##### 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

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