It's Rough Reflecting Short Wavelengths - Physics and Astronomy ...

Introduction 

Handling Surface Reections 

Details of Exact Integral Equations 

Results 

Summary 

It's Rough Reecting Short Wavelengths 

R. Steven Turley 

Department of Physics and Astronomy 

Brigham Young University 

The University of Mississippi 

April 24, 2012 

R. Steven Turley It's Rough Reecting Short Wavelengths

Outline 

Introduction 



Results 

Summary 

1 Introduction 

Why XUV is Interesting 

Challenges 

2 Handling Surface Reections 

Problem Denition 

Short and Long Wavelength Limits 

Arbitrary Accuracy 

3 Details of Exact Integral Equations 

Model Problem 

Method of Moments 

Numerical Quadrature 

Nyström Technique 

4 Results 


Introduction 



Results 

Summary 

Electromagnetic Spectrum 

XUV Interest 

Challenges 


Applications 

Introduction 



Results 

Summary 

XUV Interest 

Challenges 

smaller, faster computers 

high-resolution light microscopes 

plasma images 

fusion reactors 

stars 

Earth's magnetosphere 

nuclear weapons 


Challenges 

Introduction 



Results 

Summary 

XUV Interest 

Challenges 

air is black (strongly absorbing) 

index of refraction for everything is complex 

real part is close to one (low reectivity) 

imaginary part is signicant (high absorption) 

surface contamination is deadly 

oxidation reduces reection 

short wavelength 

uniformity of layer thicknesses is critical 

very sensitive to surface irregularities 



Introduction 



Results 

Summary 




Incident plane wave 

Interaction with optical surface 

induced surface current 

Final State 

direct interaction from incident wave 

self interaction from other parts of surface 

continuing incident wave 

scattered (reected wave) 

interested in the far eld 


Simplications 

Introduction 



Results 

Summary 




To better visualize qualitative features of process 

2d scattering 

wave in x-y plane 

wave and scatterer invariant in the z direction 

scalar wave 

no polarization 

only one eld 

perfect electrical conductor 

eld is zero at the surface 


Plane Wave 

Introduction 



Results 

Summary 




ψ(x,y) = R 

(e ) 

i(⃗ k·⃗r−ωt) 

= cos(k x x + k y y − ωt) 

k = 2πn 

λ 0 

ω = 2πν 

take real part at end 

drop time dependence 

focus on peak locations 


Scattering Process 

Introduction 



Results 

Summary 




incident 

scattered 

surface 


Introduction 



Results 

Summary 

Short Wavelength Roughness 




short wavelength limit: geometrical optics (ray tracing) 

leaves out wave nature of light (diraction) 

add diraction 

physical optics 

leaves out 

self interactions of currents on mirror 

surface traveling waves 


Ray Optics 

Introduction 



Results 

Summary 




replace phase fronts with 

vectors in direction of 

propagation 

apply Snell's Law at each 

point on reecting surface 

allow multiple bounces 

shadow regions 

unilluminated 


Surface 

Introduction 



Results 

Summary 





Introduction 



Results 

Summary 

Far Field Distributions 

Gaussian Noise in Surface Angle 





Adding Diraction 

Physical Optics 

Introduction 



Results 

Summary 




assume surface is locally smooth 

compute surface current based on what one would get from an 

innite smooth surface 

in far eld 

∫ 

φ(x,y) ∝ √ eikρ 

kρ 

S 

J(x ′ ,y ′ )e i⃗ k i ·⃗s ′ d⃗s ′ 

ρ = √ x 2 + y 2 

) 

J(⃗s) = 

(e i⃗p·⃗s + e i⃗q·⃗s sinθ 


Introduction 



Results 

Summary 

Scattering Amplitude 




φ(ρ,θ f ) ∝ eikρ 

√ 

kρ 

f (θ f ) 

f (θ f ) = 8 

kL ei3π/4 sin(θ i ) sin(δ) 

δ 

δ = kL 2 [cosθ i − cosθ f )] 


Reected Intensity 

Smooth Plate, kL=200 

Introduction 



Results 

Summary 




0.0008 

0.0006 

0.0004 

0.0002 

35 40 45 50 55 60 


Reected Intensity 

Rough Plate, kL=200 

Introduction 



Results 

Summary 




3000 

2500 

2000 

1500 

1000 

500 

35 40 45 50 55 60 

degrees 


More Roughness 

Rougher Plate, kL=200 

Introduction 



Results 

Summary 




intensity 

2500 

2000 

1500 

1000 

500 

35 40 45 50 55 60 

degrees 


Introduction 



Results 

Summary 

Dielectric Rough Plate 




more complete solution 

E s z = kg(ρ)e i 3π 4 

∫ 

r s exp { ik[x ′ (cosθ i − cosθ r ) − y ′ (sinθ i + sinθ r )] } 

dy 

× 

(S ′ ) 

sinθ i + sinθ r − cosθ r 

dx ′ dx ′ 

r s is Fresnel reection coecient 

√ 

S = 

1 + 

( 

dy ′ 

dx ′ ) 2 

R. Steven Turley 

It's Rough Reecting Short Wavelengths

Dielectric Plate 

kL = 200π, 2000π 

Introduction 



Results 

Summary 





Introduction 



Results 

Summary 

Dielectric and Conducting Plate 

kL = 200π 





Introduction 



Results 

Summary 

Intermediate Wavelengths 




diraction is a large correction, sometimes dominant 

solution 

develop integral equation using Greene's function 

more robust than dierential equation solutions 

compact domain for homogeneous materials 

solve equation numerically using Nyström technique 

ecient and high order 


Introduction 



Results 

Summary 

Exact Integral Equations 

Model Problem 




Research 

vector wave equation in 2d and 3d derived form Maxwell's 

Equations 

special care taken with singularities and boundary conditions 

Scalar analog 

no polarization 

simple Dirichlet boundary condition 

illustrates techniques and basic results 


Introduction 



Results 

Summary 

Scalar Wave Equation 

Model Problem 




( 

∇ 2 − 1 

c 2 ∂ 2 

∂ t 2 ) 

ξ (⃗r,t) = 0 

separation of variables (or Fourier Transform) 

ξ (⃗r,t) = ψ(⃗r)τ(t) 

c 2 τ(t)∇ 2 ψ(⃗r) − ψ(⃗r)τ ′′ (t) = 0 

c 2 ∇2 ψ( ⃗ r) 

ψ(⃗r) 

− τ′′ (t) 

τ(t) = 0 . 

set both terms equal to −ω 2 = −k 2 c 2 


Introduction 



Results 

Summary 

Single Frequency Solutions 

Model Problem 




( 

∇ 2 + k 2) ψ(⃗r) = ρ(⃗r) 

τ ′′ (t) = −ω 2 τ(t) 

τ(t) = e ±iωt 

use e iωt and allow negative frequencies 

restrict solution to single frequency (and wavelength) 

general time-domain solutions built by superposition 


Introduction 



Results 

Summary 

Greene's Function Solution 

Model Problem 




( 

∇ 2 + k 2) G(⃗r,⃗r ′ ) = δ(⃗r −⃗r ′ ) 

solution is Fredholm Integral Equation 

∫ 

ψ(⃗r) = φ(⃗r) + G(⃗r,⃗r ′ )ρ(⃗r ′ )d⃗r ′ 

S 

( 

∇ 2 + k 2) φ(⃗r) = 0 

choose plane wave solutions for φ for our case. 

ψ is the total eld and ξ = ψ − φ is the scattered eld 

On surface, Dirichelt Boundary condition requires ψ = 0 

∫ 

φ(⃗r) = − G(⃗r,⃗r ′ )ρ(⃗r ′ )d⃗r ′ 

S 


Introduction 



Results 

Summary 

Model Problem 




Validity of Greene's Function Solution 

( 

∇ 2 + k 2) ψ = [( ∇ 2 + k 2) φ(⃗r) ] ∫ 

+ 

( 

∇ 2 + k 2) G(⃗r,⃗r ′ )ρ(⃗r ′ )d⃗r ′ 

∫ 

= [0] + δ(⃗r −⃗r ′ )ρ(⃗r ′ )d⃗r ′ 

S 

= ρ(⃗r) 

S 

legitimacy of switching order of integration and dierentiation 

surface should be smooth 

care required for surface charge distribution 

less of a problem for Dirichlet boundary conditions 


Greene's Function 

Introduction 



Results 

Summary 

Model Problem 




nd solution to 

( 

∇ 2 + k 2) G(⃗r,⃗r ′ ) = δ(⃗r −⃗r ′ ) 

which satises boundary conditions of outgoing spherical wave 

for large r 

skipping details 

complication 

Y 0 (z) is singular at z = 0 

G(⃗r,⃗r ′ ) = i 4 H(1) 0 (kr) 

r = |⃗r −⃗r ′ | 

H (1) 

0 (z) = J 0(z) + iY 0 (z) 


Introduction 



Results 

Summary 

Solving Integral Equation 

Model Problem 




∫ 

φ(⃗r) = − G(⃗r,⃗r ′ )ρ(⃗r ′ )d⃗r ′ 

S 

φ is known (incident plane wave) 

G is known (Greene's function listed above) 

solve for ρ inside integral 

original approach: Method of Moments 

current approach: Nyström Technique 


Current Expansion 

Introduction 



Results 

Summary 

Model Problem 




Expand unknown ρ in a series of compact basis functions 

Tricks for ecient success 

b i 

b i 

−ρ(⃗r) = ∑I i b i (⃗r) 

i 

have correct symmetry 

have correct physics (reduces number of required terms) 

substitute into integral equation 

∫ 

[ ] 

φ(⃗r) = G(⃗r,⃗r ′ ) ∑I i b i (⃗r ′ ) 

S 

i 

d⃗r ′ 


Introduction 



Results 

Summary 


Model Problem 




switch order of summation and integration 

∫ 

φ(⃗r) = ∑I i G(⃗r,⃗r ′ )b i (⃗r ′ )d⃗r ′ 

i S 

unknowns now outside integral 

Galerkin: multiply both sides by same basis vectors and 

integrate 

∫ 

∫ ∫ 

φ(⃗r)b j (⃗r) = ∑I i G(⃗r,⃗r ′ )b i (⃗r ′ )b j (⃗r)d⃗r ′ d⃗r 

i S S 

this is just a matrix equation 

V j = ∑I i Z ij 

i 


Introduction 



Results 

Summary 

Performing Numerical Integrations 

Sums of Rectangles 

Model Problem 




sums of rectangles 

∫ 

f (x)dx ≈ h∑f (x i ) 

exact for piecewise constant 


Trapezoidal Rule 

Introduction 



Results 

Summary 

Model Problem 




sums of trapezoids 

∫ 

f (x)dx ≈ h 

exact for piecewise linear 

[ 

N−1 

1 

2 (f (x 1) + f (x N )) + 

∑ 

i=2 

f (x i ) 

R. Steven Turley It's Rough Reecting Short Wavelengths 

]

Introduction 



Results 

Summary 

Gaussian Quadrature 

Model Problem 




∫ 

f (x)dx ≈ 

N 

∑ 

i=1 

w i f (x i ) 

choose w i and x i so integrals are exact for polynomials up to 

order 2N − 1 

very ecient 

can be applied in piecewise fashion 

variations 

equally spaced points 

exact for products of monomials and logs 


Introduction 



Results 

Summary 


Model Problem 




In process of integrating Method of Moments found shortcut 

Consider doing integral equation numerically 

∫ 

φ(⃗r) = − G(⃗r,⃗r ′ )ρ(⃗r ′ )d⃗r ′ 

S 

≈ −∑ 

i 

w i G(⃗r,⃗r ′ 

i )ρ i 

Solve for ρ i by requiring these equations to be satised at the 

same points as the discretization 

better numerical stability 

useful for later compution of far eld integrals 


Solution 

Introduction 



Results 

Summary 

Model Problem 




straightforward linear algebra 

φ(⃗r j ) ≈ −∑w i G(⃗r j ,⃗r i )ρ i 

i 

problem when i = j: G(⃗r i ,⃗r i ) in singular 

special technique for treating matrix elements on same local 

patch 


Notes on Solutions 

Introduction 



Results 

Summary 

Model Problem 




Single 2D problem up to 1000 wavelengths can be solved on 

PC 

Complete solution requires supercomputer 

Good results for initial surface 

Improved surface model based on AFM data 


Introduction 



Results 

Summary 

Previous Predictions 

Debye-Waller Attenuation 

R = R 0 exp [ −(qh) 2] 

q =2π sinθ/λ 

h is surface roughness height 

Nevot-Croce 

Assume form 

R = R 0 exp [ −q 1 q 2 h 2] 

R = R 0 exp[f (θ,h,σ)] 


Introduction 



Results 

Summary 

Calculation Comparisons 

Conductor 


Parameterization 

Introduction 



Results 

Summary 

f = α(qh) 3 + β(qh) 2 + γ(qh) 


Future Work 

Introduction 



Results 

Summary 

periodic surfaces 

TE polarization 

3d surfaces 

parameterization to avoid need for expensive supercomputer 

calculations 

non-abrupt interfaces 

application in visible (ecient solar cells) 


Application 

Black Chrome Roughness 

Introduction 



Results 

Summary 

Groove width: 18 ± 1 µm 


Introduction 



Results 

Summary 

Nonspecular Reection 

25.6 nm at 10 ◦ 


Introduction 



Results 

Summary 

Preliminary Dielectric Data 


Summary 

Introduction 



Results 

Summary 

roughness calculation important in XUV 

computing roughness eects requires care 

integral equations with Nyström solution a good approach 

applications to acoustic waves and other wavelengths 


Acknowledgements 

Introduction 



Results 

Summary 

Hughes Research and Yale 

Students 

Stephen Wandzura 

Vladimir Rohklin 

Jed Johnson 

Todd Doughty 

Elise Martin 

Greg Hart 

Samuel Kellar 

others...

It's Rough Reflecting Short Wavelengths - Physics and Astronomy ...

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