Theory of Reflection: Reflection and Transmission of Electromagnetic, Particle and Acoustic Waves

539 Pages·2016·8.51 MB·English

by  John Lekner (auth.)

Springer Series on Atomic, Optical, and Plasma Physics 87
John Lekner
Theory of
Reflection
Reflection and Transmission of
Electromagnetic, Particle and Acoustic Waves
Second Edition
Springer Series on Atomic, Optical,
and Plasma Physics
Volume 87
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John Lekner
fl
Theory of Re ection
fl
Re ection and Transmission
of Electromagnetic, Particle
and Acoustic Waves
Second Edition
123
JohnLekner
Schoolof Chemical andPhysical Sciences
Victoria University ofWellington
Wellington
NewZealand
Previously published as Vol. 3 in “Developments in Electromagnetic Theory and
Applications”(1987), ISBN978-90-247-3418-4
ISSN 1615-5653 ISSN 2197-6791 (electronic)
SpringerSeries onAtomic, Optical, andPlasma Physics
ISBN978-3-319-23626-1 ISBN978-3-319-23627-8 (eBook)
DOI 10.1007/978-3-319-23627-8
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Preface
The second edition is an enlarged and updated version of the book I completed in
Canberra in June 1986. There are six new chapters, Uniaxial anisotropy,
Ellipsometry, Periodically stratified media, Neutron and X-ray reflection, Acoustic
waves and Chiral isotropic media. A first edition chapter has been split into two,
dealing with Pulses and Finite beams separately. The chapters on matrix methods
and on numerical methods have been combined into one. The former appendix is
nowthechapterParticlewaves,precedingthatonneutronandX-rayreflection.The
second edition contains 20 chapters, some with their own appendices, compared
with 13 chapters and one appendix in the first edition.
Theaimremainsthesame:topresentthetheoryofreflectionandtransmissionof
waves from and through (mainly) planar stratifications in a simple and physical
way, from first principles. By that I mean from the Maxwell or Schrödinger
equations, for instance. As a theorist, I have naturally favoured exact results and
haveemphasizeduniversalconservationandinvarianceproperties.However,many
particular cases are made explicit in graphs and formulae. That’s where the theory
connects with reality (as revealed by experiment), and where one gets a physical
feel for the meaning of the formulae. Applied topics do appear: two examples are
the important phenomenon of attenuated total reflection in Chap. 10, and the
reflectivity of multilayer dielectric mirrors in Chaps. 12 and 13.
Ihavetriedtomaintainalogicalprogressionthroughout,ratherthanahistorical
one. Nevertheless, due credit is given to the pioneers of the subject of wave
reflection.Rayleigh(JohnWilliamStrutt,3rdBaronRayleigh,1842–1919)features
prominently, asmay beexpected given theinfluence of his work, especially ofhis
Theoryofsound.Evenso,someofhisreflectionpapersseemtohavebeenforgotten
and his results keep being rediscovered, often in inferior form. The Rayleigh (or
weak reflection) approximation is an example, and appears frequently throughout
the book.
Rayleigh was of privileged birth and made the most of the consequent oppor-
tunities. Not so privileged was George Green (1793–1841), the baker’s and later
miller’s son. He was almost entirely self-taught, having just one year of formal
vii
viii Preface
schooling as a child, between the ages of 8 and 9, and becoming a Cambridge
undergraduatewhennearly40.Green’sfunctionsformthebasisoftheperturbation
theories for long waves in Chap. 3 and for short waves in Chap. 6. No surprise
there. But who talks of the Liouville–Green wavefunctions, or who has heard of
Green’sangle?Theformerarethehigh-frequencywaveformsdatingbackto1837.
Green’s angle, as I have called it in Sect. 1.4, is the acoustic analogue of the
Brewster angle, at which one polarization has zero reflectance from a sharp
interface.
Rayleigh’s use of k for wavenumber has become the standard, and I have built
onthattomaintainaconsistentnotationthroughoutthebook,asfaraspossible.The
normal and tangential components of the wavevector k are always labelled q and
K;thelatterisspecialinbeinganinvariantforwavesinplane-stratifiedmedia,with
the laws of reflection and transmission consequent from that invariance. Greek
letters are used (not exclusively, but in preference) for dimensionless quantities.
The book is written for scientists and engineers whose work involves wave
reflection or transmission. Most of the chapters are in the language of electro-
magnetictheory,butmanyoftheelectromagneticresultscanbeappliedtoparticle
waves, specifically to those satisfying theSchrödinger equation. The mathematical
connectionbetweenelectromagnetics(orTE)wavesandquantumparticlewavesis
established in Chap. 1. The main results for s waves are translated into quantum
mechanical language in the Chap. 15. There is also a close analogy between
acousticwavesandelectromagneticp(orTM)waves,asshowninSect.1.4,andin
detail in Chap. 17. Thus the book, though primarily intended for researchers
working in optics, microwaves or in neutron or X-ray optics, will be of use to
physicists,chemistsandelectricalengineersstudyingreflectionandtransmissionof
particlesatpotentialbarriers,andalsotothoseworkinginacoustics,oceanography
and seismology.
Chapter 1 is recommended for all readers: it introduces reflection phenomena,
definesthenotationandpreviews(inSect.1.6)thecontentsoftherestofthebook.
Thereadercanthengotoanyotherchapterinthebook,allofwhichareintendedto
be sufficiently self-contained so that only occasional reference toother parts ofthe
book is needed.
The first edition was written at the Department of Applied Mathematics of the
Australian National University, Canberra. In the Preface I had the pleasure of
thanking two Australians, Barry Ninham and Colin Pask. The second edition was
writteninNewZealand,butIagainhavepleasureofthankingtwoAustralians,this
time Tony Klein and Andrew Wildes, for their suggestions and comments on the
new chapter on X-ray and neutron reflection.
Wellington John Lekner
Contents
1 Introducing Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Electromagnetic s Wave . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Electromagnetic p Wave . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Particle Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Scattering and Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 A Look Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Exact Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1 Comparison Identities, and Conservation
and Reciprocity Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 General Expressions for rs and rp . . . . . . . . . . . . . . . . . . . . . 46
2.3 Reflection at Grazing Incidence, and the Existence
of a Principal Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 Reflection by a Homogeneous Layer. . . . . . . . . . . . . . . . . . . 55
2.5 The Tanh, Exp and Rayleigh Profiles . . . . . . . . . . . . . . . . . . 61
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3 Reflection of Long Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1 Integral Equation and Perturbation Theory for the s Wave . . . . 75
3.2 The s Wave to Second Order in the Interface Thickness . . . . . 79
3.3 Integral Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
(cid:1) (cid:1)
3.4 (cid:1)rp(cid:1)2 and rp=rs to Second Order . . . . . . . . . . . . . . . . . . . . . . 84
3.5 Reflection by a Thin Film Between Like Media . . . . . . . . . . . 88
3.6 Six Profiles and Their Integral Invariants. . . . . . . . . . . . . . . . 90
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4 Variational Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1 A Variational Expression for the Reflection Amplitude . . . . . . 95
4.2 Variational Estimate for rs in the Long Wave Case. . . . . . . . . 98
ix
x Contents
4.3 Exact, Perturbation and Variational Results
for the sech2 Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Variational Theory for the p Wave . . . . . . . . . . . . . . . . . . . . 103
4.5 Reflection by a Layer Between Like Media . . . . . . . . . . . . . . 106
4.6 The Hulthén-Kohn Variational Method Applied
to Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7 Variational Estimates in the Short Wave Case . . . . . . . . . . . . 112
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5 Equations for the Reflection Amplitudes . . . . . . . . . . . . . . . . . . . 115
5.1 A First Order Non-linear Equation for an s Wave Reflection
Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 An Example: Reflection by the Linear Profile. . . . . . . . . . . . . 117
5.3 Differential Equation for a p Wave Reflection Coefficient . . . . 120
5.4 Upper Bounds on Rs and on Rp . . . . . . . . . . . . . . . . . . . . . . 122
5.5 Long Wave Expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.6 Differential Equations for the Reflection Amplitudes. . . . . . . . 128
5.7 Weak Reflection: The Rayleigh Approximation . . . . . . . . . . . 130
5.8 Iteration of the Integral Equation for r. . . . . . . . . . . . . . . . . . 131
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Reflection of Short Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.1 Short Wave Limiting Forms for Some Solvable Profiles . . . . . 135
6.2 Approximate High-Frequency Waveforms . . . . . . . . . . . . . . . 139
6.3 Profiles of Finite Extent with Discontinuities in Slope
at the Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.4 Reflection Amplitude Estimates from
a Comparison Identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.5 Perturbation Theory for Short Waves. . . . . . . . . . . . . . . . . . . 150
6.6 Short Wave Results for rp and rp=rs . . . . . . . . . . . . . . . . . . . 152
6.7 A Single Turning Point: Total Reflection. . . . . . . . . . . . . . . . 159
6.8 Two Turning Points, and Tunneling . . . . . . . . . . . . . . . . . . . 166
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7 Simple Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.1 Anisotropy with Azimuthal Symmetry. . . . . . . . . . . . . . . . . . 175
7.2 Ellipsometry of a Thin Film on an Isotropic Substrate. . . . . . . 179
7.3 Thin Film on an Anisotropic Substrate. . . . . . . . . . . . . . . . . . 182
7.4 General Results for Anisotropic Stratifications
with Azimuthal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.5 Differential Equations for the Reflection Amplitudes. . . . . . . . 185
7.6 Reflection from the Ionosphere. . . . . . . . . . . . . . . . . . . . . . . 187
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189