑀( 𝑥′

𝐸ℎ\𝑏 𝑥′ 𝑑𝑥′="

�� (⁄𝑑𝑥"

=

�� (⁄

𝑧A = 𝑐� + 𝑐"𝑥A +12𝐸ℎ\ W W

𝑀« O + 𝑀" O + 𝑀( O

𝑏O

B

OY"

A

BY"

josep nicolas claude ruget carles...

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