NCNR Summer School '06

Reflectometry Reduction and Analysis

Paul Kienzlepaul.kienzle@nist.gov

Experimental Setup

Slit 1Slit 2

QZ

Log I Specular Scanθ2 = 2θ

QZ

Background Scanθ2 ≠ 2θ

Log I

White Beam

θ

I Rocking Curveθ or θ2 fixed

Data Reduction

Slit 3

Detector

Slit 4θ2

Sample

A=Repeat each curve for:

D=−−++

B=+−C=−+

Polarizer andFlipper (+/−) Polarizer and

Flipper (+/−)

Detector

Monochromator

QZ

I

Slit Scanθ2 = 0

Fixed slits

θ

What is it good for?

Subsurface structure up to 1μmPolymers, biofilms, magnetic surfaces, ...Determines average density at depth z

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

-100 0 100 200 300 400 500 600 700 800

rho

(num

ber d

ensi

ty)

Depth (Ang)

Optical Matrix Formalism

jQS ijii

8)(162

1 2

)cosh()sinh(/)sinh()cosh(

iiiii

iiiiii SdSdS

SSdSdM

nMMMr 21

Oscillations in reflectivity R(Q) of periodd2

d i

z

translates reflectivity into lab frame

)()( rfQR where f

Fitted Data

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Ref

lect

ivity

Q (inverse Angstroms)

data 0fit 0

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

-100 0 100 200 300 400 500 600 700 800

rho (num

ber density)

Depth (Ang)

χ2 Landscape (ρ2 vs d2)

χ2 Landscape (d2 vs d3)

Heuristics

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Ref

lect

ivity

Q (inverse Angstroms)

data 0fit 0

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

-100 0 100 200 300 400 500 600 700 800

rho (num

ber density)

Depth (Ang)

170

7100.0085 ≈2π/740

0.035 ≈2π/180

Prior Knowledge

-1e-06

0

1e-06

2e-06

3e-06

4e-06

5e-06

6e-06

7e-06

-20 0 20 40 60 80 100 120 140 160 180

rho

(num

ber d

ensi

ty)

Depth (Ang)

Current best profile for each model

rho 0rho 1rho 2

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Ref

lect

ivity

Q (inverse Angstroms)

Current best fit for each model

data 0fit 0

data 1fit 1

data 2fit 2

Simultaneous Fitting

Our Problem

Many local minima'Garden Path' fit spaceExpensive objective functionContinuous but no analytic derivativeSignificant number of parameters... but many priors

E.g., known material, known sputtering time, information from other measurements, theoretical models, bounds constraints

There is hope for ye who enter.