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TitleLight Scattering and Nanoscale Surface Roughness
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Table of Contents
                            Title Page
Copyright Page
Preface
	References
Table of Contents
Endnote
Contributors
1 Characterization of Surface Roughness
	1.1. Introduction
		1.1.1. Definition of Nanoscale Roughness
		1.1.2. Early Beginnings: Visual Surface Inspection versus Quantitative Measurements
		1.1.3. Beginnings of Quantitative Metrology in the 1940s
		1.1.4. Metrology Advances in the 1950s and 1960s
		1.1.5. Further Metrology Advances in the 1970s and 1980s
		1.1.6. Recent Developments from the 1990s to the Present
	1.2. Current Surface Metrology Techniques and Instruments
		1.2.1. Questions to Answer Prior to Taking Measurements
		1.2.2. Relations Between Surface Metrology Techniques
		1.2.3. Surface Inspection and Imaging
		1.2.4. Optical Profilers
		1.2.5. Mechanical Profilers
		1.2.6. Atomic Force Microscopes (AFM)
		1.2.7. Total Integrated Scattering and Angle-Resolved Scattering
		1.2.8. Surface Contamination and Cleaning
	1.3. Current and Future Surface Metrology Requirements
		1.3.1. General Comments
		1.3.2. Metrology of Microcomponents
		1.3.3. Metrology in the UV and Soft X-Ray Regions
		1.3.4. Metrology of Steeply-Curved Spherical or Aspheric Surfaces
		1.3.5. Polarization of Scattered Light for Target Discrimination
		1.3.6. Automated, Rapid-Response Systems with Accept-Reject Capabilities
	1.4. Summary
	References
2 The Kirchhoff and Related Approximations
	2.1. Introduction
	2.2. The Helmholtz Formula
	2.3. The Limit of the Observation Point Tending to the Surface
	2.4. Kirchhoff Approximation for the Neumann Problem
	2.5. Scattering Amplitude
	2.6. The Tangent Plane Approximation
	2.7. Scattering of Electromagnetic Waves from the Interface Between Dielectric Half-Spaces
	2.8. The Stratton-Chu Formula
	2.9. The Integral Equations for the Electromagnetic Case
	2.10. Nonlocal Small-Slope Approximation
	2.11. Relation to other Approaches
	2.12. Conclusion
	References
3 Scattering and the Spatial Frequency Representation
	3.1. Introduction
	3.2. Plane Waves
	3.3. Scattering
	3.4. Significance of the Three-Dimensional Spatial Frequencies
	3.5. Polarization Effects
	3.6. Random Surfaces
		3.6.1. Statistics of Surface Scattering
			3.6.1.1. Smooth Surface
			3.6.1.2. Rough Surface
		3.6.2. Gaussian Autocorrelation Coefficient
		3.6.3. Measurement of Surface Roughness
			3.6.3.1. Smooth Surface, n0kσ = ho<< 0.5
			3.6.3.2. Rough Surface, ho>> 0.5 sec α
			3.6.3.3. Fine Surface, Co << 1/ sin α
			3.6.3.4. Coarse Surface, Co >> 1/ sin α
		3.6.4. Imaging of Surface Roughness
		3.6.5. Inversion of Scattering Data
		3.6.6. Statistics of the Scattered Field
		3.6.7. Limitations of the Kirchhoff Approximation
	3.7. Fractal Surfaces with an Outer Scale
		3.7.1. Scattering by a Fractal Surface with an Outer Scale
	3.8. Total Integrated Scatter (TIS)
	3.9. Dielectric Medium
	3.10. Conclusions
	References
4 Rayleigh Hypothesis
	4.1. Introduction
	4.2. Is the Representation Given by Eq. (7) Fundamentally Wrong?
	4.3. Convergence of the Rayleigh Series
	4.4. Rayleigh Hypothesis and the Perturbative Expansion of the SA
	4.5. Application to Numerical Analysis
	4.6. Conclusion
	References
5 Small-Amplitude Perturbation Theory for One-Dimensionally Rough Surfaces
	5.1. Introduction
	5.2. Theory
	5.3. Gaussian Roughness Spectrum
	5.4. Enhanced Specular Peaks
		5.4.1. Gaussian Spectra
		5.4.2. Physical Origins
		5.4.3. Rectangular Spectra
			5.4.3.1. The Spectrum
			5.4.3.2. Results
	5.5. Comparisons with Experiments
		5.5.1. The 2-2 Effect
		5.5.2. The 4-4 Effect
	5.6. Conclusions
	References
6 Small-Amplitude Perturbation Theory for Two-Dimensional Surfaces
	6.1. Introduction
	6.2. Derivation of the Reduced Rayleigh Equations
		6.2.1. Propagation Equations and Boundary Conditions
		6.2.2. Field Elimination
		6.2.3. The Reduced Rayleigh Equations
	6.3. The Diffusion Matrix
	6.4. A Perturbative Development
		6.4.1. Case of One Rough Surface
			6.4.1.1. A Rough Surface Separating Two Different Media
			6.4.1.2. A Slab with a Rough Surface on the Bottom Side
			6.4.1.3. A Slab with a Rough Surface on the Upper Side
		6.4.2. Case of Two Rough Surfaces
	6.5. The Mueller Matrix Cross-Section and the Surface Statistics
		6.5.1. Case of One Randomly Rough Surface
		6.5.2. Case of Two Randomly Rough Surfaces
	6.6. Numerical Examples and Analysis of the Phenomena
		6.6.1. A Randomly Rough Surface Separating Two Different Semi-Infinite Media
		6.6.2. A Film with a Randomly Rough Surface on the Upper Side
		6.6.3. A Slab with Two Randomly Rough Surfaces
	6.7. Discussion
	Appendix
		A Expression of V h(x)
		B Perturbative Development and Reciprocity Condition
		C Scattering Matrix Coefficients
	References
7 Computer Simulations of Rough Surface Scattering
	7.1. Introduction
	7.2. Fundamental Issues
		7.2.1. One- and Two-Dimensional Surfaces
		7.2.2. Description of the Rough Surface
		7.2.3. Finite Size Surface Effects
		7.2.4. Other Physical Parameters
		7.2.5. Other "Numerical" Parameters
		7.2.6. Use of Approximate Theories
	7.3. Integral Equation Formulations
	7.4. Matrix Solution Methods
		7.4.1. Iterative Solution of Matrix Equations
		7.4.2. Preconditioning
		7.4.3. Physically Based Preconditioning
		7.4.4. Accelerating the Matrix- Vector Multiply Operation
			7.4.4.1. Canonical Grid Method
			7.4.4.2. Spectral Approach
		7.4.5. Parallelization
		7.4.6. Storage Issues
	7.5. Sample Results
	7.6. Conclusions and Recommendations for the Use of Numerical Methods
	References
8 Overview of Rough Surface Scattering
	8.1. Introduction
	8.2. Coordinate-Space Methods
		8.2.1. Scalar Problems
		8.2.2. Electromagnetic Problems
	8.3. Spectral-Space Methods
		8.3.1. Scalar Problems
		8.3.2. Electromagnetic Problems, Infinite Surface
	8.4. Surface Inversion
	8.5. Solution Methods
	8.6. Discussion
	References
9 Experimental Studies of Scattering from Weakly Rough Metal Surfaces
	9.1. Introduction
	9.2. Experimental Methods
		9.2.1. Essential Couplings
		9.2.2. Rectangular Spectra
		9.2.3. Scattering Measurements
	9.3. Experimental Results
		9.3.1. Ideal Spectrum
		9.3.2. Detuned Case
	9.4. Second Harmonic Generation
		9.4.1. Roughness Spectrum Centered on k2wsp
		9.4.2. Roughness Spectrum Centered on kwsp
	9.5. Angular Correlation Functions
	9.6. Conclusions
	References
10 Measuring Interfacial Roughness by Polarized Optical Scattering
	10.1. Introduction
	10.2. Definitions
	10.3. Measurement Methods
	10.4. Roughness of a Single Interface
		10.4.1. Theory
		10.4.2. Limitations
		10.4.3. The Inverse Problem
		10.4.4. Example
	10.5. Roughness of Two Interfaces
		10.5.1. Theory
		10.5.2. The Inverse Problem
		10.5.3. Example
	10.6. Final Comments
	References
11 Scattering of Electromagnetic Waves from Nanostructured, Self-Affine Fractal Surfaces: Near-Field Enhancements
	11.1. Introduction
	11.2. Scattering Model
		11.2.1. Scattering Geometry
		11.2.2. Scattering Equations
		11.2.3. Near Field
		11.2.4. Surface Field
		11.2.5. Self-Affine Fractals
	11.3. Near and Surface Field
		11.3.1. Surface Fields
		11.3.2. Near Field Map: Localized Surface-Plasmon Polaritons
	11.4. Surface Field Enhancement: Statistics
	11.5. Surface-Enhanced Raman Scattering
	11.6. Concluding Remarks
	References
12 Light Scattering by Particles on Substrates. Theory and Experiments
	12.1. General Introduction
	12.2. Near Field of Particles on Substrates
		12.2.1. Introduction
		12.2.2. 1D Geometry
		12.2.3. 2D Geometry
		12.2.4. Concluding Remarks
	12.3. Far Field of Particles on Substrates
		12.3.1. Introduction
		12.3.2. Regular Particles on Flat Substrates
		12.3.3. Quasi-Regular Cases: Buried Particles and Surface Defects
			12.3.3.1. Buried Particles
			12.3.3.2. Particle with a Bumped Surface Nearby
		12.3.4. Many Particles: Polydispersity, Shadowing and Multiple Scattering
			12.3.4.1. Introduction
			12.3.4.2. Polydispersity
			12.3.4.3. Multiple Scattering and Shadowing Effect
		12.3.5. Light Scattering Statistics
		12.3.6. Concluding Remarks
	References
13 Multiple Scattering of Waves by Random Distribution of Particles for Applications in Light Scattering by Metal Nanoparticles
	13.1. Introduction
	13.2. Formulation for Foldy Lax Equations
	13.3. Extinction and Absorption Efficiency of Metal Nanoparticles and Plasmon Resonance
		13.3.1. Formulations
			13.3.1.1. Extinction Cross Section
			13.3.1.2. Absorption Cross Section
		13.3.2. Results and Discussions
			13.3.2.1. Convergence Test for Numerical Parameters
			13.3.2.2. Extinction and Absorption of Two Particles with Various Orientations
			13.3.2.3. Extinction and Absorption of Gold Nanoparticles with Various Fractional Volumes
			13.3.2.4. Extinction and Absorption of Silver Nanoparticles with Different Fractional Volume
			13.3.2.5. Energy Absorption of Each Particle in the Collection
	13.4. Phase Matrix of Light Scattering by Metal Nanoparticles
		13.4.1. Formulation of Phase Matrix
			13.4.1.1. 1-2 Polarization Frame
			13.4.1.2. Scattering Cross Section
		13.4.2. Results and Discussion
			13.4.2.1. Phase Matrices of Single Realization and Average Realizations
			13.4.2.2. Phase Matrices of 1% and 5%
			13.4.2.3. Phase Matrix in Resonant Mode and Nonresonant Mode
	13.5. Optical Scattering of Nanoparticles Below or Above a Random Rough Surface
	References
14 Multiple-Scattering Effects in Angular Intensity Correlation Functions
	14.1. Introduction
	14.2. The Correlation Function C(q, k\q', k') and Its Properties
	14.3. Determination of Ĉ(q, k\q', k')
		14.3.1. Correlations in Single-Interface Systems
			14.3.1.1. The Correlation Function Ĉ(l)(q, k\q', k')
			14.3.1.2. The Correlation Function Ĉ(10)(q, k\q'k')
			14.3.1.3. The Transition from Complex Gaussian to Circular Complex Gaussian Statistics
		14.3.2. Correlations in Film Systems
	14.4. Frequency Correlation Functions
	14.5. Experimental Results
	14.6. Conclusions
	References
15 Speckle Pattern in the Near Field
	15.1. Introduction
	15.2. Role of Evanescent Waves in the Near Field
		15.2.1. Angular Spectrum
		15.2.2. Field and Intensity Correlations in the Near Field
		15.2.3. Speckle Patterns due to Random Thermal Fields
		15.2.4. Multiple Scattering
		15.2.5. Experimental Difficulties
	15.3. Nonuniversal Speckle Pattern Produced by a Slightly Rough Surface
		15.3.1. Statistical Description of a Random Rough Surface
		15.3.2. Amplitude of the Field Scattered by a Deterministic Slightly Rough Surface
		15.3.3. Speckle Pattern Generated by a Slightly Rough Surface in the Near Field
	15.4. Detection of Optical Near Fields
		15.4.1. General Expression for the Near-Field Optical Signal
			15.4.1.1. Reciprocity Theorem
			15.4.1.2. Expression of the Detected Field
			15.4.1.3. Calculation of the Response Function
		15.4.2. Polarization Response
		15.4.3. Spectral Response
	15.5. Conclusion
	Appendix: Reciprocity Theorem
		The Lorentz Reciprocity Theorem
		Another Form of the Reciprocity Theorem
	References
16 Inverse Problems in Optical Scattering
	16.1. Introduction
	16.2. The Scattering Amplitude
		16.2.1. The Thin Phase Screen Model
	16.3. Estimation of Statistical Properties of Surfaces
		16.3.1. Statistical Characterization of Random Surfaces
		16.3.2. The Random Field and Its Averages
		16.3.3. The Coherent Component
		16.3.4. The Incoherent Component
		16.3.5. Angular Correlations
	16.4. Estimation of the Surface Profile from Complex Amplitude Data
		16.4.1. Inversion Algorithm
		16.4.2. Numerical Example
	16.5. Estimation of the Surface Profile from Intensity Data
		16.5.1. Evolutionary Inversion Procedure
		16.5.2. Results of a Numerical Experiment
	16.6. Discussion and Conclusions
	References
17 The Design of Randomly Rough Surfaces That Scatter Waves in a Specified Manner
	17.1. Introduction
	17.2. A Surface That Produces a Scattered Field with a Specified Angular Dependence of Its Mean Intensity
	17.3. A Surface That Synthesizes the Infrared Spectrum of a Known Compound
	17.4. Conclusions
	References
Index
                        
Document Text Contents
Page 2

Light Scattering and
Nanoscale Surface Roughness

Page 256

9
Experimental Studies of Scattering
from Weakly Rough Metal Surfaces

K. A. O'DONNELL
Division de Fisica Aplicada, Centro de Investigacion Cient(fica y de Educacion Superior de
Ensenada, Apartado Postal 2732, Ensenada, Baja California, 22800Mexico

9.1. Introduction

The surface of a weakly rough metal, with height fluctuations of a few nanometers,
can produce remarkably strong and unusual optical effects under appropriate con-
ditions. If the random roughness of a metal surface allows an incident light wave
to launch surface plasmon polaritons, the diffuse scatter emitted by the surface
will receive contributions when these surface waves are subsequently scattered
from the surface. Under such conditions, it has been predicted that effects like
backscattering enhancement may appear in the mean diffuse scatter emitted by the
surface.I'?These theoretical works use sophisticated methods to account for mul-
tiple scattering processes involving plasmon-polariton excitation. In particular,
the incident light wave may be roughness-coupled to surface waves, which may
themselves be scattered many times within the surface, to finally be roughness-
coupled out of the surface so as to contribute to the diffusely scattered light. Other
theoretical works have used direct perturbation' or Monte Carlo" techniques to
study such effects. This line of research has been extended to include theoretical
studies of angular correlation functions' and of the generation of diffuse second
harmonic light from weakly rough metals.P'

There has been a shortage of related experimental works. Certainly, there has
been considerable experimental effort directed toward scattering from the weak
residual roughness of polished optical surfaces.f However, the intent of the work
has often been to characterize the roughness or to better understand the various
means of polishing surfaces. In any case, effects of plasmon-polariton excitation
are not commonly seen in such work even for metallic optical surfaces, presum-
ably because the surface roughness is not appropriate to produce significant sur-
face wave excitation. There have also been many experiments done with strongly
rough metal surfaces and, even though backscattering enhancement is sometimes
observed, these effects are entirely unrelated to polariton excitation." On the other
hand, there have indeed been experimental observations of diffuse scatter arising
from plasmon-polariton excitation on metallic gratings'? or metal-coated coupling
prisms 11having incidental roughness. However, here the excitation is that of a spe-
cific surface wave mode, which occurs only for a particular illumination geometry.

237

Page 257

238 O'Donnell

This physical situation is very different from that of a randomly rough free
space/metal interface where, over a wide range of incidence and scattering angles,
the roughness simultaneously provides the coupling mechanisms for excitation,
de-excitation, and multiple scattering of surface waves.

The purpose of this chapter is to describe experimental studies of the conse-
quences of plasmon-polariton excitation on rough metal surfaces. It is possible
that the lack of relevant experiments is due to difficulties in fabricating suitable
surfaces; thus the work described here begins in Sect. 9.2 with a discussion of the
essential surface wave coupling mechanisms and the lithographic surface fabrica-
tion methods employed to produce them. The fabrication method produces highly
one-dimensional surface structures, with Gaussian height statistics and root-mean-
square roughness of a few nanometers. The power spectrum of the roughness is of
rectangular form and, to produce the effects of interest, covers a range of wavenum-
ber that includes that of the surface plasmon polariton.

Section 9.3 presents results for the mean diffuse intensity scattered by these
surfaces, which exhibit backscattering enhancement under a variety of condi-
tions. This effect occurs only for p polarization, as expected for effects related
to plasmon-polariton excitation on one-dimensional surfaces. It is seen that the
rectangular spectrum allows one to either produce or suppress the backscattering
peak and its associated distribution, according to the roughness couplings allowed
by the bandwidth of the rectangular spectrum. Section 9.4 considers experiments
in which the nonlinear response of the metal surface produces a diffuse scatter-
ing distribution of second-harmonic light. Here the rectangular spectrum is again
useful in indicating the origin of the features seen in the distributions. With the
power spectrum of the roughness centered on the wavenumber of the second-
harmonic plasmon polariton, a backscattering effect is observed in the diffuse
second-harmonic light. With the spectrum centered on the wavenumber of the
fundamental plasmon-polariton, the effects are stronger and the observed features
are attributed to a variety of nonlinear wave interactions. Finally, Sect. 9.5 returns
to linear optical effects and considers the angular correlation functions of inten-
sity scattered by the rough surface. Two distinct types of angular correlations are
observed and the effects related to plasmon-polariton excitation are studied.

9.2. Experimental Methods

9.2.1. Essential Couplings

The lowest-order scattering processes producing polariton-related backscattering
enhancement are shown in Fig. 9.1. The incident light wave of frequency (J) is
incident at angle ()i upon a rough surface having dielectric constant 8 == 81 +i82.
In path A of Fig. 9.1, it launches a plasmon polariton at point 1 traveling to, for
example, the right along the surface. This polariton then reaches point 2, where
it is scattered by the roughness to produce diffuse light escaping from the surface
at an angle Os' The time-reversed process may also occur (path B), in which the

Page 512

point dipole sources,439
responsefunction(a/az)Erec, 423-25

reciprocalfield,423
spatialdependence of components,

424-425
SNOM.See scanningnear-field optical

microscopy
specklepattern at the interfacebetween

dielectricmedia,416-418
intensitycorrelationfunction, 416
speckle, contrastCI, 418
zero-orderterm, 416

spectralresponse,425-427
confocalconfiguration, 427

specklepatternin the near field
difficulties in near fieldexperiments

measurement of spectrain the near field,
413

passiveprobe,413
polarization sensitivity, 413

evanescent wavesin the electric field
angularspectrum, 409-410
cross-spectral density, 410
fieldand intensitycorrelations, 410-411
powerspectraldensity, 411

specklepatternsin the near field,412-413
imagingsurfaceplasmons,412
spatialcoherenceof the field, 412

transitionbetweenfar fieldto near field, 411
cross-spectral density, 411
polarization dependence, 412

spectral-space methods
electromagnetic problems,InfiniteSurface

direct scatteringproblem,226, 277
vectorSPectral amplitudes, 225

scalarproblems,220-225
boundedsurfacefields, 221
delta function, 221
incidentspectralamplitudeI, 223
mixedspectral-coordinate domain,224
Neumannkernel,225

statisticsof surfacescattering
mean valueof the scatteringfunction, 67, 68

statisticsof the scatteredfield,77-78
scatteringfunction, varianceof, 78

statisticalpropertiesof surfaces,439-441
angularcorrelations

specklepattern,447
coherentcomponent

angle of incidence,445
averagescatteringamplitude, 443, 444
effectof rms heightof the surface,444
phase perturbation theory,444

Index 495

incoherentcomponent, 445-446
geometrical optics approximation, 446
meandifferential reflection coefficient, 445
singlescatteringapproximation, 445

n-orderjoint probability density function,
440-441

two-pointheightcorrelationfunction, 440
randomfieldaverages

randomintensityfluctuations, 442
specklestatistics,443

storagerequirements for numericaltechniques
CAG method

store all translationally invariantterms,
198

iterativesolutionof a matrixequation, 198
NSA method

storageof "source" terms on the spectral
integration path, 198

surfacecontamination and cleaning
techniques of, 26
throughsolvents,

alcohol,26
acetone,26

uncoatedglass substrates, washing, 26
surface-enhanced Ramanscattering, 286, 291,

299-300
surfaceinversion

scalarDirichletproblem,227
normalderivative boundaryvalue,

Kirchhoffapproximation, 228
perturbation theory in surfaceheight,227
reflection spectralamplitude, 228
two-dimensional Fouriertransform, 227

surfaceof a weaklyrough metal,opticaleffects
DetunedCase, 245-247

surface's specularreflection in p
polarization, 246

diffusescatter for one-dimensional surface,
243

light scatteringin the optimalcase, 243-245
backscattering enhancement, 244

outcomeof plasmon-polaritonexcitationon
roughmetal surfaces,238

polaritonexcitation, 244, 245
lowest-order scatteringprocesses,238-239
roughness wavenumbers, 239

rectangularspectra
lithography of photoresist, 240
sinusoidalinterferencepattern,241

second-harmonic light, flat metal surface
plasmon-polaritonexcitation, 248

surfaceplasmonpolaritons. See plasmon
polaritons

Page 513

496

surfaceprofileestimationfromcomplex
amplitude data

inverse scattering procedures, 448
reconstruction of the profileof a perfectly

conducting surface, 451-453
superposition of planewave, 449
surfaceprofilefunction, 450
wavefront matching algorithm, 449

surfaceprofilefunction from far-field intensity
data

conclusion of, 461-462
correlated randomnumbers, 454
evolutionary inversion procedure, 455-457

surfacesin secondary population, 455
selectionoperator, 456

fitness (objective) functional, 453-454
inverse scatteringproblem, 453-454
one-dimensional Gaussian randomprocess,

454
solutionof direct scattering problem,

457-460
targetprofile, 458
thin phasescreenapproximation, 459
zero-mean stationary Gaussian-correlated

Gaussianrandomprocess,457
Surfacewavevectors. See wavevectors

three-dimensional (3D) spatialfrequencies
3D spatialsignificance, 65-66

scatteringdata changewithangleof
incidence, 66

BRDF. See bidirectional reflectance
distribution function

Gaussianautocorrelation coefficient, 71-73
scattering function, root-mean-square

valueof, 73
planewaves, scatteredangularspectrumof,

62
polarization effects,66-67
randomsurfaces

surfacescatteringstatistics, 67-69
roughsurface

secondmomentof the powerspectral
density, 71

scalarplanewave
amplitude of, 61-62

scattering, types of,
backward scattering, 62
forward scattering, 62

smoothsurface, 69-70
surfaceheightvariation, 70

surfacescattering, standardderivation for,63
total integrated scattering(TIS)5, 13, 25, 62, 74,

85-86
definition of, 86
roughness of surfaces, 85

wavevectors
surfacewavevectors, 268
parallelwavevectors, 410
large wavevectors, 401

white light interferometer, 13,20,22,23

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