1. Reverse Engineering Using Optical Constants
Physical Characterization of X-Ray and EUV Mirrors Using the Pearson Correlation Coefficient
Nathan Powell, Steve Turley,
David Allred, David Oliphant

2. Context
We design and manufacture thin-film extreme ultraviolet and x-ray range mirrors
Physical properties of the mirror cannot be measured directly
We must use the optical properties of the mirror to deduce the physical properties.

3. Modeling Mirror Optics
Parrat’s Recursion Formula
This formula takes into account characteristics of the mirror such as:
Layer Materials (Their Indices of Refraction at the wavelength of light that we’re using)
Layer Thicknesses
Layer Arrangements
Roughness
Light
To simulate the reflectivity of the mirror at different angles

4. Comparing Model with Physical
Compare the calculated reflectivity of the mirror with that measured using X-Ray diffraction
The Pearson Correlation Coefficient has the following Properties:
Reflects degree of linear relationship between two variables
Ranges from -1 to +1
+1 correlation corresponds to perfect linear correlation between variables
-1 correlation corresponds to perfect negative correlation between two variables
0 value corresponds to no linear correlation between variables

7. Graph Calculation Algorithm (While varying two characteristics)
Characteristic c1;
Characteristic c2;
for c1val from c1.startVal to c1.endVal:
for c2val from c2.startVal to c2.endVal:
// calculate reflectivity graph
PearsonGraph[c1val][c2val]=
PearsonCC(calculatedGraph, measuredGraph)
DisplayGraph(PearsonGraph)

8. N-Dimensional Graph Algorithm
Characteristic ca[NUM_CHARACTERISTICS];
for c1Val from ca[1].startVal to ca[1].endVal:
for c2Val from ca[2].startVal to ca[2].endVal: …
for cnVal from ca[n].startVal to ca[n].endVal:
// calculate reflectivity graph
PearsonGraph[c1val][c2val]..[cnval] =
PearsonCC( calculatedGraph, measuredGraph );
//Since higher dimensional graphs are currently difficult to display:
ListOfPatterns = DoPatternRecognitionAlgorithm(PearsonGraph);

9. Algorithm Performance
Performance of the algorithm:
Where p is the number of data points when measuring the reflectance versus angle, and di is the resolution of the ith characteristic.

10. Performance Issues
Algorithm becomes useful when the number of characteristics > 2
Computation time dramatically increases for each dimension added
Information produced by algorithm becomes less useful with time
Conclusion: we must find ways to increase performance

11. Ways to optimize algorithm
Save Intermediate Steps of Parrat’s Recusion Formula
Concentrate majority of computation near interesting features
Use more efficient math algorithms such as blas and lapack
Enhance Cache Coherency
Parallelize:
Pearson Graph Generation
Pattern Recognition

12. Current Research Using parallel processors on the super computers to:
Increase the number of characteristics explored simultaneously by increasing the number of dimensions in the search space
Increase the algorithm response time
Automate the pattern recognition algorithms to aid the researcher in characterization – especially necessary for higher dimensions, where there is no visual tool available.