Cocco Erice lecture 1 - CERN · 2 D. Cocco X-Ray optics, Erice, 6-15 April 2011 These regions are very interesting because are characterized by the presence of the absorption edges

11 D. Cocco X-Ray optics, Erice, 6-15 April 2011

Daniele Cocco

Sincrotrone Trieste ScpA, S.S. 14 Km 163.5 in Area Science Park, 34012 Trieste,ITALY

X-ray Optics

Joint US-CERN-Joint US-CERN-Japan-RussiaJapan-RussiaSchool on ParticleSchool on ParticleAcceleratorsAccelerators


These regions are very interesting because are characterized by the presence

of the absorption edges of most low and intermediate Z elements !photons with these energies are a very sensitive tool for elemental andchemical identificationBut… these regions are difficult to access.

4-30eV300-40nm

30-250eV40-5nm

250eV - several keV

few keV - 50-100 keV

Energy regionsEnergy regions


!="µl/4# µl=linear absorption coefficient

refractive index µ=1$%$i!

% (unit decrement) related to the speed in the medium

! related to the absorption

%=(e2"2/2#mc2)|N+&HNH["/"H]2ln["H2/"2-1] |

N=electron density (1023-1024 el./cm3)"H=adsorption edge’s wavelength

" far from "H' %=Ne2"2/2#mc2

Refraction IndexRefraction Index


i

(

n>1

i

n<1

(

Snell’s law: n1cos(=n2cosi

Snell LawSnell Law


i

!

n>1

i

n<1

!S

n>1

I

n<1

S

I

Fermat’s principle

42

2

22

10102

!!!"=

mc

Ne

#

$%

!

1

f= n "1( )

1

R1

+1

R2

#

$ %

&

' ( ) 1"* "1( ) +

2

R< 0

!

" #10$4

HXR % f #1m if R #1mm

Snell LawSnell Law


n<1

S

I

X-ray LensesX-ray Lenses


i

!

n>1

i

n<1

!

Snell’s law: cos(=cosi/n

(=0 n=cosic

ic critical angle: total external reflection

sinic="(e2N/#mc2)1/2

"c(min)=3.333.10-13 N-1/2 sinic

Material Density

(g/cm3)

N

(electron/cm3)

!min

nm

Pentadecane (oil) 0.77 7x1022

64.1sini

Glass 2.6 78x1022

37.9sini

Aluminum oxide 3.9 115x1022

31.2sini

Gold 19.3 466x1022

15.4sini

i=5o: "minglass=3.3nm=375 eV"mingold=1.34nm=923eV

shorter wavelength needs smaller angles of incidence

Materials with higher density (i.e. higher atomic weight)

have higher reflectivity

Snell law - Total external reflectionSnell law - Total external reflection


1.0

0.8

0.6

0.4

0.2

0.0

refle

ctivity

2000150010005000

Photon energy (eV)

1o

2o

3o

5o

10o

20o

Gold

Grazing incidence mirror reflectivityGrazing incidence mirror reflectivity

)


)

Hard X-ray reflectivityHard X-ray reflectivity


1.0

0.8

0.6

0.4

0.2

0.0

refle

ctivity

2000150010005000

Photon energy (eV)

Fused Silica

1.0

0.8

0.6

0.4

0.2

0.0

refle

ctivity

2000150010005000

Photon energy (eV)

C

1.0

0.8

0.6

0.4

0.2

0.0

refle

ctivity

2000150010005000

Photon energy (eV)

Ni

1.0

0.8

0.6

0.4

0.2

0.0

refle

ctivity

2000150010005000

Photon energy (eV)

SiC

1.0

0.8

0.6

0.4

0.2

0.0

refle

ctivity

2000150010005000

Photon energy (eV)

Al

1.0

0.8

0.6

0.4

0.2

0.0

refle

ctivity

2000150010005000

Photon energy (eV)

Au

"=2o

Other coatingsOther coatings

)


*2*r'

+s'=2 r'*)

Tangential focusing

Image

Object

r

r’ "

Effect of Defects (slope errors)Effect of Defects (slope errors)


-10

-5

0

5

10

ve

rtic

al p

ositio

n (µ

m)

-10 -5 0 5 10

horizontal position (µm)

s' = ,(Ms)2 + (2 r'*)2s' = ,(Ms)2 + (2 r'*)2

*2*r'

+s'=2 r'*)

Tangential focusing

Image

Object

r

r’ "

Image (spot) enlargementImage (spot) enlargement


+s'=2 r'*

*2*r'

50

40

30

20

10

0S

lope e

rror

contr

ibution F

WH

M (µ

m)

1086420

Slope error [rms] (µrad)

Distance mirror-image = 1m

50

40

30

20

10

0

Slo

pe e

rror

contr

ibution F

WH

M (µ

m)

1086420

Distance mirror-image (m)

Slope error = 2.5 µrad

)



Typical manufacturer capabilities (SESO, ZEISS, Winlight, Jobin Yvon)

- 2 µrad> 500 mmAspherical

- 1-2 µradUp to 500 mmAspherical

- 1-2 µrad> 500 mmToroidal

- 1 µradUp to 500 mmToroidal

1 µrad> 500 mmSpherical/flat

< 0.5 µradUp to 500 mmSpherical/flat

rms errorsLengthShape 10

8

6

4

2

0S

lope e

rror

contr

ibution F

WH

M (µ

m)

2.01.51.00.50.0

Slope error [rms] (µrad)

Distance mirror-image = 1m


r’=1m

+s'=1 µm

+s'=4 µm

+s'=10 µm


Typical manufacturer capabilities (SESO, ZEISS, Winlight, Jobin Yvon)

-10x10-6

-5

0

5

10

resid

ual heig

ht (m

m)

100806040200

mirror position (mm)

Mirror profile precisionMirror profile precision


Term F20 of the optical path function

(1/r+1/r’)cos"/2=1/R spherical mirror

Tangential focusing

Image

Object

r

r’ "

Object

Sagital focusing

Image

Term F02 of the optical path function

(1/r+1/r’)/(2cos")=1/R cylindrical/toroidal mirror

)

*t 2*t

r'

*s

image

+s't=2 r'*t

+s's=2 r' cos" *s

"cos"

Mirror surface

sin"

Tangential and Tangential and Sagital Sagital focusing geometriesfocusing geometries


Spherical mirror suffer of spherical aberration

Solution: work in 1:1 configuration

source-image

AberrationsAberrations

Deviations from perfect imaging are called aberrations


The bicycle tyre toroid is generated by rotating a circle of radius . in an arc of

radius R. In general, two non-coincident focii are produced: one in themeridional plane and one in the sagittal plane

Tangential focus:

Sagittal focus:

Stigmatic image:

J.B. West and H.A. Padmore, Optical Engineering, 1987

Torodial Torodial mirrormirror

!

1

r+1

" r

#

$ %

&

' ( cos)

2=1

R

!

1

r+1

" r

#

$ %

&

' (

1

2cos)=1

*


Tangential focus: Sagittal focus:

Stigmatic image:

Toroidal Toroidal mirror focal propertiesmirror focal properties

!

1

r+1

" r

#

$ %

&

' ( cos)

2=1

R

!

1

r+1

" r

#

$ %

&

' (

1

2cos)=1

*


For .=R ! spherical mirrorA stigmatic image can only be obtained at normal incidence.For a vertical deflecting spherical mirror at grazing incidence the horizontalsagittal focus is always further away from the mirror than the vertical tangentialfocus. The mirror only weakly focusses in the sagittal direction.

A

x

y

z

O

T

S

M N

L

K

TK

TL

SN

SM

Toroidal Toroidal mirror focal propertiesmirror focal properties


A

y

z

O

T

S

M N

L

K

TK

TL

Toroidal Toroidal mirror Tangential and mirror Tangential and Sagital Sagital focusfocus


source image

source 80 µm vertical; r=4000 mm r´=400 mm (10:1) )=88o

Beam divergence 100X100 µrad

Spherical

No slope errors

Elliptical

No slope errors

Sagittal cylinder

No slope errors

Other focusingOther focusing geometriesgeometries


Spherical

1µrad slope errors

Elliptical

5µrad slope errors

Sagittal cylinder

5µrad slope errors


Spherical

No slope errors

Elliptical

No slope errors

Sagittal cylinder

No slope errors


Spherical mirrors are good for small demagnification and/or small divergenceElliptical mirrors are better for very large demagnifications and larger divergence but..

the slope errors have to be smallToroidal / parabolic mirrors are perfect if the induced aberration are acceptable

In search of the perfect focusing geometryIn search of the perfect focusing geometry


Rays traveling parallel to the symmetry axis OX are all focused to a point A.Conversely, the parabola collimates rays emanating from the focus A.

Line equation:

Paraboloid equation:

where:

Position of the pole P:

Paraboloid equation:

J.B. West and H.A. Padmore, Optical Engineering, 1987

ParaboloidsParaboloids


Slope errors = every deviation from the ideal surface with period larger then ~ 1,2 mm

Roughness = every deviation from the ideal surface with period smaller then ~ 0.5-1 mm

Typical definition is µrad or arcsec rms.

Alternative definition is "/10 or "/20 and so on… P-V or rms

used for normal incidence mirror or “poorer” quality mirrors

Typical definition is Årms.

Alternative definition is surface quality 20-10 or 10-5 (scratch-dig)

used for normal incidence mirror or “poorer” quality mirrors

A dig is nearly equal in terms of its length and width. A scratch could be much longer then width

20-10 means 20/1000 of mm max scratch width 10/100 mm max dig dimension

Other mirror defect - RoughnessOther mirror defect - Roughness


2sin4

0

!"

#$%

&'

= (

)*+

eII

RoughnessRoughness

!

" =1

ns x( ) # s x( )[ ]

x= 0

n

$2


RoughnessRoughness


Roughness (SF>1mm-1)

Slope errors

(SF<0.5-0.2 mm-1)

Power spectral densityPower spectral density


RoughnessRoughness


0.01

0.1

1

10

Inte

nsi

ty r

educt

ion (

%)

2000180016001400120010008006004002000

Photon energy (eV)

3 Å rms

10 Å rms

3 standard 1 best

Spherical/Flat

5 standard 3 best

(1-2 if very lucky)

Roughness (Å)

Toroidal/asphericalShape

2sin4

0

!"

#$%

&'

= (

)*+

eII

Flux reductionFlux reduction


Synchrotron Radiation BeamlinesSynchrotron Radiation Beamlines

Side view

Top view

Pin

hol

e

Pre

focu

sing

Ent

ranc

e sl

it

Mon

ochr

omat

or

Exi

t sl

it

Def

lect

ion

mir

ror

Ver

tica

l fo

cusi

ng

Hor

izon

tal

focu

sing

VF

M

HF

M

Prefocusing section: Adapt the source to the monochromator requirements

Adsorb the unwanted power radiation

Monochromator:Select the proper photon energy

Refocusing section: Adapt the spot shape at the necessity of the experiment

3232 D. Cocco X-Ray optics, Erice, 6-15 April 2011D. Cocco - ASEAN Training Course on the Use of Synchrotron Radiation 23 April – 4 May 2007 NSRC

Side view

Top view

Pin

hol

e

Pre

focu

sing

Ent

ranc

e sl

it

Mon

ochr

omat

or

Exi

t sl

it

Def

lect

ion

mir

ror

Ver

tica

l fo

cusi

ng

Hor

izon

tal

focu

sing

VF

M

HF

M

There are several reasons to choose a mirrorsubstrate, one is the power arriving on it

Synchrotron Radiation BeamlinesSynchrotron Radiation Beamlines


e-

h/

Vertical

Horizontal

SR sourcesSR sources


Vertical

Horizontal

400W ! 1µm,; 10 to 100 time larger than wished

Thermal deformationsThermal deformations


Density Young’s modulus Thermal

expansion

Thermal

conductivity

Figure of

meritgm/cc GPa (!) ppm/

oC (k) W/m/

oC k/!

Fused silica 2.19 73 0.50 1.4 2.8

Zerodur 2.53 92 0.05 1.60 32

Silicon 2.33 131 2.60 156 60

SiC CVD 3.21 461 2.40 198 82

Aluminum 2.70 68 22.5 167 7.42

Copper 8.94 117 16.5 391 23.7

Glidcop 8.84 130 16.6 365 22

Molybdenum 10.22 324.8 4.80 142 29.6

Properties of typical mirrorProperties of typical mirror materialsmaterials

Bulk material

Reflecting coating


600 mm

110 mm

SiliconSilicon bulk mirrorsbulk mirrors


Direct side coolingDirect side cooling


70

60

50

40

30

20

10

0m

irro

r d

efo

rma

tio

n (

nm

)

300 250 200 150 100 50 0

mirror position (mm)

Object

Sagital focusing

Image

*s

image

+s's=2 r' cos" *s

1st mirror sagittally oriented

Direct side coolingDirect side cooling


Internally cooled mirrorsInternally cooled mirrors

5 standard 2-3best

Roughness (Å)

Metallic

3 standard 1 best

Spherical/Flat

Roughness (Å)

Glass/Silicon

Shape

0.01

0.1

1

10

Inte

nsi

ty r

educt

ion (

%)

2000180016001400120010008006004002000

Photon energy (eV)

3 Å rms

10 Å rms


3GeV Synchrotron source3GeV Synchrotron source

6.6 cm period undulator K6.6 cm period undulator Kmaxmax=5.7=5.7

BL6.1BL6.1

1.51.5o o grazing incidencegrazing incidence

++T=7.7T=7.7o o

1.51.5o o grazing incidencegrazing incidence

++h=17h=17µµmm slope 26 slope 26 µµradrad

glidcop

Internally cooled mirrors (Glidcop)Internally cooled mirrors (Glidcop)


INVAR®INVAR®

Carpenter Technology Inc.Carpenter Technology Inc.

Alloy 36 iron-nickel(36%) alloy with carbon (0.02%),Alloy 36 iron-nickel(36%) alloy with carbon (0.02%),

manganese (0.35%), Silicon (0.2%)manganese (0.35%), Silicon (0.2%)

Supernvar: iron-nickel(32%) alloy with carbonSupernvar: iron-nickel(32%) alloy with carbon

(0.02%), manganese (0.40%), Silicon (0.25%), Cobalt(0.02%), manganese (0.40%), Silicon (0.25%), Cobalt

(5.5%)(5.5%)

Invar & Invar & SuperInvarSuperInvar


++h=6h=6µµmm ++T=130T=130o o

SuperInvarSuperInvar


Contamination process:

Hydrocarbons adsorbed by the surface

Cracking induced by the incoming radiation

Formation of graphitic carbon layer (mixed C coumpond)

Effect of the contamination:

Strong adsorption at the carbon edge (0270 eV)

Reduction of reflectivity due to enanchment of the surface roughness

general deterioration of the surface

Carbon ContaminationCarbon Contamination






250 eV 260 eV 270 eV 280 eV 290 eV 300 eV 310 eV

Carbon ContaminationCarbon Contamination


Contamination process:

Hydrocarbons adsorbed by the surface

Cracking induced by the incoming radiation

Formation of graphitic carbon layer (mixed C coumpond)





+ 300-500 V (DC)

I 100 mA-1A

P 0.5-1 mbar O2

Mirr

or su

rface

UV lamp discharge

Carbon Contamination and CleaningCarbon Contamination and Cleaning



Strong adsorption at the O/Cr edge



40x10-3

30

20

10

Inte

nsity (

a.u

.)

550540530520510

Photon Energy (eV)

536.2 eV

542.8 eV

CO contamination

0.4

0.3

0.2

0.1

Inte

nsity (

a.u

.)

600580560540520500

Photon Energy (eV)

571 eV

578 eV

Cr contamination

Other ContaminationOther Contamination


SoftX-ray

I.R. U.V.VisibleMicrowave

HardX-ray

limit ~ 1-2 keV ( 1 nm)

SoftX-ray


HardX-ray

Zero order

External Orders (-)

Internal Orders (+)

d

!1

d

d sin(1) d sin(!

)

( ) ( )( )!"# sinsin $= dn

SoftX-ray


HardX-ray

Dispersive elementsDispersive elements


Laminar profile

whBlaze profile

)(

Blaze gratings: higher efficiency

Laminar gratings: Higher spectral purity

Higher resolution Blaze conditionBlaze condition

Blaze angle=(Blaze angle=(1+!1+!)/2)/2

( ) ( )( )!"# sinsin $= dnn#1#

2 #2

Zero order

External Orders (-)

Internal Orders (+)

d

!1

GratingGrating’’s profiless profiles


50

40

30

20

10

Gra

tin

g e

ffic

ien

cy (

%)

800600400200

Photon energy (eV)

Laminar grating Blaze grating

10

8

6

4

2

0

Re

lative

eff

icie

ncy (

1st

ord

/2nd

ord

)

800600400200

Photon energy (eV)

Laminar grating

Blaze grating

( ) ( )( )!"# sinsin $= dn

Laminar profile

whBlaze profile

)(

GratingGrating’’s efficiencys efficiency


Laminar profile

wh

5

4

3

2

1

0

Eff

icie

ncy (

%)

2015105

groove height (nm)

W=60% gd=2400 l/mm 200 eV

800 eV

10

8

6

4

2

0

Eff

icie

ncy (

%)

87654321

blaze angle (!) (deg)

"=90ogd=2400 l/mm

200 eV 800 eV

Blaze profile

)(

GratingGrating’’s efficiencys efficiency


Mechanical ruling blaze profile ! smaller blaze angles; higer efficiency

Holographically recording laminar and blaze profile (large blaze angle)

! higher groove density; lower spacing disomogeneity

Diamond tool

Gold (Cr) layer

Si substrate

GratingGrating’’s productions production


Exposure

Development

Ion etching

Photoresist

removal

Coating

+ ++

++

+

++

+

++

++

++ +

+ ++

+

++ +

+

+

+

+

++

+

++

+ ++

+

+

++

HolographicHolographic Recorded GratingsRecorded Gratings


Light rays choose their paths to minimize the optical length

!B

A

dlrn )(r

where is the index ofrefraction of the medium and dl isthe line segment along the path

)(rnr

A

B

!!! ==

B

A

B

A

B

A

dtcdlv

cdlrn )(

r

Fermat’s principle is also known as the principle of least time:

A

B

FermatFermat’’s principles principle


where " is the wavelength of the diffracted light, k is the order ofdiffraction (±1,±2,...), N=1/d is the groove density

For a classical grating with rectilinear grooves parallel to z with constant

spacing d, the optical path length is:

ykNPBAPF !++=z

A

OB

P

x

y

1 !

r

r’

za

zb

d

d sin(1) d sin(!

)

Optical pathOptical path


Let us consider some number of light rays starting from A and impingingon the grating at different points P. Fermat’s principle states that if thepoint A is to be imaged at the point B, then all the optical path lengthsfrom A via the grating surface to B will be the same.

B is the point of a perfect focusif:

for any pair of (y,z )

0 0 =!

!=

!

!

z

F

y

F

z

A

O BP

x

y

1 !

r

r’

Optical Path -Optical Path - Focal conditionFocal condition


can be used to decide on therequired characteristics of thediffraction grating, in particularthe shape of the surface, thegrooves density, the object andimage distances.

) 0 0 y,zpair of (for any z

F

y

F=

!

!=

!

!ykNPBAPF !++= +

Equations:

z

A

O BP

x

y

1 !

r

r’

OpticalOptical Path - Focal conditionPath - Focal condition


In general, and are functions of y and z and can not be made zero forany y,z

! when the point P wanders over the grating surface, diffracted rays fall onslightly different points on the focal plane and an aberrated image is formed

y

F

!

!

z

F

!

!

Goal: produce simpleexpressions for theintersection points inthe image planeproduced by the raysdiffracted fromdifferent points on thegrating surface

z

A

OB0

P

x

y

B

1

r

ro’

!0

za

zb0

Aberrated Aberrated imageimage


z

A

O BP

x

y

1 !

r

r’

The grating surface may in general be described by a series expansion:

a00= a10= a01= 0 because of the choiceof origin j = even if the xy plane is a symmetryplane

Giving suitable values to the coefficients aij’s we obtain the expressions forthe various geometrical surfaces.

ji

i j

ijzyax !!

"

=

"

=

=0 0

Grating surfaceGrating surface


0 ;0 ;8

1

;8

1 ;

4

1 ;

2

1 ;

2

1

3012304

3402222002

===

====

aaa

Ra

Ra

Raa

!

!!Toroid

Sphere, cylinder and plane are special cases of toroid:R=. ! sphere

R= 2 ! cylinder

R=.= 2 ! plane

Paraboloid

!

!!!

!!!

!

!!

!

33

2

043

2

40

230212

3

2

222002

cos64

sin ;

64

cossin5

8

cos sin ;

8

tan

; cos32

sin3 ;

4

cos ;

cos4

1

fa

fa

fa

fa

fa

fa

fa

==

"="=

===

Typical surfacesTypical surfaces


1

2

2

2

2

33

2

22

22

2

2

22

33

2

40

22

230

22

212

22

2

33

2

042002

11

2

cos1cos

2

3

cos16

sin

;1sin5cos sin5

cos64

; sin8

sin ;sin

cos8

tan

;1sin

cos64

;4

cos ;

cos4

1

!

"#

$%&

'(

+=

"#

$%&

'))*

+,,-

.!!=

"#

$%&

'+!=

!=!=

"#

$%&

'+===

rrfwhere

a

b

fa

aabf

ba

ef

aef

a

abf

ba

fa

fa

//

/

/

///

/

//

//

/

/

/

/

/

Ellipsoid

Typical surfacesTypical surfaces


( ) ( ) ( )222

zzyyxxAP aaa !+!+!=

( ) ( ) ( )222

zzyyxxPB bbb !+!+!=

!!

""

sin cos

sin cos

ryrx

ryrx

bb

aa

#=#=

==

z

A(xa,ya,za)

O B(xb,yb,zb)P (x,y,z)

x

y

1 !

r

r’

1 < 0; ! > 0

ykNPBAPF !++=

...2

1

4

1

2

1

8

1

4

1

8

1

2

1

2

1

2

1

2

1

211

2

202

2

102111

040

4

220

22

400

4

120

2

300

3

020

2

200

2

011100000

+++++

++++

+++++=

zFyFyyFyzF

FzFzyFyFyz

FyFzFyzFyFFji

ijk

ijk zyFF !=

Optical Path FunctionOptical Path Function


z)(y, ofpair any for 0 0 =!

!=

!

!

z

F

y

F

) 000(ijk allfor 0 !=ijkF

Each term in the series (except F000 and F100)represents a particular type of aberration

ji

ijk zyF

Perfect focal conditionPerfect focal condition


!!

! cos2cos

),( 20

2

ar

rT "=where and !! cos21

),( 02ar

rS "=

andand analogous expressions for ),( !rT " ),( !rS "

( )

( )

( )

( )!"!!

""

!"!!

""

!"

!"!"

!"#

coscos2 sin),(

sin),(

coscos2 sin),(

sin),(

coscos211

coscos2coscos

)sin(sin

12120

30300

02020

20

22

200

100

+$%&

'()

*+

++%&

'()

*=

+$%&

'()

*+

++%&

'()

*=

+$+

+=

+$,,-

.//0

1

++=

+$=

ar

rS

r

rSF

ar

rT

r

rTF

arr

F

arr

F

NkF

rrF !+=000

for r,r’ >> za,zbz

A

O BP

x

y

1 !

r

r’

AberrationsAberrations TermsTerms


Most important imaging errors:

F200 defocusF020 astigmatismF300 primary coma (aperture defect)F120 astigmatic comaF400 F220 F040 spherical aberration

There is an ambiguity in the naming of the aberrations in the grazing incidence case!

!"# Nk=+0

sinsin0100

=F grating equation

Aberrations TermsAberrations Terms


0020

=F sagittal focusing ( ) 0coscos211

02=+!

"+ #$arr

Example: toroidal mirror

Substituting in

and imposing 1 = -! = )

2

1 ;

2

12002

Raa ==

!0;

200=F 0

020=F

Rrrt

1

2

cos

'

11=!!

"

#$$%

&+

'

!"

1

cos2

1

'

11=##

$

%&&'

(+

srr

( ) 0coscos2coscos

020

0

0

22

=+!""#

$%%&

'

(+ )*

)*a

rr0

200=F tangential focusing

The tangential focal distance r’0 is obtained by setting:

The sagittal focal distance r’0 is obtained by setting:

Focal conditionsFocal conditions


)sin(sin0100 !"# $+$= DnF grating equation

!!"

#$$%

&'+'=

RrRrF

(()) cos

'

coscoscos22

200tangential focus

!!"

#

$$%

&

'(()

*++,

-.

'+(

()

*++,

-.=

rRrrRrF

///000 sincoscossincoscos22

300 primary coma

1 !r r`

Optical path function

Spherical GratingsSpherical Gratings


!!"

#$$%

&'+'=

RrRrF

(()) cos

'

coscoscos22

200tangential focus

!!"

#

$$%

&

'(()

*++,

-.

'+(

()

*++,

-.=

rRrrRrF

///000 sincoscossincoscos22

300 primary coma

0300F

200F ==

Rowland Circle Source

Gra

ting Rgrating=2Rrowland circle

!

" r = Rcos#

!

r = Rcos"

Spherical GratingsSpherical Gratings


!!"

#$$%

&'+'=

RrRrF

(()) cos

'

coscoscos22

200)sin(sin0100 !"# $+$= DnF

1 !r r`

Mantain fixed source and image in position and direction

R=30 m; gd=150 l/mm; r=4m; r’=1.5m

Variable Included Angle SphericalVariable Included Angle Spherical Grating MonochromatorGrating Monochromator


Plane mirror

Spherical/plane grating

Entrance slit/source

Exit slit/image

!!"

#$$%

&'+'=

RrRrF

(()) cos

'

coscoscos22

200)sin(sin0100 !"# $+$= DnF

"mirror=($+%)/2

Variable Included Angle Spherical Grating MonochromatorVariable Included Angle Spherical Grating Monochromator


Plane mirror



Exit slit/image

h mirror axis ~ h/2 grating center



Plane mirror



Exit slit/image



200 400 600 800 1000 1200 1400 160000

5

10

15

20

25

30

35

40

Eff

icie

ncy

(%)

Efficiency CurvesEfficiency Curves


Source

Grating

40

20

0

-20

CC

D v

ert

ica

l p

ositio

n (µ

m)

-1.0 -0.5 0.0 0.5 1.0

Photon Energy (eV)

1.0

0.8

0.6

0.4

0.2

0.0In

tensi

ty (

a.u

.)

45.00245.00145.00044.99944.998

Photon Energy (eV)

FWHM = 1.6 meV

Plane mirror

Spherical grating

Entrance slit

Exit slit

Exit slit

10 µm

Resolving PowerResolving Power


( )Nkr

sentrance

!"

cos#=$ entrance slit contribution

( )rNk

sexit

!

"!=#

$%

cosexit slit contribution

( ) ( )!"# sinsinNk $=

( )r

s

Nk=!="

#

$%&

'

(

()

)

)

* cos

( )r

s

Nk !

!="=##

$

%&&'

(

)

)*

*

*

+ cos

Resolving power = "/+" = E/+E

45/0.0016 0 28000

1.0

0.8

0.6

0.4

0.2

0.0In

tensi

ty (

a.u

.)

45.00245.00145.00044.99944.998

Photon Energy (eV)

FWHM = 1.6 meV

Plane mirror

Spherical grating

Entrance slit

Exit slit

smaller are s and s´,smaller will be the bandpass



Typical Spherical grating monochromator resolving power

( )!"

"

"

cos#$

$=

%=

% s

rNk

E

E



Typical Spherical grating monochromator resolving power +50-250 V

I(nA) Gas inlet

Photon in

4

3

2

1

Int

en

sit

y (

ar

b.

un

its

)

402.0401.5401.0400.5

Photon Energy [eV]

Gaussian Width

FWHM: 34 ± 3 meV

Experimental Data

Fitted Data

N2 1sE/+E > 11000

Resolving Power measurementResolving Power measurement


$

%

Real Source

r

Virtual Source r’

fCrr 2=!

!

"

cos

cos=fC

!!"

#$$%

&'+'=

RrRrF

(()) cos

'

coscoscos22

200

0cos

'

coscoscos22

200=!!

"

#$$%

&

'(+

'(=

))**

rrF 0

'

coscos22

=+rr

!"

!

"2

2

cos

cos' rr #=

Plane GratingPlane Grating


ellipsoidal

mirror

exit

slit

source

plane

grating

$%

Fixed virtual

source

Entrance

slit25.2=fC

r

r!

r!=2.252 r

SuperESCA at Elettra, r~4500 mm, r!~22800 mm

SX 700SX 700


Source

M1, Sagittal cylindrical collimator16m from source

Grazing incidence angle 1°

M3, Sagittal toroidal focusing mirror20m from the source

Grazing incidence angle 1°M2, Plane mirror

Plane gratings18m from source

Pin hole22m from source

0cos

'

coscoscos22

200=!!

"

#$$%

&

'(+

'(=

))**

rrF !="=+

!'0

'

coscos22

r

r

#$

One can select to work in:High resolution mode (accept to loose some flux)High efficiency mode (accept a reduction of resolution)High order suppression mode (with a typical appreciable reduction of flux)

Collimated light SX 700Collimated light SX 700


Source

M1, Sagittal cylindrical collimator16m from source

Grazing incidence angle 1°

M3, Sagittal toroidal focusing mirror20m from the source

Grazing incidence angle 1°M2, Plane mirror

Plane gratings18m from source

Pin hole22m from source

!="=+!

'0'

coscos22

r

r

#$

fCrr 2=!

In principle one can work with any Cf value, higher or lower than 1 but…

This mirror do not produce a perfectly collimated light (NEVER)

! divergence changes with Cf

This mirror is no more able to focus the radiation

Problem amplified for Cf value lower than 1

CollimatedCollimated Light SX 700 Light SX 700


!!!!++++= 3x3D2x2Dx1D0DD(x)

!

F200

= 12"n#D

1+ cos2$

r" cos$

R+

cos2%& r " cos%

R

'

(

) )

*

+

, ,

'

(

) ) )

*

+

, , ,

!

F300

="13

n#D2

+12

cos2$r

" cos$R

%

&

' '

(

)

* *

sin$r

+cos2+, r " cos+

R

%

&

' '

(

)

* *

sin+, r

-

.

/ / /

0

1

2 2 2

Groove density D varies along the grating surface:

x

!

F200

= 1

2"n#D

1+ cos2$

r+

cos2%& r

'

(

) )

*

+

, ,

'

(

) ) )

*

+

, , ,

A plane grating can focus!

Dn!"# $= sinsin

)sin(sin100 !"# $+$= DnF D> ; %<

D< ; %>

%

VariableVariable groove groove desnity desnity gratingsgratings

Documents

Cocco Erice lecture 1 - CERN · 2 D. Cocco X-Ray optics, Erice, 6-15 April 2011 These regions are very interesting because are characterized by the presence of the absorption edges