![1.3 Modeling the material response 17
is related to the electric field via the Fourier transform of [ε(ω)/ε0 − 1]. The
analyticity of this latter function in the upper ω plane allows the application
of the Cauchy theorem over a contour extending over the real axis, jumping
the pole, and closing itself at infinity in the upper plane. This direct complex
plane integration provides two relations between the real and imaginary parts
of ε(ω), known as the Kramers-Kronig relations, and expressed as (Jackson
1999)
Re(ε(ω)) − 1 =
1
π
PV
∞
−∞
dω Im(ε(ω
))
ω − ω
, (1.9a)
Im(ε(ω)) = −
1
π
PV
∞
−∞
dω Re(ε(ω
)) − 1
ω − ω
, (1.9b)
where PV denotes the Cauchy principal value. Similar relations hold for
the real and imaginary parts of the magnetic permeability μ. Consequently,
in addition to being frequency dispersive, ε and μ are also required to be
complex functions on the account of causality. The imaginary parts account
for absorption of radiation in the medium and the total absorbed energy in a
volume V is given by (Landau et al. 1984)
V
d3
r
∞
−∞
ω
Im(ε(ω))|E(r, ω)|2
+ Im(μ(ω))|H(r, ω)|2
dω
2π
. (1.10)
For example, consider a time harmonic plane wave exp[i(kz−ωt)] propagating
along the z-axis in a dissipative medium with μ = 1 and a complex ε where
Im(ε) 0. It is clear that the amplitude of the wave decays exponentially
due to absorption of the wave as it propagates, which clearly implies that
Im(k) 0. This complex wave-vector can be obtained from the Maxwell
equations as k2
= εω2
/c2
.
Eqs. (1.9) indicate that the real and imaginary parts of the permittivity (and
similarly the permeability) are Hilbert transforms of each other, as illustrated
in Fig. 1.8. These relations are derived for material media in thermodynamic
equilibrium solely on the grounds of causality. The restriction that they pro-
vide on the variation in the real and imaginary parts of material parameters
should be regarded as very fundamental. The Kramers-Kronig relations al-
low an experimentalist, for example, to measure the imaginary part of the
permittivity easily by absorption experiments at various frequencies and de-
duce the real parts of the dielectric permittivity from the imaginary part.
An example of this procedure is shown in Fig. 1.8 where the imaginary part
of the permittivity is calculated from the real parts by a Hilbert transform
with different frequency ranges for the integration. Note that the integrals in
the Kramers-Kronig relations involve frequencies all the way up to infinity,
whereas it is clear that the effective medium theories break down at high fre-
quencies. However, this does not really affect us in the case of usual optical
media since the macroscopic material response functions hold almost down to
© 2009 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/45671-250225153526-93a15aee/85/Physics-and-Applications-of-Negative-Refractive-Index-Materials-1st-Edition-S-Anantha-Ramakrishna-37-320.jpg)
![18 Introduction
0 0.5 1 1.5 2 2.5 3
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Ref
[0, 1.5]
[0, 2]
[0 3]
(a) With integration over (0, ωm).
0 0.5 1 1.5 2 2.5 3
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Ref
[−1.5, 1.5]
[−2, 2]
[−3 3]
(b) With integration over (−ωm, ωm).
Figure 1.8 Comparison between the analytic imaginary part of the permit-
tivity (thick black curve on both graphs) and the imaginary part obtained via a
Hilbert transform. The analytic expression is εr = 1−(ω2
p−ω2
o)/(ω2
−ω2
o +iγω)
with ωo = 2π × 10 GHz, ωp = 2π × 15 GHz, and γ = ω0/2. The labels 1, 2,
and 3 refer to the subscript m of ωm and correspond to ω1 = 2ωp, ω2 = 4ωp,
and ω3 = 6ωp respectively.
the level of few atomic distances. Thus, the very high frequency limit is never
really probed. In the case of metamaterials, the wavelength is usually larger
than the periodicity by only one or two orders of magnitude and this high fre-
quency cutoff, when the homogenization becomes invalid, is easily accessed.
Thus these relations should be applied cautiously to metamaterials keeping
this in mind: if the effective medium theory itself cannot describe the system,
the effective medium parameters obtained from the Kramers-Kronig relations
are not meaningful.
1.3.2 Dispersive model for the dielectric permittivity
This section briefly presents a dispersive model for the dielectric permittivity
that is due to H. A. Lorentz. The resulting expression is very general and it
has been found that many metamaterials exhibit effective constitutive param-
eters in agreement with this law. As another example, amplifying media such
as laser gain media, whose imaginary part of the permittivity is negative, are
often modeled by a generalized Lorentz model where the oscillator strength is
taken to be negative. The Lorentz model is also a good approximation to the
density matrix equations of a weakly perturbed two-level quantum system.
© 2009 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/45671-250225153526-93a15aee/85/Physics-and-Applications-of-Negative-Refractive-Index-Materials-1st-Edition-S-Anantha-Ramakrishna-38-320.jpg)
![1.4 Phase velocity and group velocity 23
over a certain frequency band. The spectrum of such a wave is never just a
Dirac delta function. The assumption of monochromatic plane waves is there-
fore a theoretical idealization, whereas in the real world, the signal is typically
composed of a slowly varying envelope confining a rapidly oscillating wave.
The simplest multi-frequency signal is composed of two closely separated fre-
quencies ω0 ± Δω, where Δω ω0, to which correspond the wave-numbers
k ± Δk. The superposition of the two waves is simply written as
Ey = cos [(k + Δk)z − (ω + Δω)t] + cos [(k − Δk)z − (ω − Δω)t] ,
= 2 cos(Δkz − Δωt) cos(kz − ωt) . (1.21)
Tracking the constant fronts of the two terms yields two velocities:
1. kz − ωt = constant yields the velocity of the rapidly oscillating wave,
which is similar to the monochromatic case discussed previously:
vp =
dz
dt
=
ω
k
. (1.22a)
2. Δkz − Δωt = constant yields the velocity of the envelope, called the
group velocity:
vg =
dz
dt
=
Δω
Δk
. (1.22b)
Intuitively, the group velocity is seen to correspond to the velocity of the
envelope or the packet, and corresponds to the velocity of propagation
of the energy in many cases.
In the limit of a very narrow-band signal, Δω → 0 and the group velocity is
expressed as
vg =
1
∂k/∂ω
. (1.23)
We can also express the group velocity in terms of the phase velocity:
1
vg
=
1
vp
+ ω
∂
∂ω
1
vp
, (1.24)
which indicates that if there is no frequency dispersion, vg = vp. In the case
of normal dispersion, ∂
∂ω (1/vp) 0 so that vg vp. We had mentioned above
that vp can be larger than the velocity of light inside the medium. It can
easily be shown that vg is in fact lower than this limit. Since vg corresponds
to the velocity at which information is carried, it is in compliance with the
principle of relativity.
In the case of anomalous dispersion relation, ∂
∂ω (1/vp) 0 so that vg vp:
the group velocity can be even larger than the speed of light in vacuum. In
this case, however, the group velocity loses its meaning as signal velocity,
which has to be defined in terms of the electromagnetic energy flow. This
issue is discussed in greater detail in Section 6.5.
© 2009 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/45671-250225153526-93a15aee/85/Physics-and-Applications-of-Negative-Refractive-Index-Materials-1st-Edition-S-Anantha-Ramakrishna-43-320.jpg)
![1.5 Metamaterials and homogenization procedure 25
Bragg scattered waves are evanescent and bound to the single scatterer. Con-
sequently, the properties of a metamaterial are not resulting from interference
between waves scattered off different points. Instead, the radiation probes the
polarizability of the structural units as it moves through the medium, inter-
acting with the polarizable objects in the same manner as in a homogeneous
medium. Note that in the limit of long wavelengths, the phase shifts for the
wave across a single structural unit are negligibly small and all units interact
with the radiation in a similar manner.
1.5.2 Negative effective medium parameters
As discussed in Section 1.3.2, there is a large amount of dispersion in the
material parameters at frequencies near the resonance. Below the resonance,
the polarization is in phase with the applied driving field, whereas it is π out
of phase above resonance. If the dissipation is sufficiently small, the resonance
can be made very sharp so as to drive the real parts of ε and μ even toward
negative values when the corresponding driving fields are the electric and the
magnetic fields, respectively. Of course, the imaginary parts of ε and μ are
also large at the resonance frequency and its immediate vicinity.
Thus, negative real parts of the material parameters should be regarded
as a natural outcome of an underdamped and overscreened response of a res-
onant medium. Fundamentally, there is no objection to negative real parts
of ε(ω) or μ(ω) as long as other physical criteria are also satisfied such as
causality. The latter implies for example that the frequency dispersive mod-
els for the permittivity and the permeability cannot be arbitrary, but should
yield constitutive parameters that satisfy the Kramers-Kronig relationship of
Eqs. (1.9).
In order to better understand the effect of negative material parameters,
consider an isotropic medium where the Im[ε(ω)] ∼ Im[μ(ω)] 0, i.e., dissi-
pation is assumed negligibly small at some frequencies (this would typically
be a good approximation at frequencies somewhat away from the resonant
frequency). We can conveniently characterize most electromagnetic materials
by the quadrant where they lie in the complex (Re(ε)−Re(μ)) plane as shown
in Fig. 1.10.
Quadrant 1: This is the realm of usual optical materials with Re(ε) 0
and Re(μ) 0. Electromagnetic radiation can propagate through these media
and the vectors E, H, and k form a right-handed triad.
Quadrant 2: The usual form of matter that has Re(ε) 0 and Re(μ) 0 is
a plasma of electric charges. It is well known that a plasma screens the interior
of a region from electromagnetic radiation. Indeed, all electromagnetic waves
are evanescent inside a plasma and no propagating modes are allowed. This
is directly expressed by the constitutive relation, which reduces to
k · k = εμω2
/c2
0 (1.26)
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![This ebook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg License included with this
ebook or online at www.gutenberg.org. If you are not located in the
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you are located before using this eBook.
Title: Transplanted
A novel
Author: Gertrude Franklin Horn Atherton
Release date: February 20, 2024 [eBook #72997]
Language: English
Original publication: New York: Dodd, Mead and Company, 1919
Credits: D A Alexander, David E. Brown, and the Online Distributed
Proofreading Team at https://www.pgdp.net (This file was
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*** START OF THE PROJECT GUTENBERG EBOOK TRANSPLANTED
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![PART ONE
CHAPTER I
MRS. HAYNE’S boarding-house stood on the corner of Market Street
and one of those cross streets which seem to leap down from the
heights of San Francisco and empty themselves into the great
central thoroughfare that roars from the sandy desert at the base of
Twin Peaks to the teeming wharves on the edge of the bay. On the
right of Market Street, both on the hills and in the erratic branchings
of the central plain, as far as the eye can reach, climbs and swarms
modern prosperous San Francisco; of what lies beyond, the less said
the better. On the left, at the far southeast, the halo of ancient glory
still hovers about Rincon Hill,[A] growing dimmer with the years: few
of the many who made the social laws of the Fifties cling to the old
houses in the battered gardens; and their children marry and build
on the gay hills across the plain. In the plain itself is a thick-set, low-
browed, dust-coloured city; “South of Market Street” is a generic
term for hundreds of streets in which dwell thousands of
insignificant beings, some of whom promenade the democratic
boundary line by gaslight, but rarely venture up the aristocratic
slopes. By day or by night Market Street rarely has a moment of
rest, of peace; it is a blaze of colour, a medley of sound, shrill,
raucous, hollow, furious, a net-work of busy people and vehicles
until midnight is over. Every phase of the city’s manifold life is
suggested there, every aspect of its cosmopolitanism.
To a little girl of eleven, who dwelt on the third floor of Mrs. Hayne’s
boarding-house, Market Street was a panorama of serious study and
unvarying interest. She knew every shop window, in all the mutable
details of the seasons, she had mingled with the throng unnumbered
times, studying that strange patch-work of faces, and wondering if](https://image.slidesharecdn.com/45671-250225153526-93a15aee/85/Physics-and-Applications-of-Negative-Refractive-Index-Materials-1st-Edition-S-Anantha-Ramakrishna-61-320.jpg)
![wouldn’t make you nervous for the world. I go way up when I want
to cross and then turn back. It’s nothing like as bad.”
“It is shocking to think that you go out at all unattended; but what
cannot be cannot, and you must have air and exercise, poor child!”
Lee, who retained a blurred, albeit rosy impression of her former
grandeur, was well pleased with her liberty; and Mrs. Tarleton was
not only satisfied that any one who could take such good care of her
mother was quite able to take care of herself, but, so dependent was
she on the capable child, that she was frequently oblivious to the
generation they rounded. Mrs. Tarleton was an invalid, and, although
patient, she met her acuter sufferings unresistingly. Lee was so
accustomed to be roused in the middle of the night that she had
learned to make a poultice or heat a kettle of water while the
receding dreams were still lapping at her brain. She dressed her
mother in the morning and undressed her at night. She frequently
chafed her hands and feet by the hour; and cooked many a dainty
Southern dish on the stove in the corner. Miss Hayne, who had a
sharp red nose and the anxious air of protracted maidenhood, but
whose heart was normal, made it her duty to fetch books for the
invalid from the Mercantile Library, and to look in upon her while Lee
was at school.
Lee brushed and mended her own clothes, “blacked” her boots with
a vigorous arm, and studied her lessons when other little girls were
in bed. Fortunately she raked them in with extreme rapidity, or Mrs.
Tarleton would have made an effort and remonstrated; but Lee
declared that she must have her afternoons out-of-doors when her
mother was well and companioned by a novel; and Mrs. Tarleton
scrupulously refrained from thwarting the girl whose narrow
childhood was so unlike what her own had been, so unlike what the
fairies had promised when Hayward Tarleton had been the proudest
and most indulgent of fathers.
FOOTNOTE:
[A] This was written before the earthquake and fire of
1906.](https://image.slidesharecdn.com/45671-250225153526-93a15aee/85/Physics-and-Applications-of-Negative-Refractive-Index-Materials-1st-Edition-S-Anantha-Ramakrishna-63-320.jpg)
Physics and Applications of Negative Refractive Index Materials 1st Edition S. Anantha Ramakrishna
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- 5. Physics and Applications of Negative Refractive Index Materials 1st Edition S. Anantha Ramakrishna Digital Instant Download Author(s): S. Anantha Ramakrishna, Tomasz M. Grzegorczyk ISBN(s): 9781420068757, 142006875X Edition: 1 File Details: PDF, 10.77 MB Year: 2009 Language: english
- 6. Physics and Applications of Negative Refractive Index Materials © 2009 by Taylor & Francis Group, LLC
- 7. Bellingham, Washington USA Physics and Applications of Negative Refractive Index Materials S. Anantha Ramakrishna Tomasz M. Grzegorczyk CRC Press is an imprint of the Taylor & Francis Group, an informa business Boca Raton London New York © 2009 by Taylor & Francis Group, LLC
- 8. Co-published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Tel.: +1 360-676-3290 Fax: +1 360-647-1445 Email: Books@SPIE.org spie.org ISBN: 9780819473998 CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-6875-7 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can- not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2009 by Taylor & Francis Group, LLC
- 9. To Kanchan, Kartik, and Kanishka To Alessandra, Eva, Davide, and to my parents © 2009 by Taylor & Francis Group, LLC
- 10. In memoriam: Jin Au Kong Jin Au Kong, Professor in the Department of Electrical Engineering and Com- puter Science at the Massachusetts Institute of Technology (MIT), contributed tremendously to the development of left-handed media from the very begin- ning, in 2001. His influence can be felt in all areas of this field, from theory and numerical simulations for which he was internationally renowned, to ex- periments that he helped conceive and carry through all around the world, notably within MIT and the MIT Lincoln Laboratory, as well as in Asia and Europe. The conference he created, Progress in Electromagnetic Research Symposium, was one of the first, if not the first, to promote technical ses- sions on left-handed media, thus contributing to the cross-fertilization of ideas among researchers worldwide. His journals, the Journal of Electromagnetic Waves and Applications and the Progress in Electromagnetic Research, con- stantly call for innovative papers on all aspects of left-handed media, and have been well regarded and well cited. Finally, his textbooks, written and pub- lished years before the advent of left-handed media, often contain remarkable ideas and concepts that were later rediscovered and accepted as pillars in this new field. Prof. Kong passed away unexpectedly on March 12, 2008, of complications from pneumonia, before this book went to press. Many of the ideas and concepts presented in the following pages have been directly inspired by him and discussed at length with him, often late into the night in his office at MIT. The course of events made it such that my efforts toward the realization of this book have become a tribute to his work during the last 7 years. The international community has lost one of its giants, which is in addition a personal loss for me. T. M. Grzegorczyk © 2009 by Taylor & Francis Group, LLC
- 11. Foreword The past ten years have seen an astonishing explosion of interest in nega- tive refractive index materials. First explored systematically by Veselago in 1968 from a theoretical point of view these materials remained without an experimental realisation for more than 30 years. That had to await devel- opment of suitable metamaterials, materials whose function is due as much to their internal sub wavelength structure as to their chemical composition. The added flexibility to create new materials enables properties unavailable in nature to be realised in practice. That opened the floodgates to a host of new experiments. Why the great interest? From its rebirth at the beginning of this century negative refraction has provoked controversy. To be consistent with the laws of causality a material has to do much more than refract negatively. For example, it must necessarily be dispersive. Thus did many misunderstandings arise and pioneers had to endure some testing assaults. Yet even that aspect can now be seen as positive because controversy drew attention to the fledgling subject and showed that negative refraction contains subtleties that even experienced scientists did not at first appreciate. Even now we as a community are learning from our errors and discovering new aspects of this long hidden subject. As work progressed and news of amazing results spread beyond the scientific community into the popular press, a broader excitement has been generated. Some of the more extraordinary results such as the prescription for a perfect lens, and particularly the possibility of making objects invisible, had already been foreseen in science fiction and fed a ready-made appetite in the popular imagination. Thus the ancient subject of classical optics has brought us new discoveries and excitement. This book, written by two leading practitioners of negative refraction, ar- rives at an opportune time because there is a substantial body of results available in the field that need to be gathered together in a systematic fash- ion sparing new arrivals hours of wasted time trawling through the very many papers in the literature. And yet new discoveries are continually reported. This is work in progress and the authors must steel themselves eventually to write a second edition! Sir John B. Pendry Imperial College London © 2009 by Taylor & Francis Group, LLC
- 12. Preface Rarely in the history of science does one have the opportunity to witness an explosion of interest for a given topic, to participate in its development from its beginning, and to witness its growth at a pace almost exponential over a period of about a decade. Yet, we believe that this is precisely what has hap- pened to us, with regard to the new development of materials that are now called metamaterials, left-handed media, or negative refractive media. Funda- mentally rooted in the electromagnetic theory and governed by the equations proposed by the Scottish physicist James Clerk Maxwell at the end of the 19th century, the development of these structured composite materials that we call metamaterials could have been another incremental step in the more general research in electromagnetics and optics. Yet, the scientific community quickly realized that the implications and applications opened by the study of metamaterials are unprecedented, potentially revolutionary, and scientifically as well as technologically highly interesting and challenging. A new paradigm of electromagnetic and optical materials has evolved today from these studies. The study of metamaterials is often thought of as being associated with neg- ative refraction. It is much more than that. Over the past decade, scientists have shown how to manipulate the macroscopic properties of matter at a level unachieved before. For decades, our world was limited to materials with pri- marily positive permittivities and permeabilities, with some exceptions such as plasmas, for example, whose permittivities can be negative. The research in metamaterials coupled with the rapid advancements in micro- and nano- fabrication technology has totally lifted this limitation, and has opened the door to almost arbitrary material properties with some extraordinary conse- quences across the electromagnetic spectrum, from radio frequencies to optical frequencies. This book is devoted to a discussion of these consequences as well as their theoretical implications and practical applications. It is inevitable that such a growing field has attracted much attention in the scientific as well as in the more popular literature: the number of scientific articles has been in constant and almost exponential growth since about the year 2000, many popular articles have been published in scientific as well as nonscientific journals, while technical reviews and a few books have already been devoted to this field. It therefore appears ambitious at best and risky at worst to attempt the publication of an additional reference in this arena. Nonetheless, we think that such an addition is necessary and was, in fact, missing. The extremely large number of scientific papers published is certainly vivid proof of the rapid evolution of this research area, but getting familiar © 2009 by Taylor & Francis Group, LLC
- 13. with and appreciating so much information also represent a daunting task for the student or researcher who is new to this field. In addition, the large number of new articles appearing on a weekly basis may also appear difficult to track, even by the expert researcher. It is with this spirit that we have targeted this book at as vast an audience as possible: the reader unfamiliar, but interested in this field, will find in the following pages the synthesis and organization of what we believe to be the most important and influential papers related to metamaterials, whereas the expert reader will hopefully find a useful viewpoint and detailed explanations of some of the most recent papers at the time of this writing, touching on as many aspects of this field as possible. An additional motivation to undertake the writing of this book was our feeling that a coherent reference presenting the history, development, and main achievements of metamaterials was missing. Although some excellent books are already available to the reader, they are usually focused on either a very specific aspect of this field, or a compilation of chapters written by renowned scientists. In the present book, we have tried to remedy what we believe are limitations of the previous two formats by offering a book covering a wide variety of topics, yet having a coherence across chapters that enables the reader to cross-reference similar topics and, hence, to delve deeper into their presentation and explanation. Naturally, it is impossible to present in a short book all aspects of a given scientific field, all the more when this field has become so vast and complex as the one the present book is devoted to. In addition, and despite our best efforts, our grasp of the field is also incomplete and is being refined by the day. We would therefore like to apologize upfront to those authors who may feel that their work is misrepresented or underrepresented in the following pages. May they put it on the account of our limited knowledge and not on our judgment of the quality of their contributions. Finally, we must remark that it has been very difficult to write a book on an emerging area: it has almost been like writing about the personality of a growing teenager. New topics of today might disappear tomorrow or, instead, might reveal unexpected promises and become the front-runners of this research field. Metamaterials of the future will necessarily be robust and reliable, multifunctional, and reconfigurable to perform satisfactorily in various demanding environments. Today’s metamaterials are quite primitive by these standards and developments are happening at breathtaking speeds. These have been the reasons why we decided not to have a concluding chapter – this book is an ongoing account of metamaterials. S. A. Ramakrishna Kanpur, India T. M. Grzegorczyk Cambridge, Massachusetts, USA © 2009 by Taylor & Francis Group, LLC
- 14. Acknowledgments This book came about not only because of our privilege to have witnessed the birth of this field, but more importantly because of our privilege to have ac- tively participated in its development from a very early date. The research we have carried out over almost an entire decade brought us in contact with many researchers and students who, in many ways, have helped us discover and learn about this exciting topic. We would like in particular to acknowledge the contributions of our most closely related colleagues: Benjamin E. Barrowes, Sangeeta Chakrabarti, Hongsheng Chen, Jianbing J. Chen, Xudong Chen (with a special thanks for proofreading parts of the manuscript), Sebastien Guenneau, Brandon A. Kemp (with a special thanks for proofreading parts of the manuscript), Jin Au Kong, Narendra Kumar, Akhlesh Lakhtakia, Jie Lu, Olivier J. F. Martin, Christopher Moss, Lipsa Nanda, Stephen O’Brien, Joe Pacheco, Jr., Sir John Pendry, Lixin Ran, Zachary Thomas, Harshawardhan Wanare, Bae-Ian Wu, and Yan Zhang. We specifically thank L. Nanda and S. Chakrabarti for their help in making some of the figures and compiling the bibliography. We thank our colleagues from across the world who have given us permission to reuse or reproduce their figures and data which, at times, might have even been original and unpublished. SAR acknowledges the support of the Centre for Development of Technical Education, IIT Kanpur via a book-writing grant and encouragement from his colleagues in the Physics Department at IIT Kanpur. Finally, for their constant support and encouragements, we would like to thank our respective families to whom we dedicate this book. © 2009 by Taylor & Francis Group, LLC
- 15. About the authors S. Anantha Ramakrishna received his M.Sc. in physics from the Indian In- stitute of Technology, Kanpur, and his Ph.D. in 2001 for his research work on wave propagation in random media at the Raman Research Institute, Banga- lore. During 2001−2003 he worked with Sir John Pendry at Imperial College London on the theory of perfect lenses made of the newly discovered negative refractive index materials. In 2003, he joined the Indian Institute of Technol- ogy, Kanpur as an assistant professor and is presently an associate professor of physics there. His research interests concern complex wave phenomena in optics and condensed matter physics. He published the first comprehensive, technical review on the development of negative refractive index materials in 2005. He is a Young Associate of the Indian Academy of Science, Banga- lore, a recipient of the Young Scientist Medal for 2007 of the Indian National Science Academy, Delhi, and was selected as an affiliate of the Third World Academy of Science, Trieste, in 2007. He was an invited professor at the Insti- tut Fresnel, Université Aix–Marseille I in May 2006, and a visiting professor at the Nanophotonics and Metrology Laboratory at the Ecole Polytechnique Federale de Lausanne during June−July 2006. He is a member of SPIE and a life member of the Indian Physics Association. Tomasz M. Grzegorczyk received his Ph.D. from the Swiss Federal Insti- tute of Technology, Lausanne, in December 2000. In January 2001, he joined the Research Laboratory of Electronics (RLE), Massachusetts Institute of Technology (MIT), U.S.A., where he was a research scientist until July 2007. Since then, he has been a research affiliate at the RLE-MIT, and founder and president of Delpsi, LLC, a company devoted to research in electromagnet- ics and optics. His research interests include the study of wave propagation in complex media and left-handed metamaterials, electromagnetic induction from spheroidal and ellipsoidal objects for unexploded ordnances modeling, optical binding and trapping phenomena, and microwave imaging. He is a senior member of IEEE, a member of the OSA, and was a visiting scientist at the Institute of Mathematical Studies at the National University of Singapore in December 2002 and January 2003. He was appointed adjunct professor of The Electromagnetics Academy at Zhejiang University in Hangzhou, China, in July 2004. From 2001 to 2007, he was part of the Technical Program Committee of the Progress in Electromagnetics Research Symposium and a member of the Editorial Board of the Journal of Electromagnetic Waves and Applications and Progress in Electromagnetics Research. © 2009 by Taylor & Francis Group, LLC
- 16. Contents 1 Introduction 1 1.1 General historical perspective . . . . . . . . . . . . . . . . . . 2 1.2 The concept of metamaterials . . . . . . . . . . . . . . . . . . 8 1.3 Modeling the material response . . . . . . . . . . . . . . . . . 14 1.3.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . 14 1.3.2 Dispersive model for the dielectric permittivity . . . . 18 1.4 Phase velocity and group velocity . . . . . . . . . . . . . . . . 22 1.5 Metamaterials and homogenization procedure . . . . . . . . . 24 1.5.1 General concepts . . . . . . . . . . . . . . . . . . . . . 24 1.5.2 Negative effective medium parameters . . . . . . . . . 25 1.5.2.1 Terminology . . . . . . . . . . . . . . . . . . 26 2 Metamaterials and homogenization of composites 29 2.1 The homogenization hypothesis . . . . . . . . . . . . . . . . . 30 2.2 Limitations and consistency conditions . . . . . . . . . . . . . 33 2.3 Forward problem . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.1 Relation between R and T and the electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.2 Determining the electromagnetic fields . . . . . . . . . 35 2.4 Inverse problems: retrieval and constitutive parameters . . . . 42 2.4.1 Standard media . . . . . . . . . . . . . . . . . . . . . . 42 2.4.2 Left-handed media . . . . . . . . . . . . . . . . . . . . 45 2.5 Homogenization from averaging the internal fields . . . . . . . 49 2.5.1 Maxwell-Garnett effective medium theory . . . . . . . 50 2.5.2 Layered media as anisotropic effective media . . . . . 52 2.5.3 Averaging the internal fields in periodic media . . . . 54 2.6 Generalization to anisotropic and bianisotropic media . . . . 57 2.6.1 Forward model . . . . . . . . . . . . . . . . . . . . . . 58 2.6.2 Inversion algorithm . . . . . . . . . . . . . . . . . . . . 65 3 Designing metamaterials with negative material parameters 77 3.1 Negative dielectric materials . . . . . . . . . . . . . . . . . . . 79 3.1.1 Metals and plasmons at optical frequencies . . . . . . 79 3.1.2 Wire mesh structures as low frequency plasmas . . . . 83 3.1.2.1 Other photonic metallic wire materials . . . . 91 3.2 Metamaterials with negative magnetic permeability . . . . . . 92 © 2009 by Taylor & Francis Group, LLC
- 17. 3.2.1 Diamagnetism in a stack of metallic cylinders . . . . . 93 3.2.2 Split-ring resonator media . . . . . . . . . . . . . . . . 95 3.2.2.1 Pendry’s split rings . . . . . . . . . . . . . . . 98 3.2.3 The Swiss Roll media for radio frequencies . . . . . . . 100 3.2.4 Scaling to high frequencies . . . . . . . . . . . . . . . . 104 3.2.5 Magnetism from dielectric scatterers . . . . . . . . . . 108 3.2.6 Arrangements of resonant plasmonic particles . . . . . 112 3.2.7 Isotropic magnetic metamaterials . . . . . . . . . . . . 116 3.3 Metamaterials with negative refractive index . . . . . . . . . 119 3.3.1 Combining the “electric” and “magnetic” atoms . . . 120 3.3.2 Negative refractive index at optical frequencies . . . . 123 3.4 Chiral metamaterials . . . . . . . . . . . . . . . . . . . . . . . 131 3.5 Bianisotropic metamaterials . . . . . . . . . . . . . . . . . . . 134 3.6 Active and non-linear metamaterials . . . . . . . . . . . . . . 137 3.6.1 Nonlinear split-ring resonators . . . . . . . . . . . . . 139 3.6.2 Actively controllable metamaterials . . . . . . . . . . . 143 4 Negative refraction and photonic bandgap materials 145 4.1 Photonic crystals and bandgap materials . . . . . . . . . . . . 146 4.1.1 One-dimensional photonic crystals: transmission lines approach . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.1.2 Two-dimensional photonic crystals: definitions and solution . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.1.2.1 Direct lattice . . . . . . . . . . . . . . . . . . 149 4.1.2.2 Reciprocal lattice . . . . . . . . . . . . . . . . 149 4.1.2.3 Brillouin zone and irreducible Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . 151 4.1.3 Bloch theorem and Bloch modes . . . . . . . . . . . . 152 4.1.4 Electromagnetic waves in periodic media . . . . . . . . 152 4.2 Band diagrams and iso-frequency contours . . . . . . . . . . . 156 4.2.1 Free-space and standard photonic crystal . . . . . . . 156 4.2.2 Iso-frequency contours . . . . . . . . . . . . . . . . . . 160 4.3 Negative refraction and flat lenses with photonic crystals . . . 164 4.3.1 Achieving negative refraction . . . . . . . . . . . . . . 164 4.3.2 Image quality and stability . . . . . . . . . . . . . . . 168 4.4 Negative refraction vs. collimation or streaming . . . . . . . . 171 5 Media with ε < 0 and μ < 0: theory and properties 175 5.1 Origins of negative refraction . . . . . . . . . . . . . . . . . . 176 5.1.1 Dispersion relation . . . . . . . . . . . . . . . . . . . . 177 5.1.2 Anisotropic media with positive constitutive parameters . . . . . . . . . . . . . . . . . . . . . . . . 180 5.1.3 Photonic crystals . . . . . . . . . . . . . . . . . . . . . 182 5.1.4 Left-handed media . . . . . . . . . . . . . . . . . . . . 183 5.1.5 Moving media . . . . . . . . . . . . . . . . . . . . . . . 183 © 2009 by Taylor & Francis Group, LLC
- 18. 5.2 Choice of the wave-vector and its consequences . . . . . . . . 185 5.2.1 Modified Snell’s law of refraction . . . . . . . . . . . . 188 5.2.2 Reversed Doppler shift . . . . . . . . . . . . . . . . . . 190 5.2.3 Reversed Goos-Hänchen shift . . . . . . . . . . . . . . 192 5.2.4 Reversed Čerenkov radiation . . . . . . . . . . . . . . 193 5.2.5 Modified Mie scattering . . . . . . . . . . . . . . . . . 198 5.3 Anisotropic and chiral media . . . . . . . . . . . . . . . . . . 201 5.3.1 Indefinite media . . . . . . . . . . . . . . . . . . . . . 202 5.3.2 Amphoteric refraction . . . . . . . . . . . . . . . . . . 204 5.3.3 Reversal of critical angle and Brewster angle . . . . . 208 5.3.4 Negative refraction due to bianisotropic effects . . . . 210 5.3.5 Flat lenses with anisotropic negative media . . . . . . 213 6 Energy and momentum in negative refractive index materials 219 6.1 Causality and energy density in frequency dispersive media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.1.1 Causality in left-handed media . . . . . . . . . . . . . 220 6.1.2 Causality and phase propagation . . . . . . . . . . . . 221 6.1.3 Energy in dispersive media . . . . . . . . . . . . . . . 227 6.2 Electromagnetic energy in left-handed media . . . . . . . . . 230 6.2.1 Erroneous concept of negative energy in lossy dispersive media . . . . . . . . . . . . . . . . . . . . . 230 6.2.2 Lossy Lorentz media . . . . . . . . . . . . . . . . . . . 231 6.3 Momentum transfer in media with negative material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 6.4 Limit of plane wave and small losses . . . . . . . . . . . . . . 236 6.4.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 6.4.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . 237 6.5 Traversal of pulses in materials with negative material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.5.1 Wigner delay time for pulses in NRM . . . . . . . . . 240 6.5.2 Traversal times based on the flow of radiative energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 6.5.2.1 Traversal times through negative refractive index media . . . . . . . . . . . . . . . . . . . 246 6.5.2.2 Traversal times for evanescent waves . . . . . 247 7 Plasmonics of media with negative material parameters 253 7.1 Surface electromagnetic modes in negative refractive materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.1.1 Surface plasmon modes on a plane interface . . . . . . 255 7.1.2 Surface plasmon polariton modes of a slab . . . . . . . 260 7.2 Waveguides made of negative index materials . . . . . . . . . 265 7.3 Negative refraction of surface plasmons . . . . . . . . . . . . . 267 © 2009 by Taylor & Francis Group, LLC
- 19. 7.4 Plasmonic properties of structured metallic surfaces . . . . . 273 7.5 Surface waves at the interfaces of nonlinear media . . . . . . . 276 8 Veselago’s lens is a perfect lens 281 8.1 Near-field information and diffraction limit . . . . . . . . . . 283 8.2 Mathematical demonstration of the perfect lens . . . . . . . . 286 8.2.1 Role of surface plasmons . . . . . . . . . . . . . . . . . 290 8.2.2 Quasi-static limit and silver lens . . . . . . . . . . . . 292 8.2.3 “Near-perfect” lens with an asymmetric slab . . . . . 294 8.3 Limitations due to real materials and imperfect NRMs . . . . 297 8.3.1 Analysis of the lens transfer function for mismatched material parameters . . . . . . . . . . . . . . . . . . . 301 8.3.2 Focussing properties of a finite slab of NRM . . . . . . 305 8.4 Issues with numerical simulations and time evolution . . . . . 311 8.4.1 Temporal evolution of the focus . . . . . . . . . . . . . 315 8.5 Negative stream of energy in the perfect lens geometry . . . . 316 8.6 Effects of spatial dispersion . . . . . . . . . . . . . . . . . . . 319 9 Designing super-lenses 323 9.1 Overcoming the limitations of real materials . . . . . . . . . . 324 9.1.1 Layering the lens . . . . . . . . . . . . . . . . . . . . . 325 9.1.2 A layered stack to direct radiation . . . . . . . . . . . 327 9.1.3 Use of amplifying media to reduce dissipation . . . . . 331 9.2 Generalized perfect lens theorem . . . . . . . . . . . . . . . . 333 9.2.1 Proof based on the symmetries of the Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . 338 9.2.2 Contradictions between the ray picture and the full wave solutions . . . . . . . . . . . . . . . . . . . . . . . 339 9.3 The perfect lens in other geometries . . . . . . . . . . . . . . 341 9.3.1 A transformation technique . . . . . . . . . . . . . . . 343 9.3.2 Perfect lenses in curved geometries: cylindrical and spherical lenses . . . . . . . . . . . . . . . . . . . . . . 344 9.3.3 Hyperlens: a layered curved lens . . . . . . . . . . . . 352 9.3.4 Perfect two-dimensional corner lens . . . . . . . . . . . 354 9.3.5 Checkerboards and a three-dimensional corner lens . . 356 10 Brief report on electromagnetic invisibility 361 10.1 Concept of electromagnetic invisibility . . . . . . . . . . . . . 361 10.2 Excluding electromagnetic fields . . . . . . . . . . . . . . . . . 364 10.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 364 10.2.2 Design procedure . . . . . . . . . . . . . . . . . . . . . 367 10.3 Cloaking with localized resonances . . . . . . . . . . . . . . . 368 A The Fresnel coefficients for reflection and refraction 373 © 2009 by Taylor & Francis Group, LLC
- 20. B The dispersion and Fresnel coefficients for a bianisotropic medium 375 C The reflection and refraction of light across a material slab 379 References 381 © 2009 by Taylor & Francis Group, LLC
- 21. 1 Introduction This book is devoted to the description of metamaterials, their origins and physical principles, their electromagnetic and optical properties, as well as to their potential applications. This field has witnessed an immense gain of in- terest over the past few years, gathering communities as diverse as those from optics, electromagnetics, materials science, mathematics, condensed matter physics, microwave engineering, and many more. The field of metamaterials being therefore potentially extremely vast, we have limited the scope of this book to those composite materials whose structures are substantially smaller, or at the least smaller, than the wavelength of the operating radiation. Such structured materials have been called metamaterials in order to refer to the unusual properties they exhibit, while at the same time being describable as effective media and characterized by a few effective medium parameters inso- far as their interaction with electromagnetic radiation is concerned. We also include in this book a chapter on photonic crystals, which work on a very different principle than metamaterials, but which have been closely connected to them and have been shown to exhibit many similar properties. The metamaterials discussed in this book are designer structures that can result in effective medium parameters unattainable in natural materials, with correspondingly enhanced performance. Much of the novel properties and phenomena of the materials discussed in this book emanate from the possibil- ity that the effective medium parameters (such as the electric permittivity and the magnetic permeability) can become negative. A medium whose dielectric permittivity and magnetic permeability are negative at a given frequency of radiation is called a negative refractive index medium or, equivalently, a left- handed medium, for reasons that will become clear shortly. In this book, we do not, however, discuss another important and powerful manner of at- taining extraordinary material properties – that of coherent control whereby atomic and molecular systems are driven into coherence by strong and coher- ent electromagnetic fields (Scully and Zubairy 1997). Due to the extremely coherent nature of the excitation and response, the quantum mechanical na- ture of the atoms and molecules is strongly manifested in these cases and the description of the atomic systems relies necessarily on quantum mechanics. In contrast, we remark that since the sizes of the metamaterial structures we are interested in are microscopically large (compared to atomic sizes) and the resonances reasonably broad, it is the classical electromagnetic properties that are apparent. Hence, we ignore the quantum mechanical nature of light 1 © 2009 by Taylor & Francis Group, LLC
- 22. 2 Introduction and matter throughout our discussions. This chapter offers a general introduction to the topic this book is de- voted to, starting with a brief description of the historical development of the subject. We give first a general account of the development of optics, electromagnetism, and the characterization of their effects by effective consti- tutive parameters. A more specific account of the development of the ideas surrounding metamaterials and negative effective medium parameters follows. We then clarify mathematically our definitions, and discuss the Lorentz model for the dispersion of the dielectric permittivity of a dispersive medium and the basic definitions for the description of negative refractive index media. In the course of this chapter, we hope to set out the basic foundations that will allow the reader to follow the book without much confusion. 1.1 General historical perspective The study of optical phenomena has accompanied the evolution of mankind from almost its origins. Astronomy, which is often said to be the oldest of all science, has led humans through an incredibly vast journey of discover- ies, turning philosophers into scientists and passive observations into active research. During the last couple of centuries, man has achieved an unprece- dented understanding and control over light thanks to one fundamental prop- erty: light exhibits just the right amount of interaction with matter. This interaction is intense enough compared to that between other particles or matter waves such as neutrons or neutrinos, and yet weak enough compared to the interaction between charged particles such as electrons or protons. The fact that light, or the photon, is one of the fundamental particles of nature and that its propagation velocity sets the ultimate limit on the speed of any signal further underlines the significance of this control. Despite being one of the oldest topics in science, Optics has remained a very fundamental area of physics and engineering because of the simplicity of its theoretical grounds. It is, for example, formidable to realize that optical properties of many materials can be characterized by a single number called the refractive index, n. This number allows one to understand refraction processes and enables the design of lenses and prisms that led to the under- standing of colors and dispersion. For a long time, this refractive index was a number that represented the optical density of a medium, a notion reasonably supported by the definition of the refractive index as n = c v , (1.1) where c is the speed of light in vacuum and v is the speed of light in the medium. © 2009 by Taylor & Francis Group, LLC
- 23. 1.1 General historical perspective 3 C i θt Crystal Air O A D B θ Figure 1.1 An adaptation of Ibn Sahl’s original drawing showing refraction at a planar interface. AO is the incident ray from inside the crystal and OB is the refracted ray. Ibn Sahl obtained the reciprocal of the refractive index as 1/n = OB OC (= OB OD OD OC = sin θi sin θt in today’s terminology). The roots of Optics as a science go as far as the ancient Greek civilization, where Aristotle, upon studying visual perception, recognized the importance of the medium in-between the eye and an object. Another Greek astronomer, Ptolemy, performed several experiments on the effects of refraction on visual perception of objects in the 2nd century AD. Despite these early works, the real credit for the association of a number to the refraction effects of a trans- parent medium is probably due to Ibn Sahl, an Arabic scholar of Catalonian origin. Ibn Sahl, who lived in Baghdad around 984 AD, wrote a treatise on Burning Instruments where he clearly stated a law of refraction for light passing across a plane interface from a material medium into air. This law, completely identical to what we now call the Snell law’s of refraction, defined the refractive index n in terms of the incident and refracted rays as shown in Fig. 1.1. Ibn Sahl further used this refractive index for a crystal to study the focusing properties of a biconvex lens and several other focusing instruments.∗ For a long time all optically transparent crystals were mainly characterized by the refractive index. Based on the experimental findings of Willebrord Snellius in 1621, the French philosopher René Descartes (1596−1650) pub- lished in his “Dioptrique” the law of refraction in the form we know it today. The refractive index was considered to be a quantification of the resistance offered by a medium to the passage of light. Based on this idea, Fermat enunciated his famous Principle of Least Action, which proved invaluable for studies of light propagation in media with spatially varying refractive index. Erasmus Bartholinus had discovered the double refraction in calc-spar in 1669 which led to the realization that there was a polarization associated with light. Malus also discovered polarization, and the rotation of polarization of light ∗The reader is referred to Rashed (1990) for a lucid description of Ibn Sahl’s work. © 2009 by Taylor & Francis Group, LLC
- 24. 4 Introduction upon passage through an anisotropic medium in 1808, again with a calc-spar crystal. This was one of the first cases when the optical material could not be characterized by a single number, but the description necessarily depended on the propagation direction and the relative orientation of the crystal. In parallel to the development of optics, the 19th century also witnessed the emergence of the theories of electricity and magnetism. A plethora of experimental observations challenged the physicists to look for underlying ex- planations and gave birth to fundamental laws such as Ampère’s, Gauss, or Faraday’s laws. Yet, electricity, magnetism, and optics were seen as indepen- dent fields, ruled by independent laws and yielding independent applications. It took the incredible insight and genius of James Clerk Maxwell (see Fig. 1.2) to first unify the former two, and then all the three fields under a uniquely simple and complete theory. With his work, Maxwell showed that electricity and magnetism are entangled phenomena, inseparable, and self-sustaining, ruled by four simple equations known today as the Maxwell equations. The concepts of dielectric permittivity and magnetic permeability, denoted by the letters ε and μ, respectively, became fundamental for the description of media and their response to electric and magnetic fields, and were called constitu- tive parameters. Moreover, upon studying the self-sustaining solutions of the electromagnetic field in vacuum, Maxwell discovered electromagnetic waves, effectively revolutionizing the field for a second time with descriptions of fre- quency, wavelength, and propagation speed, with all their fundamental and technological impacts. Finally, upon calculating the propagation speed of the newly discovered electromagnetic waves, Maxwell realized that it was very close to that of light in vacuum, which led him to bridge the two independent fields by declaring that light is an electromagnetic wave. The independent demonstrations of radio frequency waves and their propagation in vacuum by H. R. Hertz, N. Tesla, J. C. Bose, † and G. Marconi, as well as the theoretical work of Einstein obviating the need for the all permeating “aether,” made quick developments in optics possible by utilizing the Maxwell equations. The connection between the two fields, optics and electromagnetics, was summarized by the very simple equation (also known as the Maxwell relation) n2 = εμ, (1.2) relating the index of refraction, an optical quantity, to the permittivity and permeability of media, two electromagnetic quantities. It was also then re- alized that all media could be described by the concepts of permittivity and permeability, whose definitions had to be properly generalized. Hence, absorp- tion of light in materials was described by complex valued ε and μ, whereas many anisotropic crystals (where all directions are not equivalent) were de- scribed by second-rank tensors ¯ ε̄ and ¯ μ̄, effectively yielding different values in different propagation directions or for different polarization states. †J. C. Bose is credited with the discovery of millimeter waves. © 2009 by Taylor & Francis Group, LLC
- 25. 1.1 General historical perspective 5 Figure 1.2 Two giants of electromagnetism: J. C. Maxwell (left) mathe- matically unified Electricity, Magnetism, and Optics through his equations. The image is taken from the Wikipedia project, http://www.wikipedia.org. H. A. Lorentz (right) gave a microscopic model for the dispersion of the di- electric permittivity. (Courtesy of C. W. J. Beenakker, from the “Collection Instituut-Lorentz, Leiden University.”) Although dispersion of the refractive index with frequency was a well-known empirical fact by then, it was Fresnel (of the diffraction fame) who first tried to explain it in terms of the molecular structure of matter. This was also supported by Cauchy who gave the well-known dispersion formula which goes by his name. But it was essentially H. A. Lorentz (see Fig. 1.2, right) who gave a reasonably robust theory of dispersion in terms of the polarization of the basic molecules constituting a material. This Lorentz theory of dispersion (described in Section 1.3.2) has been very successful at describing the variation of the dielectric permittivity with frequency and is used as a workhorse model for describing the dispersion in resonant systems. At frequencies well away from an absorption resonance, the Lorentz theory easily approximates into the Cauchy dispersion formula. In dense media (high pressure gases, liquids, and solids), it had to be corrected for local field effects – effects of other neighboring polarized molecules, which yielded the Lorentz-Lorenz model, akin to the Clausius-Mossotti relations for the electrostatic case (Jackson 1999). Interestingly, although there was no a priori bound on the values of the constitutive parameters, all known transparent media were described by a re- fractive index between about 1.2 and 1.9 only at optical frequencies.‡ These bounds were broken for the first time when it was realized that stratified ‡Excluding semiconductors where it could be as large as 4 in the infrared regions. © 2009 by Taylor & Francis Group, LLC
- 26. 6 Introduction materials, where layers of transparent materials with different refractive in- dices are stacked together, could exhibit very different optical properties due to well-controlled interference phenomena of the multiply scattered waves at the interfaces between the different media. The most striking examples of such technology are the quarter wavelength anti-reflection coatings and high reflection thin film coatings. The theory of periodic media was later gen- eralized to higher dimensions, making the layered medium a special case of structures later to be called photonic crystals, where strong modifications of the properties of electromagnetic radiation come from multiple scattering or Bragg scattering within the structure. A drastic example is the realization of structures in which light is not able to propagate at all in any direction in a band of frequencies (bandgap) because of the proper interplay of scattering and destructive interference. Actually the realization of a one-dimensional stop-band structure should be credited to Lord Rayleigh who was probably the first to systematically investigate the wave propagation in layered mate- rials (Rayleigh 1887). Lord Rayleigh had already realized the existence of a stop-band and the fact that a layered medium would cause complete reflection of the incident light for frequencies within this band. For further reading on these topics, the reader is referred to Joannopoulos et al. (1995) and Sakoda (2005). By the middle of the 20th century, the optics of layered media had been well established, benefiting from the thrust in military requirements during World War II. Improvements were demanded in all areas of optical instrumentation, from binoculars to periscopes, and provided the impetus for industrial activity in this area. The strong modification of light propagation in such systems resulted in a variety of optical properties, ranging from highly reflecting multi- layer coatings to their opposite, the anti-reflective coatings. The reader is referred to details in the classic book by Born and Wolf (1999) for further reading on these topics. In 1987, the generalization from one-dimensional periodic media (i.e., lay- ered media) to three-dimensional periodic media was independently proposed by Yablonovitch (1987) and John (1987) who also discussed the strong mod- ification of the density of photon states in such systems. Thus, even the spontaneous emission probability for an atom within the photonic crystal, emitting at a frequency in the forbidden band (called the bandgap) was shown to be possibly strongly modulated. Yablonovitch et al. (1991) pursued this work with the demonstration of a face-centered cubic photonic crystal at mi- crowave frequencies. It was demonstrated by calculations (Ho et al. 1990) that a diamond-like lattice structure with a strong enough refractive index contrast could result in a complete bandgap for light propagating in any direction (a three-dimensional bandgap). For the last decade or so, photonic crystals with negligible absorption have become one of the most promising avenues for the development of all-optical circuits. For example, Akahane et al. (2003) have reported optical cavities using two-dimensional photonic crystals with some of the highest ever reported Q-factors (∼ 106 ) at optical frequencies. © 2009 by Taylor & Francis Group, LLC
- 27. 1.1 General historical perspective 7 An important limitation in controlling the propagation of light in matter came from the fact that the index of refraction could still take positive values only. In fact, a negative refractive index was often seen as being incompat- ible with the definition of optical density, and hence was often viewed as unphysical. However, careful theoretical considerations showed that a nega- tive refraction could indeed be physical, provided that the medium exhibits other fundamental and necessary properties. The two most important ones were shown to be frequency dispersion (where the permittivity and perme- ability are not constant with frequency) and dissipation, the two not being independent but related to one another by the necessity of causality. Despite these additional constraints, materials with a negative refractive index had no further reasons to remain hypothetical and the scientific community began a quest for their physical realization. The germs of the possibility of negative refraction probably first appeared in 1904 during discussions between Sir Arthur Schuster and Sir Horace Lamb regarding the relationship between the group velocity and the phase velocity of waves (see Boardman et al. (2005) for a detailed discussion). The negative group velocity that is possible due to anomalous dispersion at frequencies close to an absorption resonance was the point in contention. For the case of negative refraction, Schuster believed that the group velocity should have a component away from the interface while the phase velocity vector should point inward to the interface. Although Schuster’s conclusion came about from a confusion regarding negative group velocity (the energy flow need not coincide with the group velocity direction in the vicinity of a resonance), it was probably the first consideration of negative phase velocity vectors. In 1944, Mandelshtam considered the possibility of oppositely oriented phase and group velocities (Mandelshtam 1950). He noted that Snell’s law for refraction between two media admitted the mathematical solution of refraction at an angle of (π−θt) in addition to the usual angle of refraction at θt, and reconciled it with the fact that the phase velocity still tells nothing about the direction of energy flow. Mandelshtam then also presented examples of negative group velocity structures in spatially periodic dielectric media (Mandelshtam 1945) with the periodicity at wavelength scales. Sivukhin was probably the first to notice the possibility of a medium with negative ε and μ, but rejected it since the possibility of their existence was yet to be clarified. Viktor G. Veselago first formally considered media with simultaneous neg- ative ε and μ from a theoretical point of view (Veselago 1968), and concluded that the phase velocity and the energy flow in such media would point in op- posite directions. Thus, the media could be considered as having a negative refractive index. He systematically investigated several effects resulting from his conclusions, including the negative refraction at an interface, the negative Doppler shifts, an obtuse angle for Čerenkov radiation, and the possibility of momentum reversal. He also considered the behavior of convex and concave lenses made of such media and also showed that a flat slab of material with n = −1 could image a point source located on one side of the slab onto two © 2009 by Taylor & Francis Group, LLC
- 28. 8 Introduction other points, one inside the slab and one on the other side of it (provided that the thickness of the slab was sufficient). His results, however, did not spark much interest at the time and remained an academic curiosity for many subsequent years, primarily because there were no media available at the time which had both ε and μ negative at a given frequency. The realization of these media had to wait for another 30 years for the development of ideas allowing their experimental realization. Metamaterials have been the most recent development in this quest for con- trol over light via material parameters, with the recognition that engineered materials, structured in specific manners, can exhibit resonances unique to the structure at certain frequencies. The structures are engineered such that at these frequencies, the wavelength of the electromagnetic radiation is much larger than the structural unit sizes, and thus can excite these resonances while still failing to resolve the details of the structure (shape, size, etc.). Conse- quently, an array of these structural units (periodic or otherwise) appears to be effectively homogeneous to the radiation and can be well described by ef- fective medium parameters such as a dielectric permittivity ε and a magnetic permeability μ.§ 1.2 The concept of metamaterials Interestingly, the tremendous interest surrounding media with simultaneously ε < 0 and μ < 0 arose despite the fact that no natural materials have been, and still are, known to exhibit these properties and all known such media today are artificially structured metamaterials. Although Veselago speculated in his landmark paper (Veselago 1968) that some “gyrotropic substances possessing both plasma and magnetic properties” could be anisotropic examples of left- handed media, to date there is no report of a natural medium with such properties. Therefore, their realization took the path of engineered structures that have been called metamaterials.¶ The word “meta” implies “beyond” (as in “metaphysics”) and the termi- nology “metamaterials” today implies composite materials consisting of struc- tural units much smaller that the wavelength of the incident radiation and displaying properties not usually found in natural materials. Although many §It is important to note that the effective medium parameters might have little to do with the bulk material parameters of the medium making up the structures as is discussed in Chapter 3. ¶The origin of the term Metamaterial has been attributed to R. M. Walser who defined them as “Macroscopic composites having a manmade, three-dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to specific excitation” in 1999 (Walser 2003). © 2009 by Taylor & Francis Group, LLC
- 29. 1.2 The concept of metamaterials 9 of the ideas of metamaterials have their origin in the theories of homogeniza- tion of composites (see for example Milton (2002)), metamaterials differ from those in that they are crucially dependent on resonances for their properties and the nature of the bulk material of the structural units is often of marginal importance in determining the effective medium parameters in the relevant frequency bandwidth. Typically, the resonances in metamaterials can induce large amounts of dispersion (large changes with frequency) in the effective medium parameters at frequencies close to resonance. By properly driving and enhancing these resonances, one can cause the materials parameters ε or μ to become negative in a frequency band slightly above the resonance frequency. Pendry et al. (1996) first theoretically suggested and later experimentally demonstrated (Pendry et al. 1998) that a composite medium of periodically placed thin metallic wires can behave as an effective plasma medium for radia- tion with wavelength much larger than the spatial periodicity of the structure. For frequencies lower than a particular (plasma) frequency, the thin wire struc- ture therefore exhibits a negative permittivity ε. Although dense wire media had been considered with much interest as artificial impedance surfaces by electrical engineers (Brown 1960, Rotman 1962, King et al. 1983), they were usually considered when the wavelength was comparable to the period of the lattice and were therefore not really metamaterials per se, for which effective medium parameters can be defined. In 1999, Pendry et al. described how one could tailor a medium whose ef- fective magnetic permeability could display a resonant Lorentz behavior and therefore achieve negative values of the permeability within a frequency band above the resonant frequency (Pendry et al. 1999). Again, although simi- lar structures consisting of loops, helices, spirals or Omega-shaped metallic particles had been considered earlier by the electrical engineering commu- nity (Saadoun and Engheta 1992, Lindell et al. 1994) as the basis of artificial chiral and bianisotropic media, the work reported in Pendry et al. (1999) was the first to consider them as magnetizable particles that could lead to an effective negative μ. In light of the connection between (ε, μ) and the index of refraction n ex- pressed in Eq. (1.2), one should immediately wonder what happens to n when both ε and μ are negative. While in usual materials with positive consti- tutive parameters it is natural to take the positive square root in Eq. (1.2), n = √ εμ, physical and mathematical considerations lead into choosing the negative square root n = − √ εμ when ε < 0 and μ < 0. More arguments in favor of this conclusion are provided subsequently in this chapter and within the body of this book. With the basis for a negative permittivity and a negative permeability hav- ing been laid out, researchers went on to actually experimentally demonstrate the reality of a negative index medium in a prism experiment at microwave frequencies (Smith et al. 2000, Shelby et al. 2001b). A photograph of one of the original metamaterial structures possessing a negative index of refrac- © 2009 by Taylor & Francis Group, LLC
- 30. 10 Introduction Figure 1.3 One of the world’s first negative refractive index medium at mi- crowave frequencies reported in Shelby et al. (2001b). The system has negative refractive index for wave propagating in the horizontal plane with the electric field along the vertical direction. The ring-like metallic structures printed on a circuit board provide the negative magnetic permeability while metal wires make the composite acquire a negative dielectric permittivity. (Repro- duced with permission from Shelby et al. (2001b). c 2001 by the American Association for the Advancement of Science.) tion is reproduced in Fig. 1.3 and illustrates how the proposals for a negative permittivity and a negative permeability were put together in a single con- figuration. Although these initial experiments were met with some criticism, they were quickly confirmed by free-space experiments (Greegor et al. 2003, Parazzoli et al. 2003) with large sample sizes. As a consequence, at the time of the present writing, negative refractive index materials are well accepted and have become available at frequencies spanning a wide portion of the electro- magnetic spectrum, from static to microwave to optical frequencies, although the extent of homogenization and description as a homogeneous material is often questionable at the higher end of the spectrum. The realization that engineered structures can exhibit a negative index of refraction opens up several conceptual frontiers in electromagnetics and optics: several new properties become realizable while most known electromagnetic effects have to be revisited. Even pedestrian effects like refraction between two media, one of them with a negative index, are modified whereby the wave refracts on to the same side of the normal. Several other phenomena were shown to be modified, such as the Doppler shift, the Čerenkov radiation, the Goos-Hänchen shift for reflection of a beam, the radiation pressure, etc. In addition, media with negative permittivity and permeability have the ability to support surface electromagnetic modes, which has given an impetus to the new field of plasmonics (Barnes et al. 2003). The surface plasmon excitations on a metal surface (i.e., at an interface with a medium exhibiting a nega- © 2009 by Taylor Francis Group, LLC
- 31. 1.2 The concept of metamaterials 11 −ε L C C Figure 1.4 A capacitor and an inductor form a resonant circuit that can oscillate at ω0 = 1/ √ LC. A capacitor filled with a negative dielectric has negative capacitance, acts as an inductor and can resonate with another usual capacitor. (Reproduced with permission from Ramakrishna (2005). c 2005, Institute of Physics Publishing, U.K.) tive permittivity) have been well known (Raether 1986), whereas materials with negative magnetic permeability are totally novel and can be expected to support the analogous surface plasmon but of a magnetic nature. These surface plasmons on a structured metallic surface can resonantly interact with radiation and give rise to a host of novel electromagnetic effects. The origin of the surface plasmon can be simply understood as a resonance effect at the interface between two media. Let us consider, for example, the simple case of a capacitor: it is well known that a capacitor can be formed by two parallel conducting plates with an insulating dielectric placed in-between. Filling the gap with a negative dielectric material instead would lead to a capacitor with negative capacitance, which is equivalent to an inductor. Thus two capacitors in a circuit, one filled with a positive dielectric (εp) and the other filled with a negative dielectric (εm), can become resonant (see Fig. 1.4). The condition for resonance with two such capacitors turns out to be simply εm = −εp, which is exactly the condition for the excitation of a surface plasmon at the interface between a semi-infinite positive medium and a semi- infinite negative medium in the static limit. Including negative dielectric materials within regular structures of positive dielectrics can therefore yield media in which a variety of resonances can be excited and the structured media would then display many novel phenomena. The excitation of surface plasmons on small implanted metal particles has been exploited for several centuries in Europe to make brilliantly colored glass windows, and it was explained only at the beginning of the 20th century by the Mie theory of light scattering (Bohren and Huffman 1983). A direct and very novel application of these surface plasmon modes is the perfect lens, which is an imaging device that can preserve subwavelength de- tails in the image and thus overcome the classic diffraction limit (Born and Wolf 1999). It was demonstrated that not only could such a slab of nega- tive refractive medium image a point source in the sense already pointed out in Veselago (1968) for the propagating modes, but that this reconstruction © 2009 by Taylor Francis Group, LLC
- 32. 12 Introduction Figure 1.5 Imaging of an arbitrary object “NANO” by a slab of silver that acts as a super lens. The line width of the “NANO” object is 40 nm. The developed image is found to reproduce subwavelength features of the object to the extent of λ/6. The figure shows the FIB image of the actual object used at the object plane and the AFM image of the developed image on a photoresist. (Figure kindly supplied by Prof. X. Zhang and based on work published in Fang et al. (2005).) also holds for the non-propagating near-field modes of the source (Pendry 2000). Thus the imaging action is not limited by the diffraction limit and, in principle, the image can be perfect with infinite resolution. However, the conditions for a perfect resolution were shown to be highly theoretical and unphysical, and the resolution is, in fact, limited by other processes, primar- ily dissipation in the negative refractive index material (Ramakrishna et al. 2002, Smith et al. 2003). Nonetheless, even if perfect resolution is out of reach, subwavelength image resolution is still achievable and is used in op- tical lithography with subwavelength details as illustrated in Fig. 1.5 (Fang et al. 2005). This lensing effect has been generalized to the idea of comple- mentary media (Pendry and Ramakrishna 2003), which brings in a new view point on negative refractive index media as electromagnetic anti-matter that annihilates the effects of ordinary electromagnetic matter on radiation. In parallel to the development of resonant metamaterials, (Eleftheriades and Grbic 2002, Eleftheriades et al. 2002) and (Caloz and Itoh 2005) indepen- dently developed a transmission line approach with lumped circuit elements for planar metamaterials (see Fig. 1.6 for an implementation) which could support backward waves, or, in other words, an effectively planar negative re- fractive index medium. A host of effects predicted in negative refractive index materials, such as the negative refraction effect, the obtuse angle for Čerenkov radiation, and the subwavelength image resolution, were quickly realized in these transmission line systems, primarily due to the ease in implementing these designs with lumped circuit elements. © 2009 by Taylor Francis Group, LLC
- 33. 1.2 The concept of metamaterials 13 Figure 1.6 A two-dimensional transmission line system that displays a neg- ative refractive index. The transmission line has been implemented using lumped circuit elements: essentially it is a microstrip grid loaded with surface- mounted capacitors and an inductor embedded into the substrate at the cen- tral node. The figure also shows a probe to detect the near-field radiation. The inset shows the expanded unit cell of the metamaterial. (Reproduced with permission from Iyer et al. (2003). c 2003, Optical Society of America.) As the field of metamaterials grew rapidly, various communities were drawn into this research field, bringing a variety of viewpoints, expertise, and in- teresting ideas. This cross-fertilization between so many different fields of physics, mathematics, and engineering is reflected for example in the develop- ment of metamaterial antennae (Ziolkowski and Erentok 2006), optical nano- antennae for plasmonics (Muhlschlegel et al. 2005), and a new circuit element approach to the optics or plasmonics of nanosized metallic particles (Alù et al. 2006a). The emerging area of plasmonics quickly became fundamentally re- lated to metamaterials, particularly at optical frequencies. In fact, the very mechanism and designs of negative refractive index media at optical frequen- cies are, in one way, intimately related to the excitation of these plasmons in the nano-metallic particles making up the structures (Alù and Engheta 2007, Ramakrishna et al. 2007a). Surface plasmon excitations have been shown to be crucial in the mechanisms of several novel optical phenomena such as the extraordinary transmission of light (Ebbesen et al. 1998, Krishnan et al. 2001) through subwavelength-sized hole arrays in metallic films (see Fig. 1.7), large non-linearities due to local field enhancements on rough metal surfaces, sin- gle photon tunneling through subwavelength-sized holes (Smolyaninov et al. 2002), etc. © 2009 by Taylor Francis Group, LLC
- 34. 14 Introduction Figure 1.7 A 2-D array of holes (190 nm diameter and 415 nm periodic- ity) etched by focused ion beam technology in a film of gold deposited on fused silica made at IIT Kanpur. This sample shows a resonantly enhanced transmission peak for light with a wavelength of about 540 nm and 620 nm. 1.3 Modeling the material response This section reviews some fundamental concepts of continuum electromag- netism that are essential to the ideas of metamaterials. For more in-depth discussions and theoretical details, which are beyond the scope of this book, the reader is referred to standard textbooks of electromagnetic theory such as Landau et al. (1984), Jackson (1999), Kong (2000). 1.3.1 Basic equations The Maxwell equations are the fundamental equations for the understanding of all electromagnetic and optical phenomena. In their differential form, these © 2009 by Taylor Francis Group, LLC
- 35. 1.3 Modeling the material response 15 equations are written as ∇ · E = ε0 , (1.3a) ∇ · B = 0, (1.3b) ∇ × E = − ∂B ∂t , (1.3c) ∇ × B = μ0J + ε0μ0 ∂E ∂t , (1.3d) where E and B are the electric field and the magnetic induction, respectively, and and J are the volume charge and current densities, respectively. These equations are called the microscopic Maxwell equations because and J here represent the actual microscopic charge and current densities. In a mate- rial medium, for example, would describe the electronic and nuclear charge distributions. Thus and J would necessarily be complicated and vary ex- tremely fast on very small length scales. Most often, however, we are not interested in the correspondingly fast variations of the electric and magnetic fields over atomic length scales and a macroscopic description is sufficiently accurate. The fundamental Maxwell equations are therefore rewritten at the macroscopic level as ∇ · D = ρ, (1.4a) ∇ · B = 0, (1.4b) ∇ × E = − ∂B ∂t , (1.4c) ∇ × H = J + ∂D ∂t , (1.4d) where E and H are the macroscopic electric and magnetic fields, D is the dis- placement field, and B is the macroscopic magnetic induction. Similarly, ρ and J are the macroscopic net charge and current densities. Here the microscopic fields are averaged over sufficiently large volumes to yield the macroscopic field quantities wherein the fast variations over small length scales are not observable. Thus, the underlying medium appears homogeneous and shows a homogeneous response to the applied fields. We refer the reader to Jackson (1999) for an insightful derivation of these equations from the microscopic Maxwell equations. In most materials, the time domain displacement field D is directly and linearly proportional to the applied electric field E, and is a function of the material in which the field propagates. Due to the mass of the electrons in Note that the wavelength of electromagnetic radiation is of the order of 10−2 m at mi- crowave frequencies and about 10−7 m for optical (visible) radiation. In addition, the time period of the oscillations are of the order of 10−9 seconds to 10−15 seconds, respec- tively. Therefore, one usually seeks only spatially averaged and time-averaged information, averaged over much longer length scales and time scales. © 2009 by Taylor Francis Group, LLC
- 36. 16 Introduction the medium that introduce a certain inertia in the response, D does not vary instantaneously with E, but instead is a function of the entire time history of how E excited the medium. A somewhat general form for D can therefore be written in the following form: D(r, t) = t −∞ dt φ(r; t, t )E(r, t ), (1.5) where φ(r; t, t ) is called the local response function. We assume here that the polarization that sets in a medium depends on the local fields – an as- sumption that can be violated at small lengthscales due to correlations in the polarization over a given volume of the material. For stationary processes, φ(r; t, t ) = φ(r; t − t ), i.e., all physical quantities depend only on the elapsed time intervals and the above integral becomes a convolution. Frequency do- main displacement field and electric field can be defined such as E(r, t) = +∞ −∞ dωE(r, ω) e−iωt , (1.6a) D(r, t) = +∞ −∞ dωD(r, ω) e−iωt . (1.6b) Introducing these definitions into Eq. (1.5) and using the convolution theorem of Fourier transforms (Arfken 1985), it can immediately be seen that the frequency domains E(r, ω) and D(r, ω) are related by the simple linear relation D(r, ω) = ε0ε(r, ω)E(r, ω), (1.7) where ε(r, ω) is the frequency-dependent dielectric function given by ε(r, ω) = 1 ε0 ∞ −∞ dτ φ(r; τ)eiωτ . (1.8) This relation indicates that ε is dispersive, i.e., function of the frequency ω. The dispersive nature arises from the inertia of the dipoles in a causal medium (due to the mass of the electrons), which defines a material polarization that does not respond instantaneously to the applied fields, but depends on its time history as we have seen. At extremely high frequencies, for example x-rays or γ-rays, the matter cannot even respond and the “electronic” matter is almost transparent leading to the limit lim ω→∞ ε(ω) → 1. We shall see some examples of frequency-dependent dielectric functions in the next section. A similar analysis also holds true for the magnetic permeability μ(r, ω), which can be space and frequency dependent. The expression of φ(r, τ) can be obtained from an inverse Fourier transform of Eq. (1.8), and subsequently introduced in Eq. (1.5). Supposing that the or- ders of integration can be interchanged, it can be shown that the polarization © 2009 by Taylor Francis Group, LLC
- 37. 1.3 Modeling the material response 17 is related to the electric field via the Fourier transform of [ε(ω)/ε0 − 1]. The analyticity of this latter function in the upper ω plane allows the application of the Cauchy theorem over a contour extending over the real axis, jumping the pole, and closing itself at infinity in the upper plane. This direct complex plane integration provides two relations between the real and imaginary parts of ε(ω), known as the Kramers-Kronig relations, and expressed as (Jackson 1999) Re(ε(ω)) − 1 = 1 π PV ∞ −∞ dω Im(ε(ω )) ω − ω , (1.9a) Im(ε(ω)) = − 1 π PV ∞ −∞ dω Re(ε(ω )) − 1 ω − ω , (1.9b) where PV denotes the Cauchy principal value. Similar relations hold for the real and imaginary parts of the magnetic permeability μ. Consequently, in addition to being frequency dispersive, ε and μ are also required to be complex functions on the account of causality. The imaginary parts account for absorption of radiation in the medium and the total absorbed energy in a volume V is given by (Landau et al. 1984) V d3 r ∞ −∞ ω Im(ε(ω))|E(r, ω)|2 + Im(μ(ω))|H(r, ω)|2 dω 2π . (1.10) For example, consider a time harmonic plane wave exp[i(kz−ωt)] propagating along the z-axis in a dissipative medium with μ = 1 and a complex ε where Im(ε) 0. It is clear that the amplitude of the wave decays exponentially due to absorption of the wave as it propagates, which clearly implies that Im(k) 0. This complex wave-vector can be obtained from the Maxwell equations as k2 = εω2 /c2 . Eqs. (1.9) indicate that the real and imaginary parts of the permittivity (and similarly the permeability) are Hilbert transforms of each other, as illustrated in Fig. 1.8. These relations are derived for material media in thermodynamic equilibrium solely on the grounds of causality. The restriction that they pro- vide on the variation in the real and imaginary parts of material parameters should be regarded as very fundamental. The Kramers-Kronig relations al- low an experimentalist, for example, to measure the imaginary part of the permittivity easily by absorption experiments at various frequencies and de- duce the real parts of the dielectric permittivity from the imaginary part. An example of this procedure is shown in Fig. 1.8 where the imaginary part of the permittivity is calculated from the real parts by a Hilbert transform with different frequency ranges for the integration. Note that the integrals in the Kramers-Kronig relations involve frequencies all the way up to infinity, whereas it is clear that the effective medium theories break down at high fre- quencies. However, this does not really affect us in the case of usual optical media since the macroscopic material response functions hold almost down to © 2009 by Taylor Francis Group, LLC
- 38. 18 Introduction 0 0.5 1 1.5 2 2.5 3 −1 −0.5 0 0.5 1 1.5 2 2.5 3 Ref [0, 1.5] [0, 2] [0 3] (a) With integration over (0, ωm). 0 0.5 1 1.5 2 2.5 3 −1 −0.5 0 0.5 1 1.5 2 2.5 3 Ref [−1.5, 1.5] [−2, 2] [−3 3] (b) With integration over (−ωm, ωm). Figure 1.8 Comparison between the analytic imaginary part of the permit- tivity (thick black curve on both graphs) and the imaginary part obtained via a Hilbert transform. The analytic expression is εr = 1−(ω2 p−ω2 o)/(ω2 −ω2 o +iγω) with ωo = 2π × 10 GHz, ωp = 2π × 15 GHz, and γ = ω0/2. The labels 1, 2, and 3 refer to the subscript m of ωm and correspond to ω1 = 2ωp, ω2 = 4ωp, and ω3 = 6ωp respectively. the level of few atomic distances. Thus, the very high frequency limit is never really probed. In the case of metamaterials, the wavelength is usually larger than the periodicity by only one or two orders of magnitude and this high fre- quency cutoff, when the homogenization becomes invalid, is easily accessed. Thus these relations should be applied cautiously to metamaterials keeping this in mind: if the effective medium theory itself cannot describe the system, the effective medium parameters obtained from the Kramers-Kronig relations are not meaningful. 1.3.2 Dispersive model for the dielectric permittivity This section briefly presents a dispersive model for the dielectric permittivity that is due to H. A. Lorentz. The resulting expression is very general and it has been found that many metamaterials exhibit effective constitutive param- eters in agreement with this law. As another example, amplifying media such as laser gain media, whose imaginary part of the permittivity is negative, are often modeled by a generalized Lorentz model where the oscillator strength is taken to be negative. The Lorentz model is also a good approximation to the density matrix equations of a weakly perturbed two-level quantum system. © 2009 by Taylor Francis Group, LLC
- 39. 1.3 Modeling the material response 19 Within this approximation, having a negative oscillator strength corresponds to a population inversion. Note that a similar discussion would hold for mag- netizable media and the corresponding magnetic fields. The dispersion of the magnetic permeability in many magnetic materials also exhibits a Lorentz- like dispersion although the resonances usually occur at radio and microwave frequencies. Because of this fundamental importance and relevance to the spe- cific field of metamaterials, we shall introduce the derivation of the Lorentz dispersion law here in order to make it familiar to the reader as well as to bring out its generic features. The frequency dispersive nature of a medium is related to the polarizability of its basic units, viz., the atoms and molecules. Although one can give an adequate description of dispersion only by a quantum mechanical treatment, a simplified description is possible by using only a few basic results concerning the properties of atoms and molecules. One starts by noting that an applied electric field causes charge separation of the positively charged nuclei and the negatively charged electrons in an atom or molecules. Thus a dipole moment is generated and to a good approximation dominates over the other multipole moments. The induced dipole moments can be determined by the displacements of the charges from their equilibrium positions. The atoms or molecules may additionally have a permanent dipole moment in which case the equilibrium positions of the positive and negative charges do not coincide (we may ignore the motion of the nuclei due to their comparatively large mass). The force on the electrons is given by the Lorentz force: F = −e(E + v × B), (1.11) where v is the velocity of the electrons. We can usually neglect the mag- netic field effects as |B|/|E| ∼ 1/c and most of the speeds involved are non- relativistic. The electron in an atom or molecule can be assumed to be bound to the equilibrium position through an elastic restoring force. Thus, if m is the mass of the electron, the equation of motion becomes mr̈ + mγṙ + mω2 0r = −eE0 exp(−iωt), (1.12) where r is the displacement vector, ω0 is the resonance angular frequency characterizing the harmonic potential trapping the electron to the equilib- rium position, and ω is the angular frequency of the light. Here mγṙ is a phenomenological damping (viscous) force on the electron due to all inelastic processes. This damping term is extremely important as the oscillating elec- trons radiate electromagnetic waves and energy, although they can also lose energy in several other manners including collisions. Using a trial solution r = r0 exp(−iωt), the displacement of the electron is obtained as r0 = −eE0/m ω2 0 − ω(ω + iγ) . (1.13) © 2009 by Taylor Francis Group, LLC
- 40. 20 Introduction The dipole moment due to each electron is p = −er and the polarization, defined as the total dipole moment per unit volume, P, is given by the vectorial sum of all the dipoles in the unit volume. Assuming one dipole per molecule and an average number density of N molecules per unit volume, one obtains P = Np = Ne2 E/m ω2 0 − ω(ω + iγ) = ε0χeE, (1.14) where χe is the dielectric susceptibility. Hence one can write for the dielectric permittivity ε(ω) = 1 + χe(ω) = 1 + Ne2 /mε0 ω2 0 − ω(ω + iγ) . (1.15) The quantity f2 = Ne2 /mε0 is often called the oscillator strength. Eq. (1.15) is called the Lorentz formula for the dispersion of ε whose real and imaginary parts are plotted in Fig. 1.9. The imaginary part, Im(ε), is seen to strongly peak at ω0 and the full width at half maximum is determined by the levels of the dissipation parameter γ. The real part, Re(ε), changes in a characteristic manner near ω0 which is consistent with the Kramers-Kronig relations given by Eqs. (1.9). One should note that the above discussion strictly holds only for a dilute gas of the polarizable objects. In a dense material medium with a much larger concentration, the fields that arise due to nearby polarized objects affect the polarization at any given point. These fields are known as local fields and the polarization that sets in the medium is proportional to the effective field, which is the vectorial sum of the applied field and the local fields. Needless to say, the actual description of the local fields would be very complicated. On the other hand, in the spirit of homogenization, we can a think of each polarizable object to be within a small sphere surrounded by a uniformly polarized medium rather than being a set of discrete dipoles at various locations. Assuming the polarization outside to be a constant, P, one obtains the effective field as∗∗ E = Eappl + P 3ε0 . (1.16) Thus, we would have to replace the applied electric field in Eq. (1.14) with the effective field. Note that the polarizability (α) of the polarizable object is defined by p = ε0αE , where p is the induced dipole moment so that the net polarization is expressed as P = Nε0αE . From Eq. (1.14) we can write α = e2 /ε0m ω2 0 − ω(ω + γ) , (1.17) ∗∗One uses the result that the field in a uniformly polarized sphere is E = P/3ε0 in the quasi-static limit. © 2009 by Taylor Francis Group, LLC
- 41. 1.3 Modeling the material response 21 0.7 0.8 0.9 1.0 1.1 1.2 1.3 -4 -2 0 2 4 6 Re(ε) and Im(ε) ω/ω0 Re(ε1 ) Im(ε1 ) Re(ε2 ) Im(ε2 ) Figure 1.9 Real and imaginary parts of the dielectric permittivity predicted by the Lorentz model. The parameters for ε1(ω) are f2 1 = 0.03ω2 0 and γ1 = 0.025ω0 and those for ε2(ω) are f2 2 = 0.1ω2 0 and γ1 = 0.01ω0. Note that if the oscillator strength is strong and the dissipation is small enough, the real part of the permittivity can become negative at frequencies just above the resonance frequency as in the case of ε2. and the dielectric susceptibility that relates the polarization and the applied fields as χe = Nα 1 − Nα 3 . (1.18) The dielectric permittivity thus takes the form ε = 1 + χe = 1 + 2Nα 3 1 − Nα 3 , (1.19) where the local field corrections have been incorporated. This formula is known as the Lorentz-Lorenz formula after the two scientists who came to these conclusions independently and almost simultaneously. For static fields, © 2009 by Taylor Francis Group, LLC
- 42. 22 Introduction an analogous result holds and is known as the Clausius-Mossotti relation for dielectrics. Finally, we should point out that a crucial approximation made here is that the size of the polarizable objects (atoms and molecules) is very much smaller than the wavelength of radiation. This enabled us to treat all the polarizable objects in the volume as if subjected to the same field with no spatial variation (limit of infinite wavelength). The discussion, however, holds true even for more complicated but small polarizable objects, not just atoms and molecules, which is discussed subsequently. 1.4 Phase velocity and group velocity Shortly after Maxwell introduced the concept of electromagnetic waves, he immediately went about calculating the velocity of these waves and realized that, for a single frequency and in vacuum, they were propagating at the ve- locity of light (which allowed him to make the connection between the field of electromagnetics and the field of optics). The concept of velocity is funda- mental in the study of waves and signals since it provides information on how the wave evolves in space and time, and how fast information can be trans- fered from one point to another. Yet, one needs to be careful when assigning a physical significance to the various velocities that can be defined. Let us first take the case of a monochromatic plane wave propagating in the ẑ direction. In the time domain, the field is written as Ey = E0 cos(kz − ωt), where E0 is the amplitude of the wave. For a propagating wave, we can track a point of constant phase and realize that it is traveling at a velocity vp = dz dt = ω k . (1.20) Because of this definition, vp is called the phase velocity. In the case of free- space, k = ω/c so that the phase front propagates at the velocity of light. In the case of a more general lossless non-dispersive medium, k = ω √ εμ/c which is a linear function of frequency: the phase velocity is constant, typically the velocity of light in the medium. For yet more general dispersive media, the phase velocity is not a constant with frequency and the phase velocity can be typically larger than the speed of light in the medium. As we shall see subsequently, this does not violate the principle of special relativity since the phase velocity is not associated with a transport of energy, or more strictly, transmission of a signal. Nonetheless, in such a case, various components of a multi-frequency signal propagate at different velocities and cause a phase distortion. All physical signals are composed of multiple frequencies, i.e., are spread © 2009 by Taylor Francis Group, LLC
- 43. 1.4 Phase velocity and group velocity 23 over a certain frequency band. The spectrum of such a wave is never just a Dirac delta function. The assumption of monochromatic plane waves is there- fore a theoretical idealization, whereas in the real world, the signal is typically composed of a slowly varying envelope confining a rapidly oscillating wave. The simplest multi-frequency signal is composed of two closely separated fre- quencies ω0 ± Δω, where Δω ω0, to which correspond the wave-numbers k ± Δk. The superposition of the two waves is simply written as Ey = cos [(k + Δk)z − (ω + Δω)t] + cos [(k − Δk)z − (ω − Δω)t] , = 2 cos(Δkz − Δωt) cos(kz − ωt) . (1.21) Tracking the constant fronts of the two terms yields two velocities: 1. kz − ωt = constant yields the velocity of the rapidly oscillating wave, which is similar to the monochromatic case discussed previously: vp = dz dt = ω k . (1.22a) 2. Δkz − Δωt = constant yields the velocity of the envelope, called the group velocity: vg = dz dt = Δω Δk . (1.22b) Intuitively, the group velocity is seen to correspond to the velocity of the envelope or the packet, and corresponds to the velocity of propagation of the energy in many cases. In the limit of a very narrow-band signal, Δω → 0 and the group velocity is expressed as vg = 1 ∂k/∂ω . (1.23) We can also express the group velocity in terms of the phase velocity: 1 vg = 1 vp + ω ∂ ∂ω 1 vp , (1.24) which indicates that if there is no frequency dispersion, vg = vp. In the case of normal dispersion, ∂ ∂ω (1/vp) 0 so that vg vp. We had mentioned above that vp can be larger than the velocity of light inside the medium. It can easily be shown that vg is in fact lower than this limit. Since vg corresponds to the velocity at which information is carried, it is in compliance with the principle of relativity. In the case of anomalous dispersion relation, ∂ ∂ω (1/vp) 0 so that vg vp: the group velocity can be even larger than the speed of light in vacuum. In this case, however, the group velocity loses its meaning as signal velocity, which has to be defined in terms of the electromagnetic energy flow. This issue is discussed in greater detail in Section 6.5. © 2009 by Taylor Francis Group, LLC
- 44. 24 Introduction Finally, let us mention that the definition of the group velocity can be generalized to a vectorial relation as vg = ∇kω. (1.25) This gradient relationship indicates that the direction of the group velocity is normal to the iso-frequency contour in the spectral domain. This property is extensively used in Sections 5.1 and 5.2 for example. 1.5 Metamaterials and homogenization procedure 1.5.1 General concepts One of the crucial ideas in a homogenization procedure is that the wavelength of radiation is several times, preferably several orders of magnitude, larger than the underlying polarizable objects (such as atoms and molecules). In this case, the radiation is sufficiently myopic so as to not resolve the spatially fast varying structural details, but only responds to the macroscopic charge and current densities. Upon averaging in macroscopic measurements, the only remaining important parameters are the frequency-dependent polarization of the individual (atomic or molecular) oscillators driven by the applied fields. We can apply this idea to a higher class of inhomogeneous materials, such as metamaterials, where the inhomogeneities in a host background are much smaller than the wavelength of radiation, but yet much larger than the “atoms” or “molecules” that the material is composed of. Such a meso-structure would also not be resolved by the incident radiation, and the structure could be driven and polarized or magnetized by applied electromagnetic fields. Partic- ularly near the resonance frequencies (if any), the structures can have a large polarizability. An array of such structural units can then be characterized by macroscopic parameters such as ε and μ that effectively define its macro- scopic response to exciting electromagnetic fields, much like in a homogeneous material. Metamaterials, in some sense, can be strictly distinguished from other structured photonic materials such as photonic crystals or photonic bandgap materials. In the photonic crystals or bandgap materials the stop-bands or bandgaps arise as a result of multiple Bragg scattering in a periodic array of dielectric scatterers. In fact, the periodicity of the structure in these cases is of the order of the wavelength, and hence homogenization in the classical sense cannot be performed. In metamaterials, the periodicity is by comparison far less important (Chen et al. 2006a), and all the properties mainly depend on the single scatterer resonances. Alternatively, one notes that the small peri- odicity and small size of the structural units imply that all the corresponding © 2009 by Taylor Francis Group, LLC
- 45. 1.5 Metamaterials and homogenization procedure 25 Bragg scattered waves are evanescent and bound to the single scatterer. Con- sequently, the properties of a metamaterial are not resulting from interference between waves scattered off different points. Instead, the radiation probes the polarizability of the structural units as it moves through the medium, inter- acting with the polarizable objects in the same manner as in a homogeneous medium. Note that in the limit of long wavelengths, the phase shifts for the wave across a single structural unit are negligibly small and all units interact with the radiation in a similar manner. 1.5.2 Negative effective medium parameters As discussed in Section 1.3.2, there is a large amount of dispersion in the material parameters at frequencies near the resonance. Below the resonance, the polarization is in phase with the applied driving field, whereas it is π out of phase above resonance. If the dissipation is sufficiently small, the resonance can be made very sharp so as to drive the real parts of ε and μ even toward negative values when the corresponding driving fields are the electric and the magnetic fields, respectively. Of course, the imaginary parts of ε and μ are also large at the resonance frequency and its immediate vicinity. Thus, negative real parts of the material parameters should be regarded as a natural outcome of an underdamped and overscreened response of a res- onant medium. Fundamentally, there is no objection to negative real parts of ε(ω) or μ(ω) as long as other physical criteria are also satisfied such as causality. The latter implies for example that the frequency dispersive mod- els for the permittivity and the permeability cannot be arbitrary, but should yield constitutive parameters that satisfy the Kramers-Kronig relationship of Eqs. (1.9). In order to better understand the effect of negative material parameters, consider an isotropic medium where the Im[ε(ω)] ∼ Im[μ(ω)] 0, i.e., dissi- pation is assumed negligibly small at some frequencies (this would typically be a good approximation at frequencies somewhat away from the resonant frequency). We can conveniently characterize most electromagnetic materials by the quadrant where they lie in the complex (Re(ε)−Re(μ)) plane as shown in Fig. 1.10. Quadrant 1: This is the realm of usual optical materials with Re(ε) 0 and Re(μ) 0. Electromagnetic radiation can propagate through these media and the vectors E, H, and k form a right-handed triad. Quadrant 2: The usual form of matter that has Re(ε) 0 and Re(μ) 0 is a plasma of electric charges. It is well known that a plasma screens the interior of a region from electromagnetic radiation. Indeed, all electromagnetic waves are evanescent inside a plasma and no propagating modes are allowed. This is directly expressed by the constitutive relation, which reduces to k · k = εμω2 /c2 0 (1.26) © 2009 by Taylor Francis Group, LLC
- 46. Exploring the Variety of Random Documents with Different Content
- 49. The Project Gutenberg eBook of Transplanted
- 50. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Transplanted A novel Author: Gertrude Franklin Horn Atherton Release date: February 20, 2024 [eBook #72997] Language: English Original publication: New York: Dodd, Mead and Company, 1919 Credits: D A Alexander, David E. Brown, and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) *** START OF THE PROJECT GUTENBERG EBOOK TRANSPLANTED ***
- 52. TRANSPLANTED
- 53. BY MRS. ATHERTON Historical The Conqueror A few of Hamilton’s Letters California: An Intimate History War Book The Living Present Fiction California Rezánov The Doomswoman The Splendid Idle Forties (1800-46) A Daughter of the Vine (The Sixties) Transplanted (The Eighties) The Californians (Companion Volume to Transplanted) A Whirl Asunder (The Nineties) Ancestors (Present) The Valiant Runaways; A Book for Boys (1840) In Other Parts of the World The Avalanche The White Morning Mrs. Balfame Perch of the Devil (Montana) Tower of Ivory (Munich and England) Julia France and Her Times (B. W. I. and England)
- 54. Rulers of Kings (Austria, Hungary and the Adirondacks) The Travelling Thirds (Spain) The Gorgeous Isle (Nevis, B. W. I.) Senator North (Washington) Patience Sparhawk and Her Times (California and New York) The Aristocrats (The Adirondacks) The Bell in the Fog (Short Stories of various Climes and Places)
- 56. TRANSPLANTED A NOVEL By GERTRUDE ATHERTON Author of “The Conqueror,” “Tower of Ivory,” etc. NEW YORK: DODD, MEAD AND COMPANY 1919
- 57. Copyrighted, 1898 BY DODD, MEAD AND COMPANY As “American Wives and English Husbands” New and revised edition Copyrighted, 1919 By Gertrude Atherton
- 58. TO THE LADIES OF RUE GIROT BOIS GUILLAUME
- 59. TRANSPLANTED
- 60. TRANSPLANTED
- 61. PART ONE CHAPTER I MRS. HAYNE’S boarding-house stood on the corner of Market Street and one of those cross streets which seem to leap down from the heights of San Francisco and empty themselves into the great central thoroughfare that roars from the sandy desert at the base of Twin Peaks to the teeming wharves on the edge of the bay. On the right of Market Street, both on the hills and in the erratic branchings of the central plain, as far as the eye can reach, climbs and swarms modern prosperous San Francisco; of what lies beyond, the less said the better. On the left, at the far southeast, the halo of ancient glory still hovers about Rincon Hill,[A] growing dimmer with the years: few of the many who made the social laws of the Fifties cling to the old houses in the battered gardens; and their children marry and build on the gay hills across the plain. In the plain itself is a thick-set, low- browed, dust-coloured city; “South of Market Street” is a generic term for hundreds of streets in which dwell thousands of insignificant beings, some of whom promenade the democratic boundary line by gaslight, but rarely venture up the aristocratic slopes. By day or by night Market Street rarely has a moment of rest, of peace; it is a blaze of colour, a medley of sound, shrill, raucous, hollow, furious, a net-work of busy people and vehicles until midnight is over. Every phase of the city’s manifold life is suggested there, every aspect of its cosmopolitanism. To a little girl of eleven, who dwelt on the third floor of Mrs. Hayne’s boarding-house, Market Street was a panorama of serious study and unvarying interest. She knew every shop window, in all the mutable details of the seasons, she had mingled with the throng unnumbered times, studying that strange patch-work of faces, and wondering if
- 62. they had any life apart from the scene in which they seemed eternally moving. In those days Market Street typified the world to her; although her school was some eight blocks up the hill it scarcely counted. All the world, she felt convinced, came sooner or later to Market Street, and sauntered or hurried with restless eyes, up and down, up and down. The sun rose at one end and set at the other; it climbed straight across the sky and went to bed behind the Twin Peaks. And the trade winds roared through Market Street as through a mighty cañon, and the sand hills beyond the city seemed to rise bodily and whirl down the great way, making men curse and women jerk their knuckles to their eyes. On summer nights the fog came and banked there, and the lights shone through it like fallen stars, and the people looked like wraiths, lost souls condemned to wander unceasingly. When Mrs. Tarleton was too ill to be left alone, Lee amused herself watching from above the crush and tangle of street cars, hacks, trucks, and drays for which the wide road should have been as wide again, holding her breath as the impatient or timid foot-passengers darted into the transient rifts with bird-like leaps of vision and wild deflections. Occasionally she assumed the part of chorus for her mother, who regarded the prospect beneath her windows with horror. “Now! She’s started—at last! Oh! what a silly! Any one could have seen that truck with half an eye. She turned back—of course! Now! Now! she’s got to the middle and there’s a funeral just turned the corner! She can’t get back! She’s got to go on. Oh, she’s got behind a man. I wonder if she’ll catch hold of his coat-tails? There—she’s safe! I wonder if she’s afraid of people like she is of Market Street?” “If I ever thought you crossed that street at the busy time of the day, honey, I should certainly faint or have hysterics,” Mrs. Tarleton was in the habit of remarking at the finish of these thrilling interpretations. To which Lee invariably replied: “I could go right across without stopping, or getting a crick in my neck either; but I don’t, because I
- 63. wouldn’t make you nervous for the world. I go way up when I want to cross and then turn back. It’s nothing like as bad.” “It is shocking to think that you go out at all unattended; but what cannot be cannot, and you must have air and exercise, poor child!” Lee, who retained a blurred, albeit rosy impression of her former grandeur, was well pleased with her liberty; and Mrs. Tarleton was not only satisfied that any one who could take such good care of her mother was quite able to take care of herself, but, so dependent was she on the capable child, that she was frequently oblivious to the generation they rounded. Mrs. Tarleton was an invalid, and, although patient, she met her acuter sufferings unresistingly. Lee was so accustomed to be roused in the middle of the night that she had learned to make a poultice or heat a kettle of water while the receding dreams were still lapping at her brain. She dressed her mother in the morning and undressed her at night. She frequently chafed her hands and feet by the hour; and cooked many a dainty Southern dish on the stove in the corner. Miss Hayne, who had a sharp red nose and the anxious air of protracted maidenhood, but whose heart was normal, made it her duty to fetch books for the invalid from the Mercantile Library, and to look in upon her while Lee was at school. Lee brushed and mended her own clothes, “blacked” her boots with a vigorous arm, and studied her lessons when other little girls were in bed. Fortunately she raked them in with extreme rapidity, or Mrs. Tarleton would have made an effort and remonstrated; but Lee declared that she must have her afternoons out-of-doors when her mother was well and companioned by a novel; and Mrs. Tarleton scrupulously refrained from thwarting the girl whose narrow childhood was so unlike what her own had been, so unlike what the fairies had promised when Hayward Tarleton had been the proudest and most indulgent of fathers. FOOTNOTE: [A] This was written before the earthquake and fire of 1906.
- 64. CHAPTER II MARGUERITE TARLETON’S impression of the hour in which she found herself widowed and penniless was very vague; she was down with brain fever in the hour that followed. The Civil War had left her family with little but the great prestige of its name and the old house in New Orleans. Nevertheless, the house slaves having refused to accept their freedom, Marguerite had “never picked up her handkerchief,” when, in a gown fashioned by her mammy from one of her dead mother’s, she made her début in a society which retained all of its pride and little of its gaiety. Her mother had been a creole of great beauty and fascination. Marguerite inherited her impulsiveness and vivacity; and, for the rest, was ethereally pretty, as dainty and fastidious as a young princess, and had the soft manner and the romantic heart of the convent maiden. Hayward Tarleton captured twelve dances on this night of her triumphant début, and proposed a week later. They were married within the month; he had already planned to seek for fortune in California with what was left of his princely inheritance. When Tarleton and his bride reached San Francisco the fortune he had come to woo fairly leapt into his arms; in three years he was a rich man, and his pretty and elegant young wife a social power. It was a very happy marriage. Marguerite idolised her handsome dashing husband, and he was the slave of her lightest whim. Their baby was petted and indulged until she ruled her adoring parents with a rod of iron, and tyrannised over the servants like a young slave-driver. But the parents saw no fault in her, and, in truth, she was an affectionate and amiable youngster, with a fund of good sense for which the servants were at a loss to account. She had twenty-six dolls at this period, a large roomful of toys, a pony, and a playhouse of three storeys in a corner of the garden. Then came the great Virginia City mining excitement of the late Seventies. Tarleton, satiated with easy success, and longing for
- 65. excitement, gambled; at first from choice, finally from necessity. His nerves swarmed over his will and stung it to death, his reason burnt to ashes. He staggered home one day, this man who had been intrepid on the battle-field for four blood-soaked and exhausting years, told his wife that he had not a dollar in the world, then went into the next room and blew out his brains. The creditors seized the house. Two hours before Mrs. Tarleton had been carried to Rincon Hill to the home of Mrs. Montgomery, a Southerner who had known her mother and who would have offered shelter to every stricken compatriot in San Francisco if her children had not restrained her. Lee, who had been present when her father spoke his last words to his wife, and had heard the report of the pistol, lost all interest in dolls and picture-books forever, and refused to leave the sick-room. She waited on her mother by day, and slept on a sofa at the foot of the bed. Mrs. Montgomery exclaimed that the child was positively uncanny, she was so old-fashioned, but that she certainly was lovable. Her own young children, Tiny and Randolph, although some years older than Lee, thought her profoundly interesting, and stole into the sick-room whenever the nurse’s back was turned. Lee barely saw them; she retained no impression of them afterward, although the children were famous for their beauty and fine manners. When Mrs. Tarleton recovered, her lawyer reminded her that some years before her husband had given her a ranch for which she had expressed an impulsive wish and as quickly forgotten. The deeds were at his office. She gave her jewels to the creditors, but decided to keep the ranch, remarking that her child was of more importance than all the creditors put together. The income was small, but she was grateful for it. Her next of kin were dead, and charity would have been insufferable. Mrs. Hayne, a reduced Southerner, whom Tarleton had started in business, offered his widow a large front room on the third floor of her boarding-house at the price of a back one. In spite of Mrs. Montgomery’s tears and remonstrances, Mrs. Tarleton accepted the
- 66. offer, and persuaded herself that she was comfortable. She never went to the table, nor paid a call. Her friends, particularly the Southerners of her immediate circle, Mrs. Montgomery, Mrs. Geary, Mrs. Brannan, Mrs. Cartright, and Colonel Belmont were faithful; but as the years passed their visits became less frequent, and Mrs. Montgomery was much abroad with her children. Marguerite Tarleton cared little. Her interest in life had died with her husband; such energies as survived in her were centred in her child. When there was neither fog nor dust nor wind nor rain in the city, Lee dressed her peremptorily and took her for a ride in the cable-cars; but she spent measureless monotonous days in her reclining chair, reading or sewing. She did not complain except when in extreme pain, and was interested in every lineament of Lee’s busy little life. She never shed a tear before the child, and managed to maintain an even state of mild cheerfulness. And she was grateful for Lee’s skill and readiness in small matters as in great; her unaccustomed fingers would have made havoc with her hair and boots. “Did you never, never button your own boots, memmy?” asked Lee one day, as she was performing that office. “Never, honey. When Dinah was ill your father always buttoned them, and after she died he wouldn’t have thought of letting any one else touch them; most people pinch so. Of course he could not do my hair, but he often put me to bed, and he always cut up my meat.” “Do all men do those things for their wives?” asked Lee in a voice of awe; “I think they must be very nice.” “All men who are fit to marry, and all Southern men, you may be sure. I want to live long enough to see you married to a man as nearly like your father as possible. I wonder if there are any left; America gallops so. He used to beg me to think of something new I wanted, something it would be difficult to get; and he fairly adored to button my boots; he never failed to put a little kiss right there on my instep when he finished.”
- 67. “It must be lovely to be married!” said Lee. Mrs. Tarleton closed her eyes. “Was papa perfectly perfect?” asked Lee, as she finished her task and smoothed the kid over her mother’s beautiful instep. “Perfectly!” “I heard the butler say once that he was as drunk as a lord.” “Possibly, but he was perfect all the same. He got drunk like a gentleman—a Southern gentleman, I mean, of course. I always put him to bed and never alluded to it.”
- 68. CHAPTER III LEE had no friends of her own age. The large private school she attended was not patronised by the aristocracy of the city, and Mrs. Tarleton had so thoroughly imbued her daughter with a sense of the vast superiority of the gentle-born Southerner over the mere American, that Lee found in the youthful patrons of the Chambers Institute little likeness to her ideals. The children of her mother’s old friends were educated at home or at small and very expensive schools, preparatory to a grand finish in New York and Europe. Lee had continued to meet several of these fortunate youngsters during the first two of the five years which had followed her father’s death, but as she outgrew her fine clothes, and was put into ginghams for the summers and stout plaids for the winters, she was obliged to drop out of fashionable society. Occasionally she saw her former playmates sitting in their parents’ carriages before some shop in Kearney Street. They always nodded gaily to her with the loyalty of their caste; the magic halo of position survives poverty, scandal and exile. “When you are grown I shall put my pride in my pocket, and ask Mrs. Montgomery to bring you out, and Jack Belmont to give you a party dress,” said Mrs. Tarleton one day. “I think you will be pretty, for your features are exactly like your father’s, and you have so much expression when you are right happy, poor child! You must remember never to frown, nor wrinkle up your forehead, nor eat hot cakes, nor too much candy, and always wear your camphor bag so you won’t catch anything; and do stand up straight, and you must wear a veil when these horrid trade winds blow. Beauty is the whole battle of life for a woman, honey, and if you only do grow up pretty and are properly lancée, you will be sure to marry well. That is all I am trying to live for.” Lee donned the veil to please her mother, although she loved to feel the wind in her hair. But she was willing to be beautiful, as beauty
- 69. meant servants and the reverse of boarding-house diet. She hoped to find a husband as handsome and devoted as her father, and was quite positive that the kidney flourished within the charmed circle of society. But she sometimes regarded her sallow little visage with deep distrust. Her black hair hung in lank strands; no amount of coaxing would make it curl, and her eyes, she decided, were altogether too light a blue for beauty; her mother had saved Tarleton’s small library of standard novels from the wreck, and Lee had dipped into them on rainy days; the heroine’s eyes when not black “were a dark rich blue.” Her eyes looked the lighter for the short thick lashes surrounding them, and the heavy brows above. She was also very thin, and stooped slightly; but the maternal eye was hopeful. Mrs. Tarleton’s delicate beauty had vanished with her happiness, but while her husband lived she had preserved and made the most of it with many little arts. These she expounded at great length to her daughter, who privately thought beauty a great bore, unless ready-made and warranted to wear, and frequently permitted her mind to wander. “At least remember this,” exclaimed Mrs. Tarleton impatiently one day at the end of a homily, to which Lee had given scant heed, being absorbed in the adventurous throng below, “if you are beautiful you rule men; if you are plain, men rule you. If you are beautiful your husband is your slave, if you are plain you are his upper servant. All the brains the blue-stockings will ever pile up will not be worth one complexion. (I do hope you are not going to be a blue, honey.) Why are American women the most successful in the world? Because they know how to be beautiful. I have seen many beautiful American women who had no beauty at all. What they want they will have, and the will to be beautiful is like yeast to dough. If women are flap- jacks it is their own fault. Only cultivate a complexion, and learn how to dress and walk as if you were used to the homage of princes, and the world will call you beautiful. Above all, get a complexion.” “I will! I will!” responded Lee fervently. She pinned her veil all round her hat, squared her shoulders like a young grenadier, and went forth for air.
- 70. Although debarred from the society of her equals, she had friends of another sort. It was her private ambition at this period to keep a little shop, one half of which should be gay and fragrant with candies, the other sober and imposing with books. This ambition she wisely secluded from her aristocratic parent, but she gratified it vicariously. Some distance up Market Street she had discovered a book shop, scarcely wider than its door and about eight feet deep. Its presiding deity was a blonde young man, out-at-elbows, consumptive and vague. Lee never knew his name; she always alluded to him as “Soft-head.” He never asked hers; but he welcomed her with a slight access of expression, and made a place for her on the counter. There she sat and swung her legs for hours together, confiding her ambitions and plans, and recapitulating her lessons for the intellectual benefit of her host. In return he told her the histories of the queer people who patronised him, and permitted her to “tend shop.” He thought her a prodigy, and made her little presents of paper and coloured pencils. Not to be under obligations, she crocheted him a huge woollen scarf, which he assured her greatly improved his health. She also had a warm friend in a girl who presided over a candy store, but her bosom friend and confidante was a pale weary-looking young woman who suddenly appeared in a secondhand book shop in lowly Fourth Street, on the wrong side of Market. Lee was examining the dirty and disease-haunted volumes on the stand in front of the shop one day, when she glanced through the window and met the eager eyes and smile of a stranger. She entered the shop at once, and, planting her elbows on the counter, told the newcomer hospitably that she was delighted to welcome her to that part of the city, and would call every afternoon if she would be permitted to tend shop occasionally. If the stranger was amused she did not betray herself; she accepted the overture with every appearance of gratitude, and begged Lee to regard the premises as her own. For six months the friendship flourished. The young woman, whose name was Stainers, helped Lee with her sums, and had a keenly sympathetic ear for the troubles of little girls. Of herself she never
- 71. spoke. Then she gave up her own battle, and was carried to the county hospital to die. Lee visited her twice, and one afternoon her mother told her that the notice of Miss Stainers’ death had been in the newspaper that morning. Lee wept long and heavily for the gentle friend who had carried her secrets into a pauper’s grave. “You are so young, and you have had so much trouble,” said Mrs. Tarleton with a sigh, that night. “But perhaps it will give you more character than I ever had. And nothing can break your spirits. They are your grandmother’s all over; you even gesticulate like her sometimes and then you look just like a little creole. She was a wonderful woman, honey, and had forty-nine offers of marriage.” “I hope men are nicer than boys,” remarked Lee, not unwilling to be diverted. “The boys in this house are horrid. Bertie Reynolds pulls my hair every time I pass him, and calls me ‘Squaw;’ and Tom Wilson throws bread balls at me at the table and calls me ‘Broken- down-aristocracy.’ I’m sure they’ll never kiss a girl’s slipper.” “A few years from now some girl will be leading them round by the nose. You never can tell how a boy will turn out; it all depends upon whether girls take an interest in him or not. These are probably scrubs.” “There’s a new one and he’s rather shy. They say he’s English. He and his father came last night. The boy’s name is Cecil; I heard his father speak to him at the table to-night. The father has a funny name; I can’t remember it. Mrs. Hayne says he is very distingué, and she’s sure he’s a lord in disguise, but I think he’s very thin and ugly. He has the deepest lines on each side of his mouth, and a big thin nose, and a droop at the corner of his eyes. He’s the stuck- uppest looking thing I ever saw. The boy is about twelve, I reckon, and looks as if he wasn’t afraid of anything but girls. He has the curliest hair and the loveliest complexion, and his eyes laugh. They’re hazel, and his hair is brown. He looks much nicer than any boy I ever saw.”
- 72. “He is the son of a gentleman—and English gentlemen are the only ones that can compare with Southerners, honey. If you make friends with him you may bring him up here.” “Goodness gracious!” exclaimed Lee. Her mother had encouraged her to ignore boys, and disliked visitors of any kind. “I feel sure he is going to be your next friend, and you are so lonely, honey, now that poor Miss Stainers is gone. So ask him up if you like. It makes me very sad to think that you have no playmates.” Lee climbed up on her mother’s lap. Once in a great while she laid aside the dignity of her superior position in the family, and demanded a petting. Mrs. Tarleton held her close and shut her eyes, and strove to imagine that the child in her arms was five years younger, and that both were listening for a step which so often smote her memory with agonising distinctness.
- 73. CHAPTER IV LEE sat limply on the edge of her cot wishing she had a husband to button her boots. Mrs. Tarleton had been very ill during the night, and her daughter’s brain and eyes were heavy. Lee had no desire for school, for anything but bed; but it was eight o’clock, examinations were approaching, and to school she must go. She glared resentfully at the long row of buttons, half inclined to wear her slippers, and finally compromised by fastening every third button. The rest of her toilette was accomplished with a like disregard for fashion. She was not pleased with her appearance and was disposed to regard life as a failure. At breakfast she received a severe reprimand from Mrs. Hayne, who informed her and the table inclusively that her hair looked as if it had been combed by a rake, and rebuttoned her frock there and then with no regard for the pride of eleven. Altogether, Lee, between her recent affliction, her tired head, and her wounded dignity, started for school in a very depressed frame of mind. As she descended the long stair leading from the first floor of the boarding-house to the street she saw the English lad standing in the door. They had exchanged glances of curiosity and interest across the table, and once he had offered her radishes, with a lively blush. That morning she had decided that he must be very nice indeed, for he had turned scarlet during Mrs. Hayne’s scolding and had scowled quite fiercely at the autocrat. He did not look up nor move until she asked him to let her pass; he was apparently absorbed in the loud voluntary of Market Street, his cap on the back of his head, his hands in his pockets, his feet well apart. When Lee spoke, he turned swiftly and grabbed at her school- bag. “You’re tired,” he said, with so desperate an assumption of ease that he was brutally abrupt, and Lee jumped backward a foot.
- 74. “I beg pardon,” he stammered, his eyes full of nervous tears. “But— but—you looked so tired at breakfast, and you didn’t eat; I thought I’d like to carry your books.” Lee’s face beamed with delight, and its fatigue vanished, but she said primly: “You’re very good, I’m sure, and I like boys that do things for girls.” “I don’t usually,” he replied hastily, as if fearful that his dignity had been compromised. “But, let’s come along. You’re late.” They walked in silence for a few moments. The lad’s courage appeared exhausted, and Lee was casting about for a brilliant remark; she was the cleverest girl in her class and careful of her reputation. But her brain would not work this morning, and fearing that her new friend would bolt, she said precipitately: “I’m eleven. How old are you?” “Fourteen and eleven months.” “My name’s Lee Tarleton. What’s yours?” “Cecil Edward Basil Maundrell. I’ve got two more than you have.” “Well you’re a boy, anyhow, and bigger, aren’t you? I’m named after a famous man—second cousin, General Lee. Lee was my father’s mother’s family name.” “Who was General Lee?” “You’d better study United States history.” “What for?” The question puzzled Lee, her eagle being yet in the shell. She replied rather lamely, “Well, Southern history, because my mother says we are descended from the English, and some French. It’s the last makes us creoles.” “Oh! I’ll ask father.” “Is he a lord?” asked Lee, with deep curiosity.
- 75. “No.” The boy answered so abruptly that Lee stood still and stared at him. He had set his lips tightly; it would almost seem he feared something might leap from them. “Oh—h—h! Your father has forbidden you to tell.” The clumsy male looked helplessly at the astute female. “He isn’t a lord,” he asserted doggedly. “You aren’t telling me all, though.” “Perhaps I’m not. But,” impulsively, “perhaps I will some day. I hate being locked up like a tin box with papers in it. We’ve been here two weeks—at the Palace Hotel before we came to Mrs. Hayne’s—and my head fairly aches thinking of everything I say before I say it. I hate this old California. Father won’t present any letters, and the boys I’ve met are cads. But I like you!” “Oh, tell me!” cried Lee. Her eyes blazed and she hopped excitedly on one foot. “It’s like a real story. Tell me!” “I’ll have to know you better. I must be sure I can trust you.” He had all at once assumed a darkly mysterious air. “I’ll walk every morning to school with you, and in the afternoons we’ll sit in the drawing- room and talk.” “I never tell secrets. I know lots!” “I’ll wait a week.” “Well; but I think it’s horrid of you. And I can’t come down this afternoon; my mother is ill. But to-morrow I have a holiday, and if you like you can come up and see me at two o’clock; and you shall carry my bag every morning to school.” “Indeed!” He threw up his head like a young racehorse. “You must,”—firmly. “Else you can’t come. I’ll let some other boy carry it.” Lee fibbed with a qualm, but not upon barren soil had the maternal counsel fallen.
- 76. “Oh—well—I’ll do it; but I ought to have offered. Girls ought not to tell boys what to do.” “My mother always told her husband and brothers and cousins to do everything she wanted, and they always did it.” “Well, I’ve got a grandmother and seven old maid aunts, and they never asked me to do a thing in their lives. They wait on me. They’d do anything for me.” “You ought to be ashamed of yourself. Boys were made to wait on girls.” “They were not. I never heard such rot.” Lee considered a moment. He was quite as aristocratic as any Southerner; there was no doubt of that. But he had been badly brought up. Her duty was plain. “You’d be just perfect if you thought girls were more important than yourself,” she said wheedlingly. “I’ll never do that,” he replied stoutly. “Then we can’t be friends!” “Oh, I say! Don’t rot like that. I won’t give you something I’ve got in my pockets, if you do.” Lee glanced swiftly at his pockets. They bulged. “Well, I won’t any more to-day,” she said sweetly. “What have you got for me? You are a nice boy.” He produced an orange and a large red apple, and offered them diffidently. Lee accepted them promptly. “Did you really buy these for me?” she demanded, her eyes flashing above the apple. “You are the best boy!” “I didn’t buy them on purpose, but my father bought a box of fruit yesterday and I saved these for you. They were the biggest.” “I’m ever so much obliged.”
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